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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. |
Factorize:`64 a^3-27 b^3-144 a^2b+108 a b^2` |
| Answer» Correct Answer - `(4a-3b)(4a-3b)(4a-3b)` | |
| 52. |
`px-5q-5x+pq` |
| Answer» Correct Answer - `(x+q)(p-5)` | |
| 53. |
`a^3+ab(1-2a)-2b^2` |
| Answer» Correct Answer - `(a-2b)(a^2+b)` | |
| 54. |
Facorise: `125-8x^3-27y^3-90xy` |
| Answer» Correct Answer - `(5-2x-3y)(25++4x^2+9y^2+ 10x-6xy+15y)` | |
| 55. |
`8x^3-(1)/(27y^3)` |
| Answer» Correct Answer - `(2x-(1)/(3y))(4x^2+(2x)/(3y)+(1)/(9y^2))` | |
| 56. |
`8x^3+y^3+27 z^3-18 x y z` |
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Answer» We have `8x^3+y^3+27x^3-18xyz` `=(2x)^3+y^3+(3z)^3-3xx(2x)xxyxx (3z)` `=a^3+b^3+c^3-3abc, " where " 2x=a,y=b and 3z =c ` `=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)` `=(2x+y+3z)(4x^2+y^2+9z^2-2xy-3yz-6xz)`. |
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| 57. |
`8x^2y^2-x^5` |
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Answer» Correct Answer - `x^2(2y-x)(4y^2+2xy+x^2)` `8x^2y^2-x^5=x^2(8y^3-x^3)`. |
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| 58. |
factorize(a) `(3a-1)^2` - 6a + 2 |
| Answer» Correct Answer - `3(3a-1)(a-1)` | |
| 59. |
`a^2+ab(b+1)+b^3` |
| Answer» Correct Answer - `(a+b)(a+b^2)` | |
| 60. |
`(2x-3)^2-8x+12` |
| Answer» Correct Answer - `(2x-3)(2x-7)` | |
| 61. |
`ab(x^(2)+y^(2))-xy(a^(2)+b^(2))` |
| Answer» Correct Answer - `(bx-ay)(ax-by)` | |
| 62. |
`abx^2+a^2x+b^2x +ab` |
| Answer» Correct Answer - `(bx+a)(ax+b)` | |
| 63. |
If `p=2-a`; prove that `a^3+6ap+p^3-8=0` |
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Answer» We have `p=2-a` `rArr a+p-2=0` `rArr x+y+z=0, " where " a=x,y and (-2)=z` ` rArr x^3+y^3+z^3=3xyz " "[because x+y+z=0 rArr x^3+y^3+z^3=3xyz]` `rArr a^3+p^3+(-2)^3=3xx a xx pxx (-2)` `rArr a^23+6ap+p^3-8=0` Hence, `a^3+6ap+p^3-8=0`. |
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| 64. |
If `(x+1)` is a factor of the polynomial ` (2x^2+kx)` then the value of k isA. `-2`B. `-3`C. `2`D. `3` |
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Answer» Correct Answer - C Let `p(x)=2x^2+kx`. Then , `p(-1)=0`. ` therefore 2 xx (-1)^2+kxx(-1)=0rArr k=2`. |
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| 65. |
`a^3+0.008` |
| Answer» Correct Answer - `(a+0.2)(a^2-0.2a+0.004)` | |
| 66. |
`216x^3+(1)/(125)` |
| Answer» Correct Answer - `(6x+(1)/(5))(36x^2-(6x)/(5)+(1)/(25))` | |
| 67. |
Which of the following is a factor of `(x+y)^3-(x^3+y^3)?`A. `x^2+y^2+2xy`B. `x^2+y^2-xy`C. `xy^2`D. `3xy` |
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Answer» Correct Answer - D `(x+y)^3-(x^3+y^3)=3xy(x+y)`. `therefore 3xy` is a factor of `(x+y)^3-(x^3+y^3)`. |
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| 68. |
The coefficient of x in the expression of `(x+3)^3` isA. 1B. 9C. 18D. 27 |
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Answer» Correct Answer - D `(x+3)^3=x^3+3^3+3xx x xx 3 xx (x+3)=x^3+9x^2+27x+27`. So, the coefficient of x is 27. |
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| 69. |
If ` (3x+(1)/(2))(3x-(1)/(2))=9x^2-p` then the value of `p` is |
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Answer» Correct Answer - C `(3x+(1)/(2))(3x-(1)/(2))=(3x)^2-((1)/(2))^2=(9x^2-(1)/(4))`. Hence, `a=(1)/(4)`. |
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| 70. |
If `a+b+c=0` then ` (a^3+b^3+c^3)` is |
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Answer» Correct Answer - D `a+b+c=0 rArr a^3+b^3+c^3-3abc =0rArr a^3+b^3+c^3=3abc`. |
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| 71. |
If ` (x)/(y)+(y)/(x)=-1`, where` x,y ne 0` then the value of `(x^3-y^3)` isA. 1B. -1C. 0D. `(1)/(2)` |
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Answer» Correct Answer - C `(x)/(y)+(y)/(x)=-1 rArr x^2+y^2+xy=0` `rArr (x-y)(x^2-y^2+xy)=0 rArr x^3-y^3=0`. |
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| 72. |
The value of `(249)^2 -(248)^2` isA. `1^2`B. `477`C. `487`D. `497` |
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Answer» Correct Answer - D `(249)^2-(248)^2=(249-248)xx (249)+248)=1xx 497=497`. |
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| 73. |
Factroise `(a^6-b^6)` |
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Answer» We have `(a^6-b^6)` `=(a^3)^2-(b^3)^2` `=(a^3-b^3)(a^3+b^3)` `=(a-b)(a^2+ab+b^2)(a+b)(a^2-ab+b^2)` ` =(a-b) (a+b) (a^2+ab+b^2)(a^2-ab+b^2)`. `therefore (a^6-b^6)=(a-b)(a+b)(a^2+ab+b^2)(a^2-ab+b^2)`. |
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| 74. |
Factroise `(i) a^3-2sqrt(2)b^3` `(ii)1-64a^3` `(iii) 8a^3-27b^3` |
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Answer» We have `(i) (a^3-2sqrt(2)b^3)` `=(a)^3-(sqrt(2)b)^3` `=(a-sqrt(2)b)xx{a^2+(axx sqrt(2)b)+(sqrt(2)b)^2}` ` =(a-sqrt(2)b(a^2+sqrt(2)ab+2b^2)` `therefore (a^3-2sqrt(2)b^3)=(a-sqrt(2)b)(a^2+sqrt(2)ab 2b^2)`. `(ii) (1-64a^3)` `=(1)^3-(4a)^3` `=(1-4a)xx{1^2+(1xx4a)+(4a)^2} " " [because x^3-y^3=(x-y)(x^2+xy+y^2)]` `=(1-4a)(1+4a+16a^2)` `therefore (1-64a^3)=(1-4a)(1+4a+16a^2)` `(iii) (8a^3-27b^3)` `=(2a)^3-(3b)^3` `=(2a-3b)xx(4a^2+6ab+9b^2)` `therefore (8a^3-27b^3)=(2a-3b)(4a^2+6ab+9b^2)`. |
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| 75. |
`x^2-x-156` |
| Answer» Correct Answer - `(x-13)(x+12)` | |
| 76. |
`x^2-26x+133` |
| Answer» Correct Answer - `(x-19)(x-7)` | |
| 77. |
`40+3x-x^2` |
| Answer» Correct Answer - `(8-x)(5+x)` | |
| 78. |
`81x^4-y^4` |
| Answer» Correct Answer - `(3x-y)(3x+y)(9x^2+y^2)` | |
| 79. |
`6-x-x^2` |
| Answer» Correct Answer - `(2-x)(3+x)` | |
| 80. |
`sqrt(2)x^2+9x+4sqrt(2)` |
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Answer» The given expression is `sqrt(2x)^2+9x+4sqrt(2)`. Here, `sqrt(2)xx 4sqrt(2)=8` We split 9 into two parts whose sum is 9 and product 8. Clearly , `(8+1)=9 and (8xx1)=8` `therefore sqrt(2)x^2+9x+4sqrt(2)=sqrt(2)=sqrt(2)x^2+8x+x+4sqrt(2)` `=sqrt(2)x(x+4sqrt(2))+(x+4sqrt(2))` ` =(x+4sqrt(2))(sqrt(2)x +1)`. Hence, `(sqrt(2)x^2+9x+4sqrt(2))=9x+4sqrt(2))(sqrt(2)x+1)`. |
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| 81. |
Find the roots of `x^4-625`. |
| Answer» Correct Answer - `(x-5)(x+5)(x^2+25)` | |
| 82. |
`x^2-11x-80` |
| Answer» Correct Answer - `(x-16)(x+5)` | |
| 83. |
Find the roots of `x^2+11x+30` if they exist |
| Answer» Correct Answer - `(x+ 6) ( x+5)` | |
| 84. |
`a^3b-a^2 b +5ab-5b` |
| Answer» Correct Answer - `b(a-1)(x^2+5)` | |
| 85. |
Factorize`x^2+(1)/(x^2)+2-2x-(2)/(x)`. |
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Answer» `x^2+(1)/(x^2)+2-2x-(2)/(x)=(x^2+(1)/(x^2)+2)-2(x+(1)/(x))` `=(x+(1)/(x))^2-2(x+(1)/(x))` ` =(x+(1)/(x)) (x+(1)/(x)-2)` . |
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| 86. |
Factorize `3x^3-48x` |
| Answer» Correct Answer - `3x(x-4)(x+4)` | |
| 87. |
`3a^3b-243ab^3` |
| Answer» Correct Answer - `3ab(a-9b)(a+9b)` | |
| 88. |
Factorize `9x^2 -22x+8`. |
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Answer» The given expression is `9x^2-22x+8`. Here , ` 9xx 8 =72` We split -22 into two parts whose sum is -22 and product 72. Clearly, `(-18)+(-4)=-22 and (-18)xx (-4) =72` `therefore 9x^2-22x+8=9x^2-18x -4x +8` ` =9x(x-2)-4(x-2)` `=(x-2)(9x-4)`. Hence , `(9x^2-22x+8) =(x-2)(9x-4)`. |
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| 89. |
Factorize ` 1+x+y-z+xy-yz-zx-xyz` |
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Answer» We have ` 1+x+y-z+xy-yz-zx-xyz` `=(1+x+y-z+xy-yz-zx-xyz` `=(1+x)+y(1+x)-z(1+x)-yz(1+x)` ` = (1+x(1+y-z-yz)` `= 91+x)xx [(1+xy)-z(1+y)]` `(1+x)(1+y(1-z)`. |
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| 90. |
Factorize `2x^3-48x` |
| Answer» Correct Answer - `(2(x-2)(x+2)(x^2+4)` | |
| 91. |
`108a^2-3(b-c)^2` |
| Answer» Correct Answer - `3(6a+b-c)(6a-b+c)` | |
| 92. |
`20x^2-45` |
| Answer» Correct Answer - `5(2x-3)(2x+3)` | |
| 93. |
Factorize`5-20x^2` |
| Answer» Correct Answer - `5(1-2x)(1+2x)` | |
| 94. |
`9x^2+12xy` |
| Answer» Correct Answer - `3X(3X+4Y)` | |
| 95. |
Factorize `(i) (x^4+4)` `(ii) (x^2+(4)/(x^2))` `(iii) (x^4+(1)/(x^4)+1)` |
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Answer» We have `(i) (x^4+4)` `=(x^2)^2+2^2+(2xx x^2xx 2)-(2xx x^2xx2) " "["adding and subtracting" (2xx x^2xx2)]` `=(x^2+2)^2-4x^2` `=(x^2+2)^2-(2x)^2` `(x^2+2-2x)(x^2+2x+2x) =(x^2-2x+2)=(x^2-2x+2)(x^2+2x+2)` `therefore (x^4+4)=(x^2-2x+2)(x^2+2x+2)`. ` (ii) (x^2+(4)/(x^2))` `=x^2+((2)/(x))^2+(2xx x xx (2)/(x))-(2 xx x xx (2)/(x))` `=(x+(2)/(x))^2-4={(x+(2)/(x))^2-2^2}` `=(x+(2)/(x)-2)(x+(2)/(x)+2)`. ` therefore (x^2+(4)/(x^2))=(x+(2)/(x)-2)(x+(2)/(x)+2)`. `(iii) (x^4+(1)/(x^4)+1)` `=(x^4+(1)/(x^4)+2)-1=(x^2+(1)/(x^2))^2-1^2` `=(x^2+(1)/(x^2)-1){(x^2+(1)/(x^2)+2)-1}` `=(x^2+(1)/(x^2)-1){(x+(1)/(x))^2-1^2}` `=(x^2+(1)/(x^2)-1)(x+(1)/(x)-1)(x+(1)/(x)+1)` ` therefore (x^4+(1)/(x^4)+1)=(x^2+(1)/(x^2)-1)(x+(1)/(x)-1)(x+(1)/(x)=1)` |
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| 96. |
`a^2+2ab +b^2-9c^2` |
| Answer» Correct Answer - `(a+b+3c)(a+b-3c)` | |
| 97. |
Factorize `(2x-3)^2-8x+12`. |
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Answer» We have `(2x+3)^2-8x+12=(2x-3)^2-4(2x-3)` `=(2x-3)xx[(2x-3)-4]` `=(2x-3)(2x-7)`. |
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| 98. |
`18x^2y-24xyz` |
| Answer» Correct Answer - `6xy(3x-4z)` | |
| 99. |
`x^2+20x-69` |
| Answer» Correct Answer - `(x+23)(x-3)` | |
| 100. |
Factroize `(i) x^2-1-2a-a^2` `(ii) 1+2ab-(a^2+b^2)` |
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Answer» We have `(i) x^2-1-2a-a^2` ` =x^2-(1+2a+a^2)` `=x^2-(1+a)^2` `{x-(1+a)}{x+(1+a)}` `=(x-1-a)(x+1+a)`. `therefore (x^2-1-2a-a^2)=(x-1-a)(x+1+a)`. `(ii) 1+2ab-(a^2+b^2)` ` =1-(a^2+b^2-2ab)` ` =1^2-(a-b)^2` `={1-(a-b)}{(1+(a-b)}` `(1-a+b)(1+a-b)`. `therefore 1+2ab-(a^2+b^2)=(1-a+b)(1+a-b)`. |
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