**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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