InterviewSolution
Saved Bookmarks
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. |
cIf `a_1,a_2,a_3,..,a_n in R` then `(x-a_1)^2+(x-a_2)^2+....+(x-a_n)^2` assumes its least value at x=A. `a_(1) + a_(2) +....+a_(n)`B. `2(a_(1) + a_(2), a_(3) +....+a_(n))`C. `n(a_(1)+a_(2)+....+a_(n))`D. none of these |
|
Answer» Correct Answer - D We have, `(x-a_(1))^(2)+(x-a_(2))^(2)+....+(x-a_(n))^(2)` `=nx^(2)-2x(a_(1)+a_(2)+....+a_(n))+(a_(1)^(2)+a_(2)^(2)+....+a_(n)^(2))` Clearly, `y = nx^(2) - 2x (a_(1)+a_(2) +...+a_(n))+(a_(1)^(2)+a_(2)^(2)+...+a_(n)^(2))` represents a parabola which opens upward. So, it attains its minimum value at the vertex i.e. at `x = (2(a_(1)+a_(2)+....+a_(n)))/(2n)=(a_(1)+a_(2)+....+a_(n))/(n)["Using x" = (-b)/(2a)]` |
|
| 2. |
If `alpha` and `beta` are the roots of the equation `x^2-ax+b=0` and `A_n=alpha^n+beta^n`,A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.C. Statement-1 is True, Statement-2 is False.D. Statement-1 is False, Statement-2 is True. |
|
Answer» Correct Answer - D We have, `alpha + beta = a and alpha beta = b` `therefore" "alpha^(n+1) + beta^(n+1) = (alpha^(n) + beta^(n))(alpha + beta) - alpha beta(alpha^(n-1) + beta^(n-1))` `rArr" "V_(n+1) = a V_(n) - b V_(n-1)` So, statement-2 is true. Now, `V_(n+1) = a V_(n) - V_(n-1)` `rArr" "V_(2)=a V_(1) -b V_(0)=a(alpha+beta)-b(alpha^(0)+beta^(0))=a^(2)-2b` `" "V_(3)=a V_(2) -b V_(1)=a(a^(2)-2b)-ab=a^(3)-3ab` `" "V_(4)=a V_(3) -b V_(2)=a(a^(3)-3ab)-b(a^(2)-2b)=a^(4)-4a^(2)b+2b^(2)` and, `V_(5)=a V_(4)-b V_(3) = a(a^(4) -4a^(2)b + 2b^(2))-b(a^(3)-3ab)=a^(5)-5a^(3)b + 5ab^(2)` So, statement-1 is false. |
|
| 3. |
lf `alpha and beta` are the roots of the equation `x^2-ax + b = 0 and A_n = alpha^n + beta^n`, then which of the following is true ?A. `A_(n+1) = a A_(n) + b A_(n-1)`B. `A_(n+1) = b A_(n) + a A_(n-1)`C. `A_(n+1) = a A_(n) - b A_(n-1)`D. `A_(n+1) = b A_(n) - a A_(n-1)` |
| Answer» Correct Answer - C | |
| 4. |
let `alpha(a)` and `beta(a)` be the roots of the equation `((1+a)^(1/3)-1)x^2 +((1+a)^(1/2)-1)x+((1+a)^(1/6)-1)=0` where `agt-1` then, `lim_(a->0^+)alpha(a)` and `lim_(a->0^+)beta(a)`A. `-(5)/(2) and 1`B. `-(1)/(2) and -1`C. `-(7)/(2) and 2`D. `-(9)/(2) and 3` |
|
Answer» Correct Answer - B Let 1 + a = y. Then, the given equation becomes `(y^(1//3)-1)x^(2)+(y^(1//2)-1)x+(y^(1//6)-1)=0` `rArr" "((y^(1//3)-1)/(y-1))x^(2)+((y^(1//2)-1)/(y-1))x+((y^(1//6)-1)/(y-1))=0" "...(i)` Now, `a rarr 0^(+) rArr 1 + a rarr 1^(+) and 1 + a gt 1 rArr y rarr 1^(+)` Taking `underset(y rarr 1^(+))("lim")` on both sides of (i), we get `underset(y rarr 1^(+))("lim")((y^(1//3)-1^(1//3))/(y-1))x^(2)+underset(y rarr 1^(+))("lim")((y^(1//2)-1^(1//2))/(y-1))x + underset(y rarr 1^(+))("lim")((y^(1//6)-1^(6))/(y-1))=0` `rArr" "(1)/(3) x^(2) +(1)/(2)x +(1)/(6)=0` `rArr" "2x^(2)+3x + 1 = 0` `rArr" "(2x+1)(x+1)=0` `rArr" "x = -1, -(1)/(2)` `rArr" "underset(a rarr o^(+))("lim") beta(a) = -1 and underset(a rarr o^(+))("lim") alpha(a) = -(1)/(2)`. |
|
| 5. |
The roots of the equation `3x^(2) - 4sqrt(3x) + 4 = 0` are |
| Answer» Correct Answer - `(1)/(2sqrt(3)), (1)/(5sqrt(3))` | |
| 6. |
If `k_(1)` and `k_(2)` are roots of `x^(2) - 5x - 24 = 0`, then find the quadratic equation whose roots are `-k_(1)` and `-k_(2)`. |
| Answer» Correct Answer - `x^(2) + 5x - 24 = 0` | |
| 7. |
The roots of the equation `2x^(2) + 3x + c = 0` (where `x lt 0`) could be `"______"`. |
| Answer» Correct Answer - rationa or irrational, but unequal | |
| 8. |
The number of distinct real solution of `|x|^(2) - 5|x| + 6 = 0` isA. 4B. 3C. 2D. 1 |
|
Answer» Correct Answer - C (i) Solve the equation to find the number of real roots. (ii) Replace `|x|` by y and solve for y. (iii) Now, `x = +- y`. |
|
| 9. |
If the equation `3x^(2) - 2x-3 = 0`has roots `alpha`, and `beta` then `alpha.beta = "______"`. |
| Answer» Correct Answer - `-1` | |
| 10. |
Which of the following equations has roots as a,b and c ?A. `x^(3) + x^(2)(a+b+c) + x(ab+bc+ca) + abc = 0`B. `x^(3)+x^(2)(a+b+c)+x(ab+bc+ca) - abc = 0`C. `x^(3) - x^(2)(a+b+c) + x(ab+bc+ca)-abc = 0`D. `x^(3)-x^(2)(a+b+c)-x(ab+bc+ca) - abc = 0` |
|
Answer» Correct Answer - C Let `(x-a) (x-b) (x-c) = 0` and expand. |
|
| 11. |
`(x+3)(x+4)(x+6)(x+7)=1120`. |
| Answer» Correct Answer - `x = 1`, `x = -11` | |
| 12. |
If `(x^2- 3x + 2)` is a factor of `x^4-px^2+q=0`, then the values of `p and q` areA. 5, -4B. 5, 4C. `-5, 4`D. `-5, -4` |
| Answer» Correct Answer - B | |
| 13. |
The product of the roots of the equation `1/(x+1) + (1)/(x-2) = (1)/(x+2)` is `"_____"`. |
| Answer» Correct Answer - Zero | |
| 14. |
if `2x^(2) + 4x - k = 0` is same as `(x-5) (x+k/10) = 0`, then find the value of k.A. 100B. 90C. 70D. 35 |
|
Answer» Correct Answer - C `2x^(2) + 4x - k = 0 " "(1)` `:. rArr (x-5)` is a factor of Eq. (1). `rArr x - 5 = 0 rArr x = 5` `:. 2(5)^(2)+ 4(5) - k = 0` `50+20 - k = 0` `:. k = 70`. |
|
| 15. |
`x = 2` is a roots of the equation `x^(2) - 5 + 6 = 0`. Is the given statement true? |
| Answer» Correct Answer - Yes | |
| 16. |
The roots of the equation `6x^(2) - 8sqrt(2x) + 4 = 0` areA. `1/3, sqrt(2)`B. `(sqrt(2))/(1), 1`C. `(sqrt(2))/(3), sqrt(2)`D. `(3)/(sqrt(2)), sqrt(2)` |
|
Answer» Correct Answer - C Take 2 as common and then factorize. |
|
| 17. |
Two persons A and B solved a quadratic equation of the form `x^(2) + bx + c = 0`. A made a mistake in noting down the coefficient of x and obtained the roots as 18 and 2, where as B obtained the roots as `-9` and -3 by misreading the constrant term. The correct roots of the equation areA. `-6,-3`B. `-6,6`C. `-6,-5`D. `-6,-6` |
|
Answer» Correct Answer - D (i) Use the concept of sum of the roots and product of the roots of quadratic equation. (ii) The product to the roots obtained by A and sum of the roots otained by B is equal to the product and sum of the roots of the required equation respectively. |
|
| 18. |
The number of real roots of the equation `|x^2|– 5|x|+6 = 0` isA. 1B. 2C. 3D. 4 |
|
Answer» Correct Answer - D (i)Use the concept `|x|` and find the roots. (ii) Replace `|x|` by y and solve for y. (iii) Now, `x = +- y` |
|
| 19. |
If `(2x-9)` is a factor of `2x^(2) + px - 9`, then `p = "_____"`. |
| Answer» Correct Answer - `-7` | |
| 20. |
If the roots of the equation `ax^(2) + bx + c = 0` are in the ratio of `3 : 4`, |
| Answer» Correct Answer - `3b^(2) = 49 ac` | |
| 21. |
The number of real roots of the quadratic equation `(x-4)^(2)+ (x-5)^(2) + (x-6)^(2) = 0` isA. 1B. 2C. 3D. None of these |
|
Answer» Correct Answer - D (i) Use the concept of perfect square of a number. (ii) If `a^(2) + b^(2) + c^(2) = 0` is true only when `a = b = c = 0`. |
|
| 22. |
The roots of `x^(2) - 2x-1= 0` are `"_____"`.A. `sqrt(2) + 1, sqrt(2) -1`B. `1,sqrt(2)`C. `1+sqrt(2), 1-sqrt(2)`D. `2,1` |
|
Answer» Correct Answer - C Given `x^(2) - 2x-1 = 0` `x = (2+-sqrt(4-4xx1xx(-1)))/(2xx1)` `x = (2+-sqrt(8))/(2) = (2+-2sqrt(2))/(2) = 1+-sqrt(2)` `x = 1+sqrt(2)` or `1 - sqrt(2)`. |
|
| 23. |
The equation `x + 5/(3-x) = 3 + (5)/(3-x)` hasA. no real root.B. one real root.C. two equal roots.D. infinite roots. |
|
Answer» Correct Answer - A (i) Simplify the equation. (ii) A rational function `(f(x))/(g(x))` is defined only when `g(x) gt 0`. |
|
| 24. |
The roots of the equation `x^(2) + ax + b = 0` are `"______"`. |
| Answer» Correct Answer - `(-a+-sqrt(a^(2)-4b))/(2)` | |
| 25. |
If the sum of the roots of the equation `ax^2 +bx +c=0` is equal to sum of the squares of their reciprocals, then `bc^2,ca^2 ,ab^2` are inA. `c^(2) b, a^(2) c, b^(2) "a are in A.P.`B. `c^(2) b, a^(2) c, b^(2) "a are in G.P.`C. `(b)/(c), (a)/(b), (c)/(a) "are in G.P.`D. `(a)/(b), (b)/(c), (c)/(a) "are in G.P.` |
| Answer» Correct Answer - A | |
| 26. |
The quadratic equation having roots `-a,-b` is `"_____"`. |
| Answer» Correct Answer - `x^(2) + x(a+b) + ab = 0` | |
| 27. |
Find the roots of quadratic equation `ax^(2) + (a-b + c) x - b +c = 0`. |
| Answer» Correct Answer - `-1,(b-c)/(a)` | |
| 28. |
In writing a quadratic equation of the form `x^(2) + px + q = 0`,a student makes a mistake in writing the coefficientof x and gets the roots as 8 and 12. Another student makes mistake in writing the constant term and gets the roots as 7 and 3. Find the correct quadratic equation.A. `x^(2) + 10x + 96 = 0`B. `x^(2) - 20x + 21 = 0`C. `x^(2) - 21x + 20 = 0`D. `x^(2) - 96x + 10 = 0` |
|
Answer» Correct Answer - A (i) Use the concept of sum and product of the roots of a quadratic equation. (ii) The product of the roots obtained by the first student is product of the roots of the required quadratic equation. (iii) The sum of the roots obtained by the secon student of the roots of the required qua-dratic equation. |
|
| 29. |
The roots of the equation `(1)/(2x-3) - (1)/(2x+5) = 8` areA. `2-(1)/(sqrt(2)), 2 - (1)/(sqrt(2))`B. `(-1+sqrt(17))/(2),(-1-1sqrt(17))/(2)`C. `2+(1)/(sqrt(2)),2-(1)/(sqrt(2))`D. `(1+sqrt(17))/(2),(1-sqrt(17))/(2)` |
|
Answer» Correct Answer - B (i) Simplify the equation. (ii) Take LCM. (iii) Convert it into quadratic equation. (iv) Solve the equation by using formula x `= (-b+-sqrt(b^(2) - 4x))/(2a)`. |
|
| 30. |
The minimum value of `2x^(2) - 3x+ 2` is `"____"`.A. `7/8`B. `4/7`C. 4D. -3 |
|
Answer» Correct Answer - A The minimum value of `2x^(2) - 3x+2 =(4xx2xx2-(-3)^(2))/(4xx2) = 7/8`. |
|
| 31. |
If `(x-2)(x+3) =0` , then the values of x are `"______"`. |
| Answer» Correct Answer - `2,-3` | |
| 32. |
A quadratic equation whose roots are 2 moe than the roots of the quadratic equation `2x^(2) +3x + 5 = 0`, can be obtained by substituting `"_____"` for x. `[(x-2)//(x+2)]` |
| Answer» Correct Answer - `(x-2)` | |
| 33. |
Find the quadratic equation whose roots are reciprocals of the roots of the equation `7x^(2) - 2x +9 = 0`.A. `9x^(2) - 2x + 7 = 0`B. `9x^(2) - 2x - 7 = 0`C. `9x^(2) + 2x - 7 = 0`D. `9x^(2) + 2x - 7 = 0` |
|
Answer» Correct Answer - A The quadratic equation with reciprocals of the roots of the equation `f(x) = 0` is `f(1/x) = 0`. |
|
| 34. |
The discriminant of the equation `x^(2) - 7x + 2 = 0` isA. 47B. 40C. 41D. -41 |
|
Answer» Correct Answer - C Use the formula to find the discriminant. |
|
| 35. |
For which value of `p` among the following, does the quadratic equation `3x^(2) + px + 1 = 0` have real roots ? |
| Answer» Correct Answer - 4 | |
| 36. |
The number of roots of the equation `2|x|^(2) - 7|x| + 6 = 0`A. 4B. 3C. 2D. 1 |
|
Answer» Correct Answer - A `2|x|^(2) - 7 |x| + 6 = 0` Let `|x | = y ` `2y^(2) -7y + 6 = 0` `2y^(2) - 3y - 4 y + 6 = 0` `y(2y-3)-2(2y+3) = 0` `(2y-3)(y-2) = 0` `2y-3=0` or `y-2= 0` `y = 3/2` or `y = 2`. `|x| = 3/2` or `|x| = 2` `x = +- 3/2` or `x = +-2` `:.` x has `4` real solutions. |
|
| 37. |
Solve `sqrt(x+65) + sqrt(5-x) = 4` |
|
Answer» Squaring the term of the both the sides, we get `(sqrt(x+5) +sqrt(5-x))^(2) = 4^(2)` `rArr x + 5 + 5 - x + 2 sqrt((x+5)(5-x)) = 16` `rArr 10 +2 sqrt(25-x^(2)) = 16` `rArr sqrt(25-x^(2)) = 3` Squaring the term on the both the sides again, we get `25 - x^(2) = 3^(2)` `rArr x^(2) = 25 - 9` `rArr x^(2) = 16 rArr x = +- 14` `:. - 4` and 4 are the required solution of the given equation. |
|
| 38. |
If the equation `x^(3) + ax^(2) + b = 0, b ne 0` has a root of order 2, thenA. `a^(2) + 2b = 0`B. `a^(2) - 2b = 0`C. `4a^(3) + 27b + 1 = 0`D. `4a^(3) + 27b = 0` |
| Answer» Correct Answer - D | |
| 39. |
For the expression `ax^(2)+ 7x + 2` to be quadratic, the possible value of a are `"______"`. |
| Answer» Correct Answer - non-zero real numers | |
| 40. |
If the quadratic expression `x^(2) + (a-4)x + (a+4)` is a perfect square, then `a = "______"`.A. 0 and -4B. 0 and 6C. 0 and 12D. 6 and 12 |
|
Answer» Correct Answer - C (i) If the equation is a perfect square, then it has equal roots. (ii) Quadratic equation is a perfet square, if `b^(2) - 4 ac = 0` (ii) Quadratic equation is a perfect square, if `b^(2) - 4 ac = 0`. (ii) Substitute the value of b and c in the above eqution and obtained the value of a. |
|
| 41. |
If the roots of the quadratic equation `x^2-4x-log_3a=0` are real, then the least value of a isA. 81B. `1//81`C. `1//64`D. none of these |
|
Answer» Correct Answer - B Since the roots of the given equation are real. `therefore" ""Discriminant" gt 0` `rArr" "16+4 log_(3) a ge -4 rArr a ge 3^(-4) rArr a ge 1//81` Hence, the least value of a is 1/81. |
|
| 42. |
Solve `(x^(2) - 2x)^(2) - 23 (x^(2) - 2x) + 120 = 0` |
|
Answer» Let us assume that `x^(2) - 2x = y` `rArr` The given equation reduced to a quadratic equation in y. That is, `y^(2)- 23 + 120 = 0` `rArr y^(2) - 15y - 8y + 120 = 0` `rArr y(y-15) - 8 (y-5) = 0` `rArr (y-8) (y-15) = 0` `rArr y - 8 = 0` (or) `y - 15 = 0` `rArr y = 8`(or) `y = 15` But `x^(2) - 2x = y` When `y = 8, x^(2) - 2x = 8` When `y = 8, x^(2)- 2x = 8` `rArr x^(2) - 2x - 8 = 0` `rArr x^(2) - 4x + 2x - 8 = 0` `rArr x(x-4) +2(x-4) = 0` `rArr (x+2) (x-4) = 0` `rArr x+2 = ` (or) `x - 4 = 0` `rArr x = - 2` (or) `x = 4` When `y = 15, x^(2)- 2x = 15` `rArr x^(2) - 2x - 15 = 0` `rArr x^(2)- 5x + 3x - 15 = 0` `rArr x(x-5) + 3(x-5) = 0` `rArr (x-5) (x+3)= 0` `rArr x- 5 = 0` (or) `x + 3 = 0` `rArr x= 5` (or) `x =-3` `:. x = - 2,-3, 4` and 5 are the required solutions of the given equation . |
|
| 43. |
For the expression `7x^(2) + bx +4 ` to be quadratic , the possible values of b are `"______"`. |
| Answer» Correct Answer - real numbers | |
| 44. |
If `alpha` and `beta` the roots of the equation `x^(2) + 9x + 18 = 0`, then the quadratic equation having the roots `alpha + beta` and `alpha - beta` is `"_____"`. Where `(alpha gt beta)`.A. `x^(2) + 6x - 27 = 0`B. `x^(2) - 9x + 27 = 0`C. `x^(2) - 9x + 7 = 0`D. `x^(2) + 6x + 27 = 0` |
|
Answer» Correct Answer - A Find `alpha + beta, alphabeta` and using these values find `alpha - beta`. |
|
| 45. |
Verify whether `x = 2` is a solution of `2x^(2) + x - 10 =0` |
|
Answer» On substituting `x = 2`, in `2x^(3) + x - 10`, we get `2(2)^(2) + 2 - 10= 10 - 10= 0`. `:. 2` is a solution (or) root of `2x^(2) + x - 10 = 0`. |
|
| 46. |
If `alpha` and `beta` are the of the equation `x^(2) + 3x - 2 = 0` then`alpha^(2)beta + alphabeta^(2)` =A. `-6`B. -3C. 6D. 3 |
|
Answer» Correct Answer - C Find the sum and product of the root. Let `alpha^(2)beta + alphabeta^(2) = alpha beta(alpha + beta)` |
|
| 47. |
The number of real roots of the quadratic equation `3x^(2) + 4 = 0` is |
|
Answer» Correct Answer is `-2/sqrt(3)` Solve for x. |
|
| 48. |
If `alpha` and `beta`are the roots of `x^(2) - (a+1)x + 1/2 (a^(2) + a+1) = 0` then `alpha^(2) + beta^(2) = "_____"`.A. A and BB. `a^(2)`C. 2aD. 1 |
|
Answer» Correct Answer - A `x^(2) - (a+1)x + 1/2 (a^(2) + a + 1) = 0` `alpha + beta = a+1` `alpha beta = 1/2 (a^(2) + a + 1)` `alpha^(2) + beta^(2) = (alpha + beta)^(2) - 2alphabeta` `= (a+1)^(2) - 2[1/2(a^(2) +a+1)]` `= a^(2) + 2a + 1 - a^(2) - a - 1 = a`. |
|
| 49. |
If the roots of the equation `x^3+3ax^2+3bx+c=0` are in `H.P.`, then(i) `2b^2=c(3ab-c)`(ii) `2b^3=c(3ab-c)`(iii) `2b^3=c^(2)(3ab-c)`(iv) `2b^2=c^(2)(3ab-c)`A. `beta = (1)/(alpha)`B. `beta = b`C. `beta = (c)/(b)`D. `beta = (b)/(c)` |
|
Answer» Correct Answer - C Clearly, `(1)/(alpha), (1)/(beta), (1)/(gamma)` are the roots of the equation `-cx^(3) + 3bx^(2) - 3ax + 1 = 0` and are in A.P. Now, `(1)/(alpha), (1)/(beta), (1)/(gamma) = (3b)/(c)` `rArr" "(3)/(beta) = (3b)/(c)" "[therefore (1)/(alpha)+(1)/(gamma)=(2)/(beta)]` `rArr" "beta = (c)/(b)` |
|
| 50. |
If one root of the equation, `x^(2) - 11x + (p-3) = 0` is 3, then find the value of p and also its other root. |
|
Answer» Given that 3 is one of the roots of the equation `x^(2) - 11x + p - 3 = 0` `rArr x = 3` satifies the given equation . `rArr (3)^(2) - 11(3) + p - 3 = 0` `rArr p = 33 +3 - 9` `rArr p = 27`. `:.` The value of p is 27. Since the sum of the roots of the equation is 11 and one of the roots is 3, the other root of the equation of 8. |
|