Download QUESTION BANK - (IX) SUBJECT MATHS CHAPTER

QUESTION BANK SUBJECT MATHS
CHAPTER
(IX)
POLYNOMIAL
MULTIPLE CHOICE QUESTIONS
(1 MARK EACH)
1. The polynomial x2 – 2x is
a) a trinomial
b) a monomial
c) a binomial
d) none of these
2. A cubic polynomial has
a) two zeroes
b) one zero
c) three zeroes
d)none of these
2
3. The value of f(x)= 2x + 7x +3 at x = -2 is
a) -19
b) 3
c) -3
d) 0
4. Degree of the zero polynomial is
a) 1
b) 0
c) any natural number
d) not defined
3
5 The coefficient of y in the expansion of (2y – 3) is
a) 8
b) 27
c) 36
d)54
6. If a+b+c = 0, then a3 + b3 + c3 =
a) abc
b) 0
c) 3abc
d) 2abc
7. Which of the following is a zero of p(x) = 2x – 3?
a) 3
b) -3
c) 3/2
d) 2/3
8. The degree of the polynomial (x+2)(x+3) is
a) 3
b)2
c) 5
d) none of these
9. The degree of a biquadratic polynomial is
a) 2
b)3
c) 4
d) 5
10. The coefficient of a zero polynomial is
a) 0
b)1
c) 2
d) any whole number
91
11. If x + 91 is divided by x+1 then the remainder is
a) 0
b) 1
c) 90
d) none of these
2
2
12. The value of 541 - 540 is
a) 1
b) 1081
c) -1
d) 0
13. The zero of ( x – 1 ) ( x + 1 ) is
a) -2
b) 2
c) -1
d) none of these
14. √5 is a polynomial of degree
a) 2
b) 1
c) 0
d) 5
15. The exponent of any variable in a polynomial can be
a) a fraction
b) zero
c) a non negative integer d) none of these
SHORT ANSWER TYPE – I QUESTIONS
(2 MARKS EACH)
1. Expand using suitable identity (2x + 3y – 4z)2
2. Find the value of k if (x-2) is a factor of polynomial p(x) = 2x3- 6x2+5x+k
3. Factorise 25x2/4 - y2
4. Expand using suitable identity (x +y/3)3
5. . Factorise 2x2 – 7x – 15 by splitting the middle term.
6. Find x + 1/x, if x2 + 1/x2 = 62
7. If a2 + 9/a2 = 31 , what is the value of a-3/a
For what value of m is x3 – 2mx2 + 16 divisible by x+2?
By remainder theorem, find the remainder when 3x4 – 4x3 - 3x – 1 is divided by x+2.
Give one example of a binomial of degree 25 and one example of a monomial of degree 4.
Factorise x2 – x – 12
When the polynomial kx3 + 9x2 + 4x – 8 is divided by x+3, then a remainder 7 is obtained.
Find the value of k.
13.. Without finding the cubes factorise (a – b )3 + ( b – c )3 + ( c – a )3
14. If x+y = -1, then what is the value of x3 + y3 – 3xy
15. Show that p(x) is not a multiple of g(x), when p(x) = x3 + x -1 and g(x) = 3x - 1
8.
9.
10.
11.
12.
SHORT ANSWER TYPE – II QUESTIONS
( 3 MARKS EACH)
1. What must be added to 2x3 – 4x +9 to get 5x3 – 9 ?
2. Check whether the polynomial g(x) is a factor of f(x) or not?
f(x) = 4x3 – 12x2 + 14x – 3
g(x) = x – ½
3. Expand using suitable identity ( 0.1x – 0.2y )3
4. Evaluate using suitable identity 993
5. Without actually calculating the cubes , find the value of: (a-2b)3 + (2b-3c)3 + (3c-a)3
6. Factorize 25x2 + 36y2 +49z2 +60xy-84yz-70zx
7. Factorize a3 - 3√3b3
8. Factorize 9a2 – 9b2 + 6a + 1
9. Factorize x6 - y6
10. Find the remainder when x3 – ax2 + 4x – a is divided by ( x – a ).
LONG ANSWER QUESTIONS
1.
2.
3.
4.
5.
6..
7.
8.
9.
10.
( 4 MARKS EACH)
Factorize x3 + 13x2 + 32x+ 20
If x-y = 5 and xy =84, find the value of x3 – y3
Find x2 + y2 if x+y = -14 and xy = 84
Verify that x3+y3+z3-3xyz = (½)(x+y+z)[(x-y)2 + (y-z)2 + (z-x)2]
Simplify (3x-2y)3 – (3x+2y)3
If x+ 1/x = 5, find the value of x4 + 1/x4
Factorize 8x3 + 27y3 + 36x2y + 54xy2
Factorize 2√2a3 + 3√3b3 + 6√3a2b + 9√2ab2
If f(x) = x2 – 5x + 1 , evaluate f(2) – f(-1) + f(1/3)
If a +b + c = 9 and ab + bc + ca= 26 , find a2 + b2 + c2
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CHAPTER INTRODUCTION TO EUCLID’S GEOMETRY
MULTIPLE CHOICE QUESTIONS (1 MARK EACH)
1. Euclid belongs to the country:
a) Egypt
b) India
c) China
2. Which of the following needs a proof?
d)Greece
3.
4
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
a) axiom
b) theorem
c)definitions
d) postulates
According to Euclid’s definition, the ends of a line are
a) breadthless
b) points
c) lengthless
d) none of these
It is known that if x + y = 5, then x + y + z = 5 + z. The Euclid’s axiom that illustrates this
statement is
a)first axiom
b)second axiom
c)third axiom
d)fourth axiom
Boundaries of surfaces are
a)surfaces
b)curves
c) lines
d)points
Two distinct lines l and m cannot have
a)any point in common b)one point in common c)two points in common d)none of these
If AB=CD , CD=EF and EF=PQ, then which one of the following is not true ?
a)AB=PQ
b)CD=PQ
c)AB=EF
d) EF≠PQ
If a point A lies in between B and C, then
a) BC=(1/2)AC
b) AC=2BC
c) AC=BC
d) AB + AC=BC
Things which are three times of the same thing are
a)equal to each other b)not equal to each other c)half of the same thing d)double of the same thing
The number of dimensions, a point has is
a) 0
b) 1
c) 2
d) 3
Axioms are assumed
a) universal truths in all branches of Mathematics b) theorems
c) definitions
d) universal truths specific to geometry
Greeks emphasized on
a) inductive reasoning b) deductive reasoning c)both a and b
d) practical use of geometry
If equals be subtracted from equals, the remainders are
a) equal
b) unequal
c) twice of each other d) half of the other
Boundaries of solids are
a) surfaces
b) curves
c) lines
d) points
The side faces of a pyramid are
a) triangles
b) squares
c) polygons
d) trapeziums
SHORT ANSWER TYPE – I QUESTIONS (2MARKS EACH)
1.
2.
How would you rewrite Euclid’s fifth postulate so that it would appear easier to understand ?
If a point C lies between two points A and B such that AC = BC, then prove that AC = (½)AB.
Explain by drawing the figure.
3. Define: parallel lines
and
perpendicular lines
A
B
C
D
4. If AC=BD, then prove that AB=CD.
5. Solve the equation x – 2 = 8 and state the axiom that you use here.
6. If x = y and y +z = 5, then show that x + z =5
7. Ravi and Mohan have the same height. If there heights increase by 5cm each , how will there new
heights be compared?
8. If P, Q and R are three points on a line Q lies between P and R , then prove that PQ + QR= PR.
9. If a quantity A is equal to B, then A-C is equal to B-C for some quantity C. Justify your answer.
10. In geometry, we take a point, a line and a plane as defined terms. Write true or false and justify
your answer.
11. If a quantity A equals twice of quantity B and a quantity C is also equal to twice of B, then A
12.
13.
14.
15.
equals to C. Justify your answer.
In a quadrilateral ABCD, if AC is the bisector of ∟A and ∟C and ∟BAC=∟BCA, show that
∟A = ∟C.
State Euclid’s first and second postulate.
State Euclid’s third and fourth postulate.
Show that two lines parallel to the same line are parallel to each other.
SHORT ANSWER TYPE – II QUESTIONS
1.
2.
3.
4.
5.
( 3 MARKS EACH)
Prove that two distinct lines cannot have more than one point in common.
In a trianglePQR, S and T are respectively the mid points of PQ and PR. If PQ = PR, show that
SQ = TR.
Does Euclid’s fifth postulate imply the existence of parallel lines. Explain.
Prove that every line segment has one and only one mid point.
Euclidean geometry is valid only for curved surfaces. State whether this statement is true or
False. Justify your answer.
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CHAPTER COORDINATE GEOMETRY
MULTIPLE CHOICE QUESTIONS (1 MARK EACH)
1.
Abscissa of all points on the x- axis is
a) 0
b) 2
c) 1
d) any number
2. The point whose abscissa is 5 and lies on x-axis is
a) (5,0)
b) (5,5)
c) (0,5)
d) (0,-5)
3. The perpendicular distance of the point P(7,2) from the y-axis is
a) 7
b) 2
c) 5
d) 9
4. The perpendicular distance of the point A (-3,-6) from the x-axis is
a) 3 units
b) 6 units
c) 9units
d) 2 units
5. The point whose ordinate is 6 and which lies on y-axis is
a) (0,6)
b)(0,-6)
c) (6,0)
d) (-6,0)
6. Point (-4,3) lies in the quadrant
a) I
b) II
c) III
d) IV
7. Point (9,-8) lies on the line
a) 4x – 3y = 12
b) 3y – 4x = 12
c) 4x + 3y = 12
d) none of these
8. Ordinate of a point is positive in
a) I and II quadrants b) only I quadrant
c) I and IV quadrants d) only II quadrant
9. A point both of whose coordinates are negative will lie in the quadrant
a) I
b)II
c) III
d) IV
10. The point which lies on the line y = -4x is
a) (-2,-8)
b) (-2,8)
c) (3,12)
d) (-1,-4)
11.
12.
13.
14.
15.
Signs of the abscissa and ordinate of a point in the IV quadrant are respectively
a) -,+
b) -,c) +,+
d) +,Point (0,-5) lies
a) in the III quadrant b) in the IV quadrant c) on the x-axis
d) on the y-axis
The coordinates of origin are
a) (x,0)
b) (0,0)
c) (0,y)
d) 0,0
Which of the following does not lie on the x-axis?
a) (0,0)
b) (0,2)
c) (-2,0)
d) (2,0)
The x – coordinate of a point on the y –axis is always
a) 0
b) positive
c) negative
d) none of these
SHORT ANSWER TYPE –I QUESTIONS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Plot the following points on the graph: A(4,-5) , B(5,4) , C(-4, 3), D(0,4).
Plot the following points and name the figure obtained by successively joining them:
A(2,3) , B(2, -3), C(-2,-3) , D(-2,3).
Check by plotting on a graph if the points (1,3), (3,6) and (4,5) are collinear.
In which quadrant will the following points lie ? (2,-5), (5,6) ,(-5,-3), (-4,6).
What is the name of each part of the plane formed by x-axis and y-axis ? Also write the name of the
point where these two lines intersect ?
How will you describe the position of a table lamp on your study table to your friend ?
Find the coordinates of the point:
i) which lies on both x and y-axis.
ii) whose abscissa is 4 and lies on x-axis.
Plot the points (-3,0), (5,0) and (0,4) on cartesian plane. Name the figure formed by joining
these points and find its area.
Find the coordinates of the point:
i) whose ordinate is -2 and lies on y-axis. ii) whose abscissa is 5 and ordinate is -6.
Are there any points which do not lie in any of the quadrants? If yes, where do they lie?
Plot the point P (4,-6) and from it draw PM and PN perpendicular to x-axis and y-axis
respectively. Write the coordinates of the points M and N.
Three vertices of a rectangle are (-1,1) , (5,1) and (5,3). Plot these points and find the
coordinates of the fourth vertex.
Write the coordinates of the vertices of a square whose each side is 5 units, one vertex at (2,1)
and all the vertices lie in the same quadrant.
Plot the following points and write the name of the figure thus obtained:
A(2,0), B(4,0) , C(4,2) and D(2,2)
Plot the points (-3,0),(5,0) and (0,4) on Cartesian plane. Name the figure formed by joining
these points and find its area.
LONG ANSWER QUESTIONS
1.
2.
(2 MARKS EACH )
(4 MARKS EACH )
Plot the points P(2,0) , Q(5,0) and S(2,3). Find the coordinates of the point R such that PQRS
is a square.
Plot the point P(4,-6) and from it draw PM and PN perpendicular to x-axis and y-axis respectively.
3.
4.
5.
Write the coordinates of the points M and N.
Plot the points (-3,0), (5,0) and (0,4) on Cartesian plane. Name the figure formed by joining these
points and find its area.
Write the coordinates of the vertices of a square whose each side is 5 units, one vertex at (2,1) and
all the vertices lie in the same quadrant.
Draw the quadrilateral with vertices (-4,4), (-6,0), (-4,-4), (-2,0). Name the type of quadrilateral
and find its area.
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CHAPTER
NUMBER SYSTEM
MULTIPLE CHOICE QUESTIONS (1 MARK EACH)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11
12.
13.
14.
Collection of all positive integers is denoted by the symbol
i) N
ii) W
iii) Z
iv) Q
Every rational number is a
i) natural number ii) whole number iii) real number iv) none of these
The maximum number of digits in repeating block of 1/19 is
i) 19
ii) 20
iii) 18
iv) 21
Which of the following is not a rational number?
i) 2.563
ii) 4.231231123111
iii) 3.010010001…. iv) 2.401010101…..
If m and n are two natural numbers and mn = 32, then nmn is
i) 52
ii) 53
iii) 510
iv) 512
3√12/6√27 equals
i)1/2
ii)1/3
iii) √3
iv) √2
The value of 0.9999….. in the form p/q, where p and q are integers q ≠ 0 is
i) 1/9
ii) 2/9
iii) 9/10
iv) 9/100
Between two rational numbers
i) there is exactly one rational number .
ii) there are exactly two rational numbers.
iii) there are infinitely many rational numbers.
iv) there are only rational numbers and no irrational number.
If a= 2 + √3, then a – 1/a =
i) 2√3
ii) 4
iii) 2 + √3
iv) none of these
Value of ( 343)-2/3 is
i) 1/7
ii) 3/7
iii) 1/49
iv) none of these
The number (5+ √4)(5-√4) is
i) rational
ii) irrational
iii) can’t say iv) none of these
√12 x √18 is equal to
ii) 3√6
iii) 2√6
iv)5√6
i) 6√6
Value of (16/81)-3/4 is
i) 8/27
ii) 16/81
iii) 81/16
iv) 27/8
The sum of any two irrational numbers
i) is always a rational number
ii) is always an irrational number
15.
iii) is never a rational number
iv) can be a rational number
The decimal expansion of the number √5 is
i) non-terminating repeating
ii) non-terminating non- repeating
iii) a finite decimal
iv) none of these
SHORT ANSWER TYPE –I QUESTIONS
(2 MARKS EACH )
1.
Write the following in decimal form and say what kind of decimal expansion do they
have ?
11/32 ,
49/300
2.
Write two rational numbers between 2/3 and 7/9 in decimal form.
3.
Rationalize the denominator of the following expression
1/(√5 - √2)
3/4
4.
Evaluate
(625)
5.
Simplify
23/2 x 21/3
6.
Simplify the following expression…
(3 + √7)(4 + √7)
7.
Simplify the following expression
(2 √3 + 1)2
8.
Express the following in the p/q form…
___
0.162
9.
Write three irrational numbers between 1/5 and 2/7 in decimal form.
10.
Classify the following as rational or irrational:
( 3+√20) – (2+2√5)
11.
Give an example of two irrational numbers whose sum and product both are rationals.
12.
Are the square roots of all positive integers irrational? If not , give an example of the square root
of a number that is a rational number .
13. Give an example to show that the quotient of two irrational numbers need not be an irrational
.
..
number
14. Represent √2 on the number line .
15. Represent √5 on number line.
Questions carrying 3 marks each
1.
2.
3.
4.
5
6.
7.
8.
9.
10.
Represent the following on number line :
√6.8
Represent √6 on number line.
Visualize 4.237 on the number line using successive magnification.
If x = (√3+√2) / (√3-√2) , then find the value of x2.
Evaluate
21/5 x 22/3
If x = 4 - √15, then find ( x + 1/x)2
Evaluate (0.00032)-2/5
Simplify the following by rationalizing the denominator:
2√3 - √5
2√2 + √3
If x = 1 - √2, find the value of x – 1/x
Simplify the following:
(243/32)-4/5
Questions carrying 4 marks each
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Find four rational numbers between -4/7 and -3/7
If 2 + √3 = a + b√3, find the values of a and b.
2 - √3
Simplify 4 + √5
+
4 - √5
4 - √5
4 + √5
Rationalize the denominator of
1______
√5+√2-√7
Simplify
(81/16)-3/4 x (25/9)-3/2 x (2/5)-3
If x = √5 + 2, then prove that x2 + 1
= 18
x2
Taking √3=1.732 and √5=2.236, evaluate ____1______ , correct to three
4√3 - 3√5
places of decimals.
Simplify the following expression: ( 2√5+3√2)2 + (2√5-3√2)2
Visualise the representation of 1.3 on the number line upto four decimal places.
Simplify the following by rationalizing the denominator:
1
(√3-√2-√5)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CHAPTER HERON’S FORMULA
MULTIPLE CHOICE QUESTIONS (1MARK EACH)
1.
2.
3.
4.
5.
6
7.
8.
9.
If the side of an equilateral triangle is ‘a’ unit, then the area of the triangle is
a) a2 sq.unit
b) √3a2 sq.unit
c) (√3/4) a2 sq. unit
d) none of these
The area of an equilateral triangle with side 2√3 cm is
a) 5.196 cm2
b) 0.866 cm2
c) 3.496 cm2
d) 1.732 cm2
One side of an equilateral triangle is 4cm. Its area is
a) 8√3 cm2
b) 16√3 cm2
c) 4√3 cm2
d) 12√3 cm2
The altitude of an equilateral triangle is 5√3 cm. The area is
a)50√3 cm2
b) 225√3 cm2
c)40√3 cm2
d) 25√3 cm2
In a triangle, the sides are 5cm, 6cm and 7cm. The area of the triangle in sq. unit is
a) 6√6
b) 6√3
c) 6√2
d) 9√6
The length of each side of an equilateral triangle having an area of 9√3cm2 is
a) 8cm
b) 4cm
c) 6cm
d)36cm
If the area of an equilateral triangle is 16√3 cm2,then perimeter of the triangle is
a)8cm
b)12cm
c)24cm
d)30.6cm
the area of an isosceles triangle having base 2cm and the length of one of the equal sides 4cm is
a)√15sq.cm
b)√15/2sq.cm
c)2√15sq.cm
d)none of these
An isosceles right triangle has area 8sq.cm ,the length of its hypotenuse is
a)√32 cm
b)√48 cm
c)√24 cm
d)√16 cm
10. The area of a triangle ,whose size are 4cm ,13cm,and 15cm is
a)√420sq.cm
b)48sq.cm
c)56sq.cm
d)24sq.cm
11. If the sides of a triangle are doubled , then its area
a) remains the same
b) is doubled
c) becomes three times d) becomes four times
12. The base of a right triangle is 8 cm and area is 24sq.cm ,its hypotenuse will be
a) 16 cm
b) 10cm
c)9cm
d)none of these
13. The size of a triangle are in the ratio of 3:4:5.If its perimeter is 36 cm ,then its area is
a)36sq.cm
b)54sq.cm
c)72sq.cm
d)none of these
14. The size of a triangle are 2cm,5cmand 5cm,its area is
a) √6sq.cm
b)2√6sq.cm
c)24sq.cm
d)none of these
15. The hypotenuse of an isosceles right triangle is √32 m ,its area is
a)8sqcm
b)16sq.cm
c)32sq.cm
d)none of these
SHORT ANSWER TYPE –I QUESTIONS
(2 MARKS EACH )
1. The area of a parallelogram is 392m2.If its altitude is twice the corresponding base,
. . .
determine the base and height.
2. The adjacent sides of a parallelogram are 36cm and 27cm in length .If the distance
.
between the shorter sides is 12cm, find the distance between the longer sides.
3. A rectangular lawn, 75m by 60m, has two roads , each 4m wide, running through the
.
middle of the lawn, one parallel to length and other parallel to breadth. Find the cost of .
gravelling the roads at Rs 5.50 per m2
4. Using Heron’s formula, find the area of an equilateral triangle if its side is ‘a ‘units.
5. Find the percentage increase in the area of a triangle if its each side is doubled.
6. Find the area of quadrilateral ABCD whose sides in meters are 9, 40, 28 and 15 .
.
respectively and the angle between first two sides is a right angle.
7. The difference between the sides containing a right angle in a right angled triangle is
.
14cm. The area of a triangle is 120cm2.Calculate the perimeter of a triangle.
8. The area of a parallelogram is 392m2.If its altitude is twice the corresponding base,
.
determine the base and height.
9. Find the area of quadrilateral ABCD whose sides in meters are 9, 40, 28 and 15 .
.
respectively and the angle between first two sides is a right angle.
10. The difference between the sides containing a right angle in a right angled triangle is .
.
14cm. The area of a triangle is 120cm2.Calculate the perimeter of a triangle.
11. The perimeter of a right triangle is 24 cm. If its hypotenuse is 10 cm, find the other two sides. Find
.
its area by using the formula area of a right triangle. Verify your result by using Heron’s formula.
12 . The sides of a triangle are 39cm, 42cm and 45cm. A parallelogram stands on the greatest side of the
.
triangle and has the same area as that of the triangle. Find the height of the parallelogram
13. A parallelogram, the length of whose side is 60m and 25m has one diagonal 65m long. Find the
area of the parallelogram.
14. If the area of an equilateral triangle is 36√3 sq.cm. Find its height.
15.
The perimeter of an isosceles triangle is 32 cm. The ratio of the equal side to its base is 3:2. Find
the area of the triangle.
SHORT ANSWER TYPE – II QUESTIONS (3 MARKS EACH)
1.
2.
Find the area of a triangular field whose sides are 50m, 45m and 35m.
If each side of a triangle is doubled, then find the ratio of area of new triangle thus formed and
the given triangle.
3. Find the area of an isosceles triangle , whose equal sides are of length 15cm each and third side is
12 cm.
4. The sides of a triangle are in the ratio 5:12:13 and its perimeter is 150 cm . Find the area of the
triangle.
5. Find the area of a triangle, two sides of which are 8cm and 11cm and perimeter is 32cm.
6. Find the area of a triangle with longest side 15cm, perimeter 40cm and difference of the other
two sides is 1cm.
7. Find the area of a triangular field whose sides are 90m, 120m and 150m. Also find the cost of
leveling the field at the rate of Rs.12.50 per sq.m.
8. An umbrella is made by stitching 10 triangular pieces of cloth of two different designs, each
piece measuring 20cm, 50cm and 50cm. How much cloth of each design is required for the
umbrella.
9. The perimeter of an isosceles triangle is 32cm. The ratio of the equal side to its base is 3:2.
Find the area of the triangle.
10. A triangle and a parallelogram have the same base and the same area. If the sides of the
triangle are 26cm, 28cm and 30cm, and the parallelogram stands on the base 28cm, find
the height of the parallelogram
\
LONG ANSWER QUESTIONS ( 4 MARKS EACH )
1.
2.
3.
4.
5.
A floral design on a floor is made up of 16 tiles which are triangular, the sides of the triangle
.
being 9cm, 28cm and 35cm. Find the cost of polishing the tiles at the rate of 50p per cm2
A field is in the shape of a trapezium, its parallel sides are 25m and 10m and non-parallel
sides are 14m and 13m. Find the area of the field.
Find the area of a quadrilateral whose sides measure 9cm, 40cm, 28cm and 15cm. The angle
between the first two sides of the quadrilateral is a right angle.
The area of a trapezium is 475cm2 and the height is 19cm. Find the lengths of its parallel sides
If one side is 4cm greater than the other.
The lengths of the sides of a triangle are 7cm, 12cm and 13cm. Find the length of perpendicular
from the opposite vertex to the side whose length is 12cm.
XXXXXXXXXXXXXXXXXXXXXXXXXXX
CHAPTER
LINES AND ANGLES
MULTIPLE CHOICE QUESTIONS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
(1 MARK EACH )
An angle is 180 less than its complementary angle . The measure of this angle is
a) 360
b)480
c) 830
d) 810
If angles (2x-10)0 and (x-5)0 are complementary angles, the value of x is
a) 35
b)70
c)105
d) none of these
An angle is equal to five times its complement. The measure of the angle is
a) 1050
b) 900
c) 750
d) 37.50
The sum of angles at a point is
a) 900
b) 00
c) 1800
d) 3600
A triangle can have
a) two obtuse angles b) two right angles
c) two acute angles
d) none of these
An angle which is twice its supplement is
a) 1200
b) 600
c) 900
d) 300
0
The angle which exceeds its complement by 20 is
a) 450
b) 550
c) 1100
d) 700
0
The difference between two complementary angles is 15 ,the angles are
b) 37.50,52.50
c) 82.50,97.50
d)none of these
a) 500,600
The angles of a triangle are in the ratio 2:3:4.The measure of the greatest angle is
a)600
b) 800
c) 1000
d)none of these
The angles of a triangle are in the ratio 3:5:7 .the triangle is
a)a right angled triangle b)an obtused angled triangle c)an acute angled triangle d)none of these
The angles of a triangle are in the ratio 1:5:3 ,the smallest angle of the triangle is
a)200
b)600
c)1000
d)none of these
In a triangle two angles are equal to each other .Their corresponding exterior angle is 1100.
each of these angles is
a)700
b) 550
c)650
d)none of these
0
If x and y form a linear pair such that x-2y=33 then the value of x is
a) 330
b) 490
c) 1310
d) none of these
If one angle of a triangle is equal to the sum of the other two angles then the triangle is
a) an obtuse triangle b) an isosceles triangle c) an equilateral triangle d) a right triangle
If one of the angles of a triangle is 1100 ,then the angle between bisectors of the other two angles
can be
a)700
b)650
c)750
d) 1450
SHORT ANSWER TYPE-I QUESTIONS
(2 MARKS EACH )
4.
Can a triangle have all the angles less than 600? Give reason.
Can a triangle have two obtuse angles?
A transversal intersects two lines in such a way that the interior angles on the same side of
transversal are equal. Will the two lines always be parallel?
An angle is equal to 60 more than one third of its supplement. Find its measure.
5.
In the figure , if a-b = 900, find values of a and b.
1.
2.
3.
a
b
6.
Find the angle whose compliment is equal to the angle itself .
7.
Find the measure of an angle whose supplement equal to the angle itself .
8.
Two supplementary angles differ by 300 .Find the angles .
9.
Two adjacent angles on straight lines are in the ratio 3:2 . Find the measure of each angle
10. The angles of a triangle are in the ratio 6:7:2.Find the angles of the triangles.
11. Find the measure of the angle which is one fourth of its compliment .
12. find the angle whose supplement is 6 times its compliment .
13. In straight line AOB ,a ray OC stands on it .If angle AOC =(3x-10)0 and angle BOC=(2x+15)0
.
find the value of x
14. Find the measure of an angle which is 260 more than its compliment.
15. If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio
2:3,then what is the smaller angle ?
SHORT ANSWER TYPE – II QUESTIONS
1.
2.
3.
(3 MARKS EACH)
If one of the four angles formed by two intersecting lines is a right angle, show that the other
three angles will also be right angles.
It is given that ∟ABC = 560 and AB is produced to point D. If ray BE bisects∟CBD, find angle
∟ABE and reflex ∟DBE.
In the figure show that ∟DOC = ½ ( ∟AOD - ∟BOD).
C
D
900
A
4.
O
B
In the figure if ∟AOC = ∟BOD=300, find ∟COD.
D
5.
7.
C
B
O
A
Lines AB, CD and EF are concurrent at O. If ray OF bisects ∟BOD and ∟BOF = 260, find
∟BOC and ∟DOE.
If a transversal intersects two lines such that the bisectors of pair of a pair of corresponding .
8..
angles are parallel ,then prove that the two lines are parallel.
Prove that the sum of the angles of a triangle is 1800.
.In a triangle ABC ,if ∟A+∟B=1500 and ∟C+∟B=1000.Find the measure of each angle of the
triangle. .
10. Prove that if a side of a triangle is produced ,then the exterior angle so formed is equal to the sum .
.
of the two interior opposite angles .
9
LONG ANSWER QUESTIONS (4 MARKS EACH)
1.
2.
If two parallel lines are intersected by a transversal, then prove that the bisectors of interior
angles form a rectangle.
If the bisector of angles ∟B and ∟C of a triangle ABC meet at a point O, then prove that
∟BOC = 900 + 1/2∟A.
A
O
B
3.
4.
5.
C
Prove that if two lines intersect each other, then the vertically opposite angles are equal
If the arms of an angle are respectively parallel to the arms of another angle, then show that the
two angles are either equal or supplementary.
If two parallel lines are intersected by a transversal, then show that the bisectors of a pair of
alternate interior angles are parallel.
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CHAPTER
TRIANGLES
MULTIPLE CHOICE QUESTIONS (1 MARK EACH)
1.
2.
3.
4.
5.
In a triangle ABC, if ∟C is greater than ∟B, then
a)BC is greater than AC b) AB is greater than AC c)AB is less than AC d) none of these
Two sides of a triangle are 6cm and 2.5cm. The length of third side can not be
a)4.8cm
b) 5cm
c) 5.5cm
d)3.2cm
If the corresponding angles of two triangles are equal, then they are always congruent.
a) True
b) cannot be determined c) False
d) none of these
Which of the following is not a criterion for congruence of triangles ?
a) SSS
b) SSA
c) SAS
d)ASA
If r, s and t are the sides of a triangle, then
a) r = s+t
b) r = s-t
c)r+s is greater than t
d) t is greater than r+s
In triangles ABC and PQR , AB = PQ and ∟B = ∟Q. The two triangles will be congruent
by SAS axiom if
a) BC=QR
b) AC=PR
c) AB=QR
d) none of these
7.
In triangle PQR, PQ=PR and ∟Q= 650. Then ∟R is equal to
a) 1050
b) 250
c) 650
d)550
8. Which of the following set of measures can form a triangle?
a) 900,100,900
b) 600,500,900
c) 900,600,300
d) 950, 150 ,920
9.
In triangle PQR, if ∟Q= 400 and ∟R = 720, then the shortest and the largest sides of the
triangle are respectively
a) PR,PQ
b) PR,QR
c) QR,PR
d) PQ,PR
10. D is a point on the side BC of a triangleABC such that AD bisects∟BAC. Then
a) CD>CA
b) BD>BA
c)BA>BD
d) BD=CD
6.
SHORT ANSWER TYPE - I QUESTIONS
1.
2.
3.
4..
5.
6.
7.
8.
9.
10.
Show that in a right angled triangle, the hypotenuse is the longest.
Is it possible to construct a triangle with sides 5.4cm, 6.6cm and 13cm? Give reasons.
AD is a median of triangle ABC . Is it true that AB + BC + CA is greater than 2AD.
Give reasons.
If one of the angles of a triangle is 720 and the difference of other two angles is 120,
Find the other two angles.
In ABC and PQR, ∟A = ∟P, ∟B = ∟Q and AB = QR. Will the two triangles be
congruent? Give reason for your answer.
In ABC, ∟A = 650 and ∟C = 350. Which side of this triangle is the longest? Give
reason for your answer.
E and F are respectively the midpoints of equal sides AB and AC of triangleABC. Show
that BF = CE.
Prove that any point on the angle bisector of an angle is equidistant from its arms.
Prove that in an isosceles triangle, the altitude from the vertex bisects the base.
Triangle ABC is a right angled triangle in which ∟B = 900 and AB=BC. Find ∟A and ∟C.
SHORT ANSWER TYPE – II QUESTIONS
1.
2.
3.
4.
5.
6.
7.
( 2 MARKS EACH )
( 3 MARKS EACH )
Exterior angle of a triangle is 1180 and one of the interior opposite angle is 420, find
the remaining angles.
BD and CE are bisectors of ∟B and ∟C of an isosceles triangle ABC and AB = AC. Show
BD = CE.
ABCD is a quadrilateral such that AB=AD and AC is bisector of angle A . Show that
triangle ABC is congruent to triangle ADC and BC=DC.
Prove that the sides opposite to equal angles of a triangle are equal.
Prove that angles opposite to equal sides of an isosceles triangle are equal.
Show that in a right angled triangle , the hypotenuse is the longest side.
If the bisector of the exterior vertical angle of a triangle is parallel to the base. Show that the
triangle is isosceles.
Prove that the medians of an equilateral triangle are equal.
In a right triangle , prove that the line segment joining the midpoint of the hypotenuse to the
opposite vertex is half of the hypotenuse.
If two isosceles triangles have a common base, prove that the line joining the vertices bisects
the base at right angle.
8.
9.
10.
LONG ANSWER QUESTIONS
1.
( 4 MARKS EACH )
ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC
and AB respectively. Show that these altitudes are equal.
ABC is an isosceles triangle in which AB =AC. Side BA is produced to D such that AD=AB.
Show that angle BCD is a right angle.
Two sides AB and BC and median AM of one triangleABC are respectively equal to sides
PQ and QR and median PN of triangle PQR. Show that ABM is congruent to PQN.
BE and CF are two equal altitudes of triangle ABC. Using RHS congruence rule, prove
that the ABC is isosceles.
Show that of all line segments drawn from a given point not on it, the perpendicular line
segment is the shortest.
2.
3.
4.
5.
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
QUESTION BANK SECOND TERM
CLASS IX
MATHEMATICS
Ms Lily Sengupta
LINEAR EQUATIONS IN TWO VARIABLES
Multiple Choice Questions(Carrying 1 mark)
1
2
3
4
5
The linear equation 2x-5y=7 has
(a) a unique solution (b) two solution (c) infinitely many solution (d) no solution
Any point on the x-axis is of the form
(a) (x,y)
(b) (0,y)
(c) (x,0)
(d) (x,x)
The equation of x –axis is of the form
(a) x=0
(b) y=0
(c) x+y=0
(d) x=y
The point of the form (a,a) always lies on the line
(a) x-axis
(b) y-axis
(c) on the line y=x (d) on the line x+y=0
How many linear equations in x and y can be satisfied by x=1 and y=2
(a) only one
(b) two
(c) infinitely many
(d) three
6 If x=2,y=1 is a solution of the equation 2x+3k=y,then the value of k is
(a)-1
(b)2
(c)1
(d)3
7 The graph of the linear equation 4x=5 is
(a)parallel to x axis
(b)lies along x axis (c)parallel to y axis (d)passes through
Origin
8 The equation y=4x-7 has
(a) no solution
(b)unique solution
(c)infinitely many solution (d)exactly two
solution
9 The equation 5x=2 is written in two variable as
(a)5x+y=2
(b)5xy=2
(c)5x=2y
(d)5x+0y-2=0
10 The linear equation 3y-5=0, represented as ax+by+c=0 ,has
(a)a unique solution (b)infinitely many solutions
(c)two solutions
(d)no solution
11 The equation 2x+5y=7 has a unique solution,if x and y are
(a)natural numbers (b)positive real numbers
(c)real numbers
(d)rational numbers
12 The point (3,0) lies on the graph of the equation 2x+3y=k
Then the value of k is
(a)6
(b)3
(c)2
(d)5
13 The graph of the linear equation y=2x passes through the point
(a)(2,1)
(b)(2,-1)
(c)(3/2,-3)
(d)(3/2,3)
14 Any point on the line y=x,is of the form
(a)(m,0)
(b)(0, m)
(c)(m,-m)
(d)(m,m)
15 For one of the solutions of the equation ax+by+c=0, x is negative and y is positive then surely
a
portion of line lies in the
(a)first quadrant
(b)second quadrant
(c)third quadrant
(d)fourth quadrant
Short Answer Type -I Questions (carrying 2marks )
1
2
3
4
5
6
7
8
9
10
11
Give the equations of two lines passing through (2,5) .How many more such lines are there and
why?
If the point (3,4) lies on the graph of the equation 3y=ax+7,find the value of a
The cost of a note book is twice the cost of a pen .Write a linear equation in two variable to
represent this statement
Express x in terms of y, given that 3x+4y=6.check whether the point (3,2) is
On the given line
Draw the graph of the equation 6-1.5x=0
Give the geometrical representation of 2x+9=0 as an equation in
(a) one variable (b) two variable
Find three solutions for the equation 3=2x+y
When 5 times the larger of the two numbers is divided by the smaller, the
quotient and remainder are 2 and 9 respectively .Form a linear equation
in two variables for above and give its two solution
Draw the graph of x=3
Draw the graph of y=x show that the point(4,4) is on the graph
Express y in terms x,given that 3x+4y=6.check whether the point (3,2) is on the given line
12
13
14
15
Write the coordinates of any two points which lie on the line x=y=8.How many such point exists?
If x=3,y=-2 is a solution of the linear equation 3x-ky=1,then find the value of k
Find the value of k ,if line represented by the equation 2x-ky=9 passes through the point (-1,-1)
Show that the point A(1,2),B(-1,-16),C(0,-7) are on the graph y=9x-7
Short Answer type -II Question (carrying 3 marks)
1
Give the geometric representation of 2x+9=0 as an equation in one variable and equation in two
variables.
2 Find three solutions for the equation 3=2x+y
3 Draw the graph of 2x+3y=7.write the points where line meets x and y axis.
4 Draw the graph of the linear equation 2x+3y=5.Check whether(-3,4) is a solution of the given
equation
5 Draw the graph of the linear equation-3y=4.from the graph find the value of x wheny=-2
6 Draw the graph of the linear equation 3x+2y=6
7 Draw the graph of the linear equation 3x+2y=5.From the graph find the value of x,when y=4
8 Express the linear equation 6=4x in the form of ax+by+c=0 and indicate the value of a,b,c.Also
Give the geometric al representation of above equation in two variables
9 Draw the graph of the linear equation 3x+y=6 .find the points where the line meets two axes
10 Find three solution of equation 2x+3(y-1)=13how many solutions this equation has?
Long Answer type Question (carrying 4 marks)
1
Draw the graph of the linear equation x+y=7.Verify from the graph that (8,-1) is a solution of the
equation
2 Draw the graph of the equation 3x-4y=12 From the graph find the value of y, if x=8
3 The taxi fare in a city is as follows :For the first kilometer ,the fare is Rs 8 and for the subsequent
distance it is Rs 5 per km .Taking the distance covered as x km and the total fare as Rs y , Write a
linear equation for this information and draw its graph
4 Plot the graph of the following linear equation
2(x+3)-3(y+1) =0 .Also answer the following question :
(a)Write the quadrant in which the line segment intercepted between the axes
lies
(b) shade the triangular region formed by the line and the axes
(c) Write the vertices of the triangle so formed
5 Draw the graph of the linear equation 3x+4y=6.At what points ,the graph cuts the x-axis and the y axis
QUADRILATERALS
Multiple Choice Question(Carrying 1 mark)
(
1 The angles of a quadrilateral are 75◦,90◦ and 75◦. The fourth angle is
(a) 90◦ (b) 95◦ (c) 105◦ (d) 120◦
2 The figure obtained by joining the mid points of the sides of a rhombus, taken in order ,is
a. a rhombus (b) a rectangle (c) a square (d) any parallelogram
3 If angle A,B,C and D of the quadrilateral ABCD, taken in order in the
Ratio 3:7:6:4 ,then ABCD is a
b. rhombus (b) parallelogram (c) kite (d) trapezium
4 Which of the following is not true for a parallelogram?
c. opposite sides are equal (b) opposite angles are equal (c) opposite angles are bisected
by the diagonals (d) diagonals bisect each other
5 If APB and CQD are two parallel lines, then the bisectors of the angles APQ,BPQ,CQP and
PQD form
d. a square (b) a rhombus (c) a rectangle (d) any other parallelogram
5 Given four points A,B,C,D.Let these points are collinear, then joining these points in order we
get
(a) a straight line (b) a triangle (c)a quadrilateral (d) a circle
7 Which of the following is not a parallelogram ?
(a) trapezium
(b) square
(c)rectangle
(d)rhombus
8 Four points A,B,C,D are joined together in order and we noticed AB=CD=5 cm and also AB is
parallel to CD then the quadrilateral obtained is
(a) Rhombus
(b) trapezium
(c) parallelogram (d) rectangle
9 A quadrilateral, whose diagonals bisect at right angles, is called
(a )a trapezium
(b)a rectangle
(c)a parallelogram with unequal adjacent sides
(d) a rhombus
10 Two angles of a quadrilateral are 500 and 800 and other two angles are in the ratio 8:15,then the
remaining Two angles are
(a) 1400,900 (b)100 0,1300 (c)800,1500
(d)700,1600
11 Given a trapezium ABCD,in which ABIICD and AD=BC if∠D=700 then ∠C will be
(a) 700
(b)1100
(c)200
(d) none of these
12 In a parallelogram ABCD,if ∠A=600 then ∠D is equal to
(a)1100
(b)1400
(c)1200 (d)1300
13 If APB and CQD are two parallel lines,then the bisector of the angles APQ,BPQ,CQP and
PQD form
(a)a square
(b) a rhombus
(c) a rectangle
(d) any other parallelogram
14 A diagonal of a rectangle is inclined to one side of the rectangle at 250The acute angle between
The diagonals is
(a) 550
(b)500
(c)400
(d)250
15 ABCD is a rhombus such that ∠ACB =400then ∠ADB is
(b)450
(c)500
(d) 550
(a) 400
Short Answer Type -I Question (carrying 2 marks)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
In a parallelogram ABCD angle D=105◦ ,determine angle A and angleC
ABCD is a rectangle with angle BAC=42◦ determine angle DBC
PQRS is a rhombus with angle PQR=58◦ determine angle PRS
D,E and F are respectively the mid points of sides BC,CA and AB of an equilateral triangle
ABC .Prove that DEF is also an equilateral triangle.
Prove that the figure formed by joining the mid points of the pairs of consecutive sides of a
quadrilateral is a parallelogram.
The angles of a quadrilateral are in the ratio 3:5:9:13.Find all the angles of
The quadrilateral
ABCD is a parallelogram .AB is produced to E so that BE=AB. Prove that ED bisects BC
In a triangle PQR ,median PM is produced to X such that PM=MX .Prove
That PQXR is a parallelogram.
Show that the quadrilateral formed by joining the mid points of consecutive sides of a
rectangle is a rhombus.
In a parallelogram ABCD ,the bisectors of adjacent angles A and B intersect each other at P.
prove that angle APB=90◦
The diagonal AC of a parallelogram ABCD bisects∠A. Show that it bisects∠C also
Opposite angles of a quadrilateral ABCD are equal .If AB=4 cm determine CD
ABCD is a rhombus in which altitude from the point D to side AB bisects AB.find the angle
of the rhombus
One angle of a quadrilateral is of 1080and the remaining three angles are equal .Find each of
the three angles
Diagonals of a quadrilateral ABCD bisect each other .If ∠A=350determine∠B
Short Answer Type --II Question (carrying 3marks)
1 Show that a quadrilateral formed by joining the mid points of the consecutive sides of a rhombus
Is a rectangle
2 P and Q are points of trisection of the diagonal BD of a parallelogram ABCD .Prove that CQ is
parallel to AP and AC bisects PQ
3 If non parallel side of a trapezium are equal then prove that it is cyclic
4 Prove that quadrilateral formed by the internal angle bisector of any quadrilateral is cyclic
5 In a parallelogram ABCD ,AB=10cm and AD=6 cm .The bisector of angle meet s DCinE.AE
And BC produced meet at F .Find the length of CF.
6 P,Q,R and S are respectively the mid points of the sides AB,BC,CD and DA of a quadrilateral
ABCD in which AC=BD Prove that PQRS is a rhombus
7 A diagonal of a parallelogram bisects one of its angles. Show that it is a rhombus
8 E is the mid - point of a median AD of triangle ABC and BE is produced to meet AC at F.
Show that AF=1/3AC
9 Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a
rectangle
10 E and F are respectively the mid- points of the non parallel sides AD and BC of a trapezium
ABCD prove that EFII AB and EF=1/2(AB+CD)
Long Answer Type Question(carrying 4 marks)
1 Show that the quadrilateral formed by joining the mid points of the sides of a square is
Also a square
2 Prove that the diagonals of a square are equal and perpendicular to each other
3 If two parallelograms PQAD and PQBC are on the opposite sides of PQ,
Prove that ABCD is a parallelogram.
4 Prove that in a triangle ,the line segment joining the mid points of any two sides is parallel to
third side and is half of it
5 Prove that the straight line joining the mid points of the diagonals of a trapezium is parallel to
the parallel sides of the trapezium and is equal toHalf of their difference
AREAS OF A PARALLELOGRAMS AND TRIANGLES
Multiple Choice Question(Carrying1 mark)
1
2
3
4
5
6
7
The median of a triangle divides it into two
(a)triangle of equal areas (b) congruent triangles (c) right triangles (d) isosceles triangles
Two parallelograms are on equal base and between the same parallels.The ratio of their areas is
(a) 1:2 (b)1:1 (c) 2:1 (d) 3:1
If a triangle and a parallelogram are on the same base and between same parallels, then the ratio
of their area of the triangle to the area of parallelogram is
(a)1:3 (b) 1:2 (c) 3:1 (d) 1:4
ABCD is a quadrilateral whose diagonal AC divides it in two parts equal in area ,then ABCD
(a)is a rectangle (b) is always a rhombus (c) Is a parallelogram (d)none of these
Two parallelograms are on the same bases and between the same parallels. The ratio of their areas
is
(a)1:2
(b)1:1
(c)2:1
(d) 3:1
Given parallelogram ABCD and EBCF on the same base BC and between the same parallels BC
and A Give ar(EBCF)=15 square cm ,then ar(ABCD) is
(a)30 sq cm
(b)15 sq cm
(c)7.5 sq cm
(d) none of these
If ar(ABCD)=25 sq cm and on the same base CD ar(∆BCD) is given such that ,ar(∆BCD)=xcm2
then the value of x is
(a)25 cm2
(b)12.5 cm2
(c) 12,5 cm
(d)25 cm
8
If A and B are two congruent figures,then
(a) ar(A)=ar(B) (b) ar(A)>ar(B)
(c) ar(A)<ar(B) (d) none of these
9
In a parallelogram ABCD,P is a point in the interior of a parallelogram ABCD.Also
ar(parallelogramABCD)=18 cm2 .Then ar(∆APD)+ar(∆CPB) is
(a)9cm2
(b)12cm2
(c)18cm2
(d)15cm2
10 Given a quadrilateral ABCD, BE is drawn parallel to AC meeting DC produced at E.Also
ar(∆ADC)=1 cm2 ar(∆ABC)=7 cm2.Then ar(ADE) will be
(a)10 cm2
(b)7 cm2
(c)17 cm2
(d)18 cm2
11 Given a triangle ABC and E is the mid point of median AD of ∆ABC.If ar(BED)=20 cm2.Then
ar(ABC) is
(a)10 cm2
(b) 5 cm2
(c)60 cm2
(d) 80 cm2
12 D and E are the points on the side AB and AC respectively of a triangle ABC such that
DE IIBC I Ar(DBC)=15sq cm then ar(EBC) is
(a)30 sq cm
(b)7.5 sq cm (c)15 sq cm (d)none of these
13 If two triangles are on the same base and between the same parallel, then their areas are,
(a) their areas are equal (b) area of the first triangle>area of the second triangle
(c) area of second triangle>area of the first triangle (d) none of these
14 PQRS is a parallelogram in which SR=10 cm and PM ⊥SR.PM=4 cm .Then its area is
(b)40 cm2
(c)30 cm2
(d)20 cm2
(a)50cm2
15 If the base of a ∆ is 8 cm2 and altitude is 5 cm then its area is equal to
(a)15 cm2
(b) 20 cm2
(c)40 cm2
(d)10 cm
Short Answer Type -II Question (carrying 3 marks )
1 In a parallelogram ABCD ,AB=10 cm and the altitude corresponding To the side AB and AD
are respectively 7 cm and
2 AD is one of the median of a triangle ABC and X is any point on AD. Show that ar(∆ABX)=
ar(∆ACX)
3 Show that the area of a rhombus is half the product of the lengths of its diagonal
4 A quadrilateral ABCD is such that diagonal BD divides its areas in two equal parts .Prove that
BD bisects AC
5 O is any point on the diagonal BD of the parallelogram ABCD. Prove that
ar (∆OAB) =ar( ∆OBC)
6 find the area of a triangle whose perimeter is 180cm and two of its sides are 80 cm and 18 cm
7 a point O inside a rectangle ABCD is joined to the vertices .Prove that
ar (∆AOD) +ar(∆BOC)=ar(∆AOB)+ ar(∆COD)
8 If each diagonal of a quadrilateral separates it into two triangles of equal areas ,then prove that
quadrilateral is a parallelogram
9 Show that the segment joining the mid points of a pair of opposite sides
Of a parallelogram divides it into, two equal parallelograms.
10 A park is in the shape of a quadrilateral ABCD having right angle at C ,
AB=9m BC=12m ,CD=5m,and AD=8m.How much area the quadrilateral
region occupy
11 The sides of a triangular plot are in the ratio of 3:5:7 and its perimeter is 300 m .
Find its area
Long Answer Type Question(carrying 4 marks)
1
2
3
4
5
A point E is taken on the side BC of a parallelogram ABCD.AE and DC are
produced to meet at F. Prove that ar(∆ADF)=ar(ABFC)
The diagonal of a parallelogram ABCD intersect at a point O .through O , a line is drawn
to intersect AD at P and BC at Q .Show that PQ divides the parallelogram into two parts
of equal areas.
If the median of a ∆ABC intersect at G show that
ar(AGB) =ar(AGC)=ar(BGC)=1/3 ar(ABC)
prove that parallelogram on the same base and between the same parallels are equal in
areas
ABCD is a parallelogram .X and Y are the mid points of BC and Cd respectively .prove
that
ar(∆AXY) =3/8 ar(11 gmABCD)
CIRCLES
Multiple Choice Question(Carrying 1 mark)
1
AD is the diameter of a circle and AB is a chord.If AD =34cm,AB=30cm
The distance of AB from the centre of the circle is
(a) 17cm (b) 15cm (c) 4cm (d) 8cm
2
chord AB of a circle with centre O is at a distance of 4cm from the centre of the circle,another
chord CD of length 6cm is drawn. If radius of the circle is 5cm.then
(a) chord AB >chord CD (b) chord AB <chord CD (c) chord AB =chord CD (d) non of
these
3
Given chords AB and XY of a circle with centre O.Points X,O,Y are co-linear and AB=1/3XY.If
AB =3cm.Then radius of circle is
a. 9cm (b) 3cm (c) 4.5cm (d) 6cm
The region of a circle divided by a chord, which contains the centre of the circle is called a minor
segment:
(a) True (b) false
The region between a chord and either of the arcs is called
(a) an arc (b) a sector (c) a segment (d) a semi-circle
Given a circle of radius r and with centre o.A point p lies in a plane such that OP<r.Then
Point P lies
(a) In the interior of the circle (b) on the circle (c)in the exterior of circle
(d) none of these
Given a circle with centre O and chord AB,PQ and XY, points P,Q and O are
Collinear and radius of a circle is 6 cm.Then mark the correct option
(a)AB=XY=3 cm (b)AB=6 cm=XY (c)PQ=6 cm
(d)PQ=12 cm
4
5
6
7
8 Given a circle with centre o and smallest chord AB is of length 3 cm,largest
Chord CD of length 10 cm and PQ is of length 7 cm,then radius of circle is
(a)1.5 cm
(b)6 cm
(c)5 cm
(d)3.5 cm
9 Given two concentric circle with centre O .A line cuts the circles at A, B, C, D
respectively if AB=10 cm then length CD is
(a)5 cm
(b)10 cm (c)3.5 cm
(d)2.5 cm
10 Given three collinear points, then the number of circles which can be drawn
through three points are
(a) One (b) two
(c)infinite (d)none of these
11 Given a circle of radius 5 cm and centre O .OL is perpendicular to the chord
AB. If OL=3 cm then the length of the chord AB is
(a)4 cm
(b)6 cm
(c)10cm
(d)8 cm
12 In how many parts a plane can divides a circle if it intersect perpendicular?
(a) 2 parts (b)3 parts (c)4 parts (d)5 parts
13 Given a circle of radius 5 cm and centre O.OL is drawn perpendicular to the chord
AB .If OL=3 cm then length of chord AB is
(a) 4 cm
(b)8 cm
(c) 6 cm
(d)10 cm
14 Distance of a chord AB of a circle from the centre is 12 cm and length of the chord is 10 cm.
Then diameter of a circle is
(a) 26 cm
(b)13 cm
(c)√244 cm (d)20 cm
15 PQ and RS are two parallel chords of a circle on the same side of centre o and radius is 10 cm .
If PQ=16 cm and RS=12cm.The distance between the chord is
(a)6 cm
(b)2 cm
(c)8 cm
(d)4 cm
Short Answer Type -II Questions (carrying 3 marks )
1
ABCD is a quadrilateral such that A is the centre of the circle passing through B,C and D .Prove
that ∠ CBD+∠ CDB =1/2 ∠BAD
2 O is the circumcentre of the triangle ABC and D is the midpoint of the base BC .prove that
∠ BOD=∠ A
3 Two chords AB and AC of a circle subtends angles equal to 90◦ and 150◦ Respectively at the
centre .Find angle BAC, if AB and AC lie on the opposite sides of the centre .
4 If a line is drawn parallel to the base of an isosceles triangle to intersect Its equal sides, prove
that the qu adrilateral so formed is cyclic
5 A chord of a circle is equal to its radius .Find the angle subtended by this chord at a point in major
segment.
6 prove that the perpendiculars let fall from vertices of a triangle on opposite sides are concurrent
7 ABCD is a cyclic rectangle. Prove that the centre of the circle through A,B,C,D is the point of
intersection of its diagonals.
8 Prove that the quadrilateral formed by angle bisector of a cyclic quadrilateral is also cyclic.
9 Prove that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.
10 If two non parallel sides of a trapezium are equal, prove that it is cyclic.
Long Answer Type Question(carrying 4 marks)
1
2
3
4
5
Prove that angle bisector of any angle of a triangle and perpendicular
Bisector of the opposite side if intersect ,they will intersect on the circumcircle of the triangle.
If ABC is an equilateral triangle inscribed in a circle and PB any point On the minor arc BC
which does not coincide with B or C ,prove that PA Is angle bisector of angle BPC.
If bisectors of opposite angles of a cyclic quadrilateral ABCD intersect the circle ,circumscribing
it at the points P and Q ,prove that PQ is a diameter of the circle.
Two equal chords AB and CD of a circle when produced intersect at a point P.Prove that
PB=PD.
AB and AC are two chords of a circle of radius r such that AB=2AC.If p and q are the distances
of AB and AC from the centre.Prove that 4q2=p2+3r2.
CONSTRUCTIONS
Short Answer Type -II Question (carrying 3 marks )
construct a ∆ABC in which three sides are 1:3:5 and its perimeter is 13.5cm.
construct a ∆ABC if its perimeter is 10.4cm and base angles 450 and 1200.
Construct a∆PQR, given that QR=3cm,angle Q=450and PQ+QR=6cm.
Construct a triangle ABC such that BC =4cm ,angle B=600and
AC-AB =2cm.
5 Construct a right-angled triangle whose base is 12cm and the sum of its hypotenuse and other
side is 18cm.
6 By using ruler and compass draw an angle of 600and bisect it.
7 Draw a line segment of length 5.8cm.Bisect it and measure the length of each part.
8 Construct a triangle ABC in which BC=4.5 cm ,∠B =450 and AB-AC=2.5 cm
9 Construct a right triangle in which one side is 3.5 cm and sum of other two sides and hypotenuse
is 5.5 cm
10 Construct a triangle ABC in which BC=4.5 cm ∠B=450 and AB-AC=2.5 cm.
1
2
3
4
Long Answer Type Question(carrying 4 marks)
1
Construct an equilateral triangle whose altitude is 3.5cm.
Construct a triangle with perimeter 10cm and base angles 450and 600.Write the steps of
constructions.
3 Construct a quadrilateral ABCD in which AB=6.3cm,BC=5.2cm,CD=5.6cm,DA=7.1cm
and ∠B=600.Construct a triangle equal in area to this quadrilateral. Write the steps of
construction.
3 Construct a triangle ABC in which perimeter is 12cm and ∠ B=800,∠ C=600.Writr the steps
of constructions.
4 Construct a triangle ABC such that ∠B=600,∠C=450 AB+BC+CA=10 cm
2
SURFACE AREAS AND VOLUMES
Multiple Choice Questions(Carrying 1 mark)
1
The radius of a sphere is 2r, then its volume will be :
(a) 4/3∏r3 (b) 4∏r3 (c) 8∏r3/3 (d) 32/3∏r3
2
The total surface area of a cube is 96 cm2.The volume of the cube is :
(a) 8cm3 (b) 512cm3 (c) 64 cm3 (d) 27 cm2
3
The length of the longest pole that can be put in a room of dimensions(10m x 10m x 5m) is
:
(a)15m (b) 16m (c) 10m (d) 12m
4
The radii of two cylinders are in the ratio of 2:3 and their heights are in the ratio of 5:3.The
ratio of their volumes is :
(a)10:17 (b) 20:27 (c) 17:27 (d) 20:37
5
In a cylinder, radius is doubled and height halved ,curved surface area will be :
(a)halved (b) doubled (c) same (d) four times
6 Lateral surface area of a cuboid with dimension l,b h is
(a)2(lb+bh+hl)-2lb (b)2(lb+bh+hl) (c)2(l+b)h
(d)lbh
7 If volume of a cube is 216 cm3 then its edge is
(a)36 cm (b) 6 cm (c)12 cm (d)16 cm
8 Number of surfaces of the same area in a cuboid are
(a)6
(b)4
(c)2
(d)3
9 Three cubes are joined end to end forming a cuboid .If side of a cube is 2 cm then
dimension of th cuboid are
(a) l=2,b=2,h=2 (b) l=4,b=4,h=2 (c)l=4,b=2,h=4 (d)l=6,b=2,h=2
10 Given a cuboid of dimension l=3cm,b=2 cm ,h=2cm .How many cubes of 1 cm side can be
cut out of it
(a)12
(b)6
(c)4
(d)3
11 The dimensions of a cuboid are 3 cm,2 cm,1 cm then the length of diagonal the cuboid is
(a)√13 cm
(b)√5cm
(c)√10cm
(d)√14cm
12 Diagonal of a cube is √6cm.Then its lateral surface area is
(a)6√6cm2
(b)36cm2
(c)12cm2
(d)8cm2
13 Slant height of a cone is 34 cm and base diameter is 32 cm,then height of the cone is
(a) 33 cm
(b)25 cm
(c)154 cm2
(d)2156cm2
14 Diameter of earth is four times the diameter of the moon then the ratio of their surface area
is
(a) 4:1
(b)8:1
(c)16:1
(d)2:1
15 The diameter of a sphere is decreased by 25% by what percentage its volume decreases
(a)25%
(b)43.75%
(c)43.50%
(d)50%
ShortAnswer Type -II Question (carrying 3 marks )
1 Find the lateral surface area and total surface area of a cube of edge 20cm.
2 Determine the volume of a conical tin having radius of the base as 30cm and its slant height
as 50cm
(use ∏=3.14 ).
3 Find the volume of a sphere whose diameter is 7cm (take ∏=22/7).
4 The length, breadth and height of a room are 5m,4m,3m respectively. Find the cost of white
washing the walls of the room and ceiling at the rate of Rs.7.50 per m2.
5 The radius and height of a cone are in the ratio 3:4,If its volume is 301.44cm3,What is its
radius?.
6 The circumference of the base of a right circular cone is 88cm.If the height of the cone is
10cm then, Find its volume.
7 A conical tent is 10m high and the radius of its base is 24m.Find:
(a)slant height of the conical tent (b) cost of the canvas required to make the tent,If the cost
of1m2canvas is Rs.70 .
8 How many spherical lead shots each 4.2cm in diameter can be obtained from a rectangular
solid of lead with dimension 66cm,42cm,21cm? (Use ∏=22/7 ).
9 A solid metallic sphere of diameter 21cm is melted and re-casted into a number of smaller
cones, Each of diameter 7cm and height 3cm.Find number of cones so formed.
10 The diameter of a roller 120cm long is 84cm.If it takes 500 complete revolutions to level a
playground, Determine the cost of leveling it at the rate of 30paise per square meter .
Long Answer Type Question(carrying 4 marks)
1 How many cubic metres of earth must be dug out to sink a well 24m deep and of diameter
7m?Also,find the cost of plastering the inner curved surface at Rs.3 per square metre.
2 Find the cost of sinking a tubewell 280m deep having diameter 3m at the rate of Rs.3.60 per
m3.Find the cost of cementing its inner curved surface at Rs.2.50 per m2.
3 A cylindrical road roller made of iron is 1m long.Its inner diameter is 54 cm and the thickness
of the iron sheet rolled into the road roller is 9cm. Find the weight of the roller if 1cm3of iron
weighs 8g.(∏=3.14).
4 The sum of the radius of the base and height of a cylinder is 37m.If the total surface area of
the solid cylinder is 1628m2,find the volume of cylinder
5 The external and internal diameter of a hollow hemispherical vessel are 12cm and 10cm
respectively. The cost of painting is Rs.2per sq.cm.Find the cost of painting the vessel all over
6 The pillars of a temple are cylindrical shaped.If each pillar has a circular base of radius 25cm
and height 10.5cm,then find the quantity of concrete mixture used to build 30 such pillars.Also
find the cost of concrete mixture at the rate of Rs.250 per m3.Take (∏=22/7).
STATISTICS
Multiple Choice Questions(Carrying 1 mark )
1
The class mark of the class 90-120 is
(a) 90 (b) 105
(c) 115 (d) 120
2 The range of the data
25,18,20,22,16,6,17,15,12,30,32,10,19,8,11,20 is
(a) 10 (b) 15 (c) 18 (d) 26
3 In the class interval 10-20,20-30,the number 20 is included in
(a) 10-20 (b) 20-30 (c) both of the intervals (d) none of these
4 Median of the following numbers
4,4,5,7,6,7,7,12,3 is
(a) 4
(b) 5 (c) 6 (d) 7
5 Mode of the data
15, 14,19,20,14,15,16,14,15,18,14,19,15,17,15 is
(a) 14 (b) 15 (c) 16 (d) 17
6 If each observation of the data is increased by 5, then their mean is
(a) Remains the same
(b) becomes 5 times the original mean
(c) Is decreased by 5
(d) is increased by 5
7 The mean of 100 observations is 50.If one of the observation which was
50 is replaced by 150 the resulting mean will be
(a)50.5
(b)51
(c)51.5
(d)52
8 Median of the data 32,15, 27,8,15,12,9 is
(a)8
(b)21
(c)12
(d)15
9 Mean of 36 observation is 12.One observation 47 was misread as 74,then corrected mean is
(a)11.5
(b)13.5
(c)135/4
(d)45/2
10 The mean of prime number between 20 and 30 is
(a)21
(b)26
(c)25
(d)27
11 In a continuous frequency distribution ,class mark of a class is 85 and lower limit is 83,then
its upper limit is
(a)86
(b)84
(c)83
(d)87
12 In a given data ,some variables are given with particular values ,we went to represent these
Graphically ,then we can represent these ,using
(a)histogram (b)frequency polygon (c)bar graph (d)ogive
13 Mean of 20 observation is 17 .If 25 is subtracted from the sum of observation then remaining
Sum is
(a)340
(b)365
(c) 315
(d)300
14 The mean of prime numbers between 20 and 30 is
(a)21
(b)26
(c)25
(d)27
15 Given the class interval 1-10,11-20,21-30,--------then 20 is considered in class
(a)11-20
(b)21-30
(c)11-30
(d)15-25
Short Answer Type-II Question (carrying 3 marks )
1
2
3
4
5
6
7
8
9
The class marks of a distribution are 6,10,14,18,22,26,30.Find the class size and the class
interval
Find the range of the following array of data ;
70,65,71,36,55,61,62,41,40,39,35
The mean of 21 numbers is 15 .if each number is multiplied by 2 ,what will be the new
number
There are 50 students in a class of which 40 are boys and the rest are girls .The average
weight of the class is 44 kg .Find the average weight of the boys.
The mean of 10 numbers is 20.If 5 is subtracted from every number ,what will be the new
mean?
Find the arithmetic mean of first ten natural numbers?
If the median of 6,7,x-2,x,17 and 20 written in ascending order is 16,find the value of x
Mean of 25 observations was found to be 78.4.But later on it was discovered that 96 was
misread as 69.Find the correct mean
The following observations have been arranged in ascending order .If the median of the
data is 63 ,find the value of x and also find the mean
29,32,48,50,x,x+2,72,78,84,95
10 The mean of 16 observation is 8.If 2 is added to every number ,what will be the new mean?
Long Answer Type Question(carrying 4 marks)
1
For the following data ,construct a histogram;
Class interval 10-14 15-19 20-24 25-29 30-34
Frequency
300 980
800
580 290
2 construct a frequency polygon for the following data
Age in years 0-4 4-8 8-12 12-16 16-20 20-24 24-28
Number of persons 3
6
8
10
8
5
3
3 Draw a histogram and frequency polygon for the following data
Marks
0-20 20-40 40-60 60-80 80-100
No of students
10
15
40
45
40
4 Find mean, median, and mode of the following data :
78,56,22,34,45,54,39,68,54,84
5 Find the combined mean of a group of 150 students if the mean of 50 student
Is 40 and that of other 100 student is 50
PROBABILITY
Multiple Choice Questions(Carrying 1 mark)
1
Which of the words represent uncertainty?
(a) probability ( b) doubt (c) sure (d) chances
2 In an experiment ,the sum of probabilities of different events is
(a) 1
(b) 0.5 (c) -2
(d) 15/15
3 In the year 2009 ,during rainy season of 90 days, it was observed that
it rained 20 days only. Then the probability that it did not rain is
(a) 2/9 (b) 70 (c) 7/9 (d) 9/7
4 The probability of happening of an event is 37%.Then probability of the event is
(a) 37 (b) 0.037 (c) 3.7 (d) 0.37
5 In a class there are x boys and y girls ,A student is selected at random ,then probability of
selecting a girl is
(a) x/y (b)x/x+y (c) y/x+y (d) y/x
6 In an experiment ,the sum of probabilities of different event is
(a)1
(b)0.5
(c)-2
(d)15/15
7 In the year 2009 ,during rainy season of 90 days,it was observed that it rained 20 days only
Then the probability that it did not rain is
(a)2/9
(b)70
(c)7/9
(d)9/7
8 In a class of 50 students there are 12% boys ,then the number of boys in class are
(a)120
(b)60
(c)1.2
(d)none of these
9 The probability of happening of an event is 37%.Then probability of the event is
(a)37
(b)0.037
(c)3.7
(d)0.37
10 In an experiment ,probability of an event is better approximated,when an experiment is
performed
(a)10 times (b)20 times (c)30 times (d) large number of times
11 An experiment is performed and probability of an event A is recorded , probability of an event
can be
(a)0.0001
(b)0.999
(c)1.001
(d)-0.999
12 The probabilities of a student getting A,B,C ,and D grade are 0.35,0.25 ,0.35 and 0.05then the
probability that a student gets at most grade C is
(a)0.35
(b)0.40
(c)0.95
(d)0.65
13 Probability of getting a blue ball is 2/3,from a bag containing 6 blue and 3 red balls.12 red balls
are added in the bag ,then the Probability of getting a blue ball is
(a)6/12
(b)3/12
(c)2/7
(d)1/7
14 A die is thrown once, a number is noted ,then the Probability that it is a prime number is
(a)4/6
(b)3/6
(c)1/2
(d)2/6
15 In class 9th of a particular school , Probability of choosing a boy is 0.48 .If there are 96 boys in
class 9th Then the total number of students in class 9th are
(a)480
(b)192
(c)200
(d)203
Short Answer Type –II Question (carrying 3 marks )
1
2
3
4
5
6
7
8
9
10
Two coins are tossed simultaneously , find the probability of getting one or more tail
A die is thrown .Find the probability of getting an odd number
A coin is tossed 15 times and observed that 11 times head comes up. Find
the probability that a tail comes up
To know the opinion of the students about the subject statistics ,a survey of 200 students was
conducted.The data is recorded in the following table
Opinion
like
dislike
No of students 135
65
Find the probability that a student chosen at random
(a) like statistics
(b) does not like it
Two coins are tossed simultaneously 100 times and we get the following outcomes
(a) No head=30 (b)one head=20 (c)two heads=50
Find the probability of each event
What is the probability that a number selected from the number1,2,3---------,15 is a multiple of 4?
A coin is tossed 150 times with the following frequencies
Head:85, Tail:65
Compute the probability of each event A die is thrown 2000 times the following frequencies for
the outcomes 1,2,3,4,5 and 6 as given the
following tables
Outcome 1
2
3
4
5
6
Frequency 272 158 350
378
720
122
Find the probability of the happening of each outcome
The percentage of marks obtained by a student in the monthly unit test
Are given bellow
Unit test
I II III
IV
V
VI
Percentage of marks 65 55 63
70
60
68
Based on the data ,find the probability that the student gets more than 64%
Marks in a unit test
Two coins are tossed simultaneously for 400 times and we get
Two tails;128
One tail: 140
No tail:132 Find the probability of occurrence of these events
In an experiment ,a coin is tossed 500 times .if the head turns up 280 times
Find the empirical probability of getting
(a)a head
(b)a tail