Quadratic Equations

Find the quadratic equation whose roots are three times the roots of 3ax2 + 3bx +3c = 0.

1. ax2 + bx + 3x = 0
2. ax2 + 3bx + 3c = 0
3. 9ax2 + 9bx + c = 0
4. ax2 + 3bx + c = 0

Answer: (b) ax2 + 3bx + 3c = 0

If α and β are the roots of ax2 + bx + c = 0, α + k and β + k are the roots of px2 + qx + r = 0, then what is the value of k?

1. (ab - pq)
2. (a/b - p/q)
3. 1/2(b/a - q/p)
4. -1/2(a/b - p/q)

Answer: (c) 1/2(b/a - q/p)

The graphs of y = 2x2 and y = ax + b intersect at two points (2, 8) and (6, 72). Find the quadratic equation in x whose roots are a + 2 and b/4 - 1.

1. x2 - 13x - 126
2. x2 - 12x + 126
3. x2 + 11x +12
4. x2 - 11x - 126

Answer: (d) x2 - 11x - 126

Let p and q be the roots of the quadratic equation x2 - (α - 2)x - α - 1 = 0. What is the minimum possible value of p2 + q2?

1. 1
2. 2
3. 4
4. 5

Answer: (d) 5

If f(x) = -x2 + 6x - 5, then what could be the value of a if f(a) = f(1.5)?

1. 2.5, 3.5
2. 1.5, 4.5
3. 3.5, 4.5
4. 3, 4.5

Answer: (b) 1.5, 4.5

Which of the following statements about the sign of the quadratic expression E = y2 - 12y + 20 is true?

1. E ≤ 0 when 2 ≤ y ≤ 10 and E < 0 when y < 2 or y > 10.
2. E ≤ 0 when -10 ≤ y ≤ -2 and E > 0 when y < 2 or y > 10.
3. E ≤ 0 when 20 ≤ y ≤ 10 and E > 0 when y < 2 or y > 10.
4. E > 0 when 20 ≤ y ≤ 10 and E > 0 when y < 2 or y > 10.

Answer: (a) E ≤ 0 when 2 ≤ y ≤ 10 and E < 0 when y < 2 or y > 10.

A quadratic equation is given by 3cos2θ - 5cosθ + 2 = 0. How many valid solutions exist for θ¸ in the range 0° to 360°?

1. 3 solutions
2. 5 solutions
3. 4 solutions
4. 1 solution

Answer: (a) 3 solutions

In a forest, a certain number of apes equal to the square of one-eighth of the total number of their group are playing and having great fun. The rest of them are twelve in number and are on an adjoining hill. The echo of their shrieks from the hills frightens them. They come and join the apes in the forest and play with enthusiasm. What is the total number of apes in the forest?

1. 16
2. 12
3. 18
4. 16 or 48

Answer: (d) 16 or 48

Read the passage and answer the following question: A city authority in Greenfield decided to build a recreational park for the residents, promoting outdoor activities and relaxation. The park is designed in the shape of a rectangular field, with a length of 20 m and a breadth of 14 m. At the center of the park, there is a rectangular garden for flowers and small plants, with a pathway around it. The rectangular garden is surrounded by a uniformly wide pathway, and the area of the pathway is 120 m². If the width of the pathway is increased by w + 5 metres and the length and breadth of the rectangular garden is 20 and 14 m . metres, write a quadratic equation to represent the relationship between w, and the areas provided. The total area of the pathway is the same as before.

1. w2 + 27w + 80
2. w2 + 5w - 55
3. (w + 2)2 - 4w + 80
4. (w - 2)2 + 5w + 80

Answer: (a) w2 + 27w + 80

If x and y are positive real numbers satisfying the system of equations x2 + y√(xy) = 336 and y2 + x√(xy) = 112, then what is the value of x + y?

1. 15
2. 40
3. 20
4. √448

Answer: (c) 20