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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.
| 1. |
`9 x^2 -16 y^2` |
| Answer» Correct Answer - `(3x-4y)(3x+4y)` | |
| 2. |
Factorize:`x^2+(12)/(35)x+1/(35)` |
| Answer» Correct Answer - `(5x+1)((x)/(5)+(1)/(35))` | |
| 3. |
`3/2x^2+16 x+10` |
| Answer» Correct Answer - `((x)/(2)+5)(3x+2)` | |
| 4. |
If `a+b+c=0` then `((a^2)/(bc)+(b^2)/(ca)+(c^2)/(ab))=?`A. 1B. 0C. -1D. 3 |
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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`. Now , `((a^2)/(bc)+(b^2)/(ca)+(c^2)/(ab))=(a^3+b^3+c^3)/(abc)=(3abc)/(abc)=3`. |
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| 5. |
`a(a+b-c) - bc` |
| Answer» Correct Answer - `(a-c)(a+b)` | |
| 6. |
Factorise `a^3-b^3+1+3ab`. |
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Answer» We have `a^3-b^3+1+3ab` `=a^3+(-b)^3+(1)^3-3xx a xx (-b)xx 1` `=x^3+y^3+z^3-3xyz, " where" a =x, (-b) =y and 1=z` `=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)` `=(a-b+1)(a^2+b^2+1+ab+b-a)` `=(a-b+1)(a^2+b^2+ab-a+b1)`. |
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| 7. |
Factorize:`21 x^2-2x+1/(21)` |
| Answer» Correct Answer - `(x-(1)/(21))(21x-1)` | |
| 8. |
Factorize each of the following algebraic expressions :`x^8-y^8`(ii) `a^(12)x^4-a^4x^(12)` |
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Answer» We have `(x^8-y^8) =(x^4)-(y^4)^2` `=(x^4-y^4)(x^4+y^4)` `={(x^2)^2-(y^2)^2}(x^4+y^4+2x^2y^2-2x^2y^2)`. `=(x^2-y^2)(x^2+y^2)[(x^2+y^2)^2-(sqrt(2)xy)^2]` `=(x-y)(x+y)(x^2+y^2) (x^2+y^2-sqrt(2)xy)(x^2+y^2+qrt(2)xy)`. `therefore (x^8-y^8)=(x-y)(x+y)(x^2+y^2)(x^2+y^2-x sqrt(2)xy)(x^2+y^2+sqrt(2)xy)`. |
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| 9. |
Factorize `x^2+5x-24`. |
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Answer» The given expression is `x^2+5x-24`. We try to spilt 5 into two parts whose sum is 5 and product -24. Clearly , ` 8+(-3)=5 and 8 xx (-3)=-24`. `therefore x^2+5x-24=x^2+8x-3x-24` `=x(x+8)-3(x+8)` `=(x+8)(x-3)` Hence, `x^2 5x-24=(x+8)(x-3)`. |
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| 10. |
If `a+b+c=5` and `ab+bc+ca=10`, then prove that `a^3+b^3+c^3-3abc=-25`. |
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Answer» We have `(a^3+b^3+c^3-3abc)=(a+b+c)(a62+b^2+c^2-ab-bc-ca)` `=(a+b+c)[(a+b+c)^2-3(ab+bc+ca)]` `=5xx [(5)^2-(3xx10)]` `=5xx(25-30)=5xx(-5)=-25` Hence , `(a^3+b^3+c^3-3abc)=-25`. |
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| 11. |
Factorise `(p-q)^3+(q-r)^3+(r-p)^3`. |
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Answer» Putting `(p-q)=x,(q-r)=y and (r-p)=z`, we get `(p-q)^3+(q-r)^3+(r-p)^3` `=x^3+y^3+z^3, " where " (x+y+z)=(p-q)+(q-r)+(r-p)=0` `=3xyz " "[because x+y+z=0 rArr x^3+y^3+z^3=3xyz]` `=3(p-q)(q-r)(r-p)`. |
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| 12. |
Factorize `x^2-4x-21`. |
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Answer» The given expression is `x^2-4x-21`. We split -4 into two whose sum is -4 and product -21. ` therefore x^2-4x-21 =x^2-7x+3x-21`. `=x(x-7)+3(x-7)` `=(x-7)(x+3)`. Hence, `x^2-4x-21=(x-7)(x+3)`. |
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| 13. |
Factorize : `(x-2y)^3+(2y-3z)^3+(3z-x)^3` |
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Answer» Putting `(x-2y)=a,(2y-3z)=b and (3z-x)=c`, we get `(x-2y)^3+(2y-3z)^3+(3z-x)^3` `=a^2+b^3+c^3, " where " a+b+c=(x-2y)+(2y-3z)+(3z-x)=0` `=3abc [because a+b+c=0rArr a^3+b^3+c^3=3abc]` `=3(x-2y)(2y-3z)(3z-x)`. `therefore (x-2y)^3+(2y-3z)^3+(3z-x)^3=3(x-2y)(2y-3z)(3z-x)` |
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| 14. |
Factorize `(i) x^2+5sqrt(3)x +12` `(ii) x^2+3sqrt(3)x-30` |
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Answer» (i) The given expression is `x^2+5sqrt(3)x+12`. We split `5sqrt(3)` into two parts whose sum is `5sqrt(3)` and product 12. Clearly, `(4sqrt(3)+sqrt(3))=5sqrt(3) and (4sqrt(3)xx sqrt(3))=12`. `therefore x^2+5sqrt(3)x+12=x^2+4sqrt(3)x+sqrt(3)x +12` `=x(x+4sqrt(3))+sqrt(3)(x+4sqrt(3))` `=(x+4sqrt(3))(x+sqrt(3))` Hence, `x^2+5sqrt(3)x+12 =(x+4sqrt(3))(x+sqrt(3))`. (ii) The given expression is `x^2+3sqrt(3)x-30`. We split `3sqrt(3)` into two whose sum is `3sqrt(3)` and product `-30`. Clearly, `(5sqrt(3)-2sqrt(3))=3sqrt(3) and 5sqrt(3)xx {-2sqrt(3)}=-30`. `therefore x^2+3sqrt(3)x-30=x^2+5sqrt(3)x-2sqrt(3)x-30` `=x(x+5sqrt(3))-2sqrt(3)(x+5sqrt(3))` `=(x+5sqrt(3))(x-2sqrt(3))`. Hence, `x^2+3sqrt(3)x-30 =(x+5sqrt(3))(x-2sqrt(3))`. |
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| 15. |
`x^2+5sqrt(5)x+30` |
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Answer» Correct Answer - `(x+3sqrt(5))(x+2sqrt(5))` `x^2+5sqrt(5)x+30=x^2+3sqrt(5)x+2sqrt(5)x+30`. |
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| 16. |
Find the product:`(2x-y+3z)(4x^2+y^2+9z^2+2x y+3y z-6x z)` |
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Answer» Putting `2x=a,y=b and, 3z=c`, we get `(2x-y+3z)(4x^2+y^2+9z^2+2xy+3yz-6xz)` `=[2x+(-y)+(3z)]xx[(2x)^2+(-y)^2+(3z)^2-(2x)(-y)-(-y)(3z)-(2x)(3z)]` `=(a+b+c)xx(a^2+b^2+c^2-ab-bc-ca)` `=a^3+b^3+c^3-3abc` `=(2x)^3+(-y)^3+(3z)^3-3xx(2x)xx (-y)xx(3z)` `=8x^-y^3+27z^3+18zyz`. |
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| 17. |
`x^2-4x+3` |
| Answer» Correct Answer - `(x-3)(x-1)` | |
| 18. |
Factorize of theexpression:`5sqrt(5) x^2+30 x+8sqrt(5)` |
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Answer» The given expression is ` 5sqrt(5)x^2+30x+8sqrt(5)`. Here, `(5sqrt(5)xx 8sqrt(5))=200`. So, we split 30 into two parts whose sum is 30 and product 200. Clearly, `(20+10) =30 and (20xx 10) =200` `therefore 5sqrt(5)x^2+30x+8sqrt(5)=5sqrt(5)x^2+20x^2+20x+10x+8sqrt(5)`. `=5x(sqrt(5)x+4)+2sqrt(5)(sqrt(5)x+4)` `=(sqrt(5)x+4)(5x+2sqrt(5))`. Hence, `(5sqrt(5)x^2+30x+8sqrt(5))=(sqrt(5)x +4)(5x+2sqrt(5))`. |
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| 19. |
`10x^2-9x-7` |
| Answer» Correct Answer - `(5x-7)(2x+1)` | |
| 20. |
`25x^2-10x+1-36y^2` |
| Answer» Correct Answer - `(5x-1+6y)(5x-1-6y)` | |
| 21. |
If `(x+2) and (x-1) ` are factors of `(x^3+10x^2+mx+n)` thenA. `m=5,n=-3`B. `m=7,n=-18`C. `m=17,n=-8`D. `m=23,n=-19` |
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Answer» Correct Answer - B Let ` (x)=x^3+10x^2+mx+n`. Then, `p(-2)=0 and p(1)=0`. `(-2)^3+10xx (-2)^2+mxx (-2)+n=0rArr n-2m=32` `(1)^3+10xx (1)62+m xx 1 +n=0rArr n+m=-11......(i)` `therefore m=7and n=-18 ..........(ii)` |
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| 22. |
Factorize:`3sqrt(3)a^3-b^3-5sqrt(5)c^3-3sqrt(15)a b c` |
| Answer» Correct Answer - `(sqrt(a-b-sqrt(5)c)(3a^2+b^2+5c^2+sqrt(3)ab-sqrt(5)bc+sqrt(15)ac)` | |
| 23. |
Factorize:`a^6+b^6` |
| Answer» Correct Answer - `(a^2+b^2)(a^4-a^2b^2+b^4)` | |
| 24. |
`27a^3+64b^3` |
| Answer» Correct Answer - `(3a+4b)(9a^2-12ab+16b^2)` | |
| 25. |
Facorise: `a^3+8b^3+64c^3-24abc` |
| Answer» Correct Answer - `(a+2b+4c)(a^2+4b^2+16c^2-2ab-8bc-4ca)` | |
| 26. |
`4x^2-9y^2-2x-3y` |
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Answer» Correct Answer - `(2x+3y)(2x-3y-1)` `4x^2-9y^2-2x-3y=(4x^2-9y^2)-(2x+3y)`. |
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| 27. |
`x^2+(1)/(x^2)-3` |
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Answer» Correct Answer - `(x-(1)/(x)+1)(x-(1)/(x)-1)` `x^2+(1)/(x^2)-3=(x^2+(1)/(x^2)-2)-1=(x-(1)/(x))^2-1^2`. |
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| 28. |
`6x^2+17x+12` |
| Answer» Correct Answer - `(2x+3)(3x+4)` | |
| 29. |
Factorize`150-6x^2` |
| Answer» Correct Answer - `6(5-x)(5+x)` | |
| 30. |
`sqrt(3)x^2+10x+8sqrt(3)` |
| Answer» Correct Answer - `(x+2sqrt(3))(sqrt(3)x+4)` | |
| 31. |
`4(a+b)-6(a+b)^2` |
| Answer» Correct Answer - `2(a+b)(2-3a-3b)` | |
| 32. |
Expand the following `(i) (3a-2b)^(3) (ii) ((1)/(x)+(y)/(3))^(3)` (iii) `(4-(1)/(3x))^(2)` |
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Answer» (i) Putting `(1)/(x)=a and (y)/(3)=b`, we get `((1)/(x)+(y)/(3))^3=(a+b)^3` ` =a^3+b^3+3ab(a+b)` `=((1)/(x))^3+((y)/(3))^3+3xx(1)/(x)xx(y)/(3)xx((1)/((x)+(y)/(3)))` `=(1)/(x^3)+(y^3)/(27)+(y)/(x)((1)/(x)+(y)/(3))` `=(1)/(x^3)+(y^3)/(27)+(y)/(x^2)+(y^2)/(3x)=(1)/(x^3)+(y)/(x^2)+(y^2)/(3x)+(y^3)/(27)`. (ii) Putting `4=a and (1)/(3x)=b`, we get `(4-(1)/(3x))^2=(a-b)^3` `a^3-b^3-3ab(a-b)` `=(4)^3-((1)/(3x))^3-3xx4xx(1)/(3x)xx(4-(1)/(3x))` `=64-(1)/(27x^3)-(16)/(x)+(4)/(3x^2)=64-(16)/(x)+(4)/(3x^2)-(1)/(27x^3)` |
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| 33. |
Expand (i)` (4a+5b)^3` (ii) ` (3a-2b)^3` |
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Answer» (i) Putting `4a=x,and 5b=y`, we get ` (4a +5b)^3=(x+y)^3` `=x^3+y^3xy(x+y)` `=(4a)^3+(5b)^3+(3xx4a xx 5b)(4a+5b)` ` =64a^3+125b^3+60ab(4a+5b)` `=64a^3+125b^3+24a^2b+300ab^2`. (ii) Putting `3a=x and 2b =y`, we get ` 3a-2b)^3=(x-y)^3` `=x^3-y^3-3xy (x-y)` `=(3a)^3-(2b)^3-(3xx3a xx 2b) (3a- 2b)` `=27a^3=8b^3-18ab(3a-2b)` `=27 a^3-8b^3-54a^2b+36ab^2`. |
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| 34. |
Factorize `x(x-y)^3+3x^3+3x^3y(x-y)`. |
| Answer» Correct Answer - `x(x+y)(x^2+y^2-xy)` | |
| 35. |
`6(2x-(3)/(x))^2+7(2x-(3)/(x))-20` |
| Answer» Correct Answer - `(4x-(6)/(x)+5)(6x-(9)/(x)-4)` | |
| 36. |
`4x^4 +7x^2-2` |
| Answer» Correct Answer - `(x^2+2)(2x-1)(2x+1)` | |
| 37. |
`9a^2+3a-8b-64b^2` |
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Answer» Correct Answer - `(3a-8b)(3a+8b+1)` `9a^2+3a-8b-64b^2=(9a^2-64b^2)+(3aa-8b)`. |
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| 38. |
`6x^2-5x-21` |
| Answer» Correct Answer - `(3x-7)(2x+3)` | |
| 39. |
`5x^2-16x-21` |
| Answer» Correct Answer - `(5x-21)(x+1)` | |
| 40. |
Expand (i) `(a+2b+5c)^2` (ii) ` (2a+b+c)^2` (iii) ` (a-2b-3c)^2` |
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Answer» Correct Answer - (i) `a^2+4b^2+25c^2+4ab+20bc+10ac` (ii) `4a^2+b^2+c^2-2bc+4ac` (iii) ` a^2+4b^2+9c^2-4ab+12bc- 6ac` |
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| 41. |
`21x^2+5x-6` |
| Answer» Correct Answer - `(3x+2)(7x-3)` | |
| 42. |
`x^3+3xy^2-2x^2y-6y^3` |
| Answer» Correct Answer - `(x-2y)(x^2+3y^2)` | |
| 43. |
`x^3+2x^2+5x+10` |
| Answer» Correct Answer - `(x+2)(x^2+5)` | |
| 44. |
Find the solutions of `x^3-5x^2-x+5` . and the no. of roots |
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Answer» Correct Answer - `(x-5)(x-1)(x+1)` `x^3-5x^2-x+5=x^2(x-5)=(x-5)(x^2-1)` |
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| 45. |
`(a+2b)^2+101( a+2b)+100` |
| Answer» Correct Answer - `(a+2b+100)(a+2b+1)` | |
| 46. |
`8(3a-2b)^2-10(3a-2b)` |
| Answer» Correct Answer - `2(3a-2b)(12a-8b-5)` | |
| 47. |
`9a^2+6a+1-36b^2` |
| Answer» Correct Answer - `(3a+1-6b)(3a+1+6b)` | |
| 48. |
Factorize:`8a^3+27 b^3+36 a^2b+54 a b^2` |
| Answer» Correct Answer - `(2a+3b)(2a+3b)(2a+3b)` | |
| 49. |
Factorize:`a^3+3a^2b+3a b^2+b^3-8` |
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Answer» Correct Answer - `(a+b-2)[(a+b)^2+2(a+b)+4]` Given expression `=(a+b)^3-2^3`. |
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| 50. |
Factorize:`125 x^3-27 y^3-225 x^2y+135 x y^2` |
| Answer» Correct Answer - `(5x-3y)(5x-3y)(5x-3y)` | |