Question 1

For a given k, what is the minimum value of x^2+kx+k^2 ?

Accepted Answer

Let the given expression be f(x) = x^2 + kx + k^2. To find the minimum value, we can complete the square for the quadratic expression in x: f(x) = x^2 + kx + k^2 4 - k^2 4 + k^2 f(x) = (x + k 2 )^2 + 3k^2 4 Since the square of a real number is always non-negative, (x + k 2 )^2 0 for all real x. The minimum value of f(x) occurs when x = - k 2 . Therefore, the minimum value of the expression is 3k^2 4 . Answer: 3k^2 4

Question 2

If the highest degree coefficient is equal to 1, then what is the total number of quadratic equations which are unchanged on squaring their roots ?

Accepted Answer

Let the quadratic equation be x^2 + px + q = 0. Let its roots be and . Sum of roots: + = -p Product of roots: = q The roots of the new equation are ^2 and ^2. Sum of new roots: ^2 + ^2 = ( + )^2 - 2 = p^2 - 2q Product of new roots: ^2 ^2 = q^2 Since the equation remains unchanged, the sum and product of the roots must be the same as the original equation. Therefore, p^2 - 2q = -p and q^2 = q. From q^2 = q, we get q = 0 or q = 1. Case 1: q = 0 Substituting q = 0 in p^2 - 2q = -p, we get p^2 = -p

Question 3

Let and be the roots of the quadratic equation x^2-2bx+c^2=0 where b, c are positive real numbers. Let A be the arithmetic mean of and ; and G be the geometric mean of and . What are the roots of the

Accepted Answer

Given the quadratic equation x^2 - 2bx + c^2 = 0 with roots and . Sum of the roots is + = 2b. Product of the roots is = c^2. The arithmetic mean A of and is given by A = + 2 = 2b 2 = b. The geometric mean G of and is given by G = = c^2 = c (since c is a positive real number). The second quadratic equation is x^2 - (b+c)x + bc = 0. Factoring the equation, we get (x - b)(x - c) = 0. The roots are x = b and x = c. Substituting b = A and c = G, the roots are A and G. Answer: A,\ G

Question 4

Consider the following statements : I. x +x+1=0 has two irrational roots. II. 5 x -x-4=0 has two rational roots. Which of the statements given above is/are correct ?

Accepted Answer

For Statement I: Let y = x , where y 0. The equation becomes y^2 + y + 1 = 0. The discriminant is = 1^2 - 4(1)(1) = -3 Since the discriminant is negative, there are no real roots. Thus, Statement I is incorrect. For Statement II: Let y = x , where y 0. The equation becomes y^2 - 5y + 4 = 0. Factoring the quadratic gives (y - 1)(y - 4) = 0 y = 1 or y = 4. Substituting y = x , we get x = 1 or x = 16. Both 1 and 16 are rational numbers. Thus, Statement II is correct. Therefore, only Statement II is

Question 5

For the following two (02) items: Let f(x) = ax^2 + bx + c be a quadratic polynomial such that f(1) = f(4) = 2. Further, 2 is a root of f(x) = 0. What is (a + b + c) equal to?

Accepted Answer

Given f(x) = ax^2 + bx + c Substituting x = 1 in the given polynomial: f(1) = a(1)^2 + b(1) + c = a + b + c It is given in the problem that f(1) = 2. Therefore, a + b + c = 2. Answer: 2

Question 6

For the following two (02) items: Let and be the roots of the quadratic equation x^2 + ( _ 0.5 (a^2))x + ( _ 0.5 (a^2))^4 = 0, where a^2 1 and _ 0.5 (a^2) > 0. Further, ^2 = ( _ a^2 (0.5)). What is th

Accepted Answer

Let p = _ 0.5 (a^2). We are given p > 0. The given quadratic equation is: x^2 + px + p^4 = 0 Let the roots be and . Then: + = -p = p^4 We are given the relation: ^2 = _ a^2 (0.5) Using the change of base formula, _ a^2 (0.5) = 1 _ 0.5 (a^2) = 1 p . Thus, the relation becomes: ^2 = p = p ^2 Substitute = p ^2 into the product of roots: (p ^2)( ) = p^4 p ^3 = p^4 Since p > 0, we can divide by p to get: ^3 = p^3 Assuming is a real root, we have = p. Substitute = p into the sum of roots: + p = -p = -

Question 7

For the following two (02) items: Let f(x) = ax^2 + bx + c be a quadratic polynomial such that f(1) = f(4) = 2. Further, 2 is a root of f(x) = 0. What is the other root of f(x) = 0?

Accepted Answer

Given f(1) = f(4) = 2, the quadratic polynomial can be written as f(x) = a(x-1)(x-4) + 2. Since 2 is a root of f(x) = 0, we have f(2) = 0. Substituting x = 2 in the expression for f(x): a(2-1)(2-4) + 2 = 0 a(1)(-2) + 2 = 0 -2a + 2 = 0 a = 1 Thus, the polynomial is f(x) = 1(x-1)(x-4) + 2 = x^2 - 5x + 6. To find the roots of f(x) = 0, we solve x^2 - 5x + 6 = 0. (x-2)(x-3) = 0 The roots are x = 2 and x = 3. The other root is 3. Answer: 3

Question 8

If one root of the equation x^2 - kx + k = 0 exceeds the other by 2 3 , then which one of the following is a value of k?

Accepted Answer

Let the roots of the equation x^2 - kx + k = 0 be and . Given that one root exceeds the other by 2 3 , we have | - | = 2 3 . Squaring both sides, we get ( - )^2 = 12. Using the identity ( - )^2 = ( + )^2 - 4 , we can substitute the sum and product of the roots. From the given quadratic equation, + = k and = k. Substituting these values, we get k^2 - 4k = 12. k^2 - 4k - 12 = 0 (k - 6)(k + 2) = 0 Thus, k = 6 or k = -2. Among the given options, 6 is a value of k. Answer: 6

Question 9

For the following two (02) items: Let and be the roots of the quadratic equation x^2 + ( _ 0.5 (a^2))x + ( _ 0.5 (a^2))^4 = 0, where a^2 1 and _ 0.5 (a^2) > 0. Further, ^2 = ( _ a^2 (0.5)). What is eq

Accepted Answer

Let p = _ 0.5 (a^2). The given quadratic equation is x^2 + px + p^4 = 0. The product of the roots is = p^4. We are given the condition ^2 = ( _ a^2 (0.5)). Using the properties of logarithms, _ a^2 (0.5) = 1 _ 0.5 (a^2) = 1 p . Substituting this into the given condition gives ^2 = p , which implies = p ^2. Substituting = p ^2 into the product of roots equation gives: (p ^2) = p^4 p ^3 = p^4 Since p = _ 0.5 (a^2) > 0, we have p 0. Dividing by p yields: ^3 = p^3 Considering real roots, we obtain =

Question 10

If x = (1111)_2, y = (1001)_2 and z = (110)_2, then what is x^3 - y^3 - z^3 - 3xyz equal to?

Accepted Answer

Converting the given binary numbers to decimal: x = (1111)_2 = 1 2^3 + 1 2^2 + 1 2^1 + 1 2^0 = 15 y = (1001)_2 = 1 2^3 + 0 2^2 + 0 2^1 + 1 2^0 = 9 z = (110)_2 = 1 2^2 + 1 2^1 + 0 2^0 = 6 Observing the relationship between x, y, and z: y + z = 9 + 6 = 15 = x x - y - z = 0 Using the algebraic identity, if a + b + c = 0, then a^3 + b^3 + c^3 - 3abc = 0. Substituting a = x, b = -y, and c = -z: x + (-y) + (-z) = 0 x^3 + (-y)^3 + (-z)^3 - 3(x)(-y)(-z) = 0 x^3 - y^3 - z^3 - 3xyz = 0 In binary represent

Question 11

If x^2 - x + 1 = 0, then what is (x - 1 x )^2 + (x - 1 x )^4 + (x - 1 x )^8 equal to?

Accepted Answer

Given x^2 - x + 1 = 0 Dividing the entire equation by x, we get: x - 1 + 1 x = 0 x + 1 x = 1 Using the algebraic identity (x - 1 x )^2 = (x + 1 x )^2 - 4 Substituting the value of x + 1 x : (x - 1 x )^2 = (1)^2 - 4 = -3 The given expression is (x - 1 x )^2 + (x - 1 x )^4 + (x - 1 x )^8 This can be written as (x - 1 x )^2 + ( (x - 1 x )^2 )^2 + ( (x - 1 x )^2 )^4 Substituting (x - 1 x )^2 = -3 into the expression: (-3) + (-3)^2 + (-3)^4 = -3 + 9 + 81 = 87 Answer: 87

Question 12

If x + 5 y = 4 and y + 5 x = -4, then what is (x + y) equal to?

Accepted Answer

Given equations are x + 5 y = 4 and y + 5 x = -4. Adding the two equations, we get: x + y + 5 x + 5 y = 0 (x + y) + 5 ( x + y xy ) = 0 (x + y) (1 + 5 xy ) = 0 This implies either x + y = 0 or 1 + 5 xy = 0. Subtracting the second equation from the first, we get: x - y - 5 x + 5 y = 8 (x - y) + 5 ( x - y xy ) = 8 (x - y) (1 + 5 xy ) = 8 If 1 + 5 xy = 0, the second factor becomes zero, which gives 0 = 8, a contradiction. Thus, 1 + 5 xy 0. Therefore, we must have x + y = 0. Answer: 0

Question 13

For the following two (02) items: Consider the equation abx^2 + bcx + ca = cax^2 + abx + bc. If the roots of the equation are equal, then a, b, c are in

Accepted Answer

The given equation is abx^2 + bcx + ca = cax^2 + abx + bc. Rearranging the terms, we get: (ab - ca)x^2 + (bc - ab)x + (ca - bc) = 0 a(b - c)x^2 + b(c - a)x + c(a - b) = 0 The sum of the coefficients is: a(b - c) + b(c - a) + c(a - b) = ab - ac + bc - ab + ac - bc = 0 Since the sum of the coefficients is 0, one root of the equation is x = 1. It is given that the roots of the equation are equal, so the other root must also be x = 1. The product of the roots is given by c(a - b) a(b - c) . Equating

Question 14

For the following two (02) items: Consider the equation abx^2 + bcx + ca = cax^2 + abx + bc. If the roots of the equation are equal, then which one of the following is correct?

Accepted Answer

The given equation is abx^2 + bcx + ca = cax^2 + abx + bc. Rearranging the terms, we get: (ab - ca)x^2 + (bc - ab)x + (ca - bc) = 0 a(b - c)x^2 + b(c - a)x + c(a - b) = 0 The sum of the coefficients is: a(b - c) + b(c - a) + c(a - b) = ab - ac + bc - ab + ac - bc = 0 Since the sum of the coefficients is zero, one root of the equation is x = 1. It is given that the roots of the equation are equal, so the other root must also be x = 1. The product of the roots is given by c(a - b) a(b - c) . Equat

Quadratic Equation — Previous Year Questions & Solutions