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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 51. |
(a) Find the HCF of `28x^(4)` and `70x^(6)`. (b) Find the HCF of `48x^(2) (x + 3)^(2) (2x-1)^(3) (x + 1)` and `60x^(3) (x+ 3) (2x-1)^(2) (x + 2)`. |
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Answer» (a) Find f(x) = `28x^(4)` and g(x) = `70 x^(6)`. Writing f(x) and g(x) as a product of powers of irreducible factors , we have f(x) = `2^(2) xx 7 xx x^(4)` and g(x) = `2 xx 5 xx 7 xx x^(6)`. `therefore` The common factors with the least exponents are 2 , 7 and `x^(4)`. `implies` HCF = `2 xx 7 xx x^(4) = 14 x^(4)` (b) Let f(x) = `48x^(2) (x + 3)^(2) (2x - 1)^(3) (x + 1)` and g(x) = `60 x^(3) (x + 3) (2x - 1)^(2) (x + 2)`. Writing f(x) and g(x) as the product of the powers of irreducible factors , we have f(x) = `2^(4) xx 3 xx x^(2) (x + 3)^(2) (2x - 1)^(3) (x + 1)` g (x) = `2^(2) xx 3 xx 5 xx x^(3) (x + 3) (2x - 1)^(2) (x + 2)`. The common factors with least exponents are `2^(2) , 3 , x^(2) , x + 3 ` and `(2x-1)^(2)`. `therefore` HCF = `2^(2) xx 3 xx x^(2) (x + 3) (2x - 1)^(2) = 12 x^(2) (x + 3) (2x -1)^(2)`. |
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| 52. |
By using an appropriate identity , find the following : `(10(1)/(4))^(2)` |
| Answer» Correct Answer - `105 (1)/(16)` | |
| 53. |
Find the HCF of the following : 50 ab and 60 bc |
| Answer» Correct Answer - 10 b | |
| 54. |
HCF of `12x^(2)yz` and `16xyz^(2)` is 4xyz. |
| Answer» Correct Answer - True | |
| 55. |
Number of terms in the expresion (a + b) (c + d) is `"______"`.A. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - D `(a +b) (c + d) = ac + ad + bc + bd` `therefore` There are 4 terms . Hence , the correct option is (d) . |
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| 56. |
`sqrt((81(x + y)^(2))/(144(x-y)^(2))) = "_________"`A. `(9(x+y)^(sqrt2))/(12(x-y)^(sqrt2))`B. `(9(x+y)^(2))/(12(x-y)^(2))`C. `(3(x +y))/(4(x-y)) `D. `9(x+y) //12(x-y)` |
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Answer» Correct Answer - C `sqrt(([9(x + y)^(2)])/([12 (x-y)^(2)]))` `= (9(x + y))/(12(x-y))` `= (3(x+y))/(4(x-y))` Hence , the correct option is (c) . |
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| 57. |
Find the HCF of `24abc^(3) , 36 ab^(3)c`, and `48 a^(3)bc`. |
| Answer» Correct Answer - 12 abc | |
| 58. |
Factorise `12a^(3) b^(3) - 3ab`.A. `3(a^(2) b^(2) - 1)`B. `3ab(a-b)`C. `3ab(2ab + 1) (2ab -1)`D. `3ab(a + 1) (b-1)` |
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Answer» Correct Answer - C `12a^(3)b^(3) - 3ab = 3ab (4a^(2) b^(2) -1)` = `(3ab) ((2ab)^(2) - 1^(2))` `=3ab(2ab + 1) (2ab -1)`. Hence , the correct option is (c). |
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| 59. |
If `y - (1)/(y) = 3` , then find `y^(4) + (1)/(y^(4))` .A. 119B. 117C. 123D. 125 |
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Answer» Correct Answer - A `y + (1)/(y) = 3` Squaring on both sides , we get `y^(2) + (1)/(y^(2)) - 2 = 9 (because (a - b)^(2) = a^(2) + b^(2) - 2ab)` `y^(2) + (1)/(y^(2)) = 11` Again squaring on both the sides , we get `y^(4) + (1)/(y^(4)) + 2 = 121` `y^(4) + (1)/(y^(4)) = 119`. Hence , the correct option is (a) . |
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| 60. |
`(79^(2) - 9^(2))/(89^(2) - 9^(2)) = "_________"`A. `(5)/(8)`B. `(13)/(15)`C. `(6)/(9)`D. `(11)/(14)` |
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Answer» Correct Answer - D `((79 + 9) (79-9))/((89 + 9) (89-9))` `= (88 xx 70)/(98 xx 80) = (11)/(14)` Hence , the correct option is (d) . |
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| 61. |
By using an appropriate identity , find the following : `(28)^(2)` |
| Answer» Correct Answer - 784 | |
| 62. |
By using an appropriate identity , find the following : `(95)^(2)` |
| Answer» Correct Answer - 9025 | |
| 63. |
By using an appropriate identity , find the following : `(9(1)/(2))^(2)` |
| Answer» Correct Answer - `90 (1)/(4)` | |
| 64. |
Find the HCF of the following : `26x^(2)y^(2)z^(2)` and `39x^(3)y^(2)z` |
| Answer» Correct Answer - `13 x^(2) y^(2) z` | |
| 65. |
If `64x^(2) + (1)/(64 x^(2)) = 20` , then `8x - (1)/(4x)` can be `"_______"`A. 2B. 8C. 1D. 4 |
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Answer» Correct Answer - D `64x^(2) + (1)/(16x^(2)) = 20` `(8x)^(2) + ((1)/(4x))^(2) = 20` `(8x)^(2) + ((1)/(4x))^(2) - 2(8x)((1)/(4x)) + 2(8x) ((1)/(4x)) = 20` `(8x - (1)/(4x))^(2) + 4 = 20` `(because (p-q)^(2) = p^(2) + q^(2) - 2pq)` `(8x - (1)/(4x))^(2) = 16` `8x - (1)/(4x) = 4`. Hence , the correct option is (d) . |
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| 66. |
The HCF of `64x^(6) y^(4) , 48 x^(4)y^(8)` , and `54x^(5)y^(4` is `"_________"`.A. `2x^(4) y^(4)`B. `6x^(2)y^(2)`C. `8x^(4)y^(4)`D. `8x^(2)y` |
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Answer» Correct Answer - A `64x^(6) y^(4) = 2^(6)., x^(6)., y^(4)` `48x^(4)y^(8) = 2^(4) xx 3.x^(4).y^(8)` `54x^(5)y^(4) = 2 xx 3^(3).x^(5).y^(4)` HCF = `2x^(4) y^(4)` Hence , the correct option is (a) . |
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| 67. |
Divide `x^(2) + 6x + 8` by x + 2 and find the quotient . |
| Answer» Correct Answer - x + 4 | |
| 68. |
Factorise `x^(4) + x^(3) - x^(2) - x`.A. `(x^(2) + 1) (x -1) x`B. `x (x+1)^(2) (x - 1) `C. `(x^(2) +1) (x -1)^(2)`D. `x(x-1)^(2) (x + 1)` |
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Answer» Correct Answer - B `x^(4) + x^(3) - x^(2) - x` `= x^(3) (x +1) - x( x +1)` = `(x + 1) (x^(3) -x)` = `(x +1) (x) (x^(2) - 1)` `= x(x +1) (x +1) (x-1)` = `x (x-1) (x +1)^(2)` Hence , the correct option is (b) . |
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| 69. |
By using an appropriate identity , find the following : `92 xx 108` |
| Answer» Correct Answer - 9936 | |
| 70. |
By using an appropriate identity , find the following : `89 xx 111` |
| Answer» Correct Answer - 9879 | |
| 71. |
If `x + (1)/(2x) = 4`, then find `x^(2) + (1)/(4x^(2))` . |
| Answer» Correct Answer - 15 | |
| 72. |
Check whether the following are perfect squares or not : (a) `64x^(2) + 81y^(2) - 144 xy` (b) `4x^(2) + 8y^(2) + 16 xy` |
| Answer» Correct Answer - (a) Perfect square (b) Not perfect square | |
| 73. |
Find the product of (3x - 7) (4x + 6) and verify it when x = 1. |
| Answer» Correct Answer - `12 x^(2) - 10x - 42` | |
| 74. |
`(a^(4) - a^(2)) + (a^(3) + a^(2)) = "_______"`.A. `a^(2) + 1`B. `a - 1`C. `a + 1`D. `a^(2) - 1` |
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Answer» Correct Answer - B `(a^(4) - a^(2)) div (a^(3) + a^(2))` = `a^(2) (a^(2) -1) div a^(2) (a+1)` `=(a^(2) (a+1)(a-1))/(a^(2) (a+1)) = a - 1` Hence , the correct option is (b) . |
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| 75. |
`46 xx 46 + 54 xx 54 + 2 xx 46 xx 54 = "_______"`A. 9996B. 10004C. 9800D. 10000 |
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Answer» Correct Answer - D |
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| 76. |
`{:("Column" A ,, "Column" B), (225x^(2) - 625 y^(2) = ,, (a) 25(x-2) (x-2)), (x^(2) - x - y - y^(2) = ,, (b) 25(3x- 5y) (3x + 5y)), (x^(2) - x - y^(2) + y = ,, (x + y) (x - y- 1)), (25x^(2) - 100 x + 100 = ,, (d) (x - y) (x + y -1)), (,,(e) (x + y) (x + y - 1)):}` |
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Answer» Correct Answer - A::B::C::D `to` (b) : `225 x^(2) - 625 y^(2) = (15x)^(2) - (25y)^(2)` = (15x + 25y) (15x - 25y) =`25 (3x + 5y) (3x - 5y)` `to` (c) : `x^(2) - x- y - y^(2) = (x^(2) - y^(2)) - x- y` = `(x + y) (x-y) - 1(x+y)` `= (x + y) (x-y-1)` `to` (d) : `x^(2) - x - y^(2) + y = x^(2) - y^(2) - x + y =` `= (x + y) (x - y) - (x-y)` = `(x-y) ( x+ y -1)` `to` (a) : `(5x)^(2) - 2(5x)(10) + (10)^(2) = (5x - 10)^(2)` `= 25 (x -2)(x-2) ` |
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| 77. |
If `x - (1)/(2x) = 2` , then find the value of `(4x^(4) + 1)/(4x^(2))`A. 10B. 14C. 5D. 7 |
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Answer» Correct Answer - C `x - (1)/(2x) = 2` `x^(2) + (1)/(4x^(2)) - 2.x. (1)/(2x) = 4` `x^(2) + (1)/(4x^(2)) = 4 + 1` `(4x^(2) + 1)/(4x^(2)) = 5` Hence , the correct option is (c) . |
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| 78. |
Find the square root of `4^(6n^(2)) (25)^((a)/(2)) 9^(b^(4))` .A. `4^(3n^(2)) 5^((a)/(4)) 3^(b^(2))`B. `2^(6n^(2)) 5^((a)/(2)) 3^(b^(2))`C. `2^(6n^(2)) 5^((a)/(2)) 3^(b^(4)) `D. `4^(3n) 5^(a) 3^(b^(2))` |
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Answer» Correct Answer - C `sqrt(4^(6n^(2)) . (25)^((n)/(2)).9^(b^(4)))` `=[4^(6n^(2)) (25)^((a)/(2)).9^(b^(4))]^((1)/(2))` `=[2^(2 xx 6n^(2)) 5^(2 xx (a)/(2)) 3^(2xx b^(4))]^((1)/(2))` =` 2^(6n^(2)) . 5^((a)/(2)). 3^(b ^(4)) ` ltbr. Hence , the correct option is (c) . |
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| 79. |
`{:("Column" A ,, "Column" B) , ((a+b)^(2) = ,, (a) (x + 1)^(2) ( x-1)), (x^(3) - x^(2) - x + 1 = ,, (b) (a-b) + 4ab), ((a +b) (a^(2) - ab + b^(2)) = ,, (a^(3) + b^(3))), ((x + 1) (x^(2) - 1) = ,, (d) (x -1)^(2) (x + 1)), (,, (e) (a^(3) - b^(3))):}` |
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Answer» Correct Answer - b , d , c , a `to` (b) : `(a -b)^(2) + 4ab = (a + b)^(2)` `to` (d) : `x^(3) - x^(2) -x + 1 = x^(2) ( x - 1) - 1( x - 1)` `= (x -1) (x^(2) - 1)` = `(x - 1) (x-1) ( x +1)` ` (x-1)^(2) (x +1)` `to` (c) , `(a + b) (a^(2) - ab + b^(2)) = a^(3) + b^(3)` `to` (a) : `(x + 1) (x^(2) - 1) = (x +1) (x + 1) (x -1) = (x + 1)^(2) (x-1)` |
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| 80. |
`(a^(2))/(2) + (b^(3))/(3) - (3c^(3))/(4) + (a^(2))/(3) - (3b^(3))/(4) + (c^(2))/(2) - (3a^(2))/(4) + (b^(3))/(2) + (c^(3))/(3) = "______"`A. `(a^(2) + b^(3) + c^(3))/(12)`B. `(a^(2) + b^(3) - c^(3))/(6)`C. `(6a^(2) + 9b^(3) - 12c^(3))/(24)`D. `(6a^(2) + 9b^(3) - 12c^(3))/(12)` |
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Answer» Correct Answer - A `(a^(2))/(2) + (b^(3))/(3) - (3c^(3))/(4) + (a^(2))/(3) - (3b^(3))/(4) + (c^(3))/(2) - (3a^(2))/(4) + (b^(3))/(2) + (c^(3))/(3)` `= (a^(2))/(2) + (a^(2))/(3) - (3a^(2))/(4) + (b^(3))/(3)` `- (3b^(3))/(4) + (b^(3))/(2) - (3c^(3))/(4) + (c^(3))/(2) + (c^(3))/(3)` `(6a^(2) + 4a^(2) - 9a^(2))/(12) + (4b^(3) - 9b^(3) + 6b^(3))/(12) + (-9c^(3) + 6c^(3) + 4c^(3))/(12)` ` = (a^(2) + b^(3) + c^(3))/(12)` Hence , the correct option is (a) . |
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| 81. |
If `(x + (1)/(x)) = 2` , then `x^(2) + (1)/(x^(2)) = "_________"`A. 2B. 4C. 0D. 3 |
| Answer» Correct Answer - A | |
| 82. |
`(a + 3b)^(2) = "_________"` |
| Answer» Correct Answer - `a^(2) + 6ab + 9b^(2)` | |
| 83. |
`x^(2) - y^(2) = "_________"` |
| Answer» Correct Answer - `(x + y) (x - y)` | |
| 84. |
(x + a) (x + b) = `"_________"` |
| Answer» Correct Answer - `x^(2) + (a + b)x + ab ` | |
| 85. |
`(3x - 4y)^(2) - (4x + 3y)^(2) = "______"`.A. (7x - y) (7y-x)B. (x- 7y) (7x + y)C. (x + 7y) (7x - y)D. (y-7x ) (x + 7y) |
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Answer» Correct Answer - D `(3x - 4y)^(2) - (4x + 3y)^(2)` `= (3x - 4y + 4x + 3y) (3x - 4y - 4x -3y)` `(because (a^(2) - b^(2)) = (a + b) (a-b))` = `(7x - 7y) (-7y - x)` = `(y-7x) (x + 7y)` Hence , the correct option is (d) . |
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| 86. |
If `x + (1)/(x) = 2` then `x^(2010) + x^(2009) = "_________"`.A. 4019B. 2C. 0D. 1 |
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Answer» Correct Answer - B `x + (1)/(x) = 2` `implies (x^(2) + 1)/(x) = 2` `implies x^(2) - 2x + 1 = 0` `(x-1)^(2) = 0` `x - 1 = 0 implies x = 1 ` `therefore x^(2010) + x^(2009) = 1 + 1 = 2` Hence , the correct option is (b) . |
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| 87. |
If `x^(2) + 8x+ k` is a perfect square , then find the value of k .A. 1B. 4C. 16D. 64 |
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Answer» Correct Answer - C `x^(2) + 8x + k = (x)^(2) + 2(x) (4) + k` is a perfect square . `implies k = 4^(2) = 16` Hence , the correct option is (c) . |
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| 88. |
`{:( "Column" A ,, "Column" B) , ("If" a^(2) - b^(2) = 16 "and" a- b = 2 "," "then" a + b,, "(a)" 3x^(2) - x - 11), ("The degree of " (x-a) (x-b) (x - c) (x-d) ,, "(b)" 3x^(2) - x + 5), (4x^(2) + 20xy + 25y^(2) ,, (c) 8), ((5x^(2) + 7x -3)- (2x^(2) + 8x -8) ,, (d) 4), (,, (e) (4x + 5y) (x+ 5y)), (,,(f) (2x + 5y)^(2)):}` |
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Answer» Correct Answer - c, d, f, b `to (c) : a + b = (a^(2) - b^(2))/(a-b) = (16)/(2) = 8` `to (d)` : The degree of (x -a) (x -b) (x-c) (x -d) is 4 . `to ` (f) : `4x^(2) + 20xy + 25y^(2)` = `(2x)^(2) + 2(2x) (5y) + (5y)^(2)` = `(2x + 5y)^(2)` `to` (b) : `(5x^(2) + 7x - 3) - (2x^(2) + 8x - 8)` = `5x^(2) - 2x^(2) + 7x - 8x - 3 + 8` =`3x^(2) - x + 5` |
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| 89. |
The following steps are involved in finding the value of `a^(4) + (1)/(a^(4))` when `a + (1)/(a) = 1` . Arrange them in sequential order from the first to the last . (A) `a^(2) + (1)/(a^(2)) + 2 = 1 implies a^(2) + (1)/(a^(2)) = -1` (B) `(a^(2))^(2) + ((1)/(a^(2))^(2))^(2) = 1^(2)` (C) `(a + (1)/(a))^(2) = 1^(2)` (D) `(a^(2) + (1)/(a^(2)))^(2) = (-1)^(2)` E `a^(4) + (1)/(a^(4)) = -1`A. CADBEB. CDBAEC. CBADED. CEDAB |
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Answer» Correct Answer - A (C) , (A) , (D) , (B) and (E) is the sequential order from the first to the last . Hence , the correct option is (a) . |
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| 90. |
Find the degree of `(x^(2) - x)^(2)`.A. 3B. 4C. 5D. 6 |
| Answer» Correct Answer - B | |
| 91. |
The degree of `4x^(2)` isA. 1B. 2C. 3D. 4 |
| Answer» Correct Answer - B | |
| 92. |
The degree of `-3x^(3) + 5x^(2) + 4` is `"________"`.A. 1B. 2C. 3D. 4 |
| Answer» Correct Answer - C | |
| 93. |
The following steps are involved in the factorisation of `x^(2) (x - y) + (y-x) y^(2)`. Arrange them in sequential order from the first to the last . (A) `(x-y) [x^(2) - y^(2)]` (B) `x^(2)(x-y) - ( x- y) y^(2)` (C) `(x-y)^(2) (x + y)` (D) `(x-y) [(x + y)( x-y)]`A. ABDCB. BADCC. ABCDD. CDAB |
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Answer» Correct Answer - B (B) , (A) , (D) and (C) is the sequential order from the first to the last . Hence , the correct option is (b) . |
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| 94. |
The following steps are involved in finding the value of `100 (3)/(4) xx 99(1)/(4)` by using appropriate identify . Arrange them in sequential order . (A) `(100 + (3)/(4)) (100 - (3)/(4)) = (100)^(2) - ((3)/(4))^(2) [ because (a + b) (a - b) = a^(2) - b^(2)]` (B) `10000 - (9)/(16) = 9999 + 1 - (9)/(16) = 9999 (7)/(16)` (C) `(100)^(2) - ((3)/(4))^(2) = 10000 - (9)/(16)` (D) `100 (3)/(4) xx 99 (1)/(4) = (100 + (3)/(4)) (100 - (3)/(4))`A. DABCB. DACBC. DBACD. DCAB |
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Answer» Correct Answer - B (D) , (A) , (C) and (B) is the required sequential order . Hence , the correct option is (b) . |
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| 95. |
Factorise `a^(3) - 3b^(2) + 3a^(2) - ab^(2)` The following steps are involved in solving the above problem . Arrange them in sequential order . (A) `(a + 3) (a^(2) - b^(2))` (B) Rearrange the terms as `a^(3) + 3a^(2) - 3b^(2) - ab^(2)` (C) `a^(2) (a + 3) - b^(2) ( 3 + a)` (D) `(a + 3) (a + b) ( a-b)`A. BCDAB. BCADC. BDCAD. ABCD |
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Answer» Correct Answer - B (B) , (C) , (A) and (D) is the required sequential order. Hence , the correct option is (b) . |
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| 96. |
The following steps are involved in finding the value of `(97)^(2)` by using a suitable identify . Arrange them in sequential order from the first to the last . (A) 10000 - 600 + 9 = 9409 (B) Using the identify `(a - b)^(2) = a^(2) - 2ab + b^(2)`, where a = 100 and b = 3. (C) `(97)^(2) = (100 - 3)^(2)` (D) `(100-3)^(2) = (100)^(2) - 2 (100)(3) + 3^(2)`A. CABDB. CADBC. CBADD. CBDA |
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Answer» Correct Answer - D (C) , (B) , (D) and (A) is the sequential order from the first to the last . Hence the correct option is (d) . |
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| 97. |
The following steps are involved in finding the value of `10 (1)/(3) xx 9(2)/(3)` by using an appropriate indentity . Arrange them in sequential order . (A) `(10)^(2) - ((1)/(3))^(2) = 100 - (1)/(9)` (B) `10(1)/(3) xx 9(2)/(3) = (10 + (1)/(3)) (10 - (1)/(3))` (C) `(10 + (1)/(3)) (10 - (1)/(3)) = (10)^(2) - ((1)/(3))^(2) [because (a + b) (a -b) = (a^(2) - b^(2))]` (D) `100 - (1)/(9) = 99 + 1 - (1)/(9) = 99(8)/(9)`A. BCDAB. BACDC. ACBDD. BCAD |
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Answer» Correct Answer - D (B) , (C), (A) and (D) is the required sequential order . Hence , the correct option is (d) . |
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| 98. |
The degree of x + y + z is `"_______"`.A. 1B. 2C. 3D. 4 |
| Answer» Correct Answer - A | |
| 99. |
The degree of xyz is `"_______"`.A. 1B. 2C. 3D. 4 |
| Answer» Correct Answer - C | |
| 100. |
The following steps are involved in expanding `(x+ 3y)^(2)` . Arrange them in sequential order from the first to the last . (A) `(x + 3y)^(2) = x^(2) + 6xy + 9y^(2)` (B) `(x + 3y)^(2) = (x)^(2) + 2(x)(3y) + (3y)^(2)` (C) Using the identify `(a + b)^(2) = a^(2) + 2ab + b^(2)` , where a = x and b = 3y.A. ACBB. CABC. CBAD. ABC |
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Answer» Correct Answer - C (C) ,(B) and (A) is the sequential order from the first to the last . Hence , the correct option is (c) . |
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