InterviewSolution
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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. |
When the polynomial `f(x)=ax^(2)+bx+c` is divided by x , x - 2 and x +3 remainders obtained are 7 , 9 and 49 respectively . Find the value of `3a + 5b+ 2c`.A. `-2`B. 2C. 5D. `-5` |
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Answer» Correct Answer - A `f(0)=7,f(2)=9andf(-3)=49`. |
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| 2. |
If `aa n db`are distinct integers, prove that `a-b`is a factor of `a^n-b^n`, wherever `n`is a positive integer. |
| Answer» Correct Answer - n `in` N | |
| 3. |
If `f(x + 2) = x^2 + 7x-13`, then find the remainder when `f(x)` is divided by `(x +2)`.A. `-25`B. `-12`C. `-23`D. `-11` |
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Answer» Given , `f(x+2) = x^(2)+7x-13` (1) The remainder , when `f(x)` is divided by `(x+2) ` is `f(-2)`. `:.` Put `x=-4` in Eq . (1) `f(-4+2)=(-4)^(2)+7(-4)-13` `rArrf(-2)=16-28-13=-25`. |
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| 4. |
Find the value of a if x - a is a factor of the polynomial `x^(5)-ax^(4)+x^(3)-ax^(2)+2x+3a-2`. |
| Answer» Correct Answer - `(2)/(5)` | |
| 5. |
Find the remainder when `x^(3)+3px+q` is divided by `(x^(2)-a^(2))` without actual division. |
| Answer» Correct Answer - `(a^(2)+3p)x+q` | |
| 6. |
The remainder when `f(x)=x^(3)+5x^(2)+2x+3` is divided by x is _________. |
| Answer» Correct Answer - 3 | |
| 7. |
Let `f(x-(1)/(x))=x^(2)+(1)/(x2)`, find the remainder when `f(x)` is divided by ` x -3`.A. `(82)/(9)`B. `(8)/(3)`C. 10D. 11 |
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Answer» Correct Answer - D (i) `f(x-(1)/(x))=(x-(1)/(x))^(2)+2`. (ii) Replace `(x-(1)/(x))` with x. (iii) Use remainder theorem to obatin remainder. |
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| 8. |
If a polynomial f (x) is divided by `(x+a)` , then the remainder obtained is _______. |
| Answer» Correct Answer - `f(-a)` | |
| 9. |
Find the remainder when `x^999` is divided by `x^2-4x + 3`. |
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Answer» Let `q(x)` and mx + n be the quotient and the remainder respectively when `x^(999)` is divided by `x^(2)-4x+3`. `:. x^(999)=(x^(2)-4x+3)q(x)+mx+n`. If ` x= 1` , `x^(999)=(1-4+3)q(x)+m(1)+n` `rArr1=0xxq(x)+m+n` `rArrm+n=1` (1) If ` x= 3`, `3^(999)=(3^(2)-4(3)+3)q(x)+3m+n` `rArr3^(999)=0xxq(x)+3m+n` `rArr3m+n=3^(999)` (2) subtracting Eq . (1) from Eq . (2) , we get `2m=3^(999)-1` `m=(1)/(2)(3^(999)-1)` Substituting m in Eq . (1) , we have `n=1-(1)/(2)(3^(999)-1)=1-(1)/(2)3^(999)+(1)/(2)=(3)/(2)-(1)/(2)3^(999)` `n=(3)/(2)(1-3^(999))` `:.` The required remainder is `(1)/(2)(3^(999)-1)x+(3)/(2)(1-3^(999))`. |
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| 10. |
If `(x^(2)-1)` is a factor of `ax^(3)-bx^(2)-ax+d`, then find the relation between a and c . |
| Answer» Correct Answer - a = c | |
| 11. |
If `(x-1)^(2)` is a factor of `f (x)=x^(3)+bx+c` , then find the remainder when `f (x)` is divided by `(x-2)`.A. 2B. `-3`C. 4D. `-4` |
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Answer» Correct Answer - C (i) Coefficient of `x^(2)` is 0 , therefore sum of roots is 0 . `:.` Third root =-2. (ii) Apply factro theorem. (iii) To obtain the remainder , use the remainder theorem. |
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| 12. |
Let f(x) `=a_(0)x^(n)+a_(1)x^(n-1)+...+a_(n)(a_(0)ne0)` be a polynomial of degree n . If x+1 is one of its factors, then______. |
| Answer» Correct Answer - `a_(1)+a_(3)+a_(5)+...=a_(0)+a_(2)+a_(4)...` | |
| 13. |
If `(x - p) and (x-q)` are the factors of `x^(2)+px+q`, then the values of p and q are respectively _____.A. `1, -2`B. `2, -3`C. `(-1)/(3),(-2)/(3)`D. `-2,1` |
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Answer» Correct Answer - A (i) `x^(2)+px+q=(x-p)(x-q)`. (ii) compare the terms in LHS and RHS. |
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| 14. |
The remainder obtained when `x^(2)+3x+1` is divided by `(x-5)` is _________. |
| Answer» Correct Answer - 41 | |
| 15. |
If `x^(555)` is divided by `x^(2)-4x+3`, then find its remainder. |
| Answer» Correct Answer - `(1)/(2)(3^(555)-1)x+(3)/(2)(1-3^(554))` | |
| 16. |
Find the values of a if `x^(3)-5x(a-1)-3(x+1)+5a` is divisible by x - a. |
| Answer» Correct Answer - 1and 3 | |
| 17. |
The condition for which `ax^(2)+bx+a` is exactly divisible by `x-a ` is _________. |
| Answer» Correct Answer - `a=0ora^(2)+b+1=0` | |
| 18. |
If `ax^(4)+bx^(3)+cx^(2)+dx` is exactly divisible by `x^(2)-4` then `(a)/(c)` is ______.A. `(1)/(4)`B. `(-1)/(4)`C. `(-1)/(8)`D. `(1)/(8)` |
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Answer» Correct Answer - B `f(2)=0andf(-2)=0`. |
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| 19. |
Find the remainder when `x^(5)` is divided by `x^(2)-9`.A. 81xB. `81 x+10`C. `3^(5)x+34`D. 81 |
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Answer» Correct Answer - A Use division algorithm. |
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| 20. |
If `f(x+3)=x^(2)+x-6` , then one of the factors of `f(x)` is ________.A. `x - 3`B. `x - 4`C. `x - 5`D. `x - 6` |
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Answer» Correct Answer - C (i) Put `x =x-3` in `f(x+3)`to get f(x). Apply factor theorem. |
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| 21. |
If the polynomial `3x^(4)-11x^(2)+6x+k` is divided by x - 3 , it leaves a remainder 7 . Then the value of k is ______. |
| Answer» Correct Answer - `-155` | |
| 22. |
For what values of m and n is `2x^(4)-11x^(3)+mx+n` is divisible by `x^(2)-1`? |
| Answer» Correct Answer - `m=11and n =- 2` | |
| 23. |
Find the values of m and n, if `(x -m) and (x - n)` are the factors of the expression `x^2 +mx -n`.A. m = - 1 , n =- 2B. m = 0 , n n = 1C. `m=(-1)/(2),n=(1)/(2)`D. m = - 1 , n= 2 |
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Answer» Correct Answer - D (i) `x^(2)+mx-n=(x=m)(x-n)`. (ii) Fquate the corresponding terms. |
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| 24. |
A polynomial `f(x)` leaves remainders 10 and 14 respectively when divided by `(x- 3) and (x-5)` Find the remainder when `f(x)` is divided by `(x- 3) and (x-5)`.A. `2x+6`B. `2x-4`C. `2x+4`D. `2x-6` |
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Answer» Correct Answer - C (i) `f(3)=10,f(5)=14`. (ii) `"Dividend"="Divisor"xx"Quotient"+ "Remainder"`. |
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| 25. |
The remainders of a polynomial f(x) in x are 10 and 15 respectively when f (x) is divided by (x -3) and (x-4) . Find the remaider when f (x) is divided by (x-3) (x -4). |
| Answer» Correct Answer - `5(x-1)` | |
| 26. |
Find a linear polynomial which when divided by `(2x+1)and(3x+2)` leaves remainders 3 and 4, respectively. |
| Answer» Correct Answer - `-6x` | |
| 27. |
What should be added to `3x^(3)+5x^(2)-6x+3` to make it exactly divisible by x - 1 ? |
| Answer» Correct Answer - `-5` | |
| 28. |
A polynomial `p (x)` leaves remainders 75 and 15 , respectively , when divided by `(x-1)and (x+2)`. Then the remainder when `f(x)` is divided by `(x-1)(x+2)` is _______.A. `5(4x+11)`B. `5(4x-11)`C. `5(3x+11)`D. `5(3x-11)` |
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Answer» Correct Answer - A (i) `f(1)=75,f(-2)=15`. `"Dividend"="Divisor"xx"Quotient"+ "Remainder"`. |
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| 29. |
One of the factors of `2x^(17)+3x^(15)+7x^(33)` is ________. `(x^(17)//x^(15)//x^(23))` |
| Answer» Correct Answer - `x^(15)` | |
| 30. |
If the expression `6x^(2)+13x+k` is divisible by `2x + 3`. Then which of the following is the factor of the expression ?A. `3x+1`B. `3x+4`C. `3x+2`D. `3x+5` |
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Answer» Correct Answer - C Let `f(x)=6x^(2)+13x+k` Given `2x+3` is a factor of `f(x)` by factor theorem, `f(-(3)/(2))=0` `rArr6(-(3)/(2))^(2)+13(-(3)/(2))+k=0` `rArr(27)/(2)-(39)/(2)+k=0rArrk=6` `:.f(x)=6x^(2)+13x+6` `=(2x+3)(3x+2)` `:.` The other factor is `3x+2`. |
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| 31. |
If `ax^(3)-5x^(2)+x+` p is divisible by `x^(2)-3x+2`, then find the values of a and p .A. a = 2 , p = 2B. a = 2 , p = 3C. a = 1 , p = 3D. a = 1 , p = 2 |
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Answer» Correct Answer - A Let `f(x)=ax^(3)-5x^(2)+x+p` Given , `f(x)` is divisible `x^(2)-3x+2`,i.e., `rArrf(x)` is divisible by `(x-1)and(x-2)` `f(1)=0andf(2)=0rArra+p-4=0` `rArra+p=4` (1) `8a+p=18=0rArr8a+p=18` (2) On solving Eqs. (1) and (2) , we get a = p = 2 . |
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| 32. |
The value of a for which x - 7 is a factor of `x^(2)+11x-2z`, is ______. |
| Answer» Correct Answer - 63 | |
| 33. |
If `lx^(2)+mx+n` is exactly divisible by ` (x - 1 ) and (x + 1)` and leaves a remainder 1 when divided by x + 2 , then find m and n . |
| Answer» Correct Answer - `m=0,n=(-1)/(3)` | |
| 34. |
If f `(x-2)-2x^(2)-3x+4` , then find the remainder when f (x) is divided by (x-1). |
| Answer» Correct Answer - 13 | |
| 35. |
Which of the following should be added to `9x^(3)+6x^(2)+x+2`, so that the sum is divisible by `(3x+1)`?A. `-4`B. `-3`C. `-2`D. `-1` |
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Answer» Correct Answer - C Let k should be added to the given expression so that the sum is divisible by `(3x+1)`. Let `f(x)=9x^(3)+6x^(2)+x+2+k` Given , `f(-(1)/(3))=0` `rArr9(-(1)/(3))+6(-(1)/(3))^(2)-(1)/(3)+2+k=0` `rArr-(1)/(3)+(2)/(3)-(1)/(3)+2+k+0rArr2+k=0` `rArrk=-2`. |
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| 36. |
If a polynomial f (x) is divided by ` (x - 3) and (x - 4)` it leaves remainders as 7 and 12 respectively, then find the remainder when f (x) is divided by `(x-3)(x-4)`. |
| Answer» Correct Answer - `5x-8` | |
| 37. |
Factorize `x^(4)-2x^(3)-9x^(2)+2x+8` using remainder theorem. |
| Answer» Correct Answer - `(x-1)(x+1)(x+2)(x-4)`. | |
| 38. |
Which of the following is a factor of `5x^(20)+7x^(15)+x^(9)`?A. `x^(20)`B. `x^(15)`C. `x^(9)`D. `x^(24)` |
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Answer» Correct Answer - C `5x^(20)+7x^(15)+x^(9)(5x^(11)+7x^(6)+1)`. |
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| 39. |
If the polynomials f(x) `=x^(2)+5x-pandg(x)=x^(2)-2x+6p` have a common factor , then find the common factor .A. `x+2`B. xC. `x+4`D. Either (b) or (c ) |
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Answer» Given , `f(x)=x^(2)+5x-pandg(x)=x^(2)-2x+6p` Let ` x- k` be the common factor of `f (x) and g(x)`. `:.f(x)=0andg(k)=0` `rArrk^(2)+5k-p=0` (1) `k^(2)=2k+6p=0` (2) From Eqs . (1) and (2) , we get `k=p` substitute k = p in Eq . (1) , we have `p^(2)+5p-p=0` `p^(2)+4p=0rArrp=0` or `p=-4` `:.xorx+4` is a common factor of f(x) and g(x). |
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| 40. |
If `x^(2)+5x+6` is a factor of `x^(3)+9x^(2)+26x+24`, then find the remaining factor. |
| Answer» Correct Answer - `(x+4)`. | |