491. One root of the quadratic equation x2 – 12x + a = 0, is thrice the other. Find the value of a?
A. 29
B. 27
C. 28
D. 7

Explanation:
Let the roots of the quadratic equation be x and 3x.
Sum of roots = -(-12) = 12
a + 3a = 4a = 12 => a = 3
Product of the roots = 3a2 = 3(3)2 = 27.

492. The sum of the squares of two consecutive positive integers exceeds their product by 91. Find the integers?
A. 9, 10
B. 10, 11
C. 11, 12
D. 12, 13

Explanation:
Let the two consecutive positive integers be x and x + 1
x2 + (x + 1)2 – x(x + 1) = 91
x2 + x – 90 = 0
(x + 10)(x – 9) = 0 => x = -10 or 9.
As x is positive x = 9
Hence the two consecutive positive integers are 9 and 10.

493. If the roots of the equation 2×2 – 5x + b = 0 are in the ratio of 2:3, then find the value of b?
A. 3
B. 4
C. 5
D. 6

Explanation
Let the roots of the equation 2a and 3a respectively.
2a + 3a = 5a = -(- 5/2) = 5/2 => a = 1/2
Product of the roots: 6a2 = b/2 => b = 12a2
a = 1/2, b = 3.

494. The sum and the product of the roots of the quadratic equation x(power 2) + 20x + 3 = 0 are?
A. 10, 3
B. -10, 3
C. -20, 3
D. -10, -3

Explanation:
Any quadratic equation is of the form
x2 – (sum of the roots)x + (product of the roots) = 0 —- (1)
where x is a real variable. As sum of the roots is 13 and product of the roots is -140, the quadratic equation with roots as 20 and -7 is: x2 – 13x – 140 = 0.

495. If the roots of a quadratic equation are 20 and -7, then find the equation?
A. x2 + 13x – 140 = 0
B. x2 – 13x + 140 = 0
C. x2 – 13x – 140 = 0
D. x2 + 13x + 140 = 0

Explanation:
Any quadratic equation is of the form
x2 – (sum of the roots)x + (product of the roots) = 0 —- (1)
where x is a real variable. As sum of the roots is 13 and product of the roots is -140, the quadratic equation with roots as 20 and -7 is: x2 – 13x – 140 = 0.

496. The roots of the equation 3×2 – 12x + 10 = 0 are?
A. rational and unequal
B. complex
C. real and equal
D. irrational and unequal

Explanation:
The discriminant of the quadratic equation is (-12)2 – 4(3)(10) i.e., 24. As this is positive but not a perfect square, the roots are irrational and unequal.

497. Find the roots of the quadratic equation: 2×2 + 3x – 9 = 0?
A. 3, -3/2
B. 3/2, -3
C. -3/2, -3
D. 3/2, 3

Explanation:
2×2 + 6x – 3x – 9 = 0
2x(x + 3) – 3(x + 3) = 0
(x + 3)(2x – 3) = 0
=> x = -3 or x = 3/2.

498. Find the roots of the quadratic equation: x2 + 2x – 15 = 0?
A. -5, 3
B. 3, 5
C. -3, 5
D. -3, -5

499. An order was placed for the supply of a carper whose length and breadth were in the ratio of 3 : 2. Subsequently, the dimensions of the carpet were altered such that its length and breadth were in the ratio 7 : 3 but were was no change in its parameter. Find the ratio of the areas of the carpets in both the cases.
A. 4 : 3
B. 8 : 7
C. 4 : 1
D. 6 : 5

Explanation
Let the length and breadth of the carpet in the first case be 3x units and 2x units respectively.
Let the dimensions of the carpet in the second case be 7y, 3y units respectively.
From the data,.
2(3x + 2x) = 2(7y + 3y)
=> 5x = 10y
=> x = 2y
Required ratio of the areas of the carpet in both the cases
= 3x * 2x : 7y : 3y
= 6×2 : 21y2
= 6 * (2y)2 : 21y2
= 6 * 4y2 : 21y2
= 8 : 7

500. The sector of a circle has radius of 21 cm and central angle 135o. Find its perimeter?
A. 91.5 cm
B. 93.5 cm
C. 94.5 cm
D. 92.5 cm

Explanation
Perimeter of the sector = length of the arc + 2(radius)
= (135/360 * 2 * 22/7 * 21) + 2(21)
= 49.5 + 42 = 91.5 cm

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