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KENDRIYA VIDYALAYA SANGATHAN
QUESTION BANK
SUBJECT: MATHEMATICS
CLASS:X
SA-II
REGIONAL OFFICE JAIPUR
YEAR 2016-17
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 1
PREFACE
The responsibility of Review of Question Bank with key
points of class X Mathematics for the session 2016-17 has been
entrusted to KV Banar, Jodhpur.
KVS Jaipur Region acknowledges the sincere efforts of Dr.
M. M. A. USMANI, Principal KV Banar, Jodhpur.
I am confident that the question bank of class X (Mathematics)
will directly help the students to understand the concept well
and meet quality expectation.
Wish you all the best.
(Dr. JAIDEEP DAS)
Dy. Commissioner
KVS (R.O) Jaipur Region
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 2
QUESTION BANK – MATHEMATICS
FOR
CLASS – X (2016-2017)
PATRON
Dr. JAIDEEP DAS
Dy. Commissioner
KVS (R.O.) Jaipur Region
COORDINATOR
Dr. SUKRITI RAIWANI
Assistant Commissioner
KVS (R.O.) Jaipur Region
RESOURCE PERSON
Dr. M.M.A.USMANI
PRINCIPAL, KV BANAR JODHPUR
SINCERE EFFORTS: Mr MAHESH SINGH SENGAR PGT (Math)
Mr RAKESH NARAYAN TGT (Math)
Mr MAHESH GOURA TGT (Math)
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 3
SUMMATIVE ASSESSMENT-II
SECOND TERM (SA II)
UNITS
I ALGEBRA(contd)
Quadratic equations, arithmetic progressions
II GEOMETRY(contd)
Circles, constructions
III MENSURATION
Areas related to Circles, Surface Area & Volumes
IV TRIGONOMETRY(Contd)
Heights and Distances.
V COORDINATE GEOMETRY
VI PROBABILITY
TOTAL
MARKS: 90
MARKS
23
17
23
08
11
08
90
Important:
∑ Slow achiever may revise the knowledge part first.
∑ Bright students may emphasize the application part of the question Bank.
CLASS X
Design of question paper
S. No.
Types of question
Marks per question
No. of questions
Total marks
1
VSA
1
4
4
2
SA 1
2
6
12
3
SA 2
3
10
30
4
LA
4
11
44
31
90
TOTAL
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 4
DETAILS OF THE CONCEPTS TO BE MASTERED BY EVERY CHILD OF CLASS X WITH EXCERCISES AND EXAMPLES
OF NCERT TEXT BOOK
SUMMATIVE ASSESSMENT -II
01
Quadratic Equation
Standard form of quadratic
equation
Solution of quadratic
equation by factorization
Solution of quadratic
equation by completing the
square
Solution of quadratic
equation by quadratic
formula
Nature of roots
NCERT Text book
Example 5
Examples 3,4,5,
Ex. 4.2 Q. 1(iii), Q .5
Examples. 8,9 ,13
Ex. 4.3 Q. 1(iii) 3(ii),Q.8,9
Examples. 10,11,13,14,15 , Q2,3 (ii) Ex.4.3
Sum of first n terms of an
A.P.
Distance formula
Section formula
Examples 17
Q.1,2, Ex. 4.4
Exp-2, Ex. 5.1
Q.1(ii),(iv) Q.2(iv),(v)
Q.4(v),(xi),(xii),(xiv)
Exp. 4,6,8 Ex. 5.2
Q.3(iii), Q.4 Q.5(ii) Q.6,8,11,13,15,20
Examples 13,15, 16 Ex. Qs 1(iv),
2(i),3(iii),(v),(viii),Qs.4,8,9.10(ii),12,16,20
Examples 2,4 ,Ex7.1 Qs 3,5,7,8,10
Examples 7,8,9 Qs 2,3,6,8,10
Area of Triangle
Examples 14,15 Qs.2,3,5
Heights and distances
Examples- 2,3,5,6,7 Ex 9.1 3,5,5,8,10,13,15 and 16
05
Some application of
Trigonometry
Circles
Tangents to a circle
Number of tangents from a
point to a circle
Theorem 10.1 and 10.2 Examples 2 and 3
Ex. 10.2 Qs. 4,7,9,10,12 and 13
06
Constructions
Division of line segment in
the given ratio
Construction of triangle
similar to given triangle as
per given scale
Construction of tangents to
a circle
Construction 11.1
Circumference of a circle
Area of a circle
Length of an arc of a circle
Area of sector of a circle
Area of segment of a circle
Combination of figures
Example 1 Exercise 12.1 Q 2 and 4
Example 3
Exercise12.2 Q3
Example 2
Exercise 12.2Qs. 3,5,7,8,11,12 and 13
Ex 12.3 1,4,5,9,12,13 and 15
02
Arithmetic
progression
General form of an A.P.
nth term of an A.P.
03
04
07
Coordinate geometry
Area related to circles
QUESTION BANK CLASS X (MATHEMATICS)
Examples 1 and 2 Ex. 11.1 Qs.4,6 and 7
Construction 11.3 Ex. 11.2 Q 1,4,6 and 7
SA-II
Page 5
08
Surface area and
volumes
Surface area of a
combination of solids
Volume of combination of
a solid
Conversion of solids from
one shape to another
Frustum of a cone
09
Probability
Events
Probability lies between 0
and1
Performing experiment
QUESTION BANK CLASS X (MATHEMATICS)
Examples 2 and 3
Exercise 13.1 Qs. 2,5,7 and 9
Examples 6 and 7 Ex. 13.2 Qs. 2,4,6 and 7
Examples 9,10 and 111
Exercise 13.3
Qs. 2,4,5,8 and 9
Example 12& 14 Exercise 13.4 Qs. 2,4 and 5
Examples 4,6,7,9, 11,12 and 13
Ex. 15.1 Qs. 5,6,7,10,11,15,17 24 and 25
SA-II
Page 6
QUADRATIC EQUATIONS
KEY POINTS
1. The general form of a quadratic equation is ax2+bx+c=0, a≠o. a, b and c are real numbers.
2. A real number ? is said to be a root of the quadratic equation ax2+bx+c=0, a ≠ o if a ? 2+b ? +c=0. The zeroes of
the quadratic polynomial ax2+bx+c=0 and the roots of the corresponding quadratic equation ax2+bx+c=0 are the
same.
3. Discriminant: - The expression b2 - 4ac is called discriminant of the equation ax2+bx+c=0 and is usually denoted
by D. Thus discriminant D = b2-4ac.
4. Every quadratic equation has two roots which may be: distinct & real, equal & real or no real.
5. IF ? and ? are the roots of the equation ax2+bx+c=0 then
??
???√??????
??
???√??????
??
And ? =
6. Sum of the roots , ? + ? = -
?
? and product of the roots, ??
?
?
?
7. Forming quadratic equation, when the roots ? and ? are given.
x2-( ? + ?)x+ ?. ? =0
8. Nature of roots of ax2+bx+c=0
i.
If D?0, then roots are real and not equal
ii.
D=0, then the equation has equal and real roots
iii.
D<0, then the equation has no real roots
(distinct real roots)
(coincident roots)
LEVEL-1
1. If 2 is a root of the equation ?2+2?? +4 =0, then the value of ? is
Ans -2
2. If D>0, then roots of a quadratic equation ax2+bx+c=0 are
Ans
??? √?
??
3. Discriminant of ?? ? ?? ? ? = 0 is
Ans 5
4. The sum of roots of a quadratic equation ???+4 ? – 32 = 0 is
Ans -4
5. The product of roots of a quadratic equation ? ??+ 7 ? – 4 = 0 is
Ans -2
6. Values of K for which the equation ? ??+2kx - 1 = 0 has real roots are:
Ans k≥3 or K≤-3
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 7
LEVEL-2
1. For what value of k, x=a is a solution of equation ??- (a+b)x+k =0 ?
Ans. K=ab
2. Represent the situation in the form of quadratic equation:Rohan ‘s mother is 26 years older than him . The product of their ages (in years) 3 years from now will
be 360. We would like to find Rohan’s present age.
Ans .??+32x -273 = 0 where x(in years) is Rohan’s present age
3. Find the roots of ?? - 3x - 10 = 0
Ans . -2 ,5
4. Find two consecutive odd natural numbers, sum of whose squares is 130.
Ans . 7,9
−
5. Find the roots of Quadratic equation??? 2√??+2 = 0 by using the quadratic formula.
?
?
Ans . - ? ? , -? ?
6. Find the discriminant of the Quadratic equation ???-4x+3 = 0 and hence find the nature of its roots.
Ans . D= -8<0 its no real roots.
LEVEL - 3
1. If ? ? ? ??? ? ? ? are roots of the equation ???
−
−
2. Solve the equation:
?
???
??
? ? ? ?? ?? ≠ ??? ≠
???
3. Solve the equation ???
−
?
??? ? ?? ? ? find the value of k and m.
Ans. ? ?
?? ? ? ? ? by the method of completing square.
? ?
?
4. Using quadratic formula, solve the equation: ? ? ? ??
− −
??
? ?
?
?
? ? ??
Ans.? ?
?
?
Ans.? ?
Ans.? ?
5. The sum of two numbers is 15, if the sum of their reciprocals is ?? ????? ?ℎ???? ?????
??
?
?? ? ?
??? ?
?
?
??
?
??? ? ?
−
????? ?
??
??
Ans. 10 and 5
LEVEL - 4
1. By reduction of Rs 2 per kg in the price of sugar. Anita can purchase 2kg sugar more for Rs 224. Find the
original price of sugar per kg. What value of Anita is depicted in the question?
Ans: Rs. 16, Economical survey, Leadership
?
2. Two water taps together can fill a tank in ? ? hours. The tap of larger diameter takes 10 hours less than
the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
3. Find the roots of equation
?
???
?
?
+ ??? = ??? , x≠ -1,-2,-4
4. Solve the following equation for ‘x’ ??? - 3(a +b)x + a b = 0
Ans. 15 hours,25 hours.
??
Ans . 2, ?
5. If the roots of the equation (a-b) ??+ (b-c) x + (c-a) = 0 are equal, prove that 2a = b+c.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
? ?
Ans .? , ?
Page 8
Self-Evaluation
1. For what value of k are the roots of the equation ??? ? ??? ? ?? ? ? real and equal.
2. The hypotenuse of right angled triangle is 20 m. If the difference between the lengths of the other
Side is 4m find the other sides.
3. Find a and b such that x+1 and x+2 are factors of the polynomials ?? ? ???
4. Find the quadratic equation whose roots are 2 + √? and 2 - √?
−
?? ? ??.
5. Due to some technical Problems an aero plane started late by one hour from its starting point the pilot
decided to increase the speed of the aero plane by 100 km / h from its usual speed, , to cover a journey of
12000km in time. Find the usual speed of the aero plane. What value of the pilot is represented in the
question?
Ans: 300km/h, punctuality, cleverness, Leadership
−
−
6. Divide 29 into two parts so that the sum of squares of the parts is 425.
7. Solve for x: ???
??? ? ???
???? ?
ARITHMETIC PROGRESSION
∑
∑
∑
(Key Points)
Arithmetic progression (A.P.):- An A.P. is a list of numbers in which each term is obtained by adding a
fixed number to the preceding term except the first term.
This fixed number is called the common difference of the A.P.
If a is first term and d is common difference of an A.P. , then the A.P is a , a+d , a+2d , 2+3d …..
The ??? term of an a.p is denoted by ?? and ?? = a+(n-1) d , where a = first term and d = common
difference.
??? Term from the end = l – (n-1) d, where l = last term.
Three terms a-d , a , a+d are in A.P with common difference d.
Four terms a-3 d, a-d , a+d ,a+3d are in A.P with common diff. 2d .
∑
The sum of first n natural number is
∑
The sum of n terms of an A.P with first term a and common difference d is denoted by
∑
∑
∑
∑
∑
∑
?
?
??????
?
?? = ? {2a+ (n-1) d} also, ?? = ?(a+l) where, l = last term.
?? = ?? -????. Where ?? =??? term of an A.P
d = ?? -????. Where d = common difference of an A.P.
LEVEL -1
1. Find ??? term of 5, 11, 17, 23..........
2. Find the common diff. of A.P:
Ans .-4n+1)
-10 , -6 ,-2 ,2 ,………
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Ans . 4
Page 9
3. Find the A.P whose first term is 3 and common difference is – 2
Ans A.P. = 4 , 1 -2, -5, -8…………
4. In an AP, the sum of its first n terms is n2+2n. Find its 18th term.
Ans . 37
5. If 5m, 7m+10, 3m+2 are in AP then find m.
Ans . m=-10/3
6. If arithmetic mean between 3a and 2a - 7 is a+4, then find a.
Ans . a= 5
7. Find sum of all odd numbers between 50 and 100.
Ans: 1875
8. Which term of the AP: 21, 18, 15… is zero?
Ans: 8th
9. For what value of n are the ??? term of two AP , 63 , 65 , 67 ,…… and 3 , 10 , 17 ,…….equal?
Ans . n = 13.
10. If sum of n terms of an AP is ???+5n, then find its ??? term.
Ans. 4n+3
1. Find ??? term of an AP is 5-3n, find its common difference.
Ans. -4.
2. Which term of an AP 21, 18, 15… will be -81?
Ans . 35th
3. Write the next term of an AP 5, ? ? √?, ? ? ?√?…
Ans.√?.
LEVEL - 2
4. Find the value of middle most term of AP -11, -7, -3, …… ,49
5. Find the sum of all three digit numbers which are divisible by 11
6. Which term of the AP 3, 8, 13, 18… will be 55 more than its ???? term?
Ans: 17,21
Ans. 445500.
Ans: 31th term
7. Check whether -150 is a term of the A. P. 11, 8, 5, 2,
Ans No
8. How many three digit numbers are divisible by 8?
Ans. ???? term
9. Find the sum of all multiples of 7 lying between 500 and 900.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Ans: 39900
Page 10
10. In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third,
and so on. There are 5 rose plants in the last row. How many rows are there in the
Ans: 10 rows
Flower bed? What value you depict here?
LEVEL- 3
1.
Which term of AP : 38, 33, 28, 23,..... is the first negative term? Also find the sum of n terms.
Ans. 9th term, ??? (81-5n)
2. How many terms are there in the sequence 3, 6, 9, 12 …111?
Ans. 37 terms
3. The first term of an AP is -7 and the common difference 5, find its 18th term and the general term.
Ans. a18 =78n & an = 5n – 12
4. Solve the equation: - 4 + (-1) + 2+ ……x = 437
Ans. = 50
5.
If 5 times of 21st term of an AP is equal to its 5th term, show that 25th term is zero.
6. Find the sum of those integers from 1 to 500 which are multiples of 2 as well as of 5. Ans. 12750
7. Tarandeep saves Rs 2 on first day of the month, Rs 4 on the second day, Rs 6 on third day and so on.
What will be her saving in the month of feb 2012? What value is depicted by Tarandeep.
Ans: 870, saving habit
Problems for self evaluation.
1. Show that the sequence defined by ?? = 4n+7 is an AP.
2. Find the number of terms for given AP: 7, 13, 19, 25,….., 205.
3. The 7th term of an AP is 32 and it 13th term is 62. Find AP.
4. Find the sum of all two digit odd positive nos.
5. If the sum of p th term and q th term of an AP be equal to rth term and s th term and d be the common
difference of the AP, prove that p +q = r + s
6. Find the sum of 10 terms of AP: 2, 7, 12…
7. The sum of three numbers of AP is 3 and their product is -35. Find the numbers.
8. A man re-pays a loan of Rs 3250 by paying Rs 20 in the first month and then increases the payment by
Rs 15 every month. How long will it take him to clear the loan ?
9. Divide 32 into four parts which are in AP such that ratio of product of extremes to product of means is
7: 15.
10. If the m th term of A.P. IS 1/n and nth term is 1/m , then show that the sum of mn terms is 1/2( mn + 1)
11. In an AP the sum of first n terms is
???
QUESTION BANK CLASS X (MATHEMATICS)
?
?
??
?
, Find it 25th term.
SA-II
Page 11
12. Correct the following statements / formulae:
i)
Sn = a + (n- 1 ) d
ii)
a n = n/2 [ 2a + (n-1) d ]
iii)
a n – a n-1 represents the last term of an AP
iv)
The sequence of perfect square numbers is an AP.
CO-ORDINATE GEOMETRY
KEYCONCEPTS
1. Distance Formula:The distance between two points A(x1,y1) and B (x2,y2) is given by the formula.
AB=√(X2-X1)2+(Y2-Y1)2
COROLLARY:- The distance of the point P(x,y) from the origin 0(0,0) is given by
= OP= √X2+Y2
OP= √(X-0)2 + (Y-0)2
2. Section Formula :The co-ordinates of the point P(x,y) which divides the line segment joining A(x1,y1) and B(x2,y2)
internally in the ratio m:n are given by .
??
? ????? ???
??
? ??? ?
3. Midpoint Formula:-
? ????? ???
? ??? ?
If R is the mid-point, then m1=m2 and the coordinates of R are
?
=
?????
?
,
?
=
?????
?
,
4. Co-ordinates of the centroid of triangle:The co-ordinates of the Centroid of a triangle whose vertices are P(x1,y1), Q(x2,y2) and R(x3,y3) are
x1+x2+x3
3 ,
y1+y2+y3
3
5. Area of a Triangle:-
The area of the triangle formed a by the points P(x1,y1), Q(x2,y2) and R(x3,y3) is the
numerical value of the expression.
ar (∆PQR) =
?
[x (y -y )+x2(y3-y1)+x3(y1-y2)]
? 1 2 3
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 12
LEVEL- 1
1. Find the centroid of triangle whose vertices are (3, -7), (-8, 6) and ( 5, 10).
Ans. (0, 3)
?
2. If P ( ?, 4) is the midpoint of the line segment joining the points Q (-6 , 5 ) and R (-2 , 3) , then
find the value of β .
Ans . -12
3. A line intersects y –axis and x-axis at the points P and Q respectively. If ( 2 ,-5) is the midpoint of PQ ,
Then find the coordinates of P and Q respectively.
Ans. (0,-10) and (4,0)
4. Find the value of k if the point P (2,4 ) is equidistant from the point (5,k) and (k,7)
Ans: k=3
6. If the point A(x,y), B(3,6) and C(-3,4) are collinear, Show that x -3y + 15 = 0
7. Find the coordinate of the point on x-axis which is equidistant from (5,-2)and (-3,2).
Ans (1,0)
8. Find the coordinates of a point A, where AB is diameter of a circle whose centre is (2, -3) and B is (1, 4)
Ans. (3, -10)
9. The length of a line segment is 13 units. If one end point is (5, 7) and abscissa of the other point is -7,
Find the ordinate of the point.
Ans. 12 or 2
LEVEL-2
1. Point P (5, -3) is one of the two points of trisection of the line segment joining the points A (7, -2) and
B (1, -5) near to A. Find the coordinates of the other point of trisection.
Ans. (3, -4)
2. Show that the point P (-4, 2) lies on the line segment joining the points A (-4 , 6) and B (-4, -6).
3. If A (-2, 4) , B (0, 0) , C (4, 2) are the vertices of a ∆ABC, then find the length of median through the
vertex A.
Ans. 5 units
4. In what ratio does the line x – y - 2 = 0 divides the line segment joining the points A(3,-1) and
B (8, 9)?
Ans: 2:3
5. If the points A (4,3) and B (x,5) are on the circle with centre O(2,3) then find the value of x.
Ans. 2
6. What is the distance between the point A (c, 0) and B (0, -c)?
Ans. √? c
7. For what value of k, are the points (-3, 9), (2, k) and (4, -5) collinear?
Ans. ? ?
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
−
?
Page 13
LEVEL-3
1. Show that the points A(3, 0) , B(4, 5) , C (-1,4) and D (-2, -1) taken in order are the vertices of a
parallelogram.
2. Point P divides the line segment joining the points A(2,1) and B(5,-8) such that
AP: AB=1:3.If P lies on the line 2x-y+k=0, then find the value of ?.
Ans. k = -8
3. Points P, Q, R, and S in that order are dividing a line segment joining A (2, 6) and B (7, -4) in five equal
parts. Find the coordinates of point P and R?
Ans. P (3, 4) , R (5, 0)
4. Find a relation between x and y if the points (2, 1), (x, y) and (7, 5) are collinear.Ans. 4x - 5y + 3 = 0
5. find the area of the quadrilateral ABCD whose vertices are
A (-3, -1) , B (-2, -4) , C (4, -1) and D (3, 4) .
Ans. 28 sq. units
6. In a village, Sarpanch gave tender to dig a well at the centre of village passing through the houses A
(6,-6) B (3,-7) and C (3,3). Find the centre of the village. What value is depicited by Sarpanch.
Ans : (-3,2)
7. Find the point on y- axis which is equidistant from the points (5, -2) and (-3, 2)
Ans. (0, -2)
LEVEL-4
1. A (6, 1), B (8, 2), C (9, 4) are the three vertices of a parallelogram ABCD. If E is the midpoint of DC,
?
then find the area of ∆ADE.
Ans. ? ???????
2. find the value of ‘k’ for which the points are collinear (7, -2) , (5, 1) , (3, k) Ans. ? ? ?
3. 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.
Ans. 1:4
4. If the mid- point of line joining A (5,-1) and B (k,2) is ( x, y ) and 2x +2y + 1 =0 . Find the value of k.
Ans: k=-3
5. Find the area of the rhombus, if its vertices are (3,0), (4,5), (-1,4) and (-2,-1) taken in order.
Ans. 24 sq. units
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 14
SELF EVALUATION
1. Two opposite vertices of a square are (-1,2) and (3, 2). Find the coordinates of the other two vertices.
2. Find the centre of a circle passing through the points (6,-6), (3, 7) and (3, 3).
[Ans.3,-2]
3. If the points (p, q), (m, n) and (p-m, q-n) are collinear ,prove that p n =q m.
4. If A and B are (-2, -2) and (2, -4) respectively, Find the coordinates of Q such that AQ =
on the line segment AB
3
AB and Q lies
7
Ans :(2/7, -20/7)
5. The area of triangle is 5 square units. Two of its vertices are A (1, 2) and B (6, 2). If the third vertex lies
on the line x – 2y + 1 = 0, Find the third vertex.
APPLICATIONS OF TRIGONOMETRY
(HEIGHTS AND DISTANCE)
KEY POINTS
Line of sight
Line segment joining the object to the eye of the observer
is called the line of sight.
Angle of elevation
When an observer sees an object situated in upward
direction, the angle formed by line of sight with horizontal
line is called angle of elevation.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 15
Angle of depression
When an observer sees an object situated in downward
direction the angle formed by line of sight with horizontal
line is called angle of depression.
Trigonometric Ratios of some specific angles.
q
sin q
cosq
tan q
cot q
sec q
cosec q
30o
½
÷3/2
1/÷3
÷3
2/÷3
2
45o
1/÷2
1/÷2
1
1
÷2
÷2
60o
÷3/2
1/2
÷3
1/÷3
2
2/÷3
LEVEL- 1
1. A 20m long ladder rest against a wall. If the feet of the ladder is 10 m away from the wall, then find
the angle of the elevation.
Ans. 600
2. If √????? ? ???ℎ?? ???? ?ℎ???????? ?????
−
?????
Ans. -1/2
3. An observer 1.5m tall is 20.5 meters away from a tower 22m high. Determine the angle of elevation of
the top of the tower from the eye of the observer.
Ans. 45°
4. The ratio of length of rod and its shadow is 1: 3 . Find the angle of elevation of the sun.Ans: 300
5. In a rectangle ABCD, AB =20cm –BAC=600then find the length of the side AD.
Ans. 20√?cm
6. In rectangle, the angle between a diagonal and a side is 300 and the length of its diagonal is 6 cm . Find
Ans:9 3 cm2
the area of rectangle.
O
Q
7. In the adjacent figure, what are the angles of depression of the top and bottom
from the top of a tower h m high:
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
A
450
300
of a pole
M
Page 16
Ans450, 600
LEVEL -2
1. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground
making an angle 30 0 with it. The distance between the foot of the tree to the point where the top
touches the ground is 8 m. Find the height of the tree.
Ans: 8 3 m
2. A ladder 50m long just reaches the top of a vertical wall. If the ladder makes an angle of 600 with the
wall, find the height of the wall.
Ans. 25 m
3. A circus artist is climbing a 20m long rope, which is tightly stretched and tied from the top of a vertical
pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is
300.
Ans. 10 m
4. A hoarding 1.46 metretall,stands on the top of a pole, which shows the need for abolition of child labour
in the society.From a point on the ground, the angle of elevation of the hoarding is
600And from the same point the angle of elevation of the top of the pole is 450.Find the height of the
pole.
(i)
What are the ill effects of child labour on a society?
(ii)
Write any two ways by which the child labour can be abolished?
Ans: 72 m
LEVEL - 3
1. The shadow of a tower standing on a level plane is found to be 50m longer when sun’s elevation is 300
then when it is 600. Find the height of the tower.
Ans. ??√??
2. The angle of depression of the top and bottom of a tower as seen from the top of a 60 3 m high cliff
are 450 and 600 respectively. Find the height of the tower.
Ans.43.92m
3. From a window (9m above ground) of a house in a street, the angles of elevation and depression of the
top and foot of another house on the opposite side of the street are 300 and 600 respectively. Find the
height of the opposite house and width of the street.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Ans.12m,3√?m
Page 17
4. The angle of elevation of a jet fighter from a point A on the ground is 600.After a flight of 15 seconds,
the angle of elevation changes to 30◦. If the jet is flying at a speed of 720km/ hr, find the constant
height at which the jet is flying.
Ans;1500m
5. A window in a building is at a height of 10m above the ground. The angle of depression of a point P on
the ground from the window is 300. The angle of elevation of the top of the building from the point P is
600 . Find the height of the building.
Ans; 30m
6. A boy , whose eye level is 1.3m from the ground , spots a balloon moving with the wind in a
horizontal line at same height from the ground. The angle of elevation of the balloon from the eyes of
the boy at any instant is 600. After 2 seconds, the angle of elevation reduces to 300 If the speed of the
wind at that moment is 29√? m/s , then find the height of the ballon from the ground . Ans; 88.3m
7. A man on the deck on a ship 14m above water level observes that the angle of elevation of the top of
a cliff is 600and the angle of depression of the base of the cliff is 300. Calculate the distance of the cliff
from the ship and the height of the cliff.
Ans ; h= 56m , distance 24.25m
8. A straight highway 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 30°, which is approaching the foot of tower with a uniform speed
Six minutes later, the angle of depression of the car is found to be 60 . Find the time taken by the car to
reach the foot of the tower.
Ans.3 minutes
9. As observed from the top a light house, 100 m high above sea level, the angle of depression of a ship
sailing directly towards it, changes from 300 to 600. Determine the distance travelled by the ship during
the period of observation (use
3 = 1.732)
Ans115.5m
10. Two hoardings on “Save energy” are put on two poles height 10 m and 15 m standing to opposite to
each other on aground. if the distance between their feet is 5 3 m Find the distance between their
tops . Why do we need to save energy?
Ans10m
SELF EVALUATION/HOTS
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 18
1. An aero plane when flying at a height of 3125m from the ground passes vertically below another
Plane at an instant when the angles of elevation of the two planes from the same point on the
ground are 30°and 60°respectively. Find the distance between the two planes at that instant.
Ans ; 6250m
2. From the top of a building 60m high, the angels of depression of the top and bottom of a vertical lamp
post are observed to be 30° and 60°respectively. Find [i] horizontal distance between the building and
the lamp post [ii] height of the lamp post.
Ans. 34.64m, h=40m
3. If the angle of elevation of a cloud from a point h meter above a lake is a and angle of depression of its
reflection in the lake be b . Prove that the distance of the cloud from the point of observation is
2h seca
tan b - tan a
4. A boy standing on a horizontal plane finds a bird flying at a distance of 100 m from him at an elevation of
30 0. A girl standing on the 20 m high building, Find the angle of elevation of the same bird to be 450.
Both the boy and the girl are on opposite sides of the bird. Find the distance of the bird from the girl.
Ans: 30 2 m
5. The angle of elevation of a cloud from a point 60m above a lake is 30◦ and the angle of depression of the
reflection of the cloud in the lake is 60° . Find the height of the cloud from the surface of the lake.
Ans 120m
Circle
Basic concepts
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 19
1. In figure (i) line P and the circle and no common point hence ,the line P known as non intersecting line with
respect to the circle.
2. In the figure (ii) , a line P intersects the circle in two distinct points, it is called a secant of the circle .
In the figure (iii) , the lint P intersects the circle in one and only one point a, and is set to be are tangent to the circle. The
point A at which the tangent line means the circle is called point of contact.
The word tangent to a circle has been derived from the latin word “ tang ere “,which means ‘to touch’ and was
introduces by the Danish Mathematician Thomes Fineke in 1853.
Key Words
1. Circumscribed circle of a polygon: It is a circle passing through all the vertices of the polygon. The centre of
this circle is called Circumcentre and its radius is called circumradius.
2. Inscribed circle (Encircled of the polygon): It is the largest circle that can be contained in the polygon and it
touches each side of the polygon at a point. The centre of this circle is called incentre and its radius is called
inradius.
THEOREM: 10.1
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 20
Given: A circle with cente O and a tangent AB to the circle at a point P.
To prove: OP is perpendicular AB
Construction: Take any point R, other than P on the tangent AB. Join OR. Suppose OR meets the circle at Q.
Proof: OP=OQ (radius of the circle)
OP < OQ +QR
OP < OR
Thus, OP is shorter than any other line segment joining O to a any Point of AB, other than P.
We know that among all line segments joining the point O to a point AB, the shortest one is perpendicular to AB
Hence OP is perpendicular to AB
THEOREM 10.2
The length of tangents drawn from an External point P to a circle are equal
Given: AP and AQ are two tangents from a point to a circle C (O,r)
To Prove: AP = AQ
Construction: Join OA, OP and OQ.
Proof: AP is tangent at P and OP is the radius through P
Therefore OP is perpendicular to AP
Similarly AQ is a tangent at Q and OQ is the radius through Q
Therefore OQ is perpendicular to AQ
In right triangles OPA and OQA
OP =OQ (equal radii of the same circle)
AO = AO (common)
–OPA = –OQA (each 900)
∆ ??? @ ∆ OQA (RHS congruence)
AP =AQ (CPCT)
AP = AQ
LEVEL1
1. At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. Find the length of
the chord CD parallel to XY and at a distance 8 cm from A.
Ans. 8 cm
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 21
2. In a circle of a diameter 10 cm, length of each of the two equal and parallel chords is 8 cm. find the distance
between these two chords.
Ans: 6 cm
3. In the adjoining figure, ∆ ABC is circumscribing a circle, then find the length of BC.
Ans. 9cm
A
8cm
M
3 cm
N
4 cm
B
C
V
4. Find the length of the tangent drawn from a point whose distance from the centre of a circle is 25cm. Given that
the radius of the circle is 7cm.
Ans: 24cm
L
5. From a point P, two tangents PA and PB are drawn to a circle C(O,r) . If OP =2r ,then what is the type of ∆ APB.
Ans. Equilateral triangle
6. If the angle between two radii of a circle is 130?,then find the angle between the tangents at the end of the radii.
Ans. 50?.
7. ABCD is a quadrilateral. A circle centered at O is inscribed in the quadrilateral. If AB = 7cm , BC = 4cm , CD = 5cm
then find DA.
Ans. 8 cm
8. In a ∆ ABC , AB = 8cm , –ABC = 90?. Then find the radius of the circle inscribed in the triangle.
Ans. 2cm
LEVEL2
1. From an external point P, two tangents PA and PB are drawn to the circle with centre O. Prove that OP is the
perpendicular bisector of AB.
2. If PA and PB are two tangents drawn to a circle with centre O , from an external point P such that PA=5cm and
–APB = 60?, then find the length of the chord AB.
Ans. 5cm
3. CP and CQ are tangents from an external point C to a circle with centre O .AB is another tangent which touches
the circle at R and intersects PC and QC at A and B respectively . If CP = 11cm and BR = 4cm, then find the length
of BC.
4.
Ans. 7cm
If all the sides of a parallelogram touch a circle, show that the parallelogram is a rhombus.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 22
5.
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre of the
circle.
6. In adjacent figure; AB & CD are common tangents to two circles of unequal radii. Prove that AB=CD.
A
C
B
D
LEVEL3
1. Four roads have to be constructed by touching village Kanpur in circular shape of radius 1700m as shown in
figure, Sarita got contract to construct the road AB and CD while Vijay got the contract to construct AD and BC
Prove that AB + CD = AD +BC Which value is depicted by the contractor?
2. Prove that the angle between the two tangents to a circle drawn from an external point is supplementary to
the angle subtended by the line segment joining the points of contact to the centre.
3. AB is a chord of length 9.6cm of a circle with centre O and radius 6cm.If the tangents at A and B intersect at
point P then find the length PA.
Ans. 8cm
4. The in circle of a ∆ABC touches the sides BC, CA & AB at D,E and F respectively. If AB=AC, prove that BD=CD.
5. Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the
centre of the circle.
6. PQ and PR are two tangents drawn to a circle with centre O from an external point P. Prove that
–QPR=2–OQR.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 23
LEVEL4
1. Prove that the length of tangents drawn from an external point to a circle is equal. Hence, find BC, if a circle is
inscribed in a ∆ABC touching AB,BC &CA at P,Q &R respectively, having AB=10cm, AR=7cm &RC=5cm.
Ans. 8cm
2. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using the
above, do the following: If O is the centre of two concentric circles, AB is a chord of the larger circle touching the
smaller circle at C, then prove that AC=BC.
3. A circle touches the side BC of a ∆ABC at a point P and touches AB and AC when produced, at Q & R respectively.
Show that AQ=1/2 (perimeter of ∆ABC).
4. From an external point P, a tangent PT and a line segment PAB is drawn to circle with centre O, ON is perpendicular
to the chord AB. Prove that PA.PB=PN2-AN2.
5. If AB is a chord of a circle with centre O, AOC is diameter and AT is the tangent at the point A, then prove that
–BAT=–ACB.
6. The tangent at a point C of a circle and diameter AB when extended intersect at P. If –PCA=1100 , find –CBA
Ans. 700
7. If PA and PB are tangents such that–APB=600 and PB=7cm.Find the length of chord AB .
Ans AB=7 Cm
[Self Evaluation]
1. If PA and PB are tangents from an external point P to the circle with centre O, the find –AOP+–OPA.
Ans. 900
2. ABC is an isosceles triangle with AB=AC, circumscribed about a circle. Prove that the base is bisected by the point
of contact.
3. AB is diameter of a circle with centre O. If PA is tangent from an external point P to the circle with –POB=1150
Ans. 250
then find –OPA
4. PQ and PR are tangents from an external point P to a circle with centre . If –RPQ=1200, Prove that OP=2PQ.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 24
5. Show that tangent line at the end points of a diameter of a circle are parallel.
6. If a, b, c are the sides of a right triangle where c is the hypotenuse , then prove that radius r of the circle touches
the sides of the triangle is given by r= (a + b - c)/2.
0
7. A circle is inscribed in a quadrilateral ABCD where –B=90 .If AD = 24 cm ,AB=30 cm and DS =8cm .
Find the radius r of the circle .
CONSTRUCTION
KEY POINTS
1. Division of line segment in the given ratio.
2. Construction of triangles:a. When three sides are given.
b. When two sides and included angle given.
c. When two angles and one side given.
d. Construction of right angled triangle.
3. Construction of triangle similar to given similar to given triangle as per given scale.
4. Construction of triangles to a circle.
LEVEL - I
1. A farmer divides sugarcane of 7 feet length between his son and daughter equally. Divide it
geometrically considering sugarcane a line of 7 cm using construction. Find the length of each part.
Which value is depicted?
2 Draw a line segment of length 7 cm. Find a point P on it which divides it in the ratio 3:5.
3. Divide a line segment of 7cm internally in the ratio 3:4.
4. Draw a circle of radius 4 cm. Take a point P on it. Draw tangent to the given circle at P.
5. Construct an isosceles triangle whose base 7.5 cm and altitude is 4.2 cm.
6. Is it possible to divide a line segment in the ratio 3 : 1
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
3
?
Page 25
LEVEL –II
1. Construct a triangle similar to a given ∆ABC such that each of its sides is 2/3 of the corresponding sides of ∆ABC.
It is given that AB=4cm BC=5cm and AC=6cm also write the steps of construction.
2. Draw a right triangle ABC in which –B=600 AB=4.5 cm, BC=6.5cm then construct another triangle ABC whose
sides are 5/3 times the corresponding sides of ∆ABC.
3. Draw a pair of tangents to a circle of radius 5cm which are inclined to each other at an angle of 600.
4. Draw a circle of radius 5cm from a point 8cm away from its centre construct the pair of tangents to the circle
and measure their length.
5. Construct a triangle ABC in which BC=6cm –C=300 and –A=1050. Construct another triangle whose sides are
2/3 times the corresponding sides of ∆ABC
6. Draw a circle of radius 4 cm .construct a pair of tangents to it, the angle between them 600. Also justify the
construction. Measure the distance between the centre of circle and the point of intersection of tangents.
AREAS RELATED TO CIRCLES
KEY POINTS
1. Circle:
The set of points which are at a constant distance of r units from a fixed point o is called a circle with centre o.
∑
Perimeters and areas of simple closed figures.
∑
Circumference and area of a circle.
∑
Area of a circular path (i.e., ring).
∑
Sector of a circle and its central angle
∑
Major and Minor sectors.
∑
Segment of a circle – Major and Minor segments.
2. Circumference: The perimeter of a circle is called its circumference.
3. Secant: A line which intersects a circle at two points is called secant of the circle.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 26
4. Arc: A continuous piece of circle is called and arc of the circle..
5. Central angle:- An angle subtended by an arc at the center of a circle is called its central angle.
6. Semi Circle: - A diameter divides a circle into two equal arc. Each of these two arcs is called a semi circle.
7. Segment: - A segment of a circle is the region bounded by an arc and a chord, including the arc and the
chord.
8. Sector of a circle: The region enclosed by and an arc of a circle and its two bounding radii is called a sector
of the circle.
9.
Quadrant: - One fourth of a circle disc is called a quadrant. The central angle of a quadrant is 900.
Important Formula
S.N
NAME
FIGURE
PERIMETER
AREA
1.
Circle
???or ??
??2
2.
Semi- circle
??+ 2r
½ ??2
3.
Ring (Shaded
region)
2 ???+ R)
?(R2-r2)
4.
Sector of a circle
5.
???
l+2r=???° ?
Segment of a
circle
QUESTION BANK CLASS X (MATHEMATICS)
???
+2r
???°
SA-II
??
?
?
Sin
????
???°
or
?
??
?
???? ? ?
- ? sinq
???° ?
Page 27
a. Length of an arc AB=
?
2 ??
???
0
A
B
b. Area of major segment= Area of a circle – Area of minor segment
c. Distance moved by a wheel in 1 rotation=circumference of the wheel
d. Number of rotation in 1 minute =Distance moved in 1 minute / circumference
LEVEL-1
1. If the perimeter of a circle and the area of the circle are numerically equal, then the diameter of the circle is
[Ans-4]
2. The area of the square that can be inscribed in a circle of 8 cm is
[Ans-128cm2]
3. Area of a sector to circle of radius 36 cm is 54 ?cm2 . Find the length arc of the corresponding arc of the circle is
[Ans –3 πcm]
4. A wheel has diameter 84 cm. The number of complete revolution it will take to cover 792 m is.
5. The length of an arc of a circle with radius 12cm is 10 ? cm. The central angle of this arc is .
6. The area of a quadrant of a circle whose circumference is 22 cm is
[Ans-300]
[Ans-1500]
[Ans-.625cm2]
7. If the perimeter of a circle is equal to that of square, then the ratio of their areas is
LEVEL-2
1. If the diameter of a semicircular protractor is 14 cm, then find its perimeter .
[Ans-14/11]
[Ans-36 cm]
2. The radius of two circle are 3 cm and 4 cm . Find the radius of a circle whose area is equal to the sum of the areas
of the two circles.
[Ans: 5 cm]
3. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.
[Ans: 154/3 cm]
4. If a square is inscribed in a circle, what is the ratio of the areas of the circle and the square?
Ans: p : 2)
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 28
5. What is the angle subtended at the centre of a circle of radius 10 cm by an arc of length 5p cm.
LEVEL-3
Ans:900
1. The inner circumference of a circular track is 440m. The track is 14m wide. Find the diameter of the outer circle
[Ans-168m]
of the track.
[Take ? =22/7]
2. Find the area of the shaded region.
[Ans: 4.71cm2]
3. A copper wire when bent in the form of a square encloses an area of 121 cm2 . If the same wire is bent into the
[Ans 154 cm2]
form of a circle, find the area of the circle (Use ?=22/7)
4. A wire is looped in the form of a circle of radius 28cm. It is rebent into a square form. Determine the side of the
[Ans-44cm]
square (use ? ? ?????
LEVEL-4
1. In fig, find the area of the shaded region [use ? ? ?????
2. In fig find the shape of the top of a table in restaurant is that of a sector a circle with centre 0 and –BOD=900. If
OB=OD=60cm find
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 29
i.
ii.
The area of the top of the table
The perimeter of the table top (Take ? ? ?????[
[Ans 8478 cm2]
Ans 402.60 cm]
3. An arc subtends an angle of 900 at the centre of the circle of radius 14 cm. Write the area of minor sector thus
[Ans 49? cm2]
form in terms of ?.
4. The length of a minor arc is 2/9 of the circumference of the circle. Write the measure of the angle subtended by
the arc at the center of the circle.
[Ans 800]
5. The area of an equilateral triangle is 49√3 cm2. Taking each angular point as center, circle is drawn with radius
equal to half the length of the side of the triangle. Find the area of triangle not included in the circles.
[Ans 777cm2]
[Take ?√3=1.73]
SELF EVALUATION
1. If the perimeter of a semi-circular protractor is 108 cm , find the diameter of the protractor (Take π =22/7).
2. Will it be true to say that the perimeter of a square circumscribing a circle of radius a cm is 8a cm? Give
reasons for your answer.
3. Find the area of a sector of a circle of radius 28 cm and central angle 45°.
4. The wheel of a motor cycle is of radius 35 cm. How many revolutions per minute must the wheel make so as
to keep a speed of 66 km/h?
5. A cow is tied with a rope of length 14 m at the corner of a rectangular field of dimensions 20m × 16m. Find
the area of the field in which the cow can graze.
6. In Fig. ABCD is a trapezium with AB || DC, AB = 18 cm, DC = 32 cm and distance between AB and
DC = 14 cm. If arcs of equal radii 7 cm with centers A, B, C and D have been drawn, then find the area of the
shaded region of the figure.
7. A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a
5m long rope .Find
(i) the area of that part of the field in which the horse can graze.
(ii)The owner has not made any shed for the horse. What values are lacking in the owner?
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 30
Ans 19.63cm2, 58.87cm2
SURFACE AREAS AND VOLUMES
(IMPORTANT FORMULA)
SNo
NAME
1
FIGURE
LATERAL CURVED SURFACE
AREA
TOTAL SURFACE
AREA
VOLUME
NOMENCLATURE
Cuboid
perimeter of rect.x
height sq. unit
= 2(l+b) x h sq. unit
2(lxb + bxh + hx
l) sq. unit
L=length,
b=breadth,
h=height
2
Cube
= 2(l+l) x l sq. unit
=2(2l)xl sq. unit
= 4 l2 sq. unit
6 x (area of
Square) sq.unit
= 6 l2sq.unit
3
Right
Circular
Cylinder
circumference of
circle x height sq. unit
= (2 p r)x h sq.unit
4
Right
Circular
Cone
= semi perimeter of
circle x slant
heightsq.unit
=p r l sq. unit
5
Sphere
= 4X Area of circle sq.
unit
=4pr2sq. unit
= curved
surface area +
2x area of
base(circle) sq.
unit
= (2 p r)x h +
2pr2sq.unit
= curved
surface area +
area of circle
sq. unit
=p r l + pr2sq.
unit
= 4X Area of
circle sq. unit
=4pr2sq. unit
= area of
base(rec) x
height cubic
unit
= lxbx h cu unit
= area of base
(square) x
height) cubic
unit
= l2 x l cubic unit
= area of base
(circle) x height
cubic unit
= pr2 x h cubic
unit
6
Hemisph
ere
= half x area of
sphere sq. unit
=2pr2sq. unit
= 3x area of
circle sq. unit
=3pr2sq. unit
2
2
2
2
7
Spherical
shell
=2p(R + r ) sq. unit
3pR +pr sq.
unit
8
Frustum
of a cone
=pl(R+r) sq. unit
where l2=h2+(R-r)2
-p[R2 + r2 +
l(R+r)] sq. unit
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
?
l=edge of
cube
r= radius
h=height
= volume of
?
cylinder cubic
unit
?
? pr2h cubic
?
unit
r=radius of
base,
h=height ,
l=slant height
= √?? ??
= half of volume
of sphere cubic
unit
?
= pr3cubic unit
r=radius of
hemisphere
?
?
? pr3cubic unit
?
?
? p(R3 - r3)
?
cubic unit
2
2
=ph/3[R + r +
Rr] cubic unit
−
r=radius of
the sphere
R=External
radius,
r=internal
radius
R and r =
radii of the
base,
h=height,
l=slant
height.
Page 31
9. Diagonal of cuboid = √?? ? ?? ? ℎ?
10. Diagonal of Cube = ÷3l
LEVEL - 1
[1] Three cubes of volumes 64cm3 are joined end to end. Find the surface area of the resulting cuboid?
Ans: 224cm2
[2] Find the volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm?
Ans:19.4 cm3.
[3] A semicircular thin sheet of metal of diameter 28cm is bent and an open conical cup is made. Find depth. (7÷3)
[4] Find the volume of the largest right circular cone that can be cut out from a cube of edge 4.9 cm is?Ans:30.8cm3.
[5] The slant height of a frustum of a cone is 4 cm and the perimeter of its circular ends are18cm and 6cm. Find the
curved surface area of the frustum [use ? ?
??
].
?
Ans: 48cm2.
[6] A plumbline is a combination of which geometric shapes? ANS:A cone with hemisphere.
[7] What is the ratio of the volume of a cube to that of sphere which will fit inside it?
Ans : 6 : p
[8] A cylinder and a cone are of same radius and of same height. Find the ratio of the volume of the cylinder to that of
the cone.
Ans : 3 : 1
LEVEL - 2
[1] The slant height of the frustum of a cone is 5 cm .If the difference between the radii of its two circular ends is
4cm . Write the height of the frustum.
Ans:3cm
[2] A cylinder, a cone and a hemisphere are of same base and of same height .Find the ratio of their volumes?
Ans: [3:1:2].
[3] A cone of radius 4cm is divided into two parts by drawing a plane through the midpoint of its axis and parallel to its
base, compare the volume of the two parts.
Ans:1:7
[4] How many spherical lead shots each having diameter 3cm can be made from a cuboidal lead solid of dimensions
9cm X 11cm X 12cm.
Ans:84
[5] Three metallic solid cubes whose edges are 3cm, 4cm, and 5cm are melted and converted into a single cube .
Find the edge of the cube so formed?
QUESTION BANK CLASS X (MATHEMATICS)
Ans :- 6cm .
SA-II
Page 32
LEVEL-3
[1] 400 persons took dip a rectangular pool which is 40 m long and 30 m broad. What is rise in the level of water in
thepool, if the average displacement of water by a pers0n is 0.3m 3?
Ans : 10 cm
[2] Find the number of metallic circular disk with 1.5cm base diameter and of height 0.2 cm to be melted to form
a right circular cylinder of height 10cm and diameter 4.5cm ?
Ans:-450
[3] From a solid cube of side 7cm,a conical cavity of height 7cm and radius 3cm is hollowed out . Find the volume
Ans:-277cm3.
of remaining solid?
[4] A cubical block of side 7cm is surmounted by a hemisphere. what is the greatest diameter of the hemisphere
Ans:- 7cm,332.5cm2.
can have? Find the surface area of the solid?
[5] A heap of rice is in the form of a cone of diameter 9m and height 3.5m .Find the volume of the rice .How much
Ans:-74.25m3, 80.61 m2 .
canvas cloth is required to just cover the heap?
[6] A square field and an equilateral triangle park have equal perimeter .If the cost of ploughing the field at the
rate of Rs 5/m2 is Rs 720. Find the cost of maintain the park at the rate of Rs10/m2?
Ans:-Rs1108.48
[7] Student of IX class packed 500 packet of biscuits each of dimension 6 cm x 3 cm x2 cm in boxes each of
Volume 1800 cm3 to be distributed to the children of the flood victims
(i) Find the number of boxes required?
(ii) Which mathematical concept is used here?
(iii)What moral values are shown?
Ans: 10 boxes, volume concept, sharing and caring, team work etc.
[7} A Cone of maximum size is carved out from a cube of edge 14 cm. find the surface area of the cone and of
Ans: 154(÷5 +1) cm2,(1022+154)cm2
the remaining solid left out after the cone is carved out.
LEVEL -4
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 33
[1] A well in village of diameter 3cm and 14m deep in dug. The earth, taken out of it, has been evenly spread all around
it in the shape of a circular ring of width 4m to form an embankment find the height of embankment? Grampradhan
spend the money for the embankment, what value is shown by him?
?
?
ANS:- m.
[2] 21 glass spheres each of radius 2cm are packed in a cuboidal box of internal dimensions 16cmX8cmX8cmand then
ANS:-320cm3.
the box is filled with water. Find the volume of water filled in the box?
[3] The slant height of the frustum of a cone is 4cm and the circumferences of its circular ends are 18cm and 6cm. Find
. ANS:-48cm2, 76.63cm2.
curved surface area and total surface area of the frustum
[4] A farmer connects a pipe of internal diameter 25cm from a canal into a cylindrical tank in his field, which is 12m in
diameter and 2.5m deep. If water flows through the pipe at the rate of 3.6km/hr, in how much time will the tank be
filled? Also find the cost of water, if the canal department charges at the rate of Rs0.07/m3?
Ans:-96min, Rs19.80
[5] A spherical glass vessel has a cylindrical neck 7cm long and 4cm in diameter. The diameter of the spherical part is
ANS:-4939cm3.
21cm Find the quantity of water it can hold.
[6] The surface area of a solid metallic sphere is 616cm2. It is melted and recast into a cone of height 28cm. Find the
diameter of the base of the cone so formed.
ANS:-14cm
[7] “Water is precious , Save it” Rain water is collected at top of the building it flows from a pipe of internal
Diameter 25cm into a cylindrical tank which is 12m in diameter and 2.5 m deep. If water flows through
the pipe at the rate of 3.6 km /h in how much time will the tank be filled? What value is depicted by owner
of the building?
Ans: 96 minutes
SELF EVALUTION
[1] A spherical copper shell, of external diameter 18cm,is melted and recast into a solid cone of base radius 14cm
And height 4cm. Find the inner diameter of the shell.
Ans:-16cm.
[2] A bucket is in the form of a frustum of a cone with a capacity of 12308.8cm3. The radii of the top and bottom
Circular ends of the bucket are 20cm and 12cm respectively. Findthe height of the bucket and also the area
of metal sheet used in making it [take ? 3.14]?
Ans:-?? ???? ????? ? ????????? ?.
[3] The volume of a solid metallic sphere is 616cm3.its is melted and recast into a cone of height 28cm. Find the
Diameterof the base of the cone so formed?
QUESTION BANK CLASS X (MATHEMATICS)
Ans:-21cm.
SA-II
Page 34
[4] From a solid cylinder whose height is 8cm and radius 6cm , a conical cavity of height 8cm and of base radius 6cm ,
is hollowed out. Find the volume of the remaining solid correct to two places of decimals. Also find the total
Ans:-603.19cm3, 603.19cm2
Surface area of the remaining solid [take ?=3.14] ?
[5] A cylindrical vessel, with internal diameter10cm and height 10.5 cm is full of water. A solid cone of base
Diameter 7cm and height 6cm is completely immersed in water. Find the volume of:(i) Water displaced out of the cylindrical vessel.
Ans:- (i): 77cm3 , (ii) 748cm3.
(ii) Water left in the cylindrical vessel.
[6] A wooden article was made by scooping out a hemisphere from each ends of a solid cylinder. If the height of
The cylinder is 20cm, and radius of the base is 3.5cm , find the total surface area of the article.
Ans:-544cm2.
?? 3
m of
??
[7] A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains 41
If the internal diameter of the building is equal to its total height above the floor, find the height of the
building?
air.
Ans:-4m .
[8] A shuttle cock used for playing badminton has the shape of a frustum of a cone mounted on a hemisphere.
The external diameters of the frustum are 5cm and 2cm, the height of the entire shuttle cock is 7cm.
Ans:-74.38cm2
Find the external surface area.
[9] Five containers shaped like a right circular cylinder having diameter 12cm & height 15cm are full of ice cream this ice
cream is to be filled into cans of height 12cm & diameter 6cm having a hemispherical shape on the top & is to be
distributed to the children in an orphanage.
i) Find the number of such cans which can be filled with ice-cream.
ii)What values does a person doing such an act possess?
PROBABILITY
KEY CONCEPTS
1. Probability:- The theoretical probability of an event E, written as P(E) is defined as.
P (E)= Number of outcomes Favorable to E
Number of all possible outcomes of the experiment
Where we assume that the outcomes of the experiment are equally likely.
2. The probability of a sure event (or certain event) is 1.
3. The probability of an impossible event is 0.
4. The probability of an Event E is number P (E) such that 0≤P(E)≤1.
5. Elementary events:- An event having only one outcome is called an elementary event. The sum of the
probabilities of all the elementary events of an experiment is 1.
6. For any event E,P(E)+P(??)=1, where?? stands for not E, E and ?? are called complementary event.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 35
7. Performing experiments:a. Tossing a coin.
b. Throwing a die.
c. Drawing a card from deck of 52 cards.
8. Sample space:-The set of all possible outcomes in an experiment is called sample space.
LEVEL-1
1. The probability of getting bad egg in a lot of 400 is 0.035.Then find the no. of bad eggs in the lot.
[ans.14]
2. Write the probability of a sure event.
[ans.1]
3. What is the probability of an impossible event?
[ans.0]
4. When a dice is thrown, and then find the probability of getting an odd number less than 3.
[ans. ]
?
?
5. A girl calculates that the probability of her winning the third prize in a lottery is 0.08.If 6000 tickets are sold, how
many ticket has she brought.
[Ans.480]
6. What is probability that a non-leap year selected at random will contain 53 Sundays.
[Ans. ]
?
?
7. A bag contains 40 balls out of which some are red, some are blue and remaining are black. If the probability of
??
?
drawing a red ball is ?? and that of black ball is?, then what is the no. of black ball.
[Ans.10]
8. Two coins are tossed simultaneously. Find the probability of getting exactly one head.
[Ans. ]
9. A card is drawn from a well shuffled deck of 52 cards. Find the probability of getting an ace.
[Ans. ]
10. In a lottery, there are 10 prizes and 25 blanks. Find the probability of getting a prize.
[Ans. ]
?
?
?
??
?
?
LEVEL-2
1. Find the probability that a no. selected at random from the number 3, 4, 5, 6...25 is prime.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
?
??
[Ans. ]
Page 36
2. A bag contains 5 red,4 blue and 3 green balls. A ball is taken out of the bag at random. Find the probability that
? ?
[Ans.??,?]
the selected ball is (a) of red color (b) not of green color.
3. A card is drawn at random from a well-shuffled deck of playing cards. Find the probability of drawing
? ?
, ]
?? ??
(a) A face card (b) card which is neither a king nor a red card
[Ans.
?
?
[Ans. ]
4. A dice is thrown once. What is the probability of getting a number greater than 4?
5. Two dice are thrown at the same time. Find the probability that the sum of two numbers appearing on the top of the
?
?
dice is more than 9.
[Ans. ]
?
6. Two dice are thrown at the same time. Find the probability of getting different numbers on both dice. [Ans.?]
?
?
[Ans. ]
7. A coin is tossed two times. Find the probability of getting almost one head.
8. Cards with numbers 2 to 101 are placed in a box. A card selected at random from the box. Find the probability that
?
the card which is selected has a number which is a perfect square.
[Ans.???]
9. Find the probability of getting the letter M in the word “MATHEMATICS”.
[Ans. ]
?
??
LEVEL-3
1.Cards bearing numbers 3,5,…………..,35 are kept in a bag. A card is drawn at random from the bag. Find the
probability of getting a card bearing (a)a prime number less than 15 (b)a number divisible by 3 and 5.
? ?
?? ??
[Ans. , ]
2. Two dice are thrown at the same time. Find the probability of getting (a)same no. on the both side (b)different no. on
??
??
both sides.
[Ans. , ]
3. A child game has 8 triangles of which three are blue and rest are red and ten squares of which six are blue and rest
are red. One piece is lost at random. Find the probability of that is (a) A square (b) A triangle of red colour.
? ?
? ??
[Ans. , ]
4.Two dice are thrown simultaneously. What is the probability that:
(a)5 will not come up either of them? (b)5 will come up on at least one? (C)5 will come at both dice?
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 37
?? ?? ?
?? ?? ??
Ans. , ,
5. The king, queen and jack of clubs are removed from a deck of 52 playing cards and remaining cards are shuffled.
A card is drawn from the remaining cards. Find the probability of getting a card of (a)heart
?? ? ??
Ans.??,??,??
(b)Queen(c) clubs
6. A game consists of tossing a one-rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses
Give the same result, i.e., 3 heads or three tails and looses otherwise. Calculate the probability that Hanif will
?
?
Lose the game.
[Ans. ]
7. Cards bearing numbers 1,3,5,…………..,37 are kept in a bag. A card is drawn at random from the bag. Find the
probability of getting a card bearing
?
?
??
(a)a prime number less than
Ans.
(b)a number divisible by 3 and 5.
[Ans. ?
?
??
8. A dice has its six faces marked 0, 1, 1, 1, 6, 6.Two such dice are thrown together and total score is recorded.
(a) How many different scores are possible? (b) What is the probability of getting a total of seven?
[Ans. {a} 5 scores (0, 1, 2, 6, 7, 12)
?
?
{b} ]
9) In a school, 27 children out of 45 worked for making their class room clean. What is the Probability that a child
selected at random worked for the school? (i) What social values are being reflected here?
Ans27/45
10) Gopi buys a fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing 5
male fish and 8 female fish . What is the probability that the fish taken Out is a male fish? (i) Gopi does not take care of
feeding the fish & it dies after 6 days ? What values he has violated?
Ans5/13
Self Evaluation
1. Three unbiased coins are tossed together. find the probability of getting
(i)
all heads
(ii)
two heads
(iii)
one heads
(iv)
at least two heads
?
Ans. ?
?
Ans. ?
?
Ans. ?
2. Two dice are thrown simultaneously .Find the probability of getting
?
Ans. ?
?
An even number as the sum.
Ans.?
3. Cards marked with the number 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from the
box. Find the probability that the number on the card is:
(i)
An even number
QUESTION BANK CLASS X (MATHEMATICS)
Ans.
SA-II
?
?
Page 38
(ii)
A number less than 14
Ans.
(iii)
A number is perfect square
Ans.
(iv)
A prime number less than 20
Ans.
?
??
?
???
?
??
4. Out of the families having three children, a family is chosen random. Find the probability that the family has
(i)
Exactly one girl
Ans.
(ii)
At least one girl
Ans.
(iii)
At most one girl
Ans.
?
?
?
?
?
?
5. Five card the ten, jack, queen, king, and ace of diamonds are well shuffled with their face downward . One card
is picked up at random
What is the probability that the card is the queen?
(ii)
If the queen is drawn and put aside what is the probability that the second card picked up is
(a) An ace
(b)
A queen
6. In a cooperative society, 60 people go to the same office they all use their conveyance, 10 people use
Ans.
?
(i)
?
?
Ans. ??
?
Theirscooters, 10 go by their cars and the rest use their motorcycles.
(i)
What is the probability of people going by motorcycles?
(ii)
One day they all decided to go by cars but a car can accommodated only 5 people. What is the probability
of people going by a car now?
(iii)
Which value is shown in (ii) part?
SAMPLE PAPER-I FOR SA-II
CLASS – X
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 39
MATHEMATICS
Time : 3 hours 45 minutes
Maximum Marks : 90
General Instructions:
1. All questions are compulsory.
2. The question paper consists of 31 questions divided into four sections A, B, C and D. Section A
comprises of 4 questions of 1 mark each, Section B comprises of 6 questions of 2 marks each. Section
C comprises of 10 questions of 3 marks each and Section D comprises of 11 questions of 4 marks
each.
3. Use of calculator is not permitted.
4. An additional 15 minutes time has been allotted to read this question paper only.
SECTION A
1. If the quadratic equation px – 2 √5 px + 15 = 0 has two equal roots, then find the value of p.
2. In given Figure, a tower AB is 20 m high and BC, its shadow on the ground, is 20 √3 m long.
Find the Sun’s altitude.
2
3. Two different dice are tossed together. Find the probability that the product of the two numbers on the top of
the dice is 6.
4. In given Figure, PQ is a chord of a circle with centre O and PT is a tangent. If –QPT = 600, find PRQ.
SECTION B
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 40
1. In given Figure, two tangents RQ and RP are drawn from an external point R to the circle with centre O.
If –PRQ = 1200, then prove that OR = PR + RQ.
2. In given Figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that the segments BD
and DC are respectively of lengths 6 cm and 9 cm. If the area of triangle ABC is 54 cm2, then find the lengths
of sides AB and AC.
3. Solve the following quadratic equation for x :
4x2 + 4bx – (a2 – b2) = 0
4. In an AP, if S5 + S7 = 167 and S10 = 235, then find the AP, where Sn denotes the sum of its first n terms.
9. The points A(4, 7), B(p, 3) and C(7, 3) are the vertices of a right triangle, right-angled at B. Find the value of p.
10. Find the relation between x and y if the points A(x, y), B(– 5, 7) and C(– 4, 5) are collinear.
SECTION C
th
11.The 14 term of an AP is twice its 8th term. If its 6th term is – 8, then find the sum of its first 20 terms..
12. Solve for x :
√3 x2 – 2 √2 x – 2√3 = 0
13.
seconds,
0
the angle of elevation changes to 30 . If the airplane is flying at a constant height of 1500 √3 m, find the
speed of the plane in km/hr.
14. If the coordinates of points A and B are (– 2, – 2) and (2, – 4) respectively, find the coordinates of
?
P such that AP = ?AB, where P lies on the line segment AB.
?
15. The probability of selecting a red ball at random from a jar that contains only red, blue and orange balls is ? .
?
The probability of selecting a blue ball at random from the same jar is . If the jar contains 10 orange balls,
?
find the total number of balls in the jar.
16. Find the area of the minor segment of a circle of radius 14 cm, when its central angle is 600. Also find the
??
area of the corresponding major segment. [Use π = ? ]
17. Due to sudden floods, some welfare associations jointly requested the government to get 100
tents fixed immediately and offered to contribute 50% of the cost. If the lower part of each tent is of
the form of a cylinder of diameter 4.2 m and height 4 m with the conical upper part of same diameter but
of height 2.8 m, and the canvas to be used costs Rs.100 per sq. m, find the amount, the associations will
??
have to pay. What values are shown by these associations ? [Use π= ? ]
18. A hemispherical bowl of internal diameter 36 cm contains liquid. This liquid is filled into 72 cylindrical bottles of
diameter 6 cm. Find the height of the each bottle, if 10% liquid is wasted in this transfer.
19. A cubical block of side 10 cm is surmounted by a hemisphere. What is the largest diameter that the hemisphere
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 41
can have ? Find the cost of painting the total surface area of the solid so formed, at the rate of
Rs 5 per 100 sq. cm. [ Use π = 3.14 ]
20. 504 cones, each of diameter 3.5 cm and height 3 cm, are melted and recast into a metallic sphere. Find
??
the diameter of the sphere and hence find its surface area. [Use π= ? ]
SECTION D
21. The diagonal of a rectangular field is 16 metres more than the shorter side. If the longer side is 14 metres more
than the shorter side, then find the lengths of the sides of the field.
22. Find the 60th term of the AP 8, 10, 12, ..., if it has a total of 60 terms and hence find the sum of its last 10 terms.
23. A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an
average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the total journey, what is its
first speed ?
24. Prove that the lengths of the tangents drawn from an external point to a circle are equal.
25. Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points
of the arc.
26. Construct a triangle PQR with PQ= 6cm, QR=7cm and –Q=600 and then another triangle similar to
it whose sides are 5/4 of the corresponding sides of triangle PQR.
27. At a point A, 20 metres above the level of water in a lake, the angle of elevation of a cloud is 300. The angle of
depression of the reflection of the cloud in the lake, at A is 600. Find the distance of the cloud from A.
28. A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card
drawn is
(i) a card of spade or an ace.
(ii) a black king.
(iii) neither a jack nor a king.
(iv) either a king or a queen.
29. Find the values of k so that the area of the triangle with vertices (1, – 1), (– 4, 2k) and (– k, – 5) is 24 sq. units.
30. In given Figure, PQRS is a square lawn with side PQ = 42 metres. Two circular flower beds are there on the
sides PS and QR with centre at O, the intersection of its diagonals. Find the total area of the two flower beds
(shaded parts).
31. From each end of a solid metal cylinder, metal was scooped out in hemispherical form of same
diameter. The height of the cylinder is 10 cm and its base is of radius 4.2 cm. The rest of the
cylinder is melted and converted into a cylindrical wire of 1.4 cm thickness. Find the length of
??
the wire. [Use π= ? ]
EXPECTED ANSWERS/VALUE POINTS
SECTION - A
QUESTION BANK CLASS X (MATHEMATICS)
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QUESTION BANK CLASS X (MATHEMATICS)
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QUESTION BANK CLASS X (MATHEMATICS)
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QUESTION BANK CLASS X (MATHEMATICS)
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QUESTION BANK CLASS X (MATHEMATICS)
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QUESTION BANK CLASS X (MATHEMATICS)
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QUESTION BANK CLASS X (MATHEMATICS)
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QUESTION BANK CLASS X (MATHEMATICS)
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QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 50
Time : 3 hours 45 minutes
SAMPLE PAPER-II FOR SA-II
CLASS – X
MATHEMATICS
Maximum Marks : 90
General Instructions:
5. All questions are compulsory.
6. The question paper consists of 31 questions divided into four sections A, B, C and D. Section A
comprises of 4 questions of 1 mark each, Section B comprises of 6 questions of 2 marks each. Section
C comprises of 10 questions of 3 marks each and Section D comprises of 11 questions of 4 marks
each.
7. There is no overall choice. However, internal choice has been provided in 1 question of two marks, 3
questions of three marks each and 3 questions of four marks each. You have to attempt only one of
the alternatives in all such questions.
8. Use of calculator is not permitted.
9. An additional 15 minutes time has been allotted to read this question paper only.
SECTION A
1. In the Arithmetic progression :6,9,12,15,……… the common difference is
2. If a sphere and a cone of height ‘h’ have same radius and same volume then the ratio r:h is
3. The sum of the roots of y2+6y+5=0 is
4. If the radius of sphere is 8 cm , then its surface area ( in cm2) is
SECTION B
5. First term of an A.P is -5 and its last term is 45. If the sum of these terms is 120, find the number of
terms.
6. Find the value of p for which the points (-1,-1),(2,p) and (8,11) are collinear .
OR
The co-ordinates of the mid point of the line segment joining (3p,4) and (-2,2q) are (5,3).Find the
value p and q.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 51
7. In the figure ,AR=4cm,BR=3cm andAC=11cm.Find the length of BC .
8. Find the volume of a frustum of a cone whose height is 4cm and radii of the ends are 7m and 4m.
9. Determine the value of p for which 9y2-24y+p=0 has equal roots.
10. Two cubes each of side 10 cm are joined end to end.Find the surface area of the resulting cuboid.
SECTION C
11. Volume of two hemispheres are in the ratio 64:27 . Find the ratio of their curved surface areas.
12. A card is drawn at random from awell shuffied deck of 52 playing cards. Find the probability that the
card drawn is (a) ablack king
(b)ace.
13. A tent is cylindrical upto a height of 3m and conical above it .If the diameter of the base is 105 m and
slant height of the conical part is 53 cm, find the cost of canvas used to make the tent at Rs 10 per
square metre.
14. Acircle is inscribed in a quadrilateral ABCDwhere –B=90 0.If AD = 24 cm ,AB=30 cm and DS =8cm . Find
the radius r of the circle .
15. The sum of 4th and 8th term of an A.P.is 37 and the sum of 6th and 12th terms is 46. Find the firstterm of
the A.P.
16. Construct a pair of tangents to a circle of radius 3 cm from a point P at a distance of 5 cm from the
centre.
OR
Construct a triangle PQR with PQ= 6cm, QR=7cm and –Q=600 and then another triangle similar to
it whose sides are 5/4 of the corresponding sides of triangle PQR.
17. An observer 1.7m tall is 30.3m away from the foot of a tower. The angle of elevation of top of the tower
from her eyes is 450. Find the height of the tower.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 52
3a2x2+8abx+4b2=0
OR
1
2
6
+
= (x π 0,1,2)
x - 2 x -1 x
18. Solve for x:
19. Students of Class X packed 500 packets of biscuits each of dimension 6cmx3cmx2cm in boxes each of
volume 18000cm3 to be distributed to the children of flood victims.
i)
Find the number of boxes required.
ii)
Which mathematical concept is used here ?
iii)
What moral values are represented by the class X students?
20. Determine the sum of all multiples of 9 lying between 100 aand 200.
OR
th
Find the sum of first 20 terms of an A.P whose n term is 4n-1.
SECTION D
21. Prove that the lengths of the tangents drawn from an external point are equal.
22. A plane covers a distance of 1200 km at an uniform speed. Had the speed been 100 km/ hr more, it
takes 1 hour less for the journey. Find its original speed.
OR
A motor boat whose speed is 18 km/hr in still water, takes 1hour more to go 24 km upstream than
to return downstream to the same spot. Find the speed of the stream.
23. If P (2,-1), Q (3,4), R (-2,3)and S (-3,-2) are four points in a plane, show that PQRS is a rhombus but not
square.
24. Cards marked with numbers 1,2,3,……..,40 are placed in a box and mixed thoroughly. One card is
Drawn at random from the box. Find the probability that the number on the drawn card is
a) Divisible by 3 and 5
b) a prime number
c) a perfect square
25. The curved surface area of a 16 m deep cylinder is plastered with concrete at the rate of Rs 15 per m2
If the total cost of plastering the curved surface is Rs 5280, find the capacity of the cylinder.
26. Find the shaded area if ABCD is a square, region I is a semicircle with a diameter 14 cm, region II and
III are quadrants with centres at A and B respectively.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 53
OR
In D ABC, –A=900, AB=6 cm, BC=10 cm and AC=8 cm. Find the radius of the circle and the shaded area
if O is the centre of the incircle of D ABC.
27. A man saves Rs 320 during the first month, Rs 360 in the second month,Rs 400 in the third month
And so on. If he continues his savings in this squence, in how many months will he save a total of
Rs 20000?
28. The sum of the radius of the base and the height of a solid cylinder is 37 cm. If total surface area of the
cylinder is 1628 cm2. Find the radius and volume of the cylinder.(use p=22/7)
OR
The height of a cone is 32 cm .A small cone is cut off at the top by a plane parallel to its base. If
its volume is 1/64 of the volume of the given cone, at what height about the base is the cone cut?
(use p=22/7)
29. Find the ratio in which Y- axis divides the line segments joining the points (5,-6) and (-1,-4) also find the
Coordinates of the point of division.
30. Two poles are erected on either bank of a river just opposite to each other. One pole is 40 m high freom
the top and foot of this pole. The angles of elevation of the top of the other pole are 300 and 600
respectively. Find the height of the other pole and width of the river.
31. The sum of first 15 terms of an A.P is 105 and the sum of the next 15 terms is 780. Find the first three
terms of the arithmetic progression.
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 54
ANSWER KEY: (Marking scheme)
Section A
1. 3
2.
1:4
Section B
5. a=-5, an=45, Sn=120
(n/2) (a+ an) = 120
(n/2) (-5+45) = 120
n/2 =3 , n = 6
6. ½[x1(y2-y3)+x2(y3-y1)+x3(y1-y2)]=area
½[-1(p-11)+2(11+1)+8(-1-p)]=0
-p+11+24-8-8p=0
-9p=-27
p=3
Or
(3p-2)/2=5, p=4
(4+2q)/2=3, q-=1
7. BR=BP=3 cm, AR= AQ=4 cm
CQ=11-4=7
CP=7 cm
BC=7+3
=10
1
8. Volume= ph[r2+R2+rR]
3
1 22
= x x4 x[42+72+4x7]
3 7
1 22
= x x4 x[16+49+28]
3 7
=389.7 cubic cm
9. b2-4ac=0
(-242-4x9p)=0
-36p=-576
p=16
10. l=20 cm, b=10 cm, h=10 cm
Surface area=2x(lb+bh+lh) sq. unit
=2x(200+100+200)
=1000 cm2
11. 2/3pR32∏/3pr3=64/27
R3∏r3=64/27
R ∏r =4/3
2pR2∏2pr2=(4/3)2
= 16/9
3.
-3
4.
256p
½
½
½
½
½
½
½
½
1
1
1
½
½
½
½
½
½
½
½
½
½
½
½
½
½
SECTION B
1
½
½
1
12. Total Cards=52
1
a) P(black king)=2/52 or 1/26
1
b) P(an ace)=4/52 or 1/3
QUESTION BANK CLASS X (MATHEMATICS)
1
SA-II
Page 55
13. Canvas area=2pr h+pr l s q. unit
=pr(2h+l)
=(22/7)x(105/2)x (6+53)
= 9735 sq. m
cost= 9735x10=Rs 97350
1
½
½
1
14. DR=DS, AR=AQ, BQ=BP (tangents from an external point)
DS=8 cm \ DR=8 cm
AR=24-16
= 16cm, AQ=16 cm
BQ=14cm , BP=14 cm
–B=900–OQB=–OPB=900
–POQ=900 Also BQ=BP \ OPBQ is square
So, r=BP=14 cm
½
½
½
½
½
½
15. a4+a6=37
a+3d+a+7d=37, 2a+10d=37
a6+a12=46
a+5d+a+11d=46, 2a+16d=46
½
½
½
½
16. Construct a circle
Construction of tangents
Or
To draw a Triangle PQR
Construction of tangents
1
2
1
2
17.
x
.
30.3
x
1=
x=30.3
.,
30.3
Height of the tower=30.3+1.7=32
Tan450=
½
½
1
18. D=b2-4ac
= (8ab)2-4(3a2)(4b2)
QUESTION BANK CLASS X (MATHEMATICS)
SA-II
Page 56
=64a2b2-48a2b2
=16a2b2
X=(-b+√D) ∕2a
=(-8ab+√16a2b2) ∕2(3a2)
= (-8ab+4ab) ∕ 6a2
= -2b/a and -2b/3a
1
1
1
OR
???????? ?
=
1
??????? ?
????
=
?
½
??????? ?
3x2-13x+12=0
X=3 , 4/3
½
1
19. i) volume of 1packet=l x b x h cubic unit
=6x3x2
= 36 cm2
No. of packets in 1 box=1800/36=50
So, no of boxes needed =500/50=10
ii) Volume of a cuboid
½
iii)sharing and caring, empathy, teamwork etc.
½
½
½
1
20. 108, 117 ,126,…………., 198
form an A.P a=108 , d= 9, an=198
1
108+ (n-1)9=198
(n-1 )9=90
n-1 =10
n = 11
1
?
Sn= ?=(a+an)
1
??
S11= ? =(108+198)=1683
a1=4(1)-1=3,
1
Or
a2=4(20-1)=7
1
d= 4 n=20
Sn=n/2[2a+(n-1)d]
QUESTION BANK CLASS X (MATHEMATICS)
½
SA-II
Page 57
= 10x (6+19x4)
= 820
1
21. Figure, given, to prove ,construction
2
Correct proof
2
22. Let the original speed be x km/hr.
Given,
½
???? ????
?
-?????=1
1
1200 x + 120000 - 1200 x
=1
x( x + 100 )
½
X2+100x-120000=0
½
D= b2-4ac
= (100)2 – 4(1)(-120000)
= 490000
X=
???√??????
??
= - 400 , 300
1
Speed cannot be negative. So original speed = 300km/hr
½
Or
Let the original speed be x km/hr.
Given,
??
½
??
=1
???? ????
1
Solving x2+48x -324 = 0
½
D= b2-4ac
= (48)2 – 4(1)(-324)
= 3600
X=
½
???√??????
??
= - 54, 6
QUESTION BANK CLASS X (MATHEMATICS)
1
SA-II
Page 58
Speed cannot be negative. So original speed = 6 km/hr
½
23. P (2,-1), Q (3,4), R (-2,3) and S (-3,-2)
PQ2= (x2-x1)2+(y2-y1)2
PQ2 =(3-2)2+(4+1)2
=
26 1
QR2 = (3+2)2+(4-3)2
=
26
½
RS2 = (-2+3)2+(3+2)2
=
26
½
SP2 =(2+3)2+(-1+2)2
=
26
½
PR2= 32 , QS2= 72
1
Four equal sides, but diagonals are not equal. So, PQRS is Rhombus
½
24. Total number of card =40
1
(a) Numbers divisible by 3 and 5 =15,30
P(a no. divisible by 3and 5)=2/40 or1/20
1
(b) Prime numbers = 2,3,5,7,11,13,17,19,23,29,31 and 37 . P{a prime no.)= 12/40 or 3/10 1
(c) Perfect squares 1,4,9,16,25,36 P(a perfect square) =6/40 0r 3/20
1
25. Curved surface area of cylinder =5280/15
= 352 m2
1
2pr h=352
½
2 x (22/7) x r x 16 = 352
½
r=7/2 m
½
capacity of cylinder = pr2h =
22
x 3.5x3.5x16
7
= 616 m3
QUESTION BANK CLASS X (MATHEMATICS)
1½
SA-II
Page 59
26. Area of square=14 x14 =196 cm2
1
Area of semicircle= 1/2pr2 sq. unit
? ??
= ? ?? ?x7x7
= 77 cm2
1
Area of quadrant II =1/4 2pr2 sq. unit
? ??
= ? ?? ?x7x7
= 77/2 cm2
1
Area of quadrant II =1/4 2pr2 sq. unit
? ??
= ? ?? ?x7x7
= 77/2 cm2
Area of shaded region=196-(77+77) cm2
= 42 cm2
1
Or
Area of D ABC= ½ x basex height sq. unit
=1/2 x 6 x8 cm2=24 cm2
1
Join O to A, B and C
(1/2x10xr)+(1/2x6xr)+(1/2x8xr)=24
1
1/2xrx(10+6+8)=24
Radius= 2 cm
1
Area of circle= pr2 sq. unit
= 22/7x2 x2=88/7 cm2
1
27. 320, 360, 400,………. Form an A.P with a= 320, d= 40
1
?
20000=?[2a+(n-1)d]
½
n(600+40n)=40000
QUESTION BANK CLASS X (MATHEMATICS)
½
SA-II
Page 60
40n2+600n-40000=0 n2+15n-1000=0
1
n=25, -40 (rejected)
So, his saving will be Rs20000 in 25 months
½
28. r + h = 37 cm
½
Total surface area= 2pr(r + h)=1628 cm2
2x22/7 x r x 37 =1628
1
r=7cm
1
h=37-7=30cm
½
Volume =pr2h cubic unit
= (22/7) x7 x 7 x 30 cm3
= 4620 cm3
1
Or
Let radii of the cones be r and R ,their heights be h and H
1 2 1 2
pr h+ pR H=1/64
3
3
1
R2H+r2h=1/64 but r/R=h/H
1
So, h3+H3=1/64 , h3+(32)3=1/64
1
Solving, h= 8 cm
½
The cut is made 32-8=24cm above the base.
½
29. Let the ratio be k:1, Point be (0,y)
½
X= (m1x2+m2x1)/m1+m2
0=k(-1)+1(5)/ k+1, k=5
1½
Ratio = 5:1
1/2
y= (m1y2+m2y1)/m1+m2
y=5(-4)+1(-6)/ 5+1,
y=-13/2
QUESTION BANK CLASS X (MATHEMATICS)
1
SA-II
Page 61
Point of division is (0, -13/2)
½
30. Correct figure
½
Let the first pole be AB =40 m and second pole be CD. Let CP= X m PD=AB=40 m
?? ?
?
In rt. D ACP, tan300 = ?? √?=??
AP = √3x………….(equation 1)
?? √? ????
In rt. D BCD, tan600 = ??
AP = √3x = 20√3 m
=
1
??
1
3 x = x+4 x=20
1
Height of the other pole = 20+40
= 60 m
½
Width of the river = 20√3 m.
½
31. Sum of first 15 terms = 105
Sum of next 15 terms = 780
\ sum of first 30 terms = 780+105
= 885
15
 [2a+(15-1)d] =105
2
30
 [2a+(30-1)d] =885
2
1
1
 2a+ 14d = 14 and 2a + 29d =59
\ a = -14 and =3
The first three terms area, a+d, a+2d…….i.e -14,-11,-8
QUESTION BANK CLASS X (MATHEMATICS)
1
SA-II
Page 62