QUADRATIC EQUATIONS AND INEQUATIONS (CDS)

Total Questions: 67

51. If a, b and a satisfy the equation x³ - 3x² + 2x + 1 = 0 then what is the value of 1/a + 1/b + 1/c ? [2015 (II) Evening Shift]

Correct Answer: (c) -2
Solution:

52. If k = x - y + 2z , where - 2 ≤ x ≤ 1 and - 1 ≤ y ≤ 2 and 3 ≤ z ≤ 6, then which one of the following is correct? [2015 (II) Evening Shift]

Correct Answer: (c) 2 ≤ k ≤ 14
Solution:

53. If the roots of the quadratic equation x² - 4x - log₁₀ N = 0 are all real, then the minimum value of N is [2015 (II) Evening Shift]

Correct Answer: (c) 1/10000
Solution:We have,

54. The difference of maximum values of the expressions (6 + 5x - x²) and (y - 6 - y²) for any real values of x and y,is [2015 (II) Evening Shift]

Correct Answer: (c) 18
Solution:


55. The sign of the quadratic polynomial ax² + bx + c is always positive, if [2015 (I) Morning Shift]

Correct Answer: (a) a is positive and b² - 4ac ≤ 0
Solution:If a is positive and b² - 4ac ≤ 0 then the sign of quadratic polynomial ax² + bx + c is always positive.

56. For which value of k does the pair of equations x² - y² = 0 and (x - k)² + y²= 1 yield a unique positive solution of x ? [2015 (I) Morning Shift]

Correct Answer: (c) √2
Solution:Given, x² - y² = 0


57. If one of the roots of the equation px² + qx + r = 0 is three times the other, then which one of the following relations is correct? [2014 (II) Evening Shift]

Correct Answer: (a) 3q² = 16pr
Solution:Given equation is px² + qx + r = 0.

58. If the equation x² + 2(l + k)x + k² = 0 has equal roots, then what is the value of k ? [2014 (II) Evening Shift]

Correct Answer: (b) - 1/2
Solution:Given equation is x² + 2(l + k)x + k² = 0

59. If m and n are the roots of the equation x² + ax + b = 0 and m² , n² are the roots of the equation x² - cx + d = 0 then which of the following is/are correct? [2014 (II) Evening Shift]

1. 2b - a²  = c
2. b²  = d

Select the correct answer using the codes given below.

Correct Answer: (b) Only 2
Solution:1. Given, m and n are the roots of the equation x² + ax + b = 0

60. If m and n(m > n) are the roots of the equation 7(x + 2a)² + 3a² = 5a(7x + 23a) where a > 0 , then what is 3m - n equal to? [2014 (II) Evening Shift]

Correct Answer: (c) 15a
Solution:Given equation is