True Discount - Quantitative Aptitude
Q1. If the roots of the equation x2+x+1 = 0 are α,β then the value of α3+β3 is
(a) 2
(b) 3
(c) 5
(d) 6
Answer: (a) 2
Here a = 1, b = 1, c = 1
∴ α+β = - b/a = -1
αβ = c/a = 1/1 = 1
α3+β3 = (α + β)3 - 3αβ (α + β)
= (-1)3 - 3 X 1.(-1) = -1 + 3 = 2
Q2. If the roots of the equation x2+x+2 = 0 are α,β then the value of α4+α2β2+β4 is
(a) 0
(b) 2
(c) 3
(d) 5
Answer: (a) 0
α4+α2β2+β4 = (α2+β2)2 - α2β2
=[(α + β)2 - 2αβ]2 - α2β2
=[(-1)2 - 2 X 1]2 - 12
=[1-2]2 - 1 = 1 - 1 = 0
Q3. If the roots of the equation x(x - 3) = 0 are α,β then α2+β2 = ?
(a) 3
(b) 7
(c) 9
(d) 11
Answer: (c) 9
x(x - 3) = 0
⇒ x2 - 3x = 0
∴ α + β = - b/a = - (-3/1) = 3
αβ = c/a = 0/1 = 0
∴ α2 + β2 = (α + β)2 - 2αβ = 32 - 2 X 0 = 9
Q4. Find the value of K in the equation 2x2 - 5x + K = 0 if one root is double the other
(a) 13/9
(b) 17/9
(c) 23/9
(d) 25/9
Answer: (d) 25/9
Let, the roots of the equation be α, 2α
∴ α + 2β = - b/a = - (-5/2) = 5/2
⇒ 3α = 5/2
⇒ α = 5/6
α.2α = c/a = K/2
⇒ 2α2 = K/2
⇒ 2.(5/6)2 = K/2
⇒ 2.25/36 = K/2
⇒ K = (50 X 2)/36 = 25/9
Q5. Find the value of P if the equations (x+3)(x+7) = 0 and x2 + 9x + P = 0 has a common root.
(a) 8 or 12
(b) 12 or 14
(c) 12 or 16
(d) 14 or 18
Answer: (d) 14 or 18
Let, the roots of the equation be α
∴ (α + 3)(α + 7) = 0
⇒ α2 + 10α + 21 = 0
And α2 + 9α + P = 0
| Now | α2 | = | α | = | 1 |
| 10P-189 | 21-P | 9-10 |
| ⇒ | α2 | = | α | = | 1 |
| 10P-189 | 21-P | -1 |
∴ α2 = -10P + 189, α = P - 21
⇒ (P-21)2 = - 10P + 189
⇒ P2 + 441 - 42P = 189 - 10P
⇒ P2 - 32P + 252 = 0
⇒ P2 - 14P - 18P + 252 = 0
⇒ (P - 14)(P - 18) = 0
∴ P = 14 or P = 18
P = 14 or 18
Q6. Solve :- 4x-2 + 5x-1 + 1 = 0
(a) {-1, -3}
(b) {-1, -4}
(c) {-1, -5}
(d) {-1, -7}
Answer: (b) {-1, -4}
let x-1 = y
∴ 4y2 + 5y + 1 = 0
⇒ 4y2 + 4y + y + 1 = 0
⇒ 4y(y+1) + 1(y+1) = 0
⇒ (y+1)(4y+1) = 0
∴ y = -1 or 4y = -1
⇒ x-1 = -1 ⇒ y = -1/4
⇒ 1/x = -1 ⇒ x-1 = -1/4
⇒ x = -1 ⇒ 1/x = -1/4
⇒ x = -1 ⇒ x = -4
∴ x = {-1, -4}
Q7. Find the value of x in the equation 41+x + 41-x = 10
(a) ± ½
(b) ± ¼
(c) ± ¾
(d) ¼
Answer: (a) ± ½
41+x + 41-x = 10
⇒ 41.4x + 41. 4-x = 10
⇒ y.4x + 4.1/4x = 10
putting 4x = y
4y + 4/y = 10
⇒ 4y2 + 4 = 10y
⇒ 4y2 - 10y + 4 = 0
⇒ (y-2)(4y-2) = 0
∴ y-2 = 0 or 4y-2 = 0
⇒ y = 2 ⇒ y = ½
⇒ 4x = 2 ⇒ 4x = ½
⇒ 22x = 21 ⇒ 22x = 2-1
⇒ 2x = 1 ⇒ 2x = -1
⇒ x = ½ ⇒ x = -½
∴ x = ± ½
Practice Test Exam