PRACTICE QUESTIONS

CLASS IX : CHAPTER - 1
NUMBER SYSTEM

1. Prove that 5  3 is not a rational number.

8
2. Arrange the following in descending order of magnitude: 90, 4 10, 6
3. Simplify the following:
 
(i ) 4 3  2 2 3 2  4 3 
(ii )  2  3  3  5 
2
(iii )  3  2 

2 1  1 3 
(iv )  7 2  6 11    7 2  11 
3 2  3 2 
4. Rationalize the denominator of the following:
2 3 2 6 1
(i ) (ii ) (iii ) (iv)
3 5 3 2 5 2 85 2
3 2 2 3 1 4 1
(v) (vi ) (vii ) (viii)
3 2 2 3 1 7 3 53 2
5. Rationalise the denominator of the following:
2 16 5 2
(i ) (ii) (iii)
3 3 41  5 5 2
40 3 2 2 3
(iv ) (v ) (vi)
3 4 2 2 3
6 3 5 3 4 3 5 2
(vii) (viii) (ix)
2 3 5 3 48  18
1
6. If a  6  35 , find the value of a 2  .
a2
1 1
7. If x  3  8 , find the value of (i) x 2  2 and (ii) x 4 
x x4
2 6 6 2 8 3
8. Simplify, by rationalizing the denominator  
2 3 6 3 6 2
9. Simplify, by rationalizing the denominator
1 1 1 1 1
   
3 8 8 7 7 6 6 5 5 2
2 1 2 1
10. If x  and y  , find the value of x 2  y 2  xy .
2 1 2 1
3 2 3 2
11. If x  and y  , find the value of x 2  y 2 .
3 2 3 2
5 3 5 3
12. If x  and y  , find the value of x  y  xy .
5 3 5 3
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 2 5 2 5
13. If x  and y  , find the value of x 2  y 2 .
2 5 2 5
5 2 3
14. If  a  3b , find a and b where a and b are rational numbers.
7 3
43 5
15. If a and b are rational numbers and  a  b 5 , find the values of a and b.
43 5
2 3
16. If a and b are rational numbers and  a  b 3 , find the values of a and b.
2 3
11  7
17. If a and b are rational numbers and  a  b 77 , find the values of a and b.
11  7
1 1 1 1
18. Evaluate:    ............. 
2 1 3 2 4 3 9 8
1
19. If x  , find the value of 2 x3  7 x 2  2 x  1 .
2 3
1
20. If x  , find the value of x 3  2 x 2  7 x  5 .
2 3
10  5
21. If 2  1.414 and 5  2.236 , find the value of upto three places of decimals.
2 2
22. Find six rational numbers between 3 and 4.
3 4
23. Find five rational numbers between and
5 5
3 1
24. Find the value of a and b in  ab 3 .
3 1
5 2 3
25. Find the value of a and b in  ab 3
74 3
5 6
26. Find the value of a and b in  a b 6
5 6
4 5 4 5
27. Simplify  by rationalizing the denominator.
4 5 4 5
5 1 5 1
28. Simplify  by rationalizing the denominator.
5 1 5 1
3 2 3 2
29. Simplify  by rationalizing the denominator.
3 2 3 2
3 2 1 1
30. If x = , find (i) x 2  2 (ii) x 4  4 .
3 2 x x
1 1
31. If x = 4  15 , find (i) x 2  2 (ii) x 4  4 .
x x
1 1
32. If x = 2  3 , find (i) x 2  2 (ii) x 4  4 .
x x
33. Represent the real number 10 on the number line.
34. Represent the real number 13 on the number line.

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35. Represent the real number 7 on the number line.

36. Represent the real number 2, 3, 5 on a single number line.

37. Find two rational number and two irrational number between 2 and 3 .
10 7 1
38. Find the decimal expansions of , and .
3 8 7
39. Show that 3.142678 is a rational number. In other words, express 3.142678 in the form of
p
, where p and q are integers and q  0 .
q
p
40. Show that 0.3333……. can be expressed in the form of , where p and q are integers
q
and q  0 .
p
41. Show that 1.27272727……. can be expressed in the form of , where p and q are
q
integers and q  0 .
p
42. Show that 0.23535353……. can be expressed in the form of , where p and q are
q
integers and q  0 .
p
43. Express the following in the form of , where p and q are integers and q  0 .
q
(i )0.6 (ii )0.47 (iii )0.001 (iv )0.26
5 9
44. Find three different irrational numbers between the rational numbers and .
7 11
45. Visualize the representation of 5.37 using successive magnification
46. Visualize 4.26 on the number line, using successive magnification upto 4 decimal places.
47. Visualize 3.765 on the number line, using successive magnification.
48. Find the value of a and b in each of the following:
3 2 3 7 7 5
(i )  ab 2 (ii )  ab 7 (iii )  ab 5
3 2 3 7 7 5
49. Simplify each of the following by rationalizing the denominator.
64 2 5 2 52
(i ) (ii ) 
64 2 52 5 2
50. Evaluate the following expressions:
3 1
1
 256  8  343  3
(i )   (ii ) 15625  6 (iii )  
 6561   1331 
1
6561 
3
(iv) 8 (v)343
65536
32  48
51. Simplify:
8  12
7
52. Simplify:
3 32 2

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 4 3
53. Simplify: (i) 22 (ii ) 3 2. 4 2.12 32
2 1
54. If 2  1.4142 , then find the value of .
2 1
3 1
55. If 3  1.732 , then find the value of .
3 1
6
56. Find the value of a if 3 2 a 3
3 2 2 3

57. Evaluate the following expressions:

1
 2 1 4 3
 625  4 
(i )  
3
(ii)27  27  27 3 3
(iii)  6.25  2
 81 
5 1
(iv)  0.000064  6

(v) 17  8 2 2 2

p
58. Express 0.6  0.7  0.47 in the form of , where p and q are integers and q  0 .
q
7 3 2 5 3 2
59. Simplify:  
10  3 6 5 15  3 2
4 3
60. If 2  1.414, 3  1.732 , then find the value of  .
3 32 2 3 32 2
61. Simplify:
1
3
  1 1
  4
24 54

(i) 5  8  27 3 
3  (ii ) 45  3 20  4 5 (iii) 
    8 9
1
(iv) 4 12  6 7 (v) 4 28  3 7 (vi) 3 3  2 27 
3
2
(vii)  3 5  (viii) 4 81  8 3 216  15 5 32  225
2
3 1 3 3
(ix)  ( x) 
8 2 3 6

3 5 1
62. If a  then find the value of a 2  2 .
2 a
3
(4 2 )
63. Simplify:  2 5 6 
4 1 2
64. Find the value of 2
 3
 1
 216  3  256  4  243 5
1
65. If a  5  2 6 and b  then what will be the value of a2 + b2?
a

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66. Find the value of a and b in each of the following:
3 5 19
(i ) a 5
3 2 5 11
2 3
(ii )  2b 6
3 22 3
7 5 7 5 7
(iii )  a b 5
7 5 7 5 11

1
67. If a  2  3 , then find the value of a  .
a
68. Rationalise the denominator in each of the following and hence evaluate by taking
2  1.414, 3  1.732 and 5  2.236 , upto three places of decimal.
4 6 10  5 2 1
(i) (ii ) (iii ) (iv) (v )
3 6 2 2 2 3 2
69. Simplify:
2
1 4 12 6
3 8  32   1 3

(i ) 13  23  3 3 2
 (ii)    
5 5
 
 5 
(iii )   
 27 
2 1 1
1
 1 4  3 3  131 2

  8 16
(iv)   625  2   (v ) 1
(vi )64 64  64 3 
3
   3  
  32

1 1
3 2
9  27
70. Simplify: 1 2
6 3
3 3

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 PRACTICE QUESTIONS
CLASS IX : CHAPTER - 2
POLYNOMIALS
1. Factorize the following: 9x2 + 6x + 1 – 25y2.

2. Factorize the following: a2 + b2 + 2ab + 2bc + 2ca

3. Show that p(x) = x3 – 3x2 + 2x – 6 has only one real zero.

4. Find the value of a if x + 6 is a factor of x3 + 3x2 + 4x + a.

5. If polynomials ax3 + 3x2 – 3 and 2x3 – 5x + a leaves the same remainder when each is divided by
x – 4, find the value of a..

6. The polynomial f(x)= x4 – 2x3 +3x2 – ax + b when divided by (x – 1) and (x + 1) leaves the
remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when
f(x) is divided by (x – 2).

7. If the polynomials 2x3 +ax2 + 3x – 5 and x3 + x2 – 2x + a leave the same remainder when divided
by (x – 2), find the value of a. Also, find the remainder in each case.

8. If the polynomials az3 + 4z2 + 3z – 4 and z3 – 4z + a leave the same remainder when divided by
z – 3, find the value of a.

9. The polynomial p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7 when divided by x + 1 leaves the remainder
19. Find the values of a. Also find the remainder when p(x) is divided by x + 2.
1
10. If both x – 2 and x – are factors of px2 + 5x + r, show that p = r.
2
11. Without actual division, prove that 2x4 – 5x3 + 2x2 – x + 2 is divisible by x2 – 3x + 2.

12. Simplify (2x – 5y)3 – (2x + 5y)3.

13. Multiply x2 + 4y2 + z2 + 2xy + xz – 2yz by (– z + x – 2y).

a 2 b2 c2
14. If a, b, c are all non-zero and a + b + c = 0, prove that   3
bc ca ab
15. If a + b + c = 5 and ab + bc + ca = 10, then prove that a3 + b3 + c3 –3abc = – 25.

16. Without actual division, prove that 2x4 – 6x3 +3x2 +3x – 2 is exactly divisible by x2 – 3x + 2.

17. Without actual division, prove that x3 – 3x2 – 13x + 15 is exactly divisible by x2 + 2x – 3.

18. Find the values of a and b so that the polynomial x3 – 10x2 +ax + b is exactly divisible by (x – 1)
as well as (x – 2).

19. Find the integral zeroes of the polynomial 2x3 + 5x2 – 5x – 2.

 1
20. If (x – 3) and  x   are both factors of ax2 + 5x + b, then show that a = b.
 3
21. Find the values of a and b so that the polynomial x4 + ax3 – 7x2 +8x + b is exactly divisible by
(x + 2) as well as (x + 3).

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22. If x3 + ax2 + bx + 6 has (x – 2) as a factor and leaves a remainder 3 when divided by (x – 3), find
the values of a and b.

23. Find the value of x3 + y3 + 15xy – 125 if x + y = 5.

24. Without actually calculating, find the value of (25)3 – (75)3 + (50)3.

25. Factorise each of the following cubic expressions:

(i) 8x3 – y3 – 12x2y + 6xy2
(ii) 27q3 – 125p3 – 135q2p + 225qp2
(iii) 8x3 + 729 + 108x2 + 486x
1 9 1
(iv) 27 x 3   x2  x
216 2 4

26. Factorise:
(i) x3 + 216y3 + 8z3 – 36xyz
(ii) a3 – 64b3 – 27c3 – 36abc

3 3
1  3  1 
2 
 
27. Factorise:  x  3 y   3 y  3 z   3z  x 
 2 

28. Give one example each of a binomial of degree 35, and of a monomial of degree 100.

29. Find a zero of the polynomial p(x) = 2x + 1.

30. Verify whether 2 and 0 are zeroes of the polynomial x2 – 2x.

31. Find the zero of the polynomial in each of the following cases:
(i) p(x) = x + 5 (ii) p(x) = x – 5 (iii) p(x) = 2x + 5
(iv) p(x) = 3x – 2 (v) p(x) = 3x (vi) p(x) = ax, a  0
32. Find the value of each of the following polynomials at the indicated value of variables:
(i) p(x) = 5x2 – 3x + 7 at x = 1.
(ii) q(y) = 3y3 – 4y + 11 at y = 2.
(iii) p(t) = 4t4 + 5t3 – t2 + 6 at t = a.

33. Divide p(x) by g(x), where p(x) = x + 3x2 – 1 and g(x) = 1 + x.

34. Divide the polynomial 3x4 – 4x3 – 3x –1 by x – 1.

35. Find the remainder obtained on dividing p(x) = x3 + 1 by x + 1.

36. Find the remainder when x4 + x3 – 2x2 + x + 1 is divided by x – 1.

37. Check whether the polynomial q(t) = 4t3 + 4t2 – t – 1 is a multiple of 2t + 1.

38. Check whether p(x) is a multiple of g(x) or not, where p(x) = x3 – x + 1, g(x) = 2 – 3x.
x 1
39. Check whether g(x) is a factor of p(x) or not, where p(x) = 8x3 – 6x2 – 4x + 3, g(x) =  .
3 4
40. Find the remainder when x3 – ax2 + 6x – a is divided by x – a.

41. Examine whether x + 2 is a factor of x3 + 3x2 + 5x + 6 and of 2x + 4.

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42. Find the value of k, if x – 1 is a factor of 4x3 + 3x2 – 4x + k.

43. Find the value of a, if x – a is a factor of x3 – ax2 + 2x + a – 1.

44. Factorise 6x2 + 17x + 5

45. Factorise y2 – 5y + 6

46. Factorise x3 – 23x2 + 142x – 120.

47. Factorise :
(i) x3 – 2x2 – x + 2 (ii) x3 – 3x2 – 9x – 5
(iii) x3 + 13x2 + 32x + 20 (iv) 2y3 + y2 – 2y – 1

48. Factorise : 4x2 + 9y2 + 16z2 + 12xy – 24yz – 16xz

49. Expand (4a – 2b – 3c)2.

50. Factorise 4x2 + y2 + z2 – 4xy – 2yz + 4xz.

51. If x + 1 is a factor of ax3 + x2 – 2x + 4a – 9, find the value of a.

52. By actual division, find the quotient and the remainder when the first polynomial is divided by
the second polynomial : x4 + 1; x –1

53. Find the zeroes of the polynomial : p(x) = (x – 2)2 – (x + 2)2

54. Factorise :
(i) x2 + 9x + 18 (ii) 6x2 + 7x – 3
(iii) 2x2 – 7x – 15 (iv) 84 – 2r – 2r2

55. Factorise :
(i) 2x3 – 3x2 – 17x + 30 (ii) x3 – 6x2 + 11x – 6
(iii) x3 + x2 – 4x – 4 (iv) 3x3 – x2 – 3x + 1

56. Using suitable identity, evaluate the following:

(i) 1033 (ii) 101 × 102 (iii) 9992

57. Factorise the following:

(i) 4x2 + 20x + 25
(ii) 9y2 – 66yz + 121z2
2 2
 1  1
(iii)  2 x     x  
 3  2

58. Factorise the following :

(i) 9x2 – 12x + 3 (ii) 9x2 – 12x + 4

59. If a + b + c = 9 and ab + bc + ca = 26, find a2 + b2 + c2.

60. Expand the following :

(i) (4a – b + 2c)2
(ii) (3a – 5b – c)2
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 (iii) (– x + 2y – 3z)2

61. Find the value of

(i) x3 + y3 – 12xy + 64, when x + y = – 4
(ii) x3 – 8y3 – 36xy – 216, when x = 2y + 6

62. Factorise the following :

(i) 9x2 + 4y2 + 16z2 + 12xy – 16yz – 24xz
(ii) 25x2 + 16y2 + 4z2 – 40xy + 16yz – 20xz
(iii) 16x2 + 4y2 + 9z2 – 16xy – 12yz + 24 xz

63. Expand the following :

3 3
1 y  1 
(i) (3a – 2b)3 (ii)    (iii) 4 
x 3  3x 
64. Find the following products:
2
x  x 
(i)   2 y    xy  4 y 2  (ii) ( x 2  1)( x 4  x 2  1)
2  4 
65. Factorise the following :
12 2 6 1
(i) 8 p3  p  p
5 25 125
(ii) 1 – 64a3 – 12a + 48a2

66. Without finding the cubes, factorise (x – 2y)3 + (2y – 3z)3 + (3z – x)3

67. Give possible expressions for the length and breadth of the rectangle whose area is given by
4a2 + 4a –3.

68. Factorise: (i) 1  64x 3 (ii) a3  2 2b3

69. Evaluate each of the following using suitable identities:

(i) (104)3 (ii) (999)3

70. Factorise : 8x3 + 27y3 + 36x2y + 54xy2

71. Factorise : 8x3 + y3 + 27z3 – 18xyz

72. Verify : (i) x3 + y3 = (x + y) (x2 – xy + y2) (ii) x3 – y3 = (x – y) (x2 + xy + y2)

73. Factorise each of the following:
(i) 27y3 + 125z3 (ii) 64m3 – 343n3
74. Factorise : 27x3 + y3 + z3 – 9xyz
75. Without actually calculating the cubes, find the value of each of the following:
(i) (–12)3 + (7)3 + (5)3
(ii) (28)3 + (–15)3 + (–13)3

76. Find the following product :(2x – y + 3z) (4x2 + y2 + 9z2 + 2xy + 3yz – 6xz)

77. Factorise :
(i) a3 – 8b3 – 64c3 – 24abc (ii) 2 2 a3 + 8b3 – 27c3 + 18 2 abc.

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78. Give possible expressions for the length and breadth of rectangles, in which its areas is given by
35y2 + 13y –12

79. Without actually calculating the cubes, find the value of :

3 3 3
1 1 5 3 3 3
(i )         (ii )  0.2    0.3   0.1
 2  3  6
80. By Remainder Theorem find the remainder, when p(x) is divided by g(x), where
(i) p(x) = x3 – 2x2 – 4x – 1, g(x) = x + 1
(ii) p(x) = x3 – 3x2 + 4x + 50, g(x) = x – 3
(iii) p(x) = 4x3 – 12x2 + 14x – 3, g(x) = 2x – 1
3
(iv) p(x) = x3 – 6x2 + 2x – 4, g(x) = 1  x
2
81. Check whether p(x) is a multiple of g(x) or not :
(i) p(x) = x3 – 5x2 + 4x – 3, g(x) = x – 2
(ii) p(x) = 2x3 – 11x2 – 4x + 5, g(x) = 2x + 1

82. Show that p – 1 is a factor of p10 – 1 and also of p11 – 1.

83. For what value of m is x3 – 2mx2 + 16 divisible by x + 2 ?

84. If x + 2a is a factor of x5 – 4a2x3 + 2x + 2a + 3, find a.

85. Find the value of m so that 2x – 1 be a factor of 8x4 + 4x3 – 16x2 + 10x + m.

86. Show that :

(i) x + 3 is a factor of 69 + 11x – x2 + x3 .
(ii) 2x – 3 is a factor of x + 2x3 – 9x2 + 12 .
87. If x + y = 12 and xy = 27, find the value of x3 + y3.

88. Without actually calculating the cubes, find the value of 483 – 303 – 183.

89. Without finding the cubes, factorise (2x – 5y)3 + (5y – 3z)3 + (3z – 2x)3.

90. Without finding the cubes, factorise (x – y)3 + (y – z)3 + (z – x)3.

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 PRACTICE QUESTIONS
CLASS IX: CHAPTER – 4
LINEAR EQUATION IN TWO VARIABLES
1. Find the value of k, if x = 2, y = 1 is a solution of the equation 2x + 3y = k.

2. Find the points where the graph of the equation 3x + 4y = 12 cuts the x-axis and the y-axis.

3. At what point does the graph of the linear equation x + y = 5 meet a line which is parallel to the
y-axis, at a distance 2 units from the origin and in the positive direction of x-axis.
5
4. Determine the point on the graph of the equation 2x + 5y = 20 whose x-coordinate is times its
2
ordinate.

5. Draw the graph of the equation represented by the straight line which is parallel to the x-axis and
is 4 units above it.

6. Draw the graphs of linear equations y = x and y = – x on the same cartesian plane. What do you
observe?
1
7. Determine the point on the graph of the linear equation 2x + 5y = 19, whose ordinate is 1 times
2
its abscissa.

8. Draw the graph of the equation represented by a straight line which is parallel to the x-axis and at
a distance 3 units below it.

9. Draw the graph of the linear equation whose solutions are represented by the points having the
sum of the coordinates as 10 units.

10. Write the linear equation such that each point on its graph has an ordinate 3 times its abscissa.

11. If the point (3, 4) lies on the graph of 3y = ax + 7, then find the value of a.

12. How many solution(s) of the equation 2x + 1 = x – 3 are there on the : (i) Number line (ii)
Cartesian plane

13. Find the solution of the linear equation x + 2y = 8 which represents a point on (i) x-axis (ii) y-
axis

14. For what value of c, the linear equation 2x + cy = 8 has equal values of x and y for its solution.

15. Let y varies directly as x. If y = 12 when x = 4, then write a linear equation. What is the value of
y when x = 5?

16. Draw the graph of the linear equation 2x + 3y = 12. At what points, the graph of the equation
cuts the x-axis and the y-axis?

17. Show that the points A (1, 2), B (– 1, – 16) and C (0, – 7) lie on the graph of the linear equation
y = 9x – 7.

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18. The following values of x and y are thought to satisfy a linear equation :
x 1 2
y 1 3
Draw the graph, using the values of x, y as given in the above table. At what point the graph of
the linear equation (i) cuts the x-axis. (ii) cuts the y-axis.
19. The Autorikshaw fare in a city is charged Rs 10 for the first kilometer and @ Rs 4 per kilometer
for subsequent distance covered. Write the linear equation to express the above statement. Draw
the graph of the linear equation.

20. The work done by a body on application of a constant force is the product of the constant force
and the distance travelled by the body in the direction of force. Express this in the form of a
linear equation in two variables and draw its graph by taking the constant force as 3 units. What
is the work done when the distance travelled is 2 units. Verify it by plotting the graph.

21. The following values of x and y are thought to satisfy a linear equation, Write the linear equation.
x 6 –6
y –6 6
Draw the graph, using the values of x, y as given in the above table. At what point the graph of
the linear equation (i) cuts the x-axis. (ii) cuts the y-axis.

22. Draw the graph of the linear equation 3x + 4y = 6. At what points, the graph cuts the x-axis and
the y-axis.

23. The force exerted to pull a cart is directly proportional to the acceleration produced in the body.
Express the statement as a linear equation of two variables and draw the graph of the same by
taking the constant mass equal to 6 kg. Read from the graph, the force required when the
acceleration produced is (i) 5 m/sec2, (ii) 6 m/sec2.

24. If the temperature of a liquid can be measured in Kelvin units as x°K or in Fahrenheit units as
y°F, the relation between the two systems of measurement of temperature is given by the linear
9
equation y  ( x  273)  32
5
(i) Find the temperature of the liquid in Fahrenheit if the temperature of the liquid is 313°K.
(ii) If the temperature is 158° F, then find the temperature in Kelvin.

25. The linear equation that converts Fahrenheit (F) to Celsius (C) is given by the relation
5 F  160
C
9
(i) If the temperature is 86°F, what is the temperature in Celsius?
(ii) If the temperature is 35°C, what is the temperature in Fahrenheit?
(iii) If the temperature is 0°C what is the temperature in Fahrenheit and if the temperature is 0°F,
what is the temperature in Celsius?
(iv) What is the numerical value of the temperature which is same in both the scales?

26. Draw the graph of x + y = 7 and x – y = 2 on the same graph.

27. If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a.

28. The taxi fare in a city is as follows: For the first kilometre, the fare is Rs 8 and for the subsequent
distance it is Rs 5 per km. Taking the distance covered as x km and total fare as Rs y, write a
linear equation for this information, and draw its graph.

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29. Solve the equation 2x + 1 = x – 3, and represent the solution(s) on
(i) the number line,
(ii) the Cartesian plane.

30. Give the geometric representations of y = 3 as an equation

(i) in one variable (ii) in two variables

31. Give the geometric representations of 2x + 9 = 0 as an equation

(i) in one variable (ii) in two variables

32. The force applied on a body is directly proportional to the acceleration produced in the body.
Write an equation to express this situation and plot the graph of the equation.

33. Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates
of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular
region.

34. Draw the graphs of the equations y = x and y = –x in the same graph paper. Find the coordinates
of the point where two lines intersect.

35. Draw the graphs of the equations 3x – 2y = 4 and x + y – 3 = 0 in the same graph paper. Find the
coordinates of the point where two lines intersect.

36. Draw the graphs of the equations 3x – 2y + 6 = 0 and x + 2y – 6 = 0 in the same graph paper.
Find the area of triangle formed by the two lines and x – axis.

37. If the number of hours for which a labourer works is x and y are his wages (in rupees) and y = 2x
– 1, draw the graph of work – wages equation. From the graph, find the wages of the labourer if
he works for 6 hours.

38. A and B are friends. A is elder to B by 5 years. B’s sister C is half the age of B while A’s father
D is 8 years older than twice the age of B. If the present age of D is 48 years, find the present
ages of A, B and C.

39. A three-wheeler scoter charges Rs. 10 for the first km and Rs. 4.50 each for every subsequent
km. For a distance of x km, an amount of Rs. Y is paid. Write the linear equation representing
the above information.
7 3
40. Solve: 5x   x  14
2 2

6x 1 x3
41. Solve: 1 
3 6
7
42. Solve: 5 x  2(2 x  7)  2(3 x  1) 
2
3x  2 2 x  3 2
43. Solve:   x
4 3 3
3 x  2 4( x  1) 2
44. Solve:   (2 x  1)
7 5 3
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 x 1 x2
45. Solve: x  1
2 3
x 1 x 1
46. Solve:   
2 5 3 4
x 3x 5 x
47. Solve:    21
2 4 6
8 x 17 5 x
48. Solve: x7  
3 6 2
3x  4 2
49. Solve: 
2  6x 5
7 x  4 4
50. Solve: 
x2 3
51. The ages of Rahul and Haroon are in the ratio 5:7. Four years later the sum of their ages
will be 56 years. What are their present ages?

52. Baichung’s father is 26 years younger than Baichung’s grandfather and 29 years older
than Baichung. The sum of the ages of all the three is 135 years. What is the age of each
one of them?

53. Lakshmi is a cashier in a bank. She has currency notes of denominations Rs 100, Rs 50
and Rs 10, respectively. The ratio of the number of these notes is 2:3:5. The total cash
with Lakshmi is Rs 4,00,000. How many notes of each denomination does she have?

54. I have a total of Rs 300 in coins of denomination Re 1, Rs 2 and Rs 5. The number of Rs

2 coins is 3 times the number of Rs 5 coins. The total number of coins is 160. How many
coins of each denomination are with me?

55. The organisers of an essay competition decide that a winner in the competition gets a
prize of Rs 100 and a participant who does not win gets a prize of Rs 25. The total prize
money distributed is Rs 3,000. Find the number of winners, if the total number of
participants is 63.

56. The digits of a two-digit number differ by 3. If the digits are interchanged, and the
resulting number is added to the original number, we get 143. What can be the original
number?

57. Arjun is twice as old as Shriya. Five years ago his age was three times Shriya’s age. Find
their present ages.

58. A positive number is 5 times another number. If 21 is added to both the numbers, then
one of the new numbers becomes twice the other new number. What are the numbers?

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59. Sum of the digits of a two-digit number is 9. When we interchange the digits, it is found
that the resulting new number is greater than the original number by 27. What is the two-
digit number?

60. One of the two digits of a two digit number is three times the other digit. If you
interchange the digits of this two-digit number and add the resulting number to the
original number, you get 88. What is the original number?

61. Shobo’s mother’s present age is six times Shobo’s present age. Shobo’s age five years
from now will be one third of his mother’s present age. What are their present ages?

62. There is a narrow rectangular plot, reserved for a school, in Mahuli village. The length
and breadth of the plot are in the ratio 11:4. At the rate Rs100 per metre it will cost the
village panchayat Rs 75000 to fence the plot. What are the dimensions of the plot?

63. A grandfather is ten times older than his granddaughter. He is also 54 years older than
her. Find their present ages.

64. A man’s age is three times his son’s age. Ten years ago he was five times his son’s age.
Find their present ages.

65. Present ages of Anu and Raj are in the ratio 4:5. Eight years from now the ratio of their
ages will be 5:6. Find their present ages.

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 PRACTICE QUESTIONS
CLASS IX: CHAPTER – 13
SURFACE AREAS AND VOLUMES
11. The dimensions of a prayer Hall are 20m x 15m x 8m. Find the cost of painting its walls at Rs.
10 per m2.

12. Find the curved surface area of a right circular cylinder whose height is 13.5 cm and radius of tis
base is 7 cm. Find also its surface area.

13. The exterior diameter of an iron pipe is 25cm and it is one cm thick. Find the whole surface are
of the pipe it is 21cm long.

14. A roller 150 cm long has a diameter of 70 cm. To level a playground it takes 750 complete
revolutions. Determine the cost of leveling the playground at the rate of 75 paise per m2.

15. Find the total surface area of a cone, if its slant height is 21 cm and the diameter of its base is 24
cm.

16. The volume of a sphere is 4851 cm3. How much should its radius be reduced so that it volume
4312 3
becomes cm .
3

17. A river, 3 m deep and 40m wide, is flowing at the rate of 2km/hr. How much water will fall into
the sea in a minute?

18. Find the capacity in litres of a conical vessel whose diameter is 14 cm and slant height is 25 cm.

19. What is the total surface area of a hemisphere of base radius 7cm?

20. A village having a population of 4000, requires 150 litres of water per head per day. It has a tank
measuring 20m x 15m x 6m. For how many days, the water of the tank will be sufficient for the
village?
21. Mary wants to decorate her Christmas tree. She wants to place the tree on a wooden box covered
with coloured paper with picture of Santa Claus on it. She must know the exact quantity of paper
to buy for this purpose. If the box has length, breadth and height as 80 cm, 40 cm and 20 cm
respectively how many square sheets of paper of side 40 cm would she require?

22. Hameed has built a cubical water tank with lid for his house, with each outer edge 1.5 m long.
He gets the outer surface of the tank excluding the base, covered with square tiles of side 25 cm.
Find how much he would spend for the tiles, if the cost of the tiles is Rs 360 per dozen.

23. A small indoor greenhouse (herbarium) is made entirely of glass panes (including base) held
together with tape. It is 30 cm long, 25 cm wide and 25 cm high. (i) What is the area of the
glass? (ii) How much of tape is needed for all the 12 edges?

24. Shanti Sweets Stall was placing an order for making cardboard boxes for packing their sweets.
Two sizes of boxes were required. The bigger of dimensions 25 cm × 20 cm × 5 cm and the
smaller of dimensions 15 cm × 12 cm × 5 cm. For all the overlaps, 5% of the total surface area is
required extra. If the cost of the cardboard is Rs 4 for 1000 cm2, find the cost of cardboard
required for supplying 250 boxes of each kind.

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25. Parveen wanted to make a temporary shelter for her car, by making a box-like structure with
tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which
can be rolled up). Assuming that the stitching margins are very small, and therefore negligible,
how much tarpaulin would be required to make the shelter of height 2.5 m, with base dimensions
4 m × 3 m?

26. Savitri had to make a model of a cylindrical kaleidoscope for her science project. She wanted to
use chart paper to make the curved surface of the kaleidoscope. What would be the area of chart
paper required by her, if she wanted to make a kaleidoscope of length 25 cm with
a 3.5 cm radius?

27. A metal pipe is 77 cm long. The inner diameter of a cross section is 4 cm, the
outer diameter being 4.4 cm. Find its
(i) inner curved surface area,
(ii) outer curved surface area,
(iii) total surface area.

28. Find (i) the lateral or curved surface area of a closed cylindrical petrol storage tank that is 4.2 m
1
in diameter and 4.5 m high. (ii) how much steel was actually used, if of the steel actually used
12
was wasted in making the tank.

29. Find the curved surface area of a right circular cone whose slant height is 10 cm and base radius
is 7 cm.

30. The height of a cone is 16 cm and its base radius is 12 cm. Find the curved surface area and the
total surface area of the cone (Use = 3.14).

31. A corn cob shaped somewhat like a cone, has the radius of its broadest end as 2.1 cm and length
(height) as 20 cm. If each 1 cm2 of the surface of the cob carries an average of four grains, find
how many grains you would find on the entire cob.

32. In the adjoining figure you see the frame of a lampshade. It is to be covered with a decorative
cloth. The frame has a base diameter of 20 cm and height of 30 cm. A margin of 2.5 cm is to be
given for folding it over the top and bottom of the frame. Find how much cloth is required for
covering the lampshade.

33. A conical tent is 10 m high and the radius of its base is 24 m. Find (i) slant height of the tent. (ii)
cost of the canvas required to make the tent, if the cost of 1 m2 canvas is Rs 70.

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34. What length of tarpaulin 3 m wide will be required to make conical tent of height 8 m and base
radius 6 m? Assume that the extra length of material that will be required for stitching margins
and wastage in cutting is approximately 20 cm (Use = 3.14).

35. The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the
cost of white-washing its curved surface at the rate of Rs 210 per 100 m2.

36. A joker’s cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find
the area of the sheet required to make 10 such caps.

37. A hemispherical dome of a building needs to be painted. If the circumference of the base of the
dome is 17.6 m, find the cost of painting it, given the cost of painting is Rs 5 per 100 cm2.

38. A right circular cylinder just encloses a sphere of radius r. Find (i) surface area of the sphere, (ii)
curved surface area of the cylinder, (iii) ratio of the areas obtained in (i) and (ii).

39. A hemispherical bowl is made of steel, 0.25 cm thick. The inner radius of the bowl is 5 cm. Find
the outer curved surface area of the bowl.

40. A wall of length 10 m was to be built across an open ground. The height of the wall is 4 m and
thickness of the wall is 24 cm. If this wall is to be built up with bricks whose dimensions are 24
cm × 12 cm × 8 cm, how many bricks would be required?

41. A village, having a population of 4000, requires 150 litres of water per head per day. It has a tank
measuring 20 m × 15 m × 6 m. For how many days will the water of this tank last?

42. A godown measures 40 m × 25 m × 10 m. Find the maximum number of wooden crates each
measuring 1.5 m × 1.25 m × 0.5 m that can be stored in the godown.

43. A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the
new cube? Also, find the ratio between their surface areas.

44. A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will
fall into the sea in a minute?

45. The capacity of a closed cylindrical vessel of height 1 m is 15.4 litres. How many square metres
of metal sheet would be needed to make it?

46. A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior.
The diameter of the pencil is 7 mm and the diameter of the graphite is 1 mm. If the length of the
pencil is 14 cm, find the volume of the wood and that of the graphite.

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47. The pillars of a temple are cylindrically shaped. If each pillar has a circular base of radius 20 cm
and height 10 m, how much concrete mixture would be required to build 14 such

48. Monica has a piece of canvas whose area is 551 m-. She uses it to have a conical tent made, with
a base radius of 7 m. Assuming that all the stitching margins and the wastage incurred while
cutting, amounts to approximately 1 m2, find the volume of the tent that can be made with it.

49. A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. Find
the volume of the solid so obtained.

50. A heap of wheat is in the form of a cone whose diameter is 10.5 m and height is 3 m. Find its
volume. The heap is to be covered by canvas to protect it from rain. Find the area of the canvas
required.

51. A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the
cost of Rs 498.96. If the cost of white-washing is Rs 2.00 per square metre, find the (i) inside
surface area of the dome, (ii) volume of the air inside the dome.

52. Twenty seven solid iron spheres, each of radius r and surface area S are melted to form a sphere
with surface area S'. Find the (i) radius r’of the new sphere, (ii) ratio of S and S’.

53. A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine (in
mm3) is needed to fill this capsule?

54. The surface area of a sphere of radius 5 cm is five times the area of the curved surface of a cone
22
of radius 4 cm. Find the height and the volume of the cone (taking = )
7
55. The radius of a sphere is increased by 10%. Prove that the volume will be increased by 33.1%
approximately.

56. Metal spheres, each of radius 2 cm, are packed into a rectangular box of internal dimensions 16
cm × 8 cm × 8 cm. When 16 spheres are packed the box is filled with preservative liquid. Find
the volume of this liquid. Give your answer to the nearest integer. [Use = 3.14]

57. A storage tank is in the form of a cube. When it is full of water, the volume of water is 15.625
m3. If the present depth of water is 1.3 m, find the volume of water already used from the tank.

58. Find the amount of water displaced by a solid spherical ball of diameter 4.2 cm, when it is
completely immersed in water.

59. How many square metres of canvas is required for a conical tent whose height is 3.5 m and the
radius of the base is 12 m?

60. Two solid spheres made of the same metal have weights 5920 g and 740 g, respectively.
Determine the radius of the larger sphere, if the diameter of the smaller one is 5 cm.

61. A school provides milk to the students daily in a cylindrical glasses of diameter 7 cm. If the glass
is filled with milk upto an height of 12 cm, find how many litres of milk is needed to serve 1600
students.

62. A cylindrical roller 2.5 m in length, 1.75 m in radius when rolled on a road was found to cover
the area of 5500 m2. How many revolutions did it make?

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63. A small village, having a population of 5000, requires 75 litres of water per head per day. The
village has got an overhead tank of measurement 40 m × 25 m × 15 m. For how many days will
the water of this tank last?

64. A shopkeeper has one spherical laddoo of radius 5cm. With the same amount of material, how
many laddoos of radius 2.5 cm can be made?

65. A right triangle with sides 6 cm, 8 cm and 10 cm is revolved about the side 8 cm. Find the
volume and the curved surface of the solid so formed.

66. Rain water which falls on a flat rectangular surface of length 6 m and breadth 4 m is transferred
into a cylindrical vessel of internal radius 20 cm. What will be the height of water in the
cylindrical vessel if the rain fall is 1 cm. Give your answer to the nearest integer. (Take =
3.14)

67. A cylindrical tube opened at both the ends is made of iron sheet which is 2 cm thick. If the outer
diameter is 16 cm and its length is 100 cm, find how many cubic centimeters of iron has been
used in making the tube ?

68. A semi-circular sheet of metal of diameter 28cm is bent to form an open conical cup. Find the
capacity of the cup.

69. A cloth having an area of 165 m2 is shaped into the form of a conical tent of radius 5 m
5
(i) How many students can sit in the tent if a student, on an average, occupies m2 on
7
the ground?
(ii) Find the volume of the cone.

70. The water for a factory is stored in a hemispherical tank whose internal diameter is 14 m. The
tank contains 50 kilolitres of water. Water is pumped into the tank to fill to its capacity.
Calculate the volume of water pumped into the tank.

71. The volumes of the two spheres are in the ratio 64 : 27. Find the ratio of their surface areas.

72. A cube of side 4 cm contains a sphere touching its sides. Find the volume of the gap in between.

73. A sphere and a right circular cylinder of the same radius have equal volumes. By what
percentage does the diameter of the cylinder exceed its height ?

74. 30 circular plates, each of radius 14 cm and thickness 3cm are placed one above the another to
form a cylindrical solid. Find : (i) the total surface area (ii) volume of the cylinder so formed.

75. A hemispherical tank is made up of an iron sheet 1 cm thick. If the inner radius is 1 m, then find
the volume of the iron used to make the tank.

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