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Real Numbers
Key Points
1.
Euclid's division lemma :For given positive integers ‘a’ and ‘b’ there exist unique whole numbers ‘q’ and ‘r’ satisfying
the relation a = bq + r, 0 ≤r < b.
2.
Euclid’s division algorithms :
HCF of any two positive integers a and b. With a>b is obtained as follows :
Step : Apply Euclid’s division lemma to a and b to find q and r such that
a = bq + r. 0 ≤r > b
Step 2 : If r = 0, HCF (a, b), = b if r ≠ 0, apply Euclid’s lemma to b & r
3.
The Fundamental Theorem of Arithmetic :
Every composite number can be expressed (factorized) as a product of primes and this
factorizationi is unique, apart from the order in which the prime factors occur.
4.
5.
6.
p
Let x = q , q ≠ 0 to be a rational number, such that the prime factorization of ‘q’ is of the
form 2m5n, where m, n are non-negative integers. Then x has a decimal expansion which is
terminating.
p
Let x = q , q ≠ 0 be a rational number, such that the prime factorization of q is not of the
form 2m5n, where m, n are non-negative integers. Then x has a decimal expansion which is
non-terminating repeating.
√p is irrational, which p is a prime. A number is called irrational if it cannot be written in
p
the form q where p and q are integers and q ≠ 0
Real Numbers
Questions
1 Mark questions :
1.
p
A number, which we can not write in the form of q where p and q are integers and q ≠ 0
then write, what we call this number.
2.
How many prime numbers are there between 1 and 10.
3.
Write whether rational number
29
has a terminating decimal expansion or non343
terminating repeating decimal.
4.
Write the L.C.M. of the numbers 60 and 72.
(300) (Maths__Xth class)
5.
Find HCF × LCM for the numbers 100 and 190.
6.
Which of the following rational numbers have terminating decimal expansion —
37 , 43 , 28
900 2500 720
7.
p
Express 0.375 in the form of q to is lowest term.
8.
Write the H.C.F. of the numbers 3.2252 and 32235.
9.
p
Express 0.3 in the form of q
10. Find the L.C.M. of the numbers 2332 and 2233.
11. Fill in teh blank of the following
945 = 33 ×
×7
12. Write decimal representation of the rational number
46
3 2
25
13. If x = 5 + √3 and y = 5 – √3. Write whether sum of two irrational number x and y is a
rational or irrational number.
14. Whether the product of two irrational numbers (2+√5) and (2–√5) is rational or irrational
number.
15. The L.C.M. and H.C.F. of two numbers are 180 and 6 respectively. If one of the number is
30. Write the other number.
2 Marks questions –
16. State fundamental theorem of the Arithmetic.
17. Write two irrational numbers between 1 and 2.
2
1
and
.
3
2
19. Decimal expansion of two real numbers is given as (i) 0.202002000 .... (ii) 3.333..... State
whether they are rational or irrational numbers.
18. Write two rational numbers between
20. Using Euclid’s division alogrithm. Find HCF of 135 and 225.
21. Find H.C.F. and L.C.M. of numbers 40, 70 and 90.
22. State Euclid’s division Lemma.
23. Find x and y in the following diagram.
X
2
Y
3
5
(301) (Maths__Xth class)
24. Explain why 7 × 11 × 13 + 13 is a composite.
25. Find the largest number whether divides 245 and 1029 leaving remainder 5 in each case.
26. An army group of 308 members is to march behind an army band of 24 numbers in a
parade. The two groups are to march in the same number of columns. What is the maximum
number of column is which they can march.
3 Marks Questions –
27. Prove that √5 is an irrational number.
28. Prove that 2–5√3 is an irrational number.
29. Find the L.C.M. and H.C.F. of the numbers 306 and 657 and veryfy that “L.C.M. × H.C.F.
= Pruduct of two numbers.”
30. Find the HCF of 867 and 255; by using Euclid’s division alogrithm.
31. Show that 8n cannot end with the digit ‘O’ for any natural number n.
32. Divide x4 – 3x2 + 4x + 5 by x2 – x + 1 and verify the division alogrithm.
33. The length, breath and height of a room are 8m 25 cm, 6m 75 and 4 m 50 cm respectively.
Determine the longest rod wich can measure the three dimensions of the room exactly.
34. Find the largest number that will divide 398, 436 and 540 leaving remainder 7, 11 and 13
respectively.
35. Find two rational and two irrational numbers between √2 and √3.
Answers
1.
Irrational number
2.
4
3.
Non-terminating
4.
360
5.
19000
6.
43
2500
7.
3
8
8.
24
9.
7
9
10. 23.33
11. 5
12. 0.230
13. Rational
(302) (Maths__Xth class)
14. Rational
16. 36
17. Non terminating recurring decimal
(1> and <2)
18.
1 , p , 2 ; but q ≠ 0
2 q 3
19. (a) Irrational (b) Rational
20. 45
21. 2520
23. x = 30; y = 15
24. Hint : 13 (7×11+ 1)
13 is a factor except + 1
26. 16
26. 4
29. 22338, 9
30. 51
32. Q = x2 + x – 3
Remainder = 8
33. 75
34.
17
35. Hint √2 = 1.414..... and √3 = 1.732
We can take two rational number between √2 and √3
8
3
, 1.6 =
5
2
For irrational number 2<2.1<2.2<3
e.g. 1.5 =
so √2 < √2.1< √3
Polynomials
(303) (Maths__Xth class)
Key Poins
1.
Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials
respectively.
2.
A quadratic polynomial in x with real coefficient is of the form ax2 + bx + c, where a, b, c
are real number with a ≠ o.
3.
The zeroes of a polynomial p(x) are precisely the x - coordinates of the points where the
graph of y = p(x) intersectes of the x-axis i.e. x = a is a zero of polynomial p(x) if p (a) = 0.
4.
A polynomial can have at most the same number zeros as the degree of polynomial.
5.
For quadriatic polynomial ax2 + bx + c
Sum of roots = –
6.
(a ≠ 0)
b
a
c
Product of roots = a
The division alogrithm states that given any polynomial p(x) and polynomial g(x), there
are polynomials q(x) and r(x) such that :-
p(x) = g(x).q (x) + r(x), g(x) ≠ 0
wether r(x) = 0 or degree of r(x) < degree of g(x)
Polynomials
1 Mark Questions - (Q. No. 10 under HOTS)
1.
The graph of y = p(x) is shown in figure write the number of zeroes of p(x)
y
1
x
x
y
1
2.
If x = 2 is zero of polynomial, x2 – 3x + k write the value of k.
3.
If α, β are zeroes of quadratic polynomial 2x2 + 5x + 10. Write the value of α + β.
4.
Write the degree of polynomial 2x4x3 + 5x7 + 2
5.
Write the product of zeroes of the quadratic polynomial 2x2 – 2√2x + 1
(304) (Maths__Xth class)
6.
Write the polynomial p(x) whose zeroes are – 1 and 2.
7.
Write the quadratic polynomial the product and sum of zeroes are 3 and –5.
8.
How many maximum zeroes can be polynomial of degree three have.
9.
For what value of k, (–4) is zero of polynomial x2 –x – (2k+2)
10*. Write the zeroes of the polynomial 15x2 –x–6.
2 Marks question (Q. No. 18 under HOTS)
11. Find the zeroes of following quadratic polynomials and verify the relation between the
zeroes and the coefficient of the polynomials
(a)
p(x) = x2+7x + 10
(b)
q(x) = 2x2+5x+3
(c)
p(x) = 6x2 – 3 – 7x
(d)
q(s) = s2 – 3
12. Find the quadratic polynomial whose zeroes are 3+√2 and 3–√2
13. Find the zeroes of quadric polynomial x2+4√2x+6
14. Find the quadratic polynomial whose zeroes are 2 and
–3
5
15. Find the zeroes of polynomeal p(x) = x2–2.
2
1
and .
3
4
17. Find the quadratic polynomial whose product and sum of zeroes are –7, and –3.
16. Find the quadratic polynomial whose zeroes are
18*. If α, β are the zeroes of the polynomial p(x) = 2x2 – 7x +3. Find the value of α2 + β2.
3 marks questions (Q. No. 19, 23 and 26 under HOTS)
19*. Find all the zeroes of 2x4 – 3x3 – 3x2 –2. If it is given that two of its zeroes are √2 and –√2.
20. Divide 4x4 + 12x3 – x2 –3x + 4 by 2x2 + 5x – 3 and veryfy the division algorithm.
21. Find the value of p for which the polynomial x3 + 4x2 – px + 8 is exactly divisible by x–2.
22. If x+a is a factor of 2x2 + 2ax 10 find the value of a.
23*. Find all zeroes of the polynomial x4 + x3 – 7x2 – 5x + 10. If its two zeroes are √5 and –√5.
24. Find the quadratic polynomial sum of whose zeroes is 8 and product is 12. Hence find the
zeroes of the polynomial.
25. Using division algorithm find quotient and remainder on dividing p(x) by g(x) if
(a)
p(x) = 2(x)3 + 3x2 – 5x+6, g(x) = 2x–3
(b)
p(x) = x3 + 4, g (x) = x+1
26*. If two zeroes of the polynomial x4 – 6x3 – 26x2 + 138 x – 35 are 2 ±√3. Find other zeroes.
(305) (Maths__Xth class)
Answers
1.
3
2.
k=2
3.
–5
2
7
4.
6.
1
2
x2 – x – 2
7.
x2 + 3x + 5
8.
Three
9.
k=9
5.
10. 2/3, –3/5
11. (a) (–2, 5)
(b) (–1, –3/2)
–1 3
, )
3 2
12. x2 – 6x + 7
(d) ±√3
(e) (
13. –√2,
–3√2
15. ± √2
16. 12x2 – 5x – 2
17. x2 + 3x + 7
18. 37/4
1
, 1
2
20. (2x2 + 5x –3) (2x2 + x) + 4
19. √2, –√.2,
21. p = 16
22. a = 2
23. 2.1, √5, –√5
24. x2 – 8x + 12, zeroes 6, 2
25. (a) x2 + 3x + 2, 0
(b) x2 – x + 1, 3
26. –5, 7
(306) (Maths__Xth class)
Pair of Linear Equation in two variable
Key points
1.
The most general form of a pair of linear equations is :
a1x + b1y + c1 = 0
a2x ± b2y + c2 = 0
Where a1, a2, b1, b2, c1, c2 are real numbers and a12 + b12 ≠ 0, a22+b22 ≠ 0
2.
The graph of a pair of linear equations in two variables is represented by two lines ;
(i)
If the lines intersect at a point, the pair of equations is consistent. The point of
intersection gives the unique solution of the equation.
(ii)
If the lines coincide, then there are infinitely many solutions. The pair of equations
is consistent. Each point on the line will be a solution.
(iii)
If the lines are parallel, the pair of the linear equations has no solution. The pair
of linear equations is inconsistent.
3.
If a pair of linear equations is given by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
a1 b1
(i) a ≠ b
2
2
the pair of linear equations is consistent. (Unique solution)
a1 b1
c1
(ii) a = b ≠ c
2
2
2
a1 b1
c1
(iii) a = b = c
2
2
2
many solutions).
the pair of linear equations is inconsistent (No solution)
the pair of linear equations is dependent & consistent (infinitely
Pair of linear equation in two variables
1 Mark question
1.
Express y is terms of x in the equation.
3x – y = 5
2.
For what value of k the pair of linear equation ?
kx – 2y = 3
3x + y = 5 has unique solutions
3.
For what value of m the pair of linear equation and represent parallel lines ?
3x + my – 8 = 0
3x – 5y + 7 = 0
4.
For what value of k the following pair of linear equation
2x + 3y = 7
4x + ky = 14 has infinite many solutions.
(307) (Maths__Xth class)
5.
Write the condition for which pair of linear equations.
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0 has no solution
6.
The difference between two number is 36. One number is four times of other. Form pair of
linear equation of this word problem.
7.
How many solution of the equation 5x – 4y + 6 = 0 are possible.
8.
Write value of x if :
x+y=5
x–y=3
2 Marks Questions (Q. No. 17, 18 and 19 under HOTS)
Solve for x and y (qn. 9-15)
9.
x – 4y = 13
3x + 2y + 3 = 0
x+1
y+1
+
=0
2
2
11. 4x – 3y – 8 = 0
10.
6x – y –
12.
29
=0
3
3
2
+ y = 13
x
4
5
– y =–2
x
13. 31x + 43y = 117
43x + 31y = 105
4
14. 3x + y = 14
1
2x + y = 1
15.
1
1
+ 3y = 2
2x
1
1
13
+
=
3x 2y
6
16. For what value of p will be the following pair of linear equations have unique solutions.
3x = 4 – 2y
y = 3 – px
(308) (Maths__Xth class)
17*. Solve for x and y, by cross multiplication method
ax + by + a = 0
bx + ay + b = 0
Solve for x and y
18*. 3(x+y) = 7xy
3(x+3y) = 11xy
x
y
19*. a + b = 2
ax – by = a2–b2
20.
44
30
+
x+y
x – y = 10
40
55
x + y + x – y = 13
3 marks questions (Q. No. 28, 31, 32, 33 and 35 under HOTS)
21. Gaphically show that the system of linear equation
4x + 6y - 10 = 0
2x + 3y + 13 = 0 has no solution
22. Determine graphically whether the system of linear equation
3x + 2y = 5
3x – y = 2 has unique solution.
23. Show graphically that the following linear equations have infinite solution
2y = 4x – 6
2x = y + 3
24. Solve graphically for x and y, 2x – y = 4, x + y + 1 = 0
Find the points of x-axis where the lines intersect.
25. A number consists of two digits whose sum is 9. If 27 is added to the number the digit are
reversed. Find the number.
26. The ratio of income of A and B is 9:7 and the ratiio of their expenditure is 4:3 if each of
them saves Rs. 2000 yearly. Find their annual income.
27. A fraction become
1
when 1 is substracted from numerator and 2 is added in denominator.
2
1
when 7 is substracted from numerator and 2 is substracted from denominator.
3
Find the fraction.
It becomes
28*. A person travels 600 km partly by train and partly by car. He takes 8 hours, if he travels
(309) (Maths__Xth class)
120 km. by train and rest by car. He takes 20 minutes longer if he travels 200 km by train
and the rest by the car. Find the speed of the train and the car separately.
29. The taxi charges in a city comporised of a fixed charge for 1st km. together with the charge
for distance covered. For a journey of 15 km the charge paid is Rs. 115 and for a journey of
27 km the charged for paid is Rs. 199. What a person has to pay for a distance of 50 km.
30. Place A and B are 80 km a part from each other on a high way. A car starts from A and other
from B at the same time. If they move in same direction they meat in 8 hours. If they move
in opposite direction they meet in 1 hour 20 minutes. Find the speed of the cars.
31*. Solve the following pair of linear equations.
px + qx = p – q
qx – py = p + q
32*. The students of a class are made to stand in rows. If 4 students are extra in a row, there
would be 2 rows less. If 4 students are less in a row there would be 4 more rows. Find the
number of students be in the class.
33*. Solve for x and y
ax by
b – a =a+b
ax – by = 2ab
34. A father’s age is thrice the sum of ages of two children. After five year his age wil be twice
the sum of children’s ages. How old is father at present ?
35*. Sum of two numbers is 16 and the sum of their reciprocals is
1
. Find these numbers.
3
36. A boat goes 16 km. upstream and 24 km. down stream in 6 hours. Also it covers 12 km
upstream and 36 km down stream in the same time. Find the speed of boat in still water
and that of the stream.
37. 8 men and 12 boys can finish a piece of work in 5 days, while 6 men and 8 boys can finish
it in 7 days. Find the time taken by 1 man alone and that by 1 boy alone to finish the same
work.
Answers
1.
y = 3x – 5
2.
k≠–6
3.
m=–5
4.
k=6
5.
6.
a1
c1
b1
=
+
a2
c2
b2
x – y = 36
x – 4y = 0
7.
Infinite solution
(310) (Maths__Xth class)
8.
x=4
y=1
9.
x=1 y=–3
10. x = 7, y = 13
11. x = 3/2 y = –2/3
1
1
y=
3
2
13. x = 1, y = 2
12. x =
14. x = –2 y = 1/5
1
1
y
3
2
16. p ≠ 3/2
15. x =
17. x = – 1 y = 0
18. x = 1 y = 3/2
19. x = a y = b
20. x = 8 y = 3
25. Number = 36
26. (a) Rs. 18000
27.
(b) Rs. 1400
15
16
28. Speed of train 60 km/h
30. Speed of car = 80 k/b
31. x = 1 y = –1
32. Number of student 96
Here let no. of rows = y
No. of students = x
Total students = xy
(x–2)(x+4) = xy
(x+4)(x–4) = xy
33. x = b y –a
34. 45 years
35. 12 and 4
36. Speed of boat = 8 km/b
Speed of stream = 4km/b
37. Man - 70 days, Boys - 140 days
(311) (Maths__Xth class)
Quadratic Equation
Key Points
1.
The equation ax2 + bx + c = 0, a ≠ 0 is the standard form of a quadratic equation, where a,
b, c are real numbers.
2.
A real number α is said to be a root of the quadratic equation ax2 + bx + c = 0. If 9x2 + bx
+ c = 0, the zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic
equation ax2 + bx + c = 0 are the same.
3.
If we can factorize ax2 + bx + c = 0, a ≠ 0 into a product of two linear factors, then the roots
of the quadratic equation ax2 + bx + c = 0 can be found by equating each factors to zero.
4.
A quadratic equation can also be solved by the method of completing the square.
5.
A quadratic formula : the roots of a quadratic equation ax2 + bx + c = 0 are given by
–b ± √ b2 – 4ac
provided that b2 – 4ac ≥ 0
2a
6.
A quadratic equation ax2 + bx + c = 0 has :(i) Two distinct and real roots if b2 – 4ac > 0
(ii) Two equal and real roots, if b2 – 4ac = 0
(iii) Two roots are not real, if b2 – 4ac < 0
Quadratic Equations
Questions
1 Mark questions (Q. No. 9 and 10 under HOTS)
1.
The product of two consecutive odd integers is 63. Represent this in form of mathematical
equation.
2.
Write the discriminant of the quadratic equation 3x2 – 5x – 11 =0
3.
If x = –2 is a root of the equation 3x2 – 5x + 2k = 0. Write the value of k.
4.
For what value of k quadratic equation x2 – kx + 4 = 0 has equal roots.
5.
Write the nature of the roots of equation 4x2 – 12x + 8 = 0
6.
Show that x = – 3 is the solution of the quadratic equation x2 + 6x + 9 = 0
7.
Write the value of x in equation x2 – 4 = 0
8.
Form a quadratic equation whose roots are –3 and 4.
9*. For what value of m for which x = 2/3 is a solution of mx2 – x – 2 = 0
10*. For what value of p the quadratic equation
2x2 – 6x + p = 0 has real roots.
(312) (Maths__Xth class)
2 Marks questions (Q, No. 19 and 20 under HOTS)
11. Solve the following quadratic equations
3y2 + (6 + 4a) y + 8a = 0
12. Find the value of a and b such that x = 1, x = – 2 are the solution of the quadratic equation.
x2 + ax + b = 0
13. Find the roots of equation, x +
1
=3
x
14. Find the value of p if equation 2x2 + px + 3 = 0 has two equal roots.
15. Find the value of k for which equation 5kx2 + 8x + 2 = 0 has two equal roots.
16. Find the roots of equation a2b2x2 + (b2 – a2) x – 1 = 0
17. Find the roots of equation √2x2+7x + 5 √2= 0
18. Divide 51 in to two parts sucvh athat their product is 378.
19*. If the roots of the equation (b–c)x2 + (c–a)x + (a–b) = 0 are equal then prove that 2b = a+c
20*. Find k so that equation [k+4]x2 + (k+1)x + 1 = 0 has equal roots.
3 marks question (Q. No. 28, 29 and 30 under HOTS)
21. Solve the following quadratic equation by the method of completing square.
(a) 2x2 –5x + 3 = 0
(b) ax2 + bx + c = 0
22. Solve the following quadratic equation by using quadratic formula.
abx2 + (b2 – ac) x – bc = 0
23. Find the discriminant of the equation 3x2 – 2 x +
1
= 0 and hence find the nature of
3
roots, find the roots if they are real.
Solve the following equations (24 – 30)
24. p2x2 + (p2 – q2)x –q2 = 0
25.
1
x–1
x–3
+
=3
3
x–2
x–4
26.
x+1
x–1
5
+
= , x≠1
6
x–1
x+1
1
2
4
+
=
x+1
x+2
x+4
26. 2x2 + 5√3x + 6 = 0
25.
x ≠ –2, –4
1
2
6
+
=
x ≠ 0, 1.2
x
x–2
x–1
28*. 3a2x2 + 8abx + 4b2 = 0 a ≠ 0
27.
(313) (Maths__Xth class)
29*.
x
x+1
1
+
=1
x+1
x
12
30*.
1
1
1
1
=
+
+
a+b+x
a
b
x
6. Marks questions (Q. No. 31, 34, 35 adn 39 under HOTS)
31*. A two digit number is such that the product of digit is 35, when 18 is added to the number
the digits interchange their places. Find the number.
32. A train travels 360 km at uniform speed. If the speed had been 5 km/h more it would have
taken 1 hour less for the same journey. Find the speed of the train.
33. Find two numbers whose sum is 27 and product is 182.
34*. A motorboat whose speed is 9 km/h is still water goes 12 km. down stream and comes back
in a total time 3 hours. Find the speed of the stream.
35*. The hypotenuse of right angled triangle is 6cm more than twice the shortest side. If the
third side is 2 cm less than the hypotenuse find the sides of the triangle.
36. Sum of two number is 15, if sum of their recipocal is
3
. Find the numbers.
10
37. Rs. 9000 were divided equally among a certain number of students. Had there been 20
more students, each would have got Rs. 160 less. Find the original number of students.
38. In a class test sum of Kamal’s marks in Mathematics and English is 40. Had he got 3
marks more in mathematics and 4 marks less in English, the product of his marks would
have been 360. Find his marks in two subject separately.
39*. Solve for x
9x2 – 9 (a+b) x+ (2a2 + 5ab + 2b2) = 0
40. By a reduction of Rs. 2 per kg in the price of sugar. Ram Lal can 2 kg sugar more for Rs. 244.
Find the original price of sugar per kg.
41. An aeroplane takes an hour less for a journey of 1200 km. if the speed is increased by 100
km/h from its usual speed. Find the usual speed.
Answers
1.
(2x – 1)(2x+1) = 63
2.
D = 157
3.
k = – 11
4.
K==4
5.
D ≥ 0 real number
6.
– 3 is solution
7.
x = ± √2
(314) (Maths__Xth class)
8.
x2 + x – 12 = 0
9.
m=6
10. p <
9
2
11. –2,
–4a
3
12. a = 1, b = –2
13.
3 + √5 3 – √5
,
2
2
14. p = ± 2√6
15. k = 8/5
16.
–1
,
a2
1
b2
17.
–5√2
, –√2
2
18. 9, 42
19. – proof
20. (a)
3
2.1
b + √b2 – 4ac
(b) x = –
2a
x=
–b√b2 – 4ac
2a
22. c/b, –b/a
23. +
24.
1 1
,
3 3
q2
, –1
p2
25. 2+√2, 2–√2
26. –2√3, –√3/2
27. 3,
28.
4
3
–2b –2b
,
a
3a
29. –4, 3
30. x = – a, –b
(315) (Maths__Xth class)
31. 57
32. 40 km/h
33. 13, 14
34. 3 k/h
35. 26 cm, 24 cm, 10 cm
36. 5, 10
37. 25 students
38. Maths - 21
Maths - 12
39.
Eng. - 19
Eng. = 28
2a+b a+2b
,
3
3
40. Rs. 16
41. 300 km/h
(316) (Maths__Xth class)
Arithmetic Progression
Key Points
1.
Sequence : A set of numbers arranged in some definite order and formed according to
some rules is called a sequence.
2.
Progression : The sequence that follows a certain pattern is called progression.
3.
Arithmetic progression : A sequence in which the difference obtained by substracting
from any term its preceeding term is consistent throughout, is called on arithmetic sequence
or arithmetic progression (A.P.)
The general form of an A.P. is a, a+d, a+2d, ....... (a : first term, d : common difference)
4.
General Term : if ‘a’ is the first term and ‘d’ is common difference in an A.P., then nth term
(general term) is given by
an = a + (n–1)d
5.
SUM OF n TERMS OF AN A.P. : If ‘a’ is the first term and ‘d’ is the common difference of
an A.P., then sum of first n terms is given by
n
2 { 2a+(n–1)d}
If ‘l’ is the last term of a finite A.P., then the sum is given by
Sn =
6.
n
sn = 2 {a +l}
(i)
If an is given, then common difference d = an –an–1
(ii)
If sn is given, then nth term is given by an = sn – sn–1
iii)
if a, b, c are in A.P., then 2b = a + c
(iv)
If a sequence has n terms, its rth term from the end = (n – r + 1)th term from the
beginning.
1 Mark Questions
1.
If nth term of an A.P. is 5 – 3n, write the common difference of this A.P.
2.
Which term of the A.P. –7, –3, 1, 5, ............... is 73 ?
3.
If 5, 2k–3, 9 are in A.P., then write the value of ‘k’
4.
Is –10, a term of the A.P. 25, 22, 19, ....... ?
5.
Write 13th term of the A.P. 3, 8, 13 ......
6.
Write nth term of the A.P. –5, –2, 1, ......
7.
Is 7
8.
The first term of an A.P. is 3 and sixth term is 23. Write common difference o the A.P.
9.
Write the first term and common difference of the A.P. 7.3, 6.9, 6.5 .......
1
2
3
, 7 , 7 , ...... A.P. ? If yes, write the common difference.
7
7
7
10. Write first three terms of an A.P., whose second term is –4 and common difference is –1.
(317) (Maths__Xth class)
11. Write the sum of first 10 natural numbers.
12. Is √2, √8, √18, √32 ... an A.P. ? If yes, then write next two terms.
13. Write the missing terms of the A.P.
3,
, –1, –3,
14. Write 9th term from the end of the A.P.
7, 11, 15, ..........., 147
15. If the sum of n terms of an A.P. is n2, write its nth term.
1 1 m+1
,
,
are in A.P. ? (m ≠ 0)
2m m 2m
17. The sum of 6th and 7th terms of an A.P. is 39 and common difference is 3. Write its first
term.
16. For what value of ‘m’ the numbers
18. The sum of 3 numbers is A.P. is 30. If the greatest number is 13, write its common difference.
19. Write an A.P. whose third term is 16 and the difference of the 9th term from 11th term is
12.
20. Write the sum of first ‘n’ even natural number.
2 Marks Questions (Q. No. 36 to 40 under HOTS)
21. If –9, –14, –19, ............ is an A.P., then find a30 – a20.
22. Find the A.P. whose second term is 10 and the sixth term exceeds the fourth term by 12.
23. The sum of 3rd and 7th terms of an A.P. is 14 and the sum of 5th and 9th terms is 34. Find the
first term and common difference of the A.P.
24. Which term of the A.P. 41, 38, 35, ........... is the first negative term ?
25. If the sum of first n terms of an A.P. is 2n2 + n, then find nth term and common difference of
the A.P.
26. How many terms of A.P. 22, 20, 18......... should be taken so that their sum is zero ?
27. Find the sum of odd positive integers less than 199.
28. If 9 times of 9th term is equal to 8 times the 8th term of an A.P. Find its 17th term.
29. Which term of A.P. 5, 13, 21, 29 ........ will be 48 less than its 19th term.
30. How many two digits numbers between 4 and 102 are divisible by 6 ?
31. Find an A.P. whose 3rd term is –13 and 6th term is 2.
32. The angles of triangle are in A.P. If the smallest angle is one fifth the sum of other two
angles. Find the angles.
33. Nidhi, starts a game and scores 200 points in the first attempt and she increases the points
by 40 in each attempt. How many points will she score in teh 30th attempt ?
34. Find ‘k’, if the given value of x is the kth term of the A.P. 3, 7, 11, ........., x = 83
35. Anurag saves Re. 1 on day 1, Rs. 2 on day 2, Rs. 3, on day 3 and so on. How much money will
he save in the month of feb. 2010.
(318) (Maths__Xth class)
17
1
and last term is .
6
2
a4
a6
2
37*. For an A.P. a1, a2, a3, ........ if b =
, then find a .
3
7
8
36*. Find an A.P. of 8 terms, whose first term is
38*. The fourth term of an A.P. is equal to 3 times the first term and the seventh term exceeds
twice the third term by 1. Find the first term and the common difference of the A.P.
39*. If 2nd, 31st and last term of an A.P. are
31 1
13
,
and –
respectively. Find the number of
4 2
2
terms in the A.P.
40*. For what value of ‘n’, are the nth terms of two A.P.s 2, 10, 18, ...... and 68, 70, 72 ....... equal
? Also find the term.
3 Marks Questions (Q. No. 56 to 60 under HOTS)
41. Find the sum of A.P. 4 + 9 + 14 + .......... + 249
1
1
42. If pth and qth term of an A.P. are q and p respectively, then find the sum of pq terms.
43. Find the sum of the first 40 terms of an A.P., whose nth term is 3 – 2n.
44. If nth term of an A.P. is 4, common difference is 2 and sum of n terms is –14, then find first
term and number of terms.
45. Find the sum of all the three digits numbers each of which leaves the remainder 3 when
divided by 5.
46. The sum of first six terms of an A.P. is 42. The ratio of the 10th term to the 30th term is
1:3. Find first term and 11th term of the A.P.
47. The sum of three numbers in A.P. is 24 and their product is 440. Find the numbers.
48. The sum of n terms of two A.P.’s are in the ratio 3n + 8 : 7n + 15. Find the ratio of their
12th terms.
49. The sum of first 16 terms of an A.P. is 528 and sum of next 16 terms is 1552. Find the first
term and common difference of the A.P.
50. The sum of first 8 terms of an A.P. is 140 and sum of first 24 terms is 996. Find the A.P.
51. If pth, qth and rth terms of an A.P. are l,m and n respectively. then proe that
p(m–n) + q (n–l) + r(l–m) = 0
52
Find the number of terms of the A.P. 57, 54, 51....... so that their sum is 570. Expain the
double answer.
53. If the sum of first 20 terms of an A.P. is one third of the sum of next 20 terms. If first term
is 1, then find the sum of first 30 terms.
54. A picnic group for Manali consists of students whose ages are in A.P., the common diference
being 3 months. If the youngest students Rohit is just 12 years old and the sum of ages of
all the students is 375 years. Find the number of students in the group.
55. The digits of a three digits positive number are in A.P. and the sum of digits is 15. On
(319) (Maths__Xth class)
subtracting 594 from the number the digits are inerchanged, find the number.
56*. If the roots of the equation a(b–c)x2 + b(c–a)x + c (a–b) = 0 are equal, then show that
1
,
a
1 1
,
are in A.P.
b c
57*. If the sum of m terms of an A.P. is n and the sum of n terms is m, then show that sum of (m
+ n) terms is – (m +n).
58*. The sum of 5th and 9th terms of an A.P. is 8 and their product is 15. Find the sum of first 28
terms of the A.P.
59*. If mth and nth terms of an A.P. are a & b respectively, then show that the sum of its (m+n)
m+n
a–b
terms is
{a+b+
}
2
m –n
60*. Anita arranged balls in rows to form an equilateral triangle. The first row consists of one
ball, the second of two balls, and so on. If 669 more balls are added, then all the balls can be
arranged in the shape of a square and each of its sides then contains 8 ball less than each
side of the triangle. Determine the initial number of balls, Anita has.
Answers
1.
–3
2.
21
3.
k=5
4.
NO
5.
63
6.
3n=8
7.
Yes, common difference =
8.
4
9.
7.3, common difference = – 0.4
1
7
10. – 3, –4, –5.
11. 55
12. Yes, 5√2, 6√2
13. 1 and –5
14. 115
15. 2n–1
16. m = 2
17. 3
18. 7, 10 and 13
(320) (Maths__Xth class)
19. 4, 10, 16, ........
20. n2 + n
21. – 50
22. 4, 10, 16, ..........
23. First term = – 13, d = 5
24. 15th term
25. nth term = 4n–1, common difference = 4
26. 23 terms
27. 9801
28. Zero
29. 13th term
30. 15
31. – 23, –18, –13, .......
32. 300, 600 and 900
33. 1360
34. k = 21
35. Rs. 406
36.
1 5 7
, , , ......
2 6 6
37.
4
a + 3d
2
(Hint.
= )
5
a + 6d
3
38. First term = 3
Common difference = 2
39. 59.
40. n = 12, term = 90
41. 6325
1
(pq + 1)
2
43. – 1520
42.
44. First term = – 8
total terms = 7
45. 99090
46. First term = 2
11th term = 22
47. 5, 8, 11
48. 7:16
(321) (Maths__Xth class)
49. First term = 3
common difference = 4
50. 7, 10, 13, .....
51. (hint : an = a + (n–1) d)
52. 19 or 20 (20th term is zero)
53. 450
54. 25 studetns
55. 852
56. (Hint. : in quadratic equation, D = 0)
(for equal roots)
57. Hint : sn =
n
2 {2a + (n–1)d}
58. 115, 45, {d = ±
59. Hint : sn =
1
}
2
n
{2a +(n–1)d}
2
60. 1540 balls.
Trignometry
(322) (Maths__Xth class)
Key Points
Trignometrical Ratios :-
C
In ∆ABC, ∠B = 90 0 for angle ‘A’
Hy
po
te
nu
se
perpendicular
sin A = Hypotenuse
Base
cosA = Hypotenuse
Perpendicular
Base
tan A =
Base
cot A = Perpendicular
sec. A =
2.
3.
Base
Hypotenuse
cosec A = Perpendicular
Hypotenuse
Base
Reciprocal Relations :
sinθ =
1
,
cosec θ
cosecθ =
cosθ =
1
,
sec θ
sec θ =
1
cos θ'
tanθ =
1
,
cot θ
cotθ =
1
tan θ'
1
sin θ'
Quotient Relations :
tanθ =
4.
A
Sin θ
,
cos θ'
cot θ =
cos θ
sin θ'
Identities :
sin2 θ + cos2 θ = 1 ⇒sin2 θ = 1 – cos2θ and cos2 θ = 1 – sin2θ
1+ tan2 θ = sec2 θ ⇒tan2θ = sec2 θ – 1 and sec2 θ – tan2 θ = 1
1+cot2 θ = cosec2 θ ⇒cot2θ = cosec2 θ – 1 and cosec2 θ – cot2 θ = 1
5.
TRIGNOMETRIC RATIOS OF SOME SPECIFIC ANGLES :
∠A
00
300
450
600
900
sin A
0
1
2
1
√2
√3
2
1
cos A
1
√3
2
1
√2
1
2
0
(323) (Maths__Xth class)
Perpendicular
1.
B
6.
1
√3
1
√3
Not defined
cosec A Not defined 2
√2
2
√3
tan A
0
2
√3
1
sec A
1
√2
2
Not defined
cot A
Not defined √3
1
1
√3
0
Trignometric ratios of complementary angles :
sin (900 –θ) = cosθ
cos(900–θ) = sin θ
tan (900 –θ) = cot θ
cot (900 –θ) = tan θ
sec 900 –θ) = cosec θ
cosec (900 –θ) = sec θ
7.
Line of sight :- The line of sight is the line drawn from the eye of an observer to the point
in the object viewed by the observer.
8.
Angle of elevation : The angle of elevation is the angle formed by the line of sight with the
horizontal when it is above the horizontal level i.e. the case when we raise our head to
look at the object.
9.
Angle of depression : The angle of depression is the angle formed by the line of sight with
the horizontal when it is below the horizontal i.e. case when we lower our head to look at
the object.
1 mark questions
1.
Write tan A in terms of sin A
2.
In ∆PQR, ∠Q = 900 and sin R =
3.
If A and B are acute angles and sin A = Cos B, then write the value of A + B.
4.
Write the value of 7sec2620 – 7 cot2280.
5.
If sinθ =
6.
Write the value of sin620sin280–cos620cos280
3
, what is the value of cos P ?
5
1
, write the value of sinθ + cosecθ.
2
(324) (Maths__Xth class)
7.
Express cosec770 + tan 620 in terms of trignometrical ratios of angles between 00 and 450.
8.
If 4 cot θ = 3, then write the value of tanθ + cot θ
9.
If sin θ – cos θ = 0, o0< θ< 900, then write the value of ‘θ’
10. What is the value of sin2410 –cos2490 ?
11. Write the value of cot2300 + sec2450.
12. Write the value of sin2740 + sin2160.
13. If θ = 300, then write the value of sinθ + cos2θ
14
Write the value of sin(900 –θ) cosθ + cos(900 –θ) sinθ.
15. If tan (3x –150) = 1, then write the value of ‘x’.
16. In ∆ABC, write sin
A+B
in terms of angle ‘C’.
2
17. Write the value of tan (550 – θ) – cot (350 + θ)
18. If tan θ + cot θ = 5, then what is the value of tan2θ + cot2θ ?
19. If θ = 300, then write the value of 1 – tan2θ
20. If θ = 450, then what is the value of 2cosec2θ + 3sec2θ ?
2 Marks questions (Question No. 36 to 40 under HOTS)
21. If θ = 300, find the falue of
1 – tan2 θ
1 + tan2 θ
22. If sin (A+B) = 1 and cos (A–B) =
√3
, 00 ≤(A+B) ≤900, A ≤B, then find the values of A
2
and B.
23. If sin2θ = cos(θ – 360), 2θ and θ – 360 are acute angles. Find the value of ‘θ’
24. If θ = 300, then verify; sin3θ = 3sin θ – 4 sin3θ
25. If tan (320 + θ) = cotθ, θ and (320 + θ) are acute angles, find the values of ‘θ’
26. Simplify : tan2600 + 4cos2450 + 3 sec2 300 + 5cos2900
27. If tan θ = √2 – 1, then find the value of
2 tan θ
1 + tan2 θ
28. If 4 cot θ = 13, find the value of
3 cos θ + 4sin θ
5 cosθ – 3 sinθ
29. Prove that, sec4 θ – sec2 θ = tan2θ + tan4θ
30. If sin θ + sin2θ = 1, then find the value of cos2θ + cos4θ
31. Find the value of
(325) (Maths__Xth class)
sin 620
tan 730
5. sin 280. sec 620
+
3.
–
cos 280
cot 170
7sec2320 – 7 cot2580
32. Find the value of
13 sin 650
4 cos 530. cosec. 370
–
5 cos250
5 7 sec2320 – 7 cot2580
33. Find the value of sin600 geometrically
34. Find the value of
cosec2 ( 900 –θ) – tan2θ
2 tan2 300. sec2 520, sin2 380
–
3(cosec2700 – tan2200)
4 (cos2400 + cos2500)
35. If tan (A+B) = √3 and tan(A–B) =
1 0
, 0 ≤(A+B) ≤900, A>B, then find the value of cos
√3
(2A –3B)
36*. Find the value of
sin250 + sin2 100 + sin2150 + sin2200 + ............... + sin2850
37*. In ∆MNR, ∠N = 900, MN = 8cm, RN – MN = 7 cm. Find the value of sinR, tanR and secM.
38*. If sin(A+B) = sinA cosB + cosA sinB, then find the values of sin750 and cos150.
39*. If 2sin (3x –150) = √3, then find the value of sin2 (2x +100) + tan2 (x+50).
40*. If x = m sinα. cosβ, y = m sinα sinβ and z = m cos α,then prove that x2+y2+z2 = m2
3 Marks questions (question No. 56 to 60 under HOTS)
41. Prove that
cos A
cos A
+
= sin A cos A
1 – tanA
1 – cotA
42. Find the value of
2
0
2
0
3(cot 27 – sec 63 )
tan (90 –θ) cot θ – sec (90 –θ) cosec θ +
0
0
0
0
cot 26 cot41 cot45 cot49 cot64
0
0
2 sec 240 × sin 660
–
+ 3 tan2300
sin2620 + sin2 280
43. Prove that
1
1
1
1
–
=
–
cosec A – cot A sinA
sinA cosec A – cot A
44. If sec θ + tan θ = 4, then prove that cos θ =
8
17
(326) (Maths__Xth class)
sec θ – 1
+
sec θ + 1
45. Prove that
sec θ + 1
= 2 cosec θ
sec θ – 1
46. Prove that (sinθ + cosecθ)2 + (cosθ + secθ)2 = tan2θ + cot2θ + 7
47. Prove that
sec θ – cos θ+ 1
1 + sin θ
=
sec θ – cos θ – 1
cos θ
48. Pove that (1 + cot A – cosec A) (1 + tan A + Sec A) = 2
49. Prove that (1 +
1
1
1
) (1 +
)=
2
2
2
tan θ
cot θ
sin θ – sin4θ
50. Prove that cos8θ – sin8θ = (cos2θ – sin2θ) (1–2 sin2θ. cos2θ)
51. If a sinA = b cos A and a sin3A + b cos3A = sinAcosA, then prove that a2+b2 = 1
52. Prove that 2(sin6A + cos6A) – 3 (sin4A + cos4A) + 1 = 0
53. If cosθ – sinθ = √2 sinθ, then prove that :
cosθ + sinθ = √2 cosθ
54. If secθ = x +
55. Prove that
1
1
, then prove that secθ + tanθ = 2x or
4x
2x
cosec A – 1
cot A – cos A
=
cosec A + 1
cot A + cos A
56*. If cos2 α – sin2α = tan2β, then prove that
cos β =
1
√2 cosα
57*. If cosecθ - sinθ = m3 and secθ – cosθ = n3, then prove that m4n2 + m2n4 = 1
58*. If x = tanA + sinA and y = tanA – sinA, then prove that x2 – y2 = 4√xy
59*. If sinα = α sinβ and tan α = b tan β, then prove that
cos2α =
a2 – 1
2
b –1
60*. If sinθ + sin2θ = 1, then prove that
cos12θ + 3cos10 θ + 3 cos8 θ + cos6θ + 2 cos4θ + cos2 θ = 2 + sin2θ
6 MARKS QUESTIONS (Question No. 76 to 80 under HOTS)
61. The shadow of a tower standing on a level ground is found to be 60 m shorter when the
Sun’s altitude is 600 then when it is 300, find the height of the tower.
62. The angles of elevation of a bird from a point on the ground is 600, after 50 seconds flight
the elevation changes to 300. If the bird flying at the height 500√3m. Find the speed of the
bird.
63. From a point on the ground the angles of elevation of the bottom and the top of a water
tank kept at the top of 30 m. high building are 450 and 600 respectively. Find the height of
(327) (Maths__Xth class)
the water tank.
64. A tree breaks due to storm and the broken part bends so that the top of the tree touches
the ground making an angle 600 with the ground. The distance from the foot of the tree to
the point where the top touches the ground is 5 m. Find total height of the tree.
65. From a window (20 m high above the ground) o a house in a street. The angles of elevation
and depression of the top and the foot of an other house opposite side of street are 600 and
450 respectively. Find the height of the opposite house.
66. A light pole 4 m high is fixed on the top of a building, the angle of elevation of the top of the
pole observed from a point ‘p’ on the ground is 600 from the top of the building is 450. Find
the height of the building.
67. An aeroplane at an altitude of 1200 meters observes the angles of depression of opposite
points on the two banks of a river to be 600 and 450, find the width of the river.
68. The angles of elevation of the top of a pole from two points P and Q at distances of ‘x’ and
‘y’ respectively from the base and in the same straight line with it, are complementary.
Prove that the height of the pole is √xy.
69. The angle of elevation of a bird from a point 12 metres above a lake is 300 and the angle of
depression of its reflection in the lake is 600. Find the distance of the bird from the point of
observation.
70. The angle of elevation of the top of a 10 metres tall building from a point P on the ground
is 300. A flag is hoisted at the top of the building and the angle of elevation of the flag staff
from P is 450. Find the length of flag staff and the distance of the building from P.
71. Nikita standing on a bank of a river observes that the angle subtended by a tree on the
opposite bank is 600, when she retires 30 metres from the bank, she finds the angle to be
300, find the breadth of the river and height of the tree.
72. A man, on a cliff, observes a boat at an angle of depression of 300, which is approaching the
shore to the point ‘A’ on the immediately beneath the observer with a uniform speed, 12
minutes later, the angle of depression of the boat is found to be 600. Find the time taken by
the boat to reach the shore.
73. A man on the deck of a ship, 18 metres above water level, observes that the angle of elevation
and depression respectively of the top and bottom of a cliff are 600 and 300. Find the distance
of the cliff from the ship and height of the cliff.
74. The angle of depression of the top and bottom of a 10 metres tall building from the top of
a tower are 300 and 450 respectively. Find the height of the tower and distance between
building and tower.
75. An aeroplane when 3000 metres high, passes vertically above another aeroplane at an
instant when the angle of elevation of two aeroplanes from the same point on the ground
are 600 and 450 respectively. Find the vertical distance between the two planes.
76*. At the foot of the mountain the elevation of its summit is 450. After ascending 1000 metres
towards the mountain at an inclination of 300, the elevation is 600. Calculate the height of
the mountain. (√3 = 1.732)
77*. From an aeroplane vertically above a straight horizontal plane, the angle of depression of
two consecutive kilometre stones on the opposite sides of the aeroplane are found to be θ
(328) (Maths__Xth class)
and α. Show that the height of the aeroplane is
tanθ. tanα
tanθ + tanα
78*. At a point ‘P’ on level ground, the angle of elevation of a vertical tower is found to be such
3
5
that its tangent is . On walking 192 metres away from P the tangent of the angle is .
4
12
Find the height of the tower.
79*. A round balloon of radius ‘r’ subtends on an angle ‘θ’ at the eye of the observer while the
angle of elevation of its centre is α. prove that the height of the centre of the balloon is r
θ
sinα cosec
2
80*. A boy standing on a horizontal plane, finds a bird flying at a distance of 100 metres from
him at an elevation of 300. A girl, standing on the roof of 20 metres high building, finds the
angle of elevation of the same bird to be 450. Both the boy and girl are on opposite sides of
the bird. Find the distance of the bird from the girl.
Answers
1.
tan A =
sin A
2
1 – sin A
2.
cos p =
3
5
3.
A + B = 900
4.
7
6.
5
2
Zero
7.
sec130 + cot280
5.
8.
9.
25
12
θ = 450
10. Zero
11. 5
12. 1
13. 1
14. 1
15. x = 20
16. cos
c
2
(329) (Maths__Xth class)
17. Zero
18. 23
19. –2
20. 10
1
2
22. A = 600, B = 300
21.
23. θ = 420
24. (Hint. : 3θ = 900)
25. θ = 290
26. 9
27.
1
√2
28.
55
53
29. Hint : sec2θ – 1 = tan2θ
30. 1
31.
23
7
32.
9
5
33.
√3
2
34.
1
36
35.
1
√2
36.
17
2
37. sin R =
sec M =
38.
8
8
, tan R =
17
15
17
8
√3+1 √3+1
,
(hint : A = 450, B = 300 or A = 300, B = 450)
2√2 2√2
(330) (Maths__Xth class)
13
12
42. – 5
39.
2
cos θ
57. (Hint : m =
sinθ
n=
sin2θ
cosθ
1
3
1
3
58. (Hint : m + n = 2 tanθ)
m – n = 2sinθ)
60. (Hint : sinθ = cos2θ
(a+b)3 = a3 + 3a2b + 3ab2 + b3)
61. 30√3 metres
62. 20 metres/sec.
63. 30(√3–1) metres
64. 5(2+√3) metres
65. 20(√3+1) metres
66. 2(√3+1) metres
67. 400 (3+√3) metres
68. (Hint : complementary angles)
69. 24√3 metres
70. Length of flag staff = 10 (√2–1) metres
Distance of the building = 10√3 metres
71. 15 metres, 15√3 metres
72. 18 minutes
73. 18√3 metres, 72 metres
74. 5(3+√3) metres, 5(3+√3) metres
75. 1000 (3–√3) metres
76. 1366 metres
78. 180 metres
80. 30 metres
(331) (Maths__Xth class)
Co-ordinate Geometry
Key Points
1.
The length of a line segment joining A & B is the distance between two points A(x1, y1) and
B (x2, y2) is {√(x2–x1)2 + (y2 –y)2}
2.
The distance of a point (x, y) from the origin is √(x2+y2). The distance of P from x-axis is y
units and from y-axis is x-units.
3.
The co-ordinates of the points p(x, y) which divides the line segment joining the points
A(x1, y1) and B(x2,y2) in the ratio m1 : m2 are
m1x2+m2x1 m1y2+m2y1
( m +m , m +m )
1
2
1
2
m1
we can take ratio as k:1, k = m
2
x1+x2
,
2
4.
The mid-points of the line segment joining the points P(x1, y1) and Q(x2, y2) is (
5.
y1+y2
)
2
The area of the triangle formed by the points (x1, y1), (x2, y2) and (x3, y3) is the numeric
1
value of the expressions [x1(y2–y3) + x2(y3–y1) + x3 (y1–y2)]
2
6.
If three points are collinear then we can not draw a triangle, so the area will be zero i.e.
x1(y2–y3) + x2(y3–y1) + x3(y1 –y2) = 0
1 Mark questions (Question 8-11 are under HOTS)
1.
What is the distance between (a1o) and (o1 b).
2.
What is the midpoint of the line segment joining the points (3, 4) and (11, 6) ?
3.
What is the value of a and b if (2, –3) is the mid point of the line segment joining (2, a) and
(b, –1) ?
4.
What is the area of the triangle joining the points (2, 4), (2, 0) and (2, –11) ?
5.
AB is the diameter of a circle with centre at origin. What are
the co-ordinates of B if co-ordinates of point A are (3, –4) ?
6.
What is the length of the side of the rhombus A(1, –2),
B(2, 5), C(–5, 4) and D(–6, –3) ?
7.
B
(–3, 3)
In the adjoining figure what is the length of AB ?
A(1, 0)
(332) (Maths__Xth class)
8.
What is the value of x if (3, 5) and (7, 1) are equidistant from T(x, o) ?
9.
What is the value of y if ar (∆ABC) = 0 and co-ordinates of vertices are A(1, 2), B(y, 6), C(1,
3) ?
10. Given a circle with centre at origin and radius 5√2 units. State where the point (5, –7)
lies?
11. A line is drawn through p(4, 6) parallel to x-axis what is the distance of the line from xaxis?
2 marks questions - (Question 26-28 are under HOTS)
12. Find x if the distance between the points (x, 2) and (3, 4) be √8 units.
13. Find the point on y-axis which is equidistant from the points (–2, 5) and (2, –3).
14. Find the co-ordinates of the point which divides the line segment joining the points (1, 3)
and (2, 7) in the ratio 3:4.
15. A and B are the points (1, 2) and (2, 3). Find the co-ordinates of a point G on the
AG
4
line-segment AB such that
= .
GB
3
16. The mid point of the line segment joining the points (5, 7) and (3, 9) is also the mid point
of the line segment joining the points (8, 6) and (a, b). Find a, b.
17. Find the distance between the points A(a, b) and B(b, a), if a – b = 4
18. Find the ratio in which the point (11, 15) divides the line segment joining the points (15,
5) and (9, 20).
19. Find the point of trisection of the line segment joining the points (–3, 4) and (1, –2).
20. NICE is a parallelogram whose three vertices taken in order are (–3, 1), (1, 1) and (3, 3).
Find the co-ordinate of the fourth vertex.
3
of the way from (3, 1) to (–2, 5).
4
22. Prove that we can draw the line passing through the points (0, 1), (3, 5) and (6, 9). (Show
that points are collinear).
21. Find the point which is
23. Find the area of the triangle whose vertices are (1, –1), (–3, 5) and (2, 7).
24. Find the value of k if (k, 1), (5, 5) and (10, 7) are Collinear.
25. The vertex of the triangle ABC are A(–1, 3), B(1, –1) and C(5, 1). Find the length of the
median drawn from the vertex A.
26. ∆ABC is an isosceles triangle with AB = AC and vertex A is on y-axis. If the co-ordinates of
vertex B and C are (–5, –2) and (3, 2) respectively then find the co-ordinates of vertex A.
27. The point K(1, 2) lies on the perpendicular bisector of line segment joining the points E(6,
8) and F(2, 4). What is the distance of the point K from the line segment EF.
28. Point P(k, 3) is the mid point of AB. If the distance AB = √52 units and co-ordinates of A
are (–3, 5) then find the value of k.
3 marks questions - (Questions 40-42 are under HOTS)
(333) (Maths__Xth class)
29. Find the abscissa of a point whose ordinate is 4 and which is at a distance of 5 units from
(5, 0).
A (5, 3)
30. In figure, CD is a median from the vertex C on the side AB of
DP
.
∆ABC, P is the point on CD such that DP = 1 unit. Find
PC
D
P
C
B(3, –1)
(7, –3)
31. If A(–3, 2), B(x, y) and C(1, 4) are the vertices of an isosceles
triangle with AB = BC. Find the value of (2x+y).
32. Find the ratio in which the line 3x + y = 12 divides the line segment joining the points
(1, 3) and (2, 7).
33. Prove that the figure obtained on joining the mid points of parallelogram PQRS is a square
where P(1, 0). Q(5, 3), R(2, 7) and S(–2, 4). Also find the sum of the diagonals.
34. A point P on the x-axis divides the line segment joining the points (4, 5) and (1, –3) in
certain ratio. Find the co-ordinates of point P.
35. In right angled triangle ABC, ∠B= 900 and AB = √34 unit. The co-ordinates of points B
and C are (4, 2) and (–1, y) respectively. If the ar (∆ABC) = 17 unit2 then find the value of
y.
36. If the point (6, 4) divides the line segment joining L (a,b) and M (8,5) in the ratio 2:5 then
find the value of a and b. Also find the co-ordinates of the mid point of ML.
37. The vertices of quadrilateral ABCD are A(–5, 7), B(–4, –5), C(–1, –6) and D(4, 5). Find the
area of the quadrilateral ABCD.
A (2, 1)
38. In figure D and E are the mid-points of the side BC
and AB respectively. Find the length of DE.
E
B (–6,–1)
D
C (4, –2)
39. Find the value of y such that ar (∆ABC) = 4 units2
and co-ordinates of the vertices are A(1, 2), B(3, y) and C(5, 2)
40. If the point P(3, 4) is equidistant from the points A(a+b, b–a) and B(a–b, a+b) then prove
that 3b–4a = 0
41. If the area of the quadrilateral PQRS is zero where P(1, –2), Q(–5, 6) R(7, –4) and 5(h, –2)
are the vertices then find the value of h. State are the points really making quadrilateral.
42. In figure find the radius of the circle.
(334) (Maths__Xth class)
y
2
1
–1
x
1
–2
1
2 3
–1
2
y1
ANSWERS
1.
√(a2 + b2)
2.
(7, 5)
3.
a=–5
b=2
4.
Zero
5.
(–3, 4)
6.
√50 unit
7.
5 unit
8.
x=2
9.
y =1
10. Outside
11. 6 unit
12. x = 1, 5
13. (0, 1)
14. (
10 33
,
)
7 7
15. (
11 18
,
)
7 7
16. a = 0, b = 10
17. 4√2 unit
(335) (Maths__Xth class)
x
18.
2:1
19. (
–1
–5
, 2), ( , 0)
3
3
20. (–1, 3)
–3
, 4)
4
23. 5 Sq. units
21. (
24. k = –5
25. 5 unit
26. (0, –2)
27. 5 unit
28. k = 0, –6
29. 2, 8
30. 1:4
3.1 1
32. 6:1
33. 1 0 √2
34. (
17
, 0)
8
35. – 1
36. a =
25
5
b=
18
5
(
66 43
,
)
10 10
37. 72 square unit
1
√13 unit
2
39. y - 0
38.
41. h = 3,
NO
42.
5√2
units
7
Triangles
(336) (Maths__Xth class)
Key Points
1.
Similar triangles :
Two triangles are said to be similar in their corresponding angles are equal and their
corresponding sides are proportional.
2.
Criteria for Similarity :in ∆ABC and ∆DEF
(i) AAA similarity ∆ABC ~ ∆DEF when ∠A = ∠D, ∠B = ∠E and ∠C = ∠F
(ii) SAS similarity : ∆ABC ~ ∆DEF when
AB BC AC
=
=
AND ∠A = ∠D
DE EF
DF
AB AC BC
=
=
DE
DF
EF
The proofs of the following theorems can be asked in the examination :(iii) SAS similarity : ∆ABC ~ ∆DEF
3.
(i)
Basic proportionality Theorems : If a line is drawn parallel to one side of a triangle to
intersect the other sides in distinct points, the other two sides are divided in the
same ratio.
(ii) The ratio of the area of two similar triangles is equal to the square of the ratio of their
corresponding sides.
(iii) Pythagoras theorem : In a right triangle, the square of the hypotenuse is equal to the
sum of the squares of the other two sides.
(iv) Converse of Pythagoras Theorem : In a triangle, if the square of ne side is equal to
the sum of the squares of the other two sides then the angle opposite to the first side
is a right angle.
Similar Triangles
P
1 Mark Questions
1.
PS
In fig(1) what SQ if ST||QR, PT = 8 cm and PR = 10 cm.
S
2.
In fig (2) is ∆ABC ~ ∆AQP
3.
In fig (2) if QC = 10 cm is it possible
that PQ||BC ?
(Fig. 1)
Q
(Fig. 2)
(337) (Maths__Xth class)
cm
12
P
B
Q
m
15c
18
cm
10
cm
A
T
C
R
4.
In ∆XYZ, P&Q are points on XY and XZ respectively
2
XP
= , XQ = 6 cm, QZ = 3 cm.
XY
3
What type of linesegments PQ & YZ are.
5.
In ∆PQR, S&T are points on PQ and PR such that ST||QR, PS = 2 cm, SQ = 3 cm, PR =
15 cm. What is the length of TR.
6.
An isoscels triangle ABC is similar to ∆PQR. AC = AB = 4 cm, PQ = 10 cm and BC = 6 cm.
What is the length of PR ?
l
13
5
0
x
y
In fig (3), l||m. what is the measurement of x ?
12
0
0
7.
m
(Fig. 3)
R
A
8.
In fig (4) ∆ABC ~ ∆PQR. What is the value of x ?
5
4
B
7.5
6
Q
x
CP
6
(Fig. 4)
A
x
8
9.
In fig (5), if DE||BC, what is the values of x ?
D
E
2
x
B
P
10. If the ratio of the corresponding sides of two similar triangles
in 4:5. What is the ratio of their areas ?
E
D
1
11. In fig (6) DE||QR and DE =
QR. How many times is PQ of
4
PD. (This can be asked as write PQ : QD)
Q
(338) (Maths__Xth class)
C
(Fig. 5)
(Fig. 6)
R
12. The length of median of an equilateral riangle is 3 cm. What is the length of its sides ?
PR
AB
1
13. In two triangles ABC and PQR if ∠B = ∠Q and PQ = , what is the value of QR ?
2
14. Measurement of three sides of a triangle are a, √10 a, 3a. What is the measurement of the
A
angle opposite to the longest side ?
c
10
m
15. If fig (7), DE||BC what is the value of DE.
D
m
6c
2 Marks Questions
2c
3 cm
B
P
E
x
C
(Fig. 7)
m
16. In fig (8) find SR.
Q
S
(Fig. 8)
R
9 cm
17. In ∆PQR, RS⊥PQ, ∠QRS = ∠P, PS = 5 cm, SR = 8cm. Find PQ
18. Two similar triangles ABC and PBC are made on opposite sides of the same base BC.
Prove that AB = BP
F
19. In fig (9) ABCD is a rectangle. ∆ADE and ∆ABF are two triangles.
D
Such that ∠E = ∠F.
Prove that
C
E
AD
AB
=
AE
AF
A
B
(Fig. 9)
A
20. In fig. (10) DE||BC.
If
ar (∆ADE)
3
AD
= . Find ar (∆ABC)
8
AB
E
D
21. In figure (10) DE||BC, DE = 3 cm, BC = 90 cm and ar
(∆ADE) = 30 cm2. Find ar (trap BCED)
(339) (Maths__Xth class)
B
(Fig. 10)
C
22. Amit is standing at a point on the ground 8 m away from a house. A mobile network tower
is fixed on the root of the house. Finds that the top and bottom of the tower are 17 m and
10 m away from the point. Find the heights of the tower and house.
AB
BC
= √3. Find
.
AC
AB
24. In a right angled triangle PRO, PR is the hypotanous and the other two sides are of length
6 cm and 8 cm. Q is a point outside the triangle such that PQ = 24 cm, PQ = 26 cm. What
is the measure of ∠RPQ ? How many such triangles PQR are possible ?
23. In a right angled triangle right angle at B,
25. ∠B and ∠ACD of a quadrilateral ABCD and right angles. Prove that AD2 = AB2 + BD2 +
CD2.
A
26. In figure (11) ∆ABC is isosceles with AB = AC.
Prove taht
BM
MP
=
NP
CN
N
M
B
P
(Fig. 11)
C
E
D
B
27. Find the length of the diagonal of rectangle BCDE (fig 12 (i))
if ∠BCA = ∠DCF.
A
C
5 cm
(Fig. 12 (i))
F
10 cm
A
28. In fig. (12 (ii) ∆BEF is a rectangle. C is the mid point of BD.
If AB = 16 cm, De = 9 cm, BD = 24 cm. AE = 25 cm.
F
E
Prove that ∠ACE = 900
B
A
D
2
P
C
2x+
x+2
(Fig. 12 (ii))
Q
3x
4 x+
29. In fig. (13) Find the value of x if PQ||BC
1
B
(Fig. 13)
C
30. PQRS is a trapezium. SQ is a diagonal. E & F are two points on PQ and RS respectively
interesting SQ at G. Prove that SG × QE = QG × SF.
(340) (Maths__Xth class)
A
31. In figure (14) prove that DE||BC.
ar (∆ADE)
Also find the ratio of or (trap BCED) .
4
Where D is the mid point of BC.
6
D
E
x
B
1.5
C
(Fig. 14)
32. In ∆ABC, EF||BC, such that EF passes through the controid G. Find
is the mid point of BC.
, where D
C
D
AD
CE
33. In fig. (15) D||BE and
=
.
DC
BE
Prove that PDCE is a parallelogram.
A
E
B
P
(Fig. 15)
34. In a quadrilateral ABCD, ∠A + ∠D = 900. Prove that AC2 + BC2 = AD2 + BC2.
S
A
35. In fig. 15(i) PQR and S are points on the sides of quadrilateral ABCD
such that these points divides the sides AB, CB, CD and AD in the
ratio 2:1.
Prove that PQRS is a parallelogram.
D
R
R
B
Q
(Fig. 15 (i))
C
36. Equiangular triangles are drawn on sides of right angled triangle in
which perpendicular is double of the base. Show that the area of the triangle on the
hypotenuse is the sum of the areas of the other two triangles.
37. In a rhombus prove that four times the square of any sides is equal to sum of squares of its
diagonals.
38. ABCD is a rectangle in which length is double of its breadth two equilateral triangles are
drawn one each on length and breadth. Find the ratio of their areas.
A
39. In fig. (16) ∠AEF = ∠AFE E is the mid point of CA.
B
Prove that BD ×CE = BF ×CD.
E
(Fig. 16)
C
D
40. Prove that if a line is drawn parallel to one side of a triangle, it divides the other two sides
in the same ratio. Rider’s based on above theorem
(341) (Maths__Xth class)
C
(i) in the adj. fig. (18) AB||DE, BD||EF. Find CD2.
F
E
D
A
D
(Fig. 18)
B
C
(ii) ABCD is a parallelogram (see fig.19. )
P
DP
DC
Prove that PQ = BQ
A
Q
B
(Fig. 19)
A
9
–1
3x
x–4
E
D
(iii) Find x, if DE||BC (fig. 20)
4
8
B
(iv) ABCD is a trapezium. Find value of x (fig. 21)
C
(Fig. 20)
D
C
3
3x
A
x–
–1
9
5
x–3
B
(Fig. 21)
41. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares
of their corresponding sides.
Use above result to prove the following
A
B
Riders :
ar ∆ABC
16
ar (trap BCED) = 49 , BC||DE
D
C
(Fig. 22)
E
What is the length of the altitudes to the bigger triangles if length of altitude to smaller
triangle is 8 cm (fig. 22)
(342) (Maths__Xth class)
42. In a triangle, if the square of one side is equal to sum of the squares on the other two sides,
the angles opposite to the first side is a right triangle.
Use above result to prove the following Rider;s
(i)
Dinesh, Naresh and Ashu are standing in such a way that distance between Dinesh
and Naresh is p meter . Naresh and Ashu are at a distance √(q2 –2) m from ech other.
Dinesh and Ashu are √(p2 + q2 –2) m apart. What type of triangle they are forming.
(ii) Three sticks of length (a–1) cm, 2√a cm and (a+1) cm are joined with their end pts. to
from a triangle. Do they form a right triangle. Show.
43. In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the
other two sides.
Rider based on aboe theorm.
Amar and Ashok are two friends standing at a corner of a
rectangular garden. They wanted to drink wter. Amar goes due
north at a speed of 50 m/min. and Ashok due west at a speed of
60 m/min. They travel for 5 minutes. Amar reaches the tap
and drink water. But somebody told Ashok to go towards point
C. As there was to other tap.
C
Find the min. distance Ashok has to travel to reach point C.
(fig. 23)
A
B
(Fig. 23)
Answers
2.
4
1
NO
3.
No
4.
Parrallel
5.
9 cm
6.
10 cm.
7.
750
8.
9 cm
9.
4 cm
1.
10. 16:25
11. 4
12. 2 √3
13.
1
2
(343) (Maths__Xth class)
14. Right angle
15. 2.5
16. 4
17. 12.8 cm
18. ---19. ---20.
9
64
21. 60 cm2
22. 9 m, 6 m
23.
√3
2
24. Right angle, Two
25. ---26. ----27. 5 √1 0
28. ---29. 2
30. ---31.
16
9
32.
4
9
33. --34. --35. --36. --37. --38. 1:4
39. --40. (i) CF × CA, (ii) ---, (iii) –11, (iv) 8 or 9
41. 14 cm
42. (i) right angled triangle (iv) yes
43. 50 √61 m
(344) (Maths__Xth class)
Statistics
Key Points
1.
The mean for grouped data can be found by :
(i) The direct method = X =
2.
fixi
fi
(ii) The assumed mean method = X = a +
fidi
, where di = xi – a
fi
(iii) The step deviation method = X = a +
fiui
x–a
× h , where ui = i
h
fi
The mode for the grouped data can be found byusing the formula :mode = l +
f1 – f0
×h
2f1 – f0 – f2
l = lower limit of the model class.
f1 = frequency of the model class
f0 = frequency of the proceeding class of the model class.
f2 = frequency of the succeeding class of the model class
h = size of the class interval.
Model class - class interval with highest frequency.
3.
The median for the grouped data can be found by using the formula :median = l +
n –Cf
2
f
×h
l = lower limit of the median class.
n = number of observations
Cf = cumulative frequency of class interval preceeding the median class.
f = frequency of median class.
h = class size.
4.
Imperical Formula :Mode = 3 median - 2 mean
5.
Cumulative frequency curve or an Ogive :(i) Ogive is the graphical representation of the cumulative frequency distribution.
(345) (Maths__Xth class)
(ii) Less than type Ogive :* Construct a cumulative frequency table
* Mark the upper class limit on the x = axis.
(iii) More than type Ogive :* Construct a frequency table
* Mark the lower class limit on the x-axis.
(iv) To obtain the median of frequency distribution from the graph :* Locate point of intersection of less than type Ogive and more than type Ogive :Draw a perpendicular from this point to x-axis.
* The point at which it cuts the x-axis gives us the median.
Statistics
I mark Questions (Question12-15 are under HOTS)
1
What is the median of the following distribution
2,3,6,0,1,4,8,2,5
2
What is the Mean if Median= 14 and Mode=12
3
What is the mean of x, x+1, x+2, x+3, x+4
4
The following table shows the frequenay distribution of the marks of 50 students What is
the value of y.
class interval
frequency
0-5
5-10
10-15
15-20
20-25
8
11
13
y
10
5
Write the class mark of the class interval 16.5-21.5
6
A teacher ask the student to find the average marks obtauned by most of the Students.
What the student will find: Mean, Mode or Median.
7
What is the mode of the following data.
1, 0, 2, 2, 3, 1, 4, 5, 1, 0,
8
In the following distribution, Write the modal class
Class interval 10-15
Frequency
4
15-20
20-25
25-30
30-35
7
20
8
1
9
For the frequency distribution ∑fi=40 and ∑fixi=2440 What is the mean of the
distribution.
10
A teacher ask the student to find the average marks obtained by the class student in
Mathematics What the student will find: Mean Mode or Median.
(346) (Maths__Xth class)
11
The following data is arranged in the ascending order
11,12,12,13,7x,7x+1,16,16,18,20 if the median of data is 14.5,what is the value of x
12
What is the value of the median of the data using the following qraph of “less then ogive’’
and “ More than ogive’’
13
The following “ More than ogive’’ Shows the weight of 40 Student of a class. What is the
lower limit of the Median class.
(347) (Maths__Xth class)
14. From the cumulative frequency table Write the frequency of the class 20-30
Marks
Number
of student
less than 10
1
less than 20
14
less than 30
36
less than 40
59
less than 50
60
15. Following is a commulative frequency curve for the marks obtained by 20 Student as shown
in find the Median marks obtained by the student.
6 Marks Questions (Question No.24, 25, 26, 27 are under HOTS)
16. Find the mean of the following frequency distribution:
Class interval 18-24
Frequency
6
24-30
30-36
36-42
42-48
48-54
10
12
8
4
2
17. Find the value of p, if the mean of the following distribution is 20
x
15
17
19
20+p
23
f
2
3
4
5p
6
18. The mean of the following frequency distribution is 62.8 and the sum of all the frequencies
is 50. Find the values of p and q.
Class-Interval
Frequency
0-20
20-40
40-60
60-80
80-100
100-120
5
p
10
q
7
8
(348) (Maths__Xth class)
19. The following frequency distribution gives the daily wage of a worker of a factory: Find the
mean daily wage of a worker.
Daily wage
No. of
(in Rs)
Workers
More than 300
0
More than 250
12
More than 200
21
More than 150
44
More than 100
53
More than 50
59
More than 0
60
20. A retailer stocks car batteries of a particular type. The lives (in years) of 40 such batteries
are recorded in the following frequency distribution:
Life of
2.0-2.5
2.5-3.0
3.0-3.5
3.5-4.0
4.0-4.5
4.5-5.0
2
6
14
11
4
3
batteries in Yrs.
Number
find the modal life of a battery in years.
21. The following frequency distribution shows the marks obtained by 100 students in a school.
Find the mode.
Marks
Number of students
less than 10
10
less than 20
15
less than 30
30
less than 40
50
less than 50
72
less than 60
85
less than 70
90
less than 80
95
less than 90
100
22. The following frequency distribution gives the weights of 30 students of a class. Find the
median Weight of the students.
Weight (in kg)
40-45
45-50
50-55
55-60
60-65
65-70
70-75
No.of persons
2
3
8
6
6
3
2
23. A life insurance agent found the following data for distribution of ages of 150 policy holders.
calculate the median aqe. if policies are only given to persons having age 18 years. Onwards
(349) (Maths__Xth class)
but less than 60 year .
Age (in years)
number of policy holder
Below 20
8
Below 25
10
Below 30
32
Below 35
59
Below 40
105
Below 45
125
Below 50
138
Below 55
148
Below 60
150
24. The median of the following frequency distribution is 28.5 and sum of all the frequencies
is 60. Fund the value of x and y.
Class interval
frequency
0-10
10-20
20-30
30-40
40-50
50-60
Total
5
x
20
15
y
5
60
25. Draw “More than ogive” for the following distribution:
Class Interval 50-55
Frequency
2
55-60
60-65
65-70
70-75
75-80
8
12
24
38
16
Also find median from the graph of ogive verify that by using the formula.
26. Draw “less than” ogive for the following distribution.
Marks
0-10
Number
5
10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
3
4
3
3
4
7
9
7
8
of Students
Find the median from the graph.
27. A survey regarding the heigh (in cm) of 50 girls of class × of a school was conducted and
regarding the followig data was obtained.
Height in cm
Number of
120-130
130-140
140-150
150-160
160-170
Total
2
8
12
20
8
50
girls
Find the mean, median and mode of the above data.
(350) (Maths__Xth class)
ANSWERS
1.
I
2.
15
3.
x+2
4.
y=8
5.
19
6.
Mode
7.
1
8.
20-25
9.
61
10. Mean
11. x=2
12. 40
13. 147.5
14. 22
15. 20
16. 33
17. p=1
18. P=8 q=12
19. Rs 182.50
20. 3.364 years
21. 41.82
22. 56.67 Kg
23. 35.93 years
24. x=8,y=7
25. 71.58
26. 66
27. Mean=149.8 cm
Median=151.5 cm
Mode=154 cm
(351) (Maths__Xth class)
Probablity
Key points
1.
The Theoretical probablity of an event E written as (E) is .
Number of outcomes favourable to E
P(E)=
Number of all possible outcomes of the experiment
2.
The sum of the probability of all the elementary events of an experiment is 1.
3.
The probability of a sure event is 1 and probability of an impossible event is o.
4.
if E is an event, in general, it is true that P(E) + P(E) = 1
5.
From the definition of the probability, the numerator is always less than or equal to the
denominator therefore O ≤P(E) ≤ 1
PROBABILITY
1 mark questions (* questions are under HOTS)
1.
Two coins are tossed once. Write the sample space (possible out comes)
2.
E is an event such that P(E) = 0.32 What is P(E)?
3.
If probability of success is 52% What is the probability of failure?
4.
A bag contains 4 blue and 5 green balls. What is the probability of getting a red ball if a ball
is drawn randomly from the bag.
5.
E is an event such that P(E) =1/5 what is P(E).
6.
A bag contain 9 Red and 6 Blue marbles A marble is taken out randomly,, What is the
probability of getting red marble.
7.
A game of chance of a spinning wheel has numbers 1 to 10 What is the probability of
getting a number less than and equal to 7 when wheel comes to rest.
8.
In a class of 40 students there are 25 girls and 15 boys. What is the probability of getting a
boy selected as class monitor.
9.
In a survey it is found that every fourth person possess a vehicle what is the probability of
person not possessing the vehicle.
10. During IPL Cricket tournament a match is played between Delhi dare Devils and Knight
Riders If probability of Dare Devils winning the match is 0.579 What is the probability of
Knight Riders Winning the match.
11. Only face cards are well shuffled A card is drawn at random What is the probability of
getting a red queen? (there are 12 face cards)
12. Humanshu and Sachin are friends What is the probability that they both have birthday
on 13th April (ignoring leap year)
(352) (Maths__Xth class)
13. A dice is thrown once. What is the probability of getting an “even prime number”?
14. In the coloured water 3 coke ,5 orange and 7 Nimbuz bottles are kept. What is the probability
of getting a bottle of coke?
15. Two dice are rolled once What is the probability of getting a doublet.
16. In a family there are one son and two daughters What is the probability of son being the
youngest?
17. In the word “success” What is the probability of getting the letter ‘S’?
18. A dice is rolled once. What is the probability of getting a prine number?
19. A garden has different plants. Every sixth plant is a rose plant. A child plucks a flower.
What is the probability of getting a flower other than rose.
20. A bank ATM gas notes of denomination 100, 500, 1000 in equal numbers What is the
probability of getting a note of Rs 1000.
21. In a film fare award, SHAHRUKH KHAN is nominated in the Best Actor category If there
are five nominations then what is his probability of winning an award.
22*. What is the probability of getting a number greater than 6 in a single throw of dice.
23. A selection commiltee interviewed 50 people for the post of sales Manager. Out of which
35 are males and 15 are females. What is the probability of a female candidate being selected.
2 mark questions (Questions 34-40 are under HOTS)
24. Two dice are rolled simultaneously. Find the probability that the sum is more than and
equal to 10.
25. From the well shuffled pack of 52 cards, Two Black Kunqe and two red Ace ‘s are reinoved
What is the probability of getting a face card.
26. A box contains green and red balls. The probability of drawing green ball is thrice the
probability of drawing red balls. Find the number of green balls it the total number of
balls in box is 28.
27. A box contain cards marked with the number 2to17. A card is drawn randomly. What is
the probability that the card contains the number between 8 and 14.
28. A box contain card marked with the number 2 to 101. One card is drawn at random. What
is the probability of getting a number which is a perfect square.
29. A carton consist of 100 pens, out of which 65 are good one, 15 have minor defects and
remaining have major defects what is the probability that a man will pick up a pen with
major defects.
30. A box contains orange, mango and lemon flavoured candies A candy is drawn randomly. If
P(not lemon) = 11/15 and P(mango)=1/3 then what is P(orange)?
31. A box contain cards which are numbered from 1to 20 The cards containing prime numbers
are removed from the box. A card is drawn randomly What is the probability of getting a
card containing even number.
32. From the well shuffled pack of 52 cards . few cards of same colour are missing et P(Red
(353) (Maths__Xth class)
card)=1/3 and P(Black card) =2/3 then which colour of cards are missing and how many?
33. In a beginning of session there were 40 students in Class X out of which 19 were girls and
21 were boys. 7 students got promoted after compartmental exam.out of which 4 were
girls and 3 boys who gas the more probability of getting selected as the class monitor?
What is the difference between their probabilities.
BLUE
34. An archery target marked with its three scoring are as from
the centre outward as Gold, Red, Blue The diameter of
the region representing Gold score is 21 cm and each of
the other band as 10.5cm wide it the archer hit the gold
area he is declared winner. What is the probability of
winning the game.
RED
GOLD
35. Four Jacks are missing in a well shuffted pack of 52 cards.
A child could find only 2 red jack and reshuffled it back.
what is the probability of getting a face card.
36. A meeting of officials is to last for 2 hrs. What is the probability of getling it over in 1 hr. 20
min?
D
C
LAND
A
2 KM
JUNGLE
6 KM
38. A child organises the game in which if you drop a coin
in the circle, you win the game as shown in the figure.
What is the probability of winning the game?
1
39. A bag contains 12 balls out of which x are black. If 6
more black balls are put in the box, the probability of
drawing a black ball is double of what it was before. Find x.
(354) (Maths__Xth class)
3 UNIT
B
2 UNIT
4.5 KM
37. An aeroplane has to do an emergency landing in
the rectangular area as shown in the fiqure. What
is the probability that it lands safely?
4 KM
40. A bag contain few black and red marbles The number of red marbles are two more than
half the black marbles. If total number of marbles are 14, then find the probability of black
and red marble.
ANSWERS
1)
HH,HT,TH.TT
2)
o.68
3)
48% or 12/25
4)
zero
5)
4/5
6)
3/5
7)
7/10
8)
3/8
9)
3/4
10) 0.421
11) 1/6
12) 1/365
13) 1/6
14) 1/5
15) 1/6
16) 1/3
17) 3/7
18) 1/2
19) 5/6
20) 1/3
21) 1/5
22) 0
23) 3/10
24) 1/12
25) 5/24
26) 21
27) 5/16
28) 9/100
29) 1/5
30) 2/5
(355) (Maths__Xth class)
31) 3/4
32) Red,13
33) Boys,1/47
34) 1/9
35) 1/5
36) 2/3
37) 5/27
38) 11/21
39) 3
40) P(B)=4/7
P(R)=3/7
(356) (Maths__Xth class)
Mensuration (Continued)
Surface Areas and Volumes
Key Points
1.
Total Surface area of cube of side a units = 6a2 units.
2.
Volume of cube of side a units = a3 cubic units.
3.
Total surface area of cuboid of dimensions l, b & h = 2(l ×b+b×h+h×l) square units
4.
Volume of cuboid of cylinder of dimensions l, b and h = l × b ×h cubic units.
5.
Curved surface area of cylinder of radius r and height h = 2πrh square units.
6.
Total surface area of cylinder of radius r and height h = 2πr(r+h) square units.
7.
Volume of cylinder of radius r and height h = πr2h cubic units.
8.
Curved surface area of cone of radius r height h and slant height l = πrl square units
where
l = √r2 + h2
9.
Total surface area of cone = πr(l+r) sq. units.
10. Volume of cone =
1 2
πr h
3
11. Total curved surface area of sphere of radius r units = 4πr2 sq. units.
12. Curved surface area of hemisphere of radius r units 2πr2 sq. units.
13. Total surface area of hemisphere of radius r units = 3πr2 sq. units
14. Volume of sphere of radius r units
4 3
πr cubic units.
3
15. Volume of hemisphere of radius r units
2 3
πr cubic units.
3
16. Curved surface of frustum = πl(r+R) sq. units, where l slant height of frustum and radii of
circular ends are r & R.
17. Total surface area of frustum = πl(r+R) + π(r2 + R2) sq. units.
18. Volume of Frustum =
1
πh(r2 + R2 + rR) cubic units
3
(357) (Maths__Xth class)
MENSURATION
QUESTION
1 mark questions (use, π =
22
)
7
1.
The Perimeter of a protractor (semi- circle) is 66 cm. What is its radius.
2.
A wire is in the form of a circle of diameter 21cm. It is cut and bent to form a square. Write
the measure of the side of square.
3.
The numerical difference between circumference and diameter is 30cm. What is the radius
of the circle.
4.
Write the Perimeter of the adjoining figure; Where AED is a semicircle and ABCD is a
rectangle.
D
A
E
20 cm
B
14 cm
C
5.
What is the Perimeter of a sector of angle 450of a circle With radius 7cm.
6.
A chord of a circle of diameter 28cm subtends an angle of 600. at the center. Write the area
of the major sector.
7.
A Pendulam of Wall clock swings through an angle of 300. describes an arc of length 8.8cm.
Write length of Pendulam.
8.
From each vertex of trapezium a sector of radius 7cm has been cut off. Write the total area
cut off.
9.
Write the ratio of the areas of two sectors I & 111.
III
120
150
10. If an are forms 450 at the centre ‘
circumference of the circle?
0
0
O’ of the circle. What is the ratios of its length to the
11. How many cubes of side 4cm can be cut from a cuboid measuring (16×12×8) cm3 ?
(358) (Maths__Xth class)
12. The diameter and heigth of a cylinder and a cone are equal. What is the ratio of their
volume?
13. A cylindrical tank has a capacity of 6160m3. What is its depth, it the radius of the base is
14cm?
14. A cylinder, a cone and a hemisphere are of equal base and have the same height. What is
the ratio in their volumes?
15. A rectangular sheet of Paper 66cm.×18cm. is rolled along its length and a cylinder is
formed. Write the curved surface area of the cylinder.
16. A metallic sphere of total volume πcm3, is melted and recast in to the shape of a cylinder of
radius 0.5cm. What is the height of cylinder?
17. The length of minor are is 7-20 of the cirumference of the circle. Write the angle subtended
by the arc at the centre.
18. A bicycle Wheel make s 5000 revolutions in_ moving 10km. Write the perimeter of wheel.
19. Curved surface area of a right circular cylinder is obtained on multi plying volume by ‘K’.
Write the value of K.
20. The sum of the radius of the base and the height of a solid cylinder is 15cm. If total surface
area is 660m2, Write the radius - of the base of cylinder.
21. Find the height of largest right circular cone that can be cut out of a cube whose volume is
729 cm3.
22. What is the ratio of the areas of a circle and an equilateral triangle Whose diameter and a
side are respectively equal?
23. A solid sphere of radius ‘s’ is melted and cast in to the shape of a solid cone of height ‘s’
What is the radius of the base of the cone.
24. If the circumference of the circle exceeds its diameter by 30cm. What is the diameter of
the circle.
25. The length of an are of a circle of radius 12cm is 10πcm. Write the angle measure of this
are.
(3 Marks Questions ) : (Que 42-51 under HOTS)
26. Find area of the shaded region shown in the adjoining
figure where ABCD is a square of side 10cm and semi
circles are drawn with each side of the square as
diameter. (useπ=3.14)
D
C
10 cm
A
(359) (Maths__Xth class)
10 cm
B
27. Ι n the figure given along side, Find the area of the shaded portion?
A
B
5 cm
12 cm
C
D
28. Find the area of the shaded portion; If side of square is 28cm?
A
B
12 cm
C
D
29. Find the shaded area of the figure- along side et diagonal of square is 28cm?
7 cm
7 cm
7 cm
7 cm
7 cm
30. If AOB is a quadrant of a circle of radius 14cm. Find the area of shaded region?
14 cm
A
O
B
(360) (Maths__Xth class)
31. Find the area of the shaded region when PQ=QR=RS=14cm(student can find Perimeter
of this figure also)
P
R
S
Q
32. Find the volume of the largest right circular cone that can be cut out of a cube whose
volume is 729 cm2?
33. A hemi spherical bowl of internal radius 15cm contains a liquid. The liquid is to be filled
in to cylindrical shaped bottles of diameter 5cm and height 6cm. How many bottles are
necessary to empty the bowl?
34. The base radii of two right circular cones, of the same height are in the ratio 3:5 Find the
ratio of their volumes?
35. The dimensions of a metallic caboid are 100cm×80cm×64cm.If it is melted and recast in
to a cube; Find the surface area of the cube.
36. Three equal cubes are placed adjacently in a row. Find the ratio of the total surface area of
the new cuboid to that of the sum of the surface zreas of the three cubes.
37. A sheet of paper is in the form of a reetangle ABCD in which AB=40cm and AD= 28cm. A
semicircular portion with BC as diameter is cut off. find the area of the remaining paper.
38. A bicycle wheel makes 500 revloutions in moving 11km. find the diameter of the wheel.
39. Two circles touch externally. The sum of their areas is 130πsq cm and the distance between
their centres is 14cm. find the radii of the circles.
40. AB is a chord of a circle of radius 10cm. The chord subtends a right angle at the centre of
the circle. find the area of the major segment.(take π=3.14)
41. The inner cirumference of a circular track is 440cm. The track is 14cm wide. find the
castof leveling it at 20 paise/ sam. Also find the cast of putting up a fencing along outer
circle at Rs2/ metre.
42. A hemispherical bowl of internal diameter 36 cm is full of liquid. Thus liquid is to be filled
in cylindrical bottles of radius 3 cm and height 6 cm. How many bottles are required to
empty the bowl?
43. The area of equilateral triangle is 49√3 cm2. Taking each angular Points as centre, a circle
is described with radius equal to half the lenght of the side of the triangle. find the area of
triangle not included in the circle?
44. AB is a chord of a circle of radius 10cm. The chord subtends a right angle at the centre of
the circle. find the area of the minor segment?
(361) (Maths__Xth class)
45. AB and CD are two perpendicular diameters and CD= 8cm. find the shaded region?
A
D
O
C
B
46. If the area of four sectors having same radius is 616 sq cm. What is the radius?
r
r
120
0
0
60
r
r
47. A cylindrical pipe has inner diameter of 4cm and water flows through it at the rate of 20m.
per minute. How long would it take to to fill a conical tank. whose diameter of base is 80cm
and depth 72cm?
48. Sum of the areas of two squares is 468πm2. If the difference of their perimeter is 24cm.
find the side of the two squares.
49. PQRS is a diameterof a circle of radius 6cm, The lengths
PQ, QR and RS are equal, semi circles are drawn on PQ
and QS as diameter as shown in the figure. Find the
perimeter of the shaded region.
P
R
Q
S
A
3
(362) (Maths__Xth class)
cm
B
4
cm
50. In the adjoining figure ABC is a right angled
triangle, right angled at A. semi circle are drawn
on AB;AC and BC as diameters. Find the area
of shaded region.
C
51. A small torch can spread its light over a seetor of angle 900 up to a distance of 14cm. Vishal
Raj put two torches of same type at opposite vertices of a square of side14 cm as shown in
figure. find the area which get light from both sides.
6 Marks Question (Question, 70-74 under HOTS)
52. The rain water from a roof 22m×20m drains into a cylindrical vessel having diameter by
base 2m and height 3.5m et the vessel is just full; find the rainfall in cm.
53. A cylinder whose height is two third of its diameter has the same volume as a sphere of
radius 4cm. calculate the radius of the radius of the base of the cylinder.
54. 50 circular plates, each of radius 4 cm, and thickness 1/2 cm. are placed one above the
other to form a solid right circular cylinder. find the total surface area and volume of the
cylinder so formed.
55. Marbles of diameter 1.4cm are dropped in to a cylindrical beaker of diameter 7cm containing
some water find the number of Marbles dropped so that water level rises by 5.6cm.
56. A bucket is in the form of a frustum of a cone and holds 28.49 litres of milk. The radii of the
top and bottom are 28cm and 21cm respectively find the height of the bucket.
57
A student Navneet, developed a model of a hut. He made a cylinder of radius 6cm and
height 6 cm and attached a hemisphore of radius new at its top.He left an open area for
door on the cylinder of length 4cm and breath 3cm. find the surface area of the hut.
58. The slant height of right circular cone is 10cm and its height is 8cm. It is cut by a plane
parallel to its base passing through the mid point of the height find ratio of the volume of
two parts.
59. A cylindrical piece of wax of radius 2.8cm and height 2cm is placed in a metallic
hemispherical bowl of diameter 7cm. If the wax melts into the bowl, will the bowl over
flow. If yes, how much wax over flows. If not how much more can be accomodated.
60. The interior of a building is in the form of a right circular cylinder of radius 7cm and
height 6cm, surmounted by a right circular cone of same radius and of vertical angle 600.
Find the cast of painting the building from inside at the rate of Rs30/m2.
61. A container, shaped like a right circular cylinder, having diameter 12cm. and height 15cm
is full of ice-cream. This ice-cream is to be filled in to cones of height 12cm and diameter
6cm, having a hemi spherical shap on the top, find the number of such cones which can be
filled with ice-cream.
62. A solid spherical ball of the metal is divided in to two hemispheres and joined as shown in
the figure,This solid is placed in a cylindrical tub, full of water in such a way that the
(363) (Maths__Xth class)
whole solid is sub merged in water. the radius and height of cylindrical tub are 4cm and
5cm respectively.the radius of spherical ball is 3cm. find the volume of water left in the
cylindrical tub.
63. A gulab jamun, contains sugar syrup up to about 30% of its volume. find approximately
how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder, with two
hemispherical shaped like a cylinder, with two hemispherical ends, with lenght 5cm and
its diameter 2.8cm.
64. Water flows out through a circular pipe whose internal radius is 1 cm, at the rate of 80cm/
second in to an empty cylindrical tank, the radius of whose base is 40cm. By how much
will the level of water rise in the tank in half an hour?
65. A solid is in the form of a right circular cone mounted on a hemisphere. The radius of the
hemisphere is 2.1cm and height of cone is 4cm. The solid is placed in a cylindrical tab, full
of water,in such a way that the whole solid is submerged in water. If the radius of cylinder
is 5cm and height 9.8 cm. find the volume of water left in the tub.
66. The height of a solid cylinder is 15 cm and diameter 7cm. Two equal conical holes of radius
3 cm and height 4cm are cut and surface area of remaining solid.
67. Three cubes of metal whose edge are in the ratio 3:4:5 are melted in to a single cube whose
diagonal is 12√3cm find the edge of the three cubes.
68. A toy is in the form of a cone mounted on a cone frustum.et
the radius of the top and bottom are 14cm and 7cm. find the
volume and height of cone is 5.5 cm. find the volume of the
where the height of toy is 10.5cm.
10.5 cm
(364) (Maths__Xth class)
69. A tent is in the shape of a cylinder surmounted by a conical top. it the height and diameter
of the cylinder part are 2.1m and 4m respectively and the slant height of the top is 2.8m,
find the area of the canvas used for making the tent find the cast of the canvas of the tent
at the rate of Rs 500 per m2 Also find the volume of air enclosed in the tent.
70. In the adjoining figure, ABC is a right triangle, right anglod at A. find the area of shaded
region it AB = 6cm, BC= 10cm and o is the centre of the incircle of ∆ABC.(Take π= 3.14)
A
C
71. From solid cylinder of height 28cm and radius 12cm, a conical cavity of height 16cm, and
radius 12cm, is drilled out. find (a) the volume (b) total surface area of remaining solid.
72. A conical vessel of radius 6cm and height 8cm is completely filled with water. A solid sphere
is immersed in to it and its size is such that when it touches the sides, it is just immerged.
Find the quantity of water over flows?
6 cm
8 cm
1m
73. Two maths teacher observed six pillars to support shades of metro
station at Kashmiri Gate, which are of the shape as given in the figure.
One of them said that the approximately height and radius of
cylindrical portion are 1 m and 1/2 m respectively.The total height of
the pillar is approximately 3-1/2 m She asked the other teacher. how
much money is required to construet one such pillar it the current
rate of construction is Rs 3000 per cubic metre.
2½ m
½m
74. An orange contains juice about 15% of its volume. Find approximately how many dozen
oranges are required for a gathering of 50 people of each guest is to be served with 250 ml
juice. Assuming that radius of each orange is 3.5m.
(365) (Maths__Xth class)
MENSURATION (Answers)
1.
21cm
2.
16.5cm
3.
r=cm
4.
76cm
5.
19.5 cm
6.
513 1/3sq.cm
7.
16.8cm
8.
154 cm2
9.
4:3
10. 1:8
11. 24
12. 3:1
13. 10m
14. 3:1:2
15. 1188 sq.cm
16. 4cm
17. 126
18. 2m
19. 2/r
20. 7cm
21. 27cm
22. π : √3
23. 2 r
24. 14cm
25. 150
26. 57 cm2
27. 1019/14 sq cm
28. 154 sq cm
29. 308 sq cm
30. 77sq.cm
31. 462 sq cm
32. 190 13/14 cm2
33. 60
34. 9:25
(366) (Maths__Xth class)
35. 38400 sq cm
36. 7:9
37. 812 sq cm
38. 70cm
39. 11cm,3cm
40
28.5 sq cm
285.5sq cm
41. Rs.1355.20
Rs. 1056
42. 72
43. (49 √3 – 77) cm2
44. 28.50 sq cm
45. 108/7 dq cm
46. 14cm
47. 4 minute 48 secend
48. 18,12
49. 12π
50. 6 sq cm
51. 112 sq cm
52. 2.5 cm
53. 4 cm
54. 1408 sq cm, 3850 cm3
55. 150
56. 15cm
57. 563.14 sq cm
[HINT: total Area= 2πR2 + 2πrh + π(R2– r2) – L×B
58. 8:7
59. No, 39.55 cm3
60. 17160
[HINT: to Find slant height of cone use sin 300]
61. 10
[HINT: volume of ice cream in one cone=volume of cone+ volume of hemi sphere]
62. 138.29 cm2
63. 338.17 cm2
(367) (Maths__Xth class)
[HINT. volume of Gulab jamun= volume of cylindrical postion+2× volume of hemisphere]
64. 98m
65. 732.12 cm3
66. 502.07cu.cm, 444.17 sq cm
67. 6cm,8cm,10cm
68. 29 26 cu cm
69. 44 sq m Rs 22000,34.57 cu.m
70. 11.44 sq.cm
HINT: join o to A, B and C are of ABC= ar OAB+ ar. OBC+ ar.OAC 24=1/2 AB×r +1/2 BC×r+
1/2 × AC×r [r=2]
71. 10258 2/7 cu.cm, 3318 6/7 cm3
72. 113.14 cu.cm
[HINT: AB=10cm
AD× AC=AE2
73. Rs16 110
74. 39 Dozen
(368) (Maths__Xth class)
Circle
Key points
1.
Tangent to a circle : It is a line that intersects the circle at only one point.
2.
There is only one tangent at a point of the circle.
3.
The proofs of the following theorems can be asked in the examination :
(i)
The tangent at any point of a circle is perpendicular to the radius through the point
of contact.
(ii) The lenghts of tangents drawn from an external point to a circle are equal.
1 mark questions
1.
In fig(i) PS = 8 cm, SR = 6 cm. What is the value of PQ ?
2.
For fig (i) which one is true
(i)
PS2 + SR2 = PR2
(ii)
SR2+ RP2 = SP2
(iii)
RQ2 + PQ2 = PR2
(iv)
RQ2 + PS2 = SR2
S
R
P
Q
Fig. (1)
3.
In fig(i) if ∠RPS = y + 180 & ∠PRS = 24, what is the value of y.
4.
A tangent is drawn to a circle of radius 8 cm from a
point which is 17 cm from the centre. What is the
length of the tangent.
C
B
D
5.
A
If perimeter of DABC is 16 cm, what is the value of
AC + AE. (fig. 2)
F
E
Fig. (2)
6.
In fig. (2) what is the perimeter of quadrilateral ACDE,
if CD = 8 cm, AD = 17 cm.
7.
In fig (3) PR = 7.6 cm. What is the length of QS.
8.
Two tangents AB and AC are drawn to a circle from
outside point A. If AB + AC = 14.8 cm. What is the
length of AB.
(369) (Maths__Xth class)
P
Q
S
Fig. (3)
R
9.
A
In fig (4) if ÐB = 300 what statement is true
(a) AC > CB, (ii) AC = CB, (iii) AC<CB
0
30
D
B
C
Fig. (4)
B
10. In fig. (5) ∠BCD = 1000, what is the measure of x
x
C
11. In fig. (5) if ∠BCA = 2x and ∠BAC = x, what is the
measure of x.
A
D
Fig. (5)
12. AB is tangent to O circle of length equalto radius BC. What is the measure of ∠BCA.
P
y
y
13. In fig (6) PB is diameter and Qp is tangent. What is the
value of X ?
A
Q
x
350
B
Fig. (6)
14. In fig (7), what is the length of
AD.
17 cm
B
A
Fig. (7)
m
F
D
8c
3cm
4cm
C
E
2 marks question
15. An incircle is drown touching the equal sides of an isosceles triangle at E&F. Show that
the point D. Whee the circle touches the third side is the mid point of that side.
16. The length of tangent to a circle of radius 2.5 cm from external point P is 6cm. Find the
distance of P from the nearest point of the circle.
(370) (Maths__Xth class)
17. TP & TQ are the tangents from the external point of circle with centre O. If ∠DPQ = 300,
then find the measure of ∠TQP
A
4cm
18. In fig. (8) AP = 4 cm, BQ = 6 cm and AC = 9 cm. Find the semi
perimeter of ∆ABC
4.2 cm
P
R
C
S
19. In fig. (9) find SR
9 cm
A
Q
R
B
6 cm
(Fig. 8)
D
B
7.3 cm
C
6.5 cm
Q
(Fig. 9)
20. In fig. (10) A semi circle is drown outside the
semicircle. Diameter BE of the smaller semi circle is
half the radius BF of the bigger semi circle. If radius
of bigger semi circle is 4√3 cm. Find the length of
tangent AC from A on a smaller semi circle.
B
D
F
E
C
A
(Fig. 10)
A
21. If ∠P = 1200. Prove that OP = 2AP (fig. 11)
P
O
Fig. (11)
A
x–1
3 Marks Question
x
22. In fig. (12) find x, if PA and PB are tangents from P.
x+1
O
B
Fig. (12)
(371) (Maths__Xth class)
P
23. PA & PB are two tangents to a circle from an external
point
P.
Prove
that
perimeter
(∆PAB) = 2PA or 2 PB
B
P
24. In fig. (13) the radii of two concentric circles are 5
cm and 7 cm. Lengthof tangent from P to bigger circle
is 15 cm. Find length of tangent to smaller circle.
O
A
A
Fig. (13)
R
B
25. In fig (14) AB = 13 cm, BC = 7 cm, AD = 15 cm. Find
PC.
4 cm
S
Q
C
P
Fig. (14)
D
26. An incircle in drawn touching the sides of a right angled
triangle the base and perpendicular of the triangle are 6 cm and 2.5 cm respectively. Find
radius of the circle.
27. AP and AP are tangents to a circle with centre O from external point A of ∠OPQ = 300,
prove that ∆APQ is equiangular triangle.
28. On the side AB as diameter of a right angled triangle ABC a circle is drawn intersecting
hypotenuse AC in P. Prove that PB = PC
A
E
29. In fig. (15) AB = AC, D is the mid point of AC and BD is the
diameter of the circle.
Prove that AE =
D
B
1
AC
4
Q
30. In fig. (16), PQ is tangent AP = 8 cm,
A
O
length of tangent exceeds the radius by 1.
Find radius of the circle.
Fig. (16)
(372) (Maths__Xth class)
C
Fig. (15)
B
P
31. A chord AB of 8 cm is drawn in a circle with centre O of radius 5 cm. Find length of
tangents from external point P to A & B.
32. PQ is diameter of a circle and PR is the chord S.T. ∠RPQ = 300. Tangent at R intersects PQ
produced at S. Prove that RQ = QS.
A
23
cm
R
B
29 cm
5 cm
33. Find r in fig. 17.
r
S
Q
C
P
Fig. (17)
D
A
34. PA and PB are two tangents to a circle with centre with centre O from
exernal point P such that OP = 2 radius prove that ∆ABP is
equilateral triangle
3
C
cm
5
3√
35. In fig. (18) if r = 30 cm. Find
perimeter (∆ABC)
Fi
3√5
7)
(1
g.
B
6 Marks questions
36. Prove that the tangents at any point of a circle is perpendicular to the radius through the
point of contact.
Rider
1.
Prove that in two concentric circles the chord of the
larger circle which to includes the smaller circle is
bisected at the point of contact.
P
y
2.
If PQ is tangent from Q and PB in diameter what is
the value of x (fig. 19).
x
A
350
B
(373) (Maths__Xth class)
Fig. (19)
y
Q
3.
In fig. (20) prove that 2a + b = 900.
B
O
A
f
D
4.
C
o
z
x
y
Fig. (20)
0
In fig. (21) PAT is tangent. Find value of x.
55
A
Fig. (21)
P
B
T
37. Prove that the length of tangents, drawn from an external point to a circle are equal. Use
the above result in the following :
A
K
1.
In fig. (22) PA & PB are tangents from point P.
P
O
Prove that KN = AK +BN.
B
N
Fig. (22)
2.
AB is diameter of a circle. P is a point on the circle through which a tangent L is
drawn. AC ⊥L and BD⊥L. Prove that AC = BD = AB
Answers
1.
4 cm
2.
(i)
3.
24
4.
15
5.
8
6.
46 (15+8+8+15)
7.
3.8 cm.
8.
7.4 cm.
9.
(ii)
10. 400
11. 300
12. 450
(374) (Maths__Xth class)
13. 350
14. 33 cm (5+3+8+17)
15. ----16. 4 cm.
17. 600
18. 15 cm.
19. 5 cm.
20. 12 cm.
21. Hint - Use trignometry
22. 4 cm.
23. ---24. 2 √6 6
25. 5 cm
26. r = 1 cm (h = 6.5 cm)
27. ---28. ---29. ---30. r = 3 cm
31.
20
cm.
3
32. ----33. 11 cm.
34. ----35. 32 cm.
36. (2) 350,
(4) 550
37. -----
(375) (Maths__Xth class)
Chapter
Constructions
Key Points
1.
Construction should be neat and clean and as per scale given in question.
2.
Steps of construction should be provided only to those questions where it is mentioned.
3. Marks Questions
1.
Draw a line segment AB = 8 cm. Take a point P on AB such that AP : PB = 3:5 (This
3
AP
=
or 5×AP = 3×PB.
question can be asked by giving
5
PB
2.
Draw a line segment xy = 10 cm. Take a point A on xy such that
2
XA
=
(This question
5
XY
2
3
XA
AY
=
or
= .
3
5
AY
XY
Draw a triangle PQR such that PQ = 5 cm, QR = 7.4 cm and ∠PQR = 740. Now construct
3
3
a ∆ P’QR’ ~ ∆ PQR with its sides
times of the corresponding sides of DPQR. ( times
2
2
can be written as 1.5 times.)
can be asked by giving
3.
4.
Draw an isoceles ∆ ABC with AB = AC and base AC = 7 cm and vertical angle 1200. Construct
1
∆AB'C'~∆ ABC with its sides 1 times of the corresponding sides of ABC.
3
5.
Construct ∆D'EF'~∆DEF with its sides equal to
where ∠E = 900, DE = 6 cm, EF = 8 cm.
1
rd of the corresponding sides of ∆DEF
3
6.
Draw a ∆PQR in which RQ = 7.0cm, ∠P = 600 altitude from P on RQ is 5 cm. Draw a
similar ∆PQ'R' such that RQ = 8.5 cm.
7.
Draw a right angled triangle in which base is 2 times of the perpendicular. Now construct
a triangle similar to it with base 1.5 times of the original triangle.
8.
Draw an equilateral triangle ABC with side 5.5 cm. Now draw ∆AB'C' such that
AB
1
= .
A'B
2
Measure A’B. What type of triangle is AB'C'.
9.
Draw circle of radius 4 cm with centre O. Take a point P such that OP = 6 cm. Draw two
tangents PX and PY to the circle from P. Find perimeter of the triangle PXY.
10. Draw a circle of radius 3.5 cm. Now draw a set of tangents from external point P such that
the angle between the two tangents is half of the central angle made by joining the point of
contacts to the centre.
11. Draw a circle with centre P and radius 3 cm. Now draw tne tangents OA and OB from
external point O such that ∠AOB = 450. What is the value of ∠AOB + ∠APB.
12. Draw a line segment AB = 9 cm. Draw circles of radius 5 cm and 3 cm from A and B
(376) (Maths__Xth class)
respectively. Draw tangents to each circle from the centre of the other.
13. Draw a diameter AB of a circle of any radius with centre O. Now draw a radius OP⊥AB.
Though P draw a tangent. Is the tangent parallel to the diameter.
14. Draw a circle of radius OP = 3 cm. Draw ∠POQ = 550. Such that OQ = 5 cm. Now draw
two tangent from Q to the given circle. If P one of the point of contact of the tangent to the
circle.
15. Draw a circle with centre O and radius 3.5 cm. Take a horizontal diameter. Extend it to
both the sides to point P&Q such that OP = OQ = 7 cm. Draw tangents PQ and QB one
above the diameter and the other below the diameter. Is PQ//BQ.
16. Draw a right angled isocles triangle with equal side 5 cm. Draw a circle such that the two
equal sides of the triangle are tangents to the circle and the end poins of the hypotenuse
are points of contacts.
17. Draw any triangle ABC a point P outside it. Join P to vertices of the triangleby dotted
lines. On these doted lines draw ∆DEF similar to ∆ABC.
18. Draw a circle of radius 3 cm with centre O. Take point P such that OP = 5 cm. PO cuts the
circle at T. Draw two tangents PQ and PR. Join Q to R through T. Draw AB parallel to QR
such that A and B are points on PQ and PR respectively. Check whether PQ + PR =
perimeter (∆PAB)
PRACTICE PAPER I
(377) (Maths__Xth class)
Time : 3 hours
M.M. : 80
General Instructions :(i)
All questions are compulsory.
(ii) The question paper consists of thirty questions divided into 4 section – A, B, C and D.
Section A comprises of ten questions of 01 mark each. Section B comprises of five
questins of 02 marks each, Section - C comprises of ten questions of 03 marks each
and section D comprises of five questions of 06 marks each.
(iii) All questions in Section A are to be answered in one word, one sentence or as per the
exact requirement of the question.
(iv) There is no overall choice. However, internal choice has been provided in one question
of 02 marks each, three questions of 03 marks each and two questions of 06 marks
each. You have to attempt only one of the alternatives in all such questions.
(v)
In question on construction, drawings should be neat and exactly as per the given
measurements.
(vi) Use of calculators is not permitted.
1.
Write 3276 as a product of prime factors.
2.
For what value of ‘k’, the quadratic equation
2x2 + kx+ 2 = 0, has equal roots?
3.
Write a polynomial, whose zeros are 2 and -3.
4.
If nth term of an A.P. is 2–5n, write its common difference.
5.
If sin(2x–150) =
6.
From a well shuffled pack of cards, a card is drawn at random, write the probability of
getting a face card.
7.
A cone and a hemisphere have equal bases and equal volumes. Write the ratio of their
heights.
8.
The perimeter of two similar triangles ∆PQR and ∆ABC are respectively 36 cm and 24 cm.
If AB = 10 cm, then write the value of PQ.
9.
In fig., diameter AB = 12 cm & AP = PQ = QB.
1
, then write the value of 'x'.
2
Write the area of shaded region.
A
(378) (Maths__Xth class)
P
Q
B
10. In fig., PQ || BC and
write the value of
A
1
AP
= ,
PB
3
area of ∆APQ
.
are of ∆ABC
P
B
Q
C
Section-B (Two marks each)
11. Evaluate
2sin2620 + 2sin2 280
5(sec2720 – cot2180)
–
sec580. sin320
3 tan150tan350 tan450 tan 550 tan 750
12. Find the ratio in which the point (x, 2) divides the join of p(–3, 5) and Q (5, 1). Also find
the value of 'x'
13. Find all the zeros of the polynomial x4 + x3 – 9x2 – 3x + 18, if its two zeros are √3 and –√3.
14. T is a point on the side QR of a ∆PQR, such that ∠PTR = ∠QPR, show that RP2 = RQ. RT.
or
∆ABC is right angled at B and D is mid point of BC, prove that
AC2 = 4AC2 – 3AB2
15. A die thrown once. Find the probability of getting
(i) a number less than 4
(ii) a prime number.
Section - 'C' (Three mark each)
16. Prove that
1 + cosθ
= cosecθ + cotθ
1 – cosθ
or
Prove that
2sec2θ – sec4θ – 2 cosec2θ + cosec4θ = cot4θ – tan4θ
17. How many terms of the AP,, 47, 43, 39, ........ be taken so that their sum is 299 ?
18. Show that 3 + 2 √5 is an irrational number.
19. Draw the graphs o the equation 3x + 2y –12 = 0 and x - y + 1 = 0. Find the coordinates of
the vertices of the triangle formed by the lines and the y-axis. Also find the area of the
triangular region.
20. Construct a ∆PQR in which PQ = 6.5 cm, ∠Q = 600 and QR = 5.6 cm. Also construct a
(379) (Maths__Xth class)
triangle PQ'R' similar to ∆PQR, whose each side is
3
times the corresponding side of the
2
triangle PQR.
21. Show that the points A(2, –2), B(14, 10), C(11, 13) and D(–1, 1) are the vertices of a
rectangle.
22. Two poles of height 'a' and 'b' (a>b) are 'c' metres apart. Prove that the height of the point
of intersection of the lines joining the top of each pole to the foot of the opposite pole is
ab
.
a+b
or
Two triangle ∆ABC and ∆DBC are on the same base BC and on the same side of BC in
which ∠A = ∠D= 900. If CA and BD meet each other at E, show that
AE. EC = BE. ED
23. Find the area of the shaded part of figure. Where PQRS is a square of side 28 cm. Region X
is a semi circle on RS as diameter. Region Y and Z are quadrants of circles with centres at
P and Q respectively. T is the mid point of PQ.
Q
T
P
Y
Z
X
S by completing the perfect square.
R
24. Solve the following equation
2x2 – 3x + 1 = 0
25. Find the area of the triangle formed by joining the mid-points of the sides of the triangle,
whose vertices are (0, –1), (2, 1) and (0, 3). Find the ratio of this area to the area of the
given triangle.
Section-D (Six marks each)
26. Prove that the ratio of the areas of two smilar triangles is equal to the ratio of squares of
their corresponding sides. Using the above result, prove the following.
In a ∆ABC, XY is parallel to BC and it divides ∆ABC in to two equal parts of equal area.
Prove that
√2 –1
BX
=
.
AB
√2
Or
(380) (Maths__Xth class)
The lenghts of tangents drawn from an external point to a circle are equal. Prove it.
Use the above theorem.
Prove that, if all sides of a parallelogram touches a circle, then it is a rhombus.
27. A straight high way leads to the foot of a tower. A man standing at the top of the tower
observes a car at an angle of depression of 300, which is approaching the foot of the tower
with a uniform speed. After 6 seconds, the angle of depression of the car is found to be 600.
Find the time taken by the car to reach the foot of the tower.
or
From the top of 12 metres high building the angle of elevation of the top of a tower is 600
and angle of depression of its foot is 450. Find the height of the tower.
28. The mean of the following frequency distribution is 62.8 and the sum of all frequencies is
50. Find the values of missing frequencies x and y.
Class
Frequency
0-20
20-40
40-60
60-80
80-100
100-120
5
8
x
12
y
8
29. A bucket made up of a metal is in the form of a frustum of a cone of height 16 cm with
diameters of its lower and upper ends as 16 cm and 40 cm respectively. Find the volume of
the bucket. Also find the cost of the bucket if the cost of material sheet used is Rs. 40 per
100 cm2. (Use π = 3.14)
30. A fast train takes 3 hours less a slow train for a journey of 600 km. If the speed of the slow
train is 10km/hr. less than that of the fast train, find the speed of the two trains.
Answers
1.
22× 32 × 7 ×13
2.
k=±4
3.
x2 + x – 6
4.
–5
5.
x = 22.5
6.
3
13
7.
2:1
8.
15 cm
9.
12πcm 2
10.
1
16
11.
1
3
12. 3:1, x = 3
(381) (Maths__Xth class)
13. √3, –√3, –3, 2
15.
1 1
,
2 2
17. 13
19. (–1, 0), (2, 3), (0, 6) 5 square units
23. 168 cm2
24. 1,
1
2
25. 1:4
27. 3 sec. or 12 (√3+1)
28. x = 10, y = 7
29. Rs. 766.46 (Approx.)
30. 50 km/h, 40 km/m
Practice Paper - 2 (With Solution)
(382) (Maths__Xth class)
Section - A
77
.
210
1.
Write down the decimal representation of
2.
Write a quadratic equation whose sum of roots is zero and product is –15.
3.
Which term of an A.P. 13, 20, 27, ......... is 35 more than 19th term.
4.
Write the zero of the polynomial p(x) whose graph is given below.
5.
Write the value of cotθ from the adjoining figure.
24 cm
7 cm
6.
In the word "MISSIPPI". What is the probability of getting60the
letter 'I'.
cm
7.
Write the median if the mode and mean of the data is 36.8 and 35.3 respectively.
8.
Write the perimeter of quadrant of a circle of radius 7 cm.
9.
In the figure, ∠Q = ∠R = 500. Express x in
terms of a, b and c ? Where a, b and c are the
length of QP, QR and RS respectively.
(383) (Maths__Xth class)
10. If two tangents of 7 cm each are inclined at an angle of 900. Write the radius of that circle;
which have that tangents.
Section-B
11. AB is a line segment, where is A(7,1) and B(3, –4), intersected by x – axis at point P. Find
AP:PB.
12. In given figure DE||BC, find the value x
D
x–
2
B
E
x–1
13. Find the value of m if following equations have no solutions.
x+2
x
A
C
(m+1) x + 2y = 5
3x + y = 3
14. Find the value of tan600 geometrically.
or
If √3 tanθ = 3 sinθ
Find the value of sin2θ – cos2θ
15. A box contain some cards bearing the numer 3, 4, 5, 6 ...... 19. If a card is taken out from the
box at random. Find the brobability that the card drawn bears –
(i) an even number
(ii) a number divisible by 3 or 5
Section - C
16. Prove that √11 is an irrational number.
17. Prove that :
Secθ(1–sinθ) (secθ + tanθ) = 1
18. Solve the system of linear equation
ax + by = c
bx + ay = 1 + c
or
(384) (Maths__Xth class)
Draw the graph of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Calculate the area
bounded by these lines and x – axis.
19. The mid point of the sides of a triangle are (1, 2), (3, –1) and (5, 0). Find the vertices of
triangle.
20. Find the zeroes of the polynomial if two of its zeroes are –√3 and √3 and the given polynomial
is 2x4 –3x3 –5x2 + 9x – 3
21. How many terms of the A.P.
43, 39, 35, ..........
Must be taken to give a sum of 252 ?
22. If A(4, –8), B(3, 6) and C (5, –4) are the vertices of ∆ABC AD is the median and P is a point
AP
on AD such that
= 2. Find the co-ordinate of point P.
PD
or
Prove that only one line passes through (7, –2), (5, 1) and (3, 4)
A
23. In the figure a circle is inscribed in a quadrilateral ABCD in which
∠B = 900. If AD = 23 cm, B = 29 cm and DS = 5 cm, find the
radius of the circle.
R
O
24. Construct a ∆ABC in which AB = 4 cm, BC = 6 cm and ∠ABC =
3
times the
600. Construct ∆A'BC'~∆ABC such that sides are
4
corresponding sides o given triangle.
25. ABCD is a rectangle in which AB:BC = 2:1 M is
the mid point of AB MD and MC are one fourth
of the circles with centres A and B respectively.
Find the ratio of the area of the rectangle ABCD
and the shaded portion.
or
The given figure, OACB represent a quadrant of a circle of
radius 3.5 cm with centre at O.
Calculate the area of the shaded region if OD = 2 cm.
(385) (Maths__Xth class)
Q
r
r
D
S
B
P
C
SECTION - D
26. An aeroplane with flying at a height of 4000 m from the ground passes vertically above
another aeroplane at an instant when the angles of elevation of the two aeroplanes from
the same point on the ground are 600 and 450 respectively. Find the vertical distance
between the two aeroplanes at that instant.
27. A piece of cloth costs Rs. 4000. If the piece was 5 m longer and each metre of cloth costs Rs.
4 less, than cost of the piece will remain unchanged. How long is the piece and what is the
original rate per metre ?
or
An eagle is sitting on the top of a tower, which is 9 m high. From a point 27 m away from
the bottom of the tower, a snake is coming towards his hole which is at the base of the
tower. Eagle flew from the top to catch the snake. If their speed are equal, at what distance
from the hole is snake caught ?
28. The weight of 60 students taken at random from a school is represented in the following
table.
Weight
Frequency
0-10
10-20
20-30
30-40
40-50
50-60
Total
3
p
20
15
q
5
60
If the medium weightis 28.5, find values of p and q.
29. In a triangle of the square on one side is equal to the sum of squares on the other two
sides, prove that the angle opposite to the first side is a right angle. Using this prove the
following.
In a quadrilateral PQRS, if ∠Q = 900, PS2 = PQ2 + QR2 + RS2. Prove that ∠PRS = 900
30. A conical hole is drilled in a circular cylinder of height 15 m and radius 8 cm. The height
and base radius of the cone are also the same as of cylinder. Find
(i)
Volume of the remaining solid figure.
(ii)
Total surface area of the remaining solid figure after removing cone,.
or
A nurse fills syrup in an inverted frustum shaped flask up to the level. The radii and slant
height of the flask are 4 cm, 10cm and 10 cm respectively. She has to give this syrup in
cylindrical shaped cups of height and radius 1.3 cm and 1 cm respectively to the children
in children ward of a hospital. If each child has to take two cups. How many children can
drink the syrup at a time.
(386) (Maths__Xth class)
SECTION - A
Answer (with solution)
Note : A student can write; ony the answer for 1 mark questions to get full one mark.
1.
11
77
=
210 30
by long division :
11
= 0.36666.......
30
77
= 0.36 Ans.
210
2.
Quadratic equation :
x2 – (sum of roots) x + product of roots = 0
x2 – (o) x + (–15) = 0
x2 – 15 = 0
3.
d = 20 – 13 = 7
7 × 5 = 35
19th term + next 5 term = 24th term.
Ans. : 24th term.
4.
x=2
5.
AB = √242 + 72
AB = √625 = 25
6.
7.
Cot θ =
60 12
25 5
Cot θ =
12
5
3
8
Mode = 3 median – 2 mean
P (I) =
36.8 = 3 median – 2× 35.3
107.4
= median
3
Median = 35.8 Ans.
(387) (Maths__Xth class)
8.
7 cm
7 cm
Perimeter of the quadrant
=
1
(2πr) + r + r
4
=
1 22
× × 7 + 7+ 7
7
2
= 11 + 14 = 25 cm. Ans.
9.
∠Q = ∠R, TR||PQ and
So ∠P = ∠T
∆STR ~ ∆SPQ
SR
TR
PQ = (QR + RS) (KM=
QR + RS)
x
a
=
c
b +c
ac
x = b +c Ans.
10. Quadrilateral formed by two radius (from the point of contact tangents) and two tangents
(inclined at an angle 900 is a square so radius = length of tangents
r - 7 cm
SECTION - B
(7, 1)
11.
A
K
:
1
p (x, o)
Let P (x, 0) divides the AB in K : I
y=
ky2 + y1
k+1
–k+1
k+1
o = – 4k + 1
o=
4k = 1
k=
1
4
AP : PB :: 1 : 4 Ans.
(388) (Maths__Xth class)
(3, –4)
B
12. DE || BC
According to basic proportionality theorem –
AE
AD
=
BD
EC
x
(x + 2)
(x –2) – (x –1)
x2 – x = x 2 – 4
x = 4 Ans.
13. Equations are in the form of ax + by + c = o as follows :
(m +1) x + 2y – 5 = 0
3x + y – 3 = 0
condition for no solution :
a1
c1
b1
=
≠
a2
c2
b2
a2 = m + 1
b1 = 2 c1 = –5
a2 = 3
b2 = 1 c2 = –3
m+1
3
2
1
=
m+1=6
m = 5 Ans.
P
14. Let ∠XOP = 600
PM⊥ OX and MQ = OM = a
⇒∆OMP ≅ ∆QMP (SAS)
2a
so ∠MOP = ÐMQP = 600
DPOQ is an equilateral triangle
OP = OQ = 2a
MP = √OP2 – OM2
= √4a2 – a2
P
= √3 a
tan 600 =
MP
√3a
=
= √3 Ans.
OM
a
or
√3 + tanθ = 3 sinθ
(389) (Maths__Xth class)
a
M
Q X
√3
sinθ
= 3 sinθ
cosθ
cosθ =
√3
3
Now, sin2θ – cos2θ
= 1 – cos2θ – cos2θ
= 1– 2 cos2θ
= 1– 2 (
√3 2
)
3
=1–2×
=1–
3
9
1
3
2
3
1
Ans.
3
15. Total No. of cards = 17
=
(i) Even number (from 3 to 19)
= (4, 6, 8, 10, 12, 14, 16, 18)
⇒8 = No. of favourable outcomes.
8
17
(ii) Number divisible by 3 or 5
p(even number)
(3, 5, 6, 9, 10, 12, 15, 18)
⇒8 = No. of favourable outcomes.
p ( no divisible by 3 or 5) =
8
17
SECTION - C
16. Let √11 is rational
that is, we can find integers a & b (C ≠ o)
such that √11 =
a
, a and b are coprime.
b
or b√11 = a
Squaring both sides, we get 11 b2 = a2 .............. (i)
(390) (Maths__Xth class)
∴a2 is divisible by 11 ⇒a is also divisible by 11 so we can write a = 11x ................ (2)
⇒11 b2 = 121x2
⇒ b2 = 11x2
⇒b2, is divisible by 11
⇒b, is divisible by 11
⇒11, is divisible common factor of a & b.
But this contradicts the fact that a & b are coprime.
∴√11 is not rational
so √ is irrational
17-
L.H.S. =
secθ(1–sinθ)(secθ+tanθ)
=
sin θ
1
1
(1–sinθ) (
+
)
cosθ
cosθ
cosθ
=
(1– sin θ) (1+sinθ)
(1+ sin θ)
1
(1 – sinθ)
=
cosθ
cosθ
cos2θ
1– sin 2θ
cos2θ
=
= 1 RHS
2
2
cos θ
cos θ
=
18. By cross multiplication method.
b
c
a
b
a
1+c
b
a
x
y
–1
=
b(1+c) – ac
cb – a (1+c) = a 2 – b2
(i)
(ii)
x
–1
= 2
b + bc – ac
a–b
from (i) and (iii)
x=
=
(iii)
2
– b – bc + ac
–b
c (a – b)
= 2
2
2
2 +
a –b
a –b
a 2 – b2
c
–b
2 =
a+b
a –b
2
y
–1
from (ii) and (iii) = cb – a (1+c) = 2
a–b
2
(391) (Maths__Xth class)
y=
x
a
– bc + a + ac
c ( a–b)
= 2
2 +
2
2
a –b
a –b
a 2 – b2
=
a
c
2 +
a –b
a +b
=
c
–b
,
2 +
a +b
a –b
2
2
y=
a
c
2 +
a –b
a +b
2
or
From x – y + 1 = 0
x
–1
0
2
y
0
1
3
From 3x + 2y – 12 = 0
Area ∆ACF =
19
x
0
2
4
y
6
3
3
1
× 5 × 3 = 7.5 square units.
2
In ∆ABC
P is the mid point of AB
1=
a +x
⇒a + x = 2
2
(i)
2=
b +y
⇒b + y = 4
2
(ii)
Q, is the mid point of AC
a +p
=3
2
⇒a + b = 6
(iii)
b+q
= –1
2
⇒b + q = – 2
(iv)
R, is mid point of BC
x+p
=5
2
⇒ x + p = 10
(v)
y+p
=0
2
⇒y + q = 0
(vi)
Adding (i), (iii) and (v)
(392) (Maths__Xth class)
a + x + a + p + x + p = 18 ⇒a + x + p = q
(vii)
Adding (ii), (iv) and (vi)
b+y+b+q+y+q=2
from (i) and (vii)
p=7
from (iii) and (vii)
x=3
from (v) and (vii)
a = –1
from (ii) and (viii)
q=3
from (iv) and (viii)
y=3
from (vi) and (viii)
b=1
⇒b + y + q = 1
(viii)
∴Vertices of ∆ABC are A(–1, 1), B(3, 3), C (7, –3)
20. –√3 and √3 are zeros
∴(x + √3) (x – √3) = x2 – 3 is a factor of the polynomial.
∴
x2 – 3) 2x4 – 3x2 – 5x2 + 9x – 3 (2x2 – 3x + 1
2x2
– 6c2
– 3 x3 + x2 + 9x – 3
± 3 x3
± 9x
x2 – 3
– x2 ± 3
x
2x4 – 3c3 – 5x2 + 9x – 3
= (x2 –3) (2x2 – 3x + 1)
= (x2 – 3)(2x2 – 2x – x +1)
= (x2 –3) [2x–1) (x–1)]
∴Zeros of polynomial are √3, – √3,
21. Sn
Sn
252
252
1
,1
2
= 252, a = 43, d = – 4
n
[2a + (n–1) d]
2
n
=
[2×43 + (n–1) (–4)]
2
n
=
[86 – 4n +4)
2
=
(393) (Maths__Xth class)
252
252
n
[90 –4n)
2
= n (45 – 2n)
=
or 2n2 – 45n + 252 = 0
(2n –21) (n –12) = 0
n = 12, =
21
2
rejecting n =
21
, We get n = 12
2
22. AD is median
⇒D, is the mid point of BC
∴Co-ordinate of D is (4, 1)
But
2
AP
=
PD
1
∴coordinate of P
=(
2 × 4 + 1× 4 2 × 1 + 1× –8
]
)
2 +1
2 +1
=¼
8+4 2–8
] 3 ½
3
=
¼
12 –6
] 3½
3
= ¼4]
&2½
or
If a line passes through three points, then
area of the triangle = 0
Area of the triangle
=
1
[x (y – y ) + x2 (y3 – y1) + c3 (Y1 – y2)]
2 1 2 3
=
1
[7, (1–4) + 5 (4– (–2) + 3 (–2–1)]
2
=
1
[7×–3 + 5 × 6 + 3 × – 3)
2
=
1
[–21 + 30 – 9] = 0
2
(394) (Maths__Xth class)
∴Points are collinear.
⇒A line passes through ¼7]
23-
&2½] ¼5] 1½
and
¼3] 4½
In Fig. OP⊥BC and OQ ⊥BA
∠B = 900 , OP = OQ = r
⇒OPBQ is a square
BP = BQ = r
DR = DS = 5 cm
AR = AD – DR = 23 – 5 = 18 cm
AQ = AR = 18 cm
BQ = AB – AQ = 29 – 18 = 11 cm
Y
A
∴r - 11 cm.
A'
cm
24.
4l
600
B
B1
C
C'
6 cm
B2
B3
B4
25. Let
X
BC = x
AB = 2 x
∴M is the mid point of AB
AM = MB - x
Area of the quadrant of the circle =
1
πx2
4
Area of two quadrants of teh circle =
1
πx2
2
Aera of sheded rigion = Area of the rectangle – Area of two quadrants
= 2x × x –
1 2
πx
2
(395) (Maths__Xth class)
= 2x2 –
1 22 2
× x
7
2
= 2x2 –
3
11 2
x = x2
7
7
Ratio is = 2x2 :
3 2
x or 14:3
7
or
∠AOB = 900 radious of OACB = 3.5 cm.
Area of OACB =
90 22
×
× 3.5 × 3.5
360
7
= 9.625 cm2
∆AOD is right angle :
∴Area of ∆AOD =
1
1
× AO × OD =
× 3.5 × 2
2
2
= 3.5 cm2
Area of shaded region
= 9.625 – 3.5
= 6.125 cm2
SECTION - D
A
26. In ∆ADC
AC
DC
⇒ DC =
4000√3
3
B
4000
tan600 =
In ∆BDC
tan450 =
BC
DC
0
C
⇒BC = DC
BC = 2306.7 m
⇒ AB = (4000 – 2306.7) m
= 1693.3 m
27. Total cost = Rs. 400
(396) (Maths__Xth class)
60 45
0
D
Let length of the cloth piece = x m.
Cost / m
400
x
= Rs.
New Cost / m = Rs.
New length
& 4½
∴¼ 400
x
¼
400
x
& 4½
= (x + 5) m
(x+5) = 400
x2 + x – 500 = 0
A
x = – 25, 20
rejected = – 25, length of the piece = 20 m.
400
= Rs. 20
20
27
–x
or
9m
and cost /m = Rs.
–b
2
a –b
2
A : position of eagle
B : hole
B
C : point, where eagle caught snake
D : Position of snake
In ∆ABC
AB2 + BC2 = AC2
⇒(9)2 + x2 = (27 – x)2
⇒81 + x2 = 729 + x2 – 54x
⇒ 54 x = 729 – 81
⇒
28-
x = 12 m
Class-internal
Frequency
Cummulative frequency
0&10
5
5
10&20
p
5+p
20&30
20
25 + p
30&40
15
40 + p
40&50
q
40 + p + q
50&60
5
45 + p + q
(397) (Maths__Xth class)
27–x
x
27 m
60
45 + p + q
= 60
p+q
= 15
Median class is 20 – 30
l = 20, f = 20, cf = 5 + p, h = 10, N = 60
N
– cf
×h
Median = l + 2
f
60
– (5 + p)
median = 20 + [ 20
] × 10
20
8.5
17
25 – p
2
= 25 – p
p = 8, q = 7
=
S
P
29. Figure, given and proof
PS2 = PQ2 + QR2 + RS2 .............. (i)
In ∆PQR
PQ2 + QR2 = PR2
from (i)
PS2 = PR2 + RS2
Q
∴∠PRS = 90
0
30. Slant height of the cone
= √82 + 152
= 17
¼1½
= πr2h
Volume of the cylinder
= 960π cm3
Volume of conical hole
=
1
πr2h
3
= 320 π cm3
= 960 π – 320 π
Volume of remaining solid
= 640 π cm3
¼2½
Curned surface area of the cylinder
=2πrh
=
240
π cm2
(398) (Maths__Xth class)
R
= πrl
Curved surface area of cone
=
136 π cm2
= πr2 cm2
Area of base of the cylinder
= 64πcm2
Surface area of remaining solid = Curved surface area of the cylinder + area of the
base + curved surface area of the cone
= 240π + 64π + 136π cm2
= 440 πcm2
or
Given r = 4 cm, R = 10 cm l = 10 cm
∴l
= √h2 + (R–r)2
⇒h = 8 cm
Volume of the flask
=
1
πh (R2 + r2 + Rr)
3
= 416π cm3
Volume of cylinderical cup = πr2h
= 1-3 π cm3
∴No. of children
Volume of flask
=
=
2 × volume of cylinderical cup
416 π
2 × 1.3π
= 160
(399) (Maths__Xth class)