Vardaan Learning Institute

1. Which of the following is a quadratic equation?

1. $x^2 + 2x + 1 = (4-x)^2 + 3$
2. $-2x^2 = (5-x)(2x - 2/5)$
3. $(k+1)x^2 + 3/2 x = 7$ (where k = -1)
4. $x^3 - x^2 = (x-1)^3$

2. If the discriminant of $ax^2 + bx + c = 0$ is zero, then the roots are:

1. Real and distinct
2. Real and equal
3. Imaginary
4. Zero

3. The roots of the quadratic equation $x^2 - 0.04 = 0$ are:

1. $\pm 0.2$
2. $\pm 0.02$
3. $0.4$
4. $0.2$

4. Values of k for which the quadratic equation $2x^2 - kx + k = 0$ has equal roots is:

1. 0 only
2. 4
3. 8 only
4. 0, 8

5. If 1/2 is a root of the equation $x^2 + kx - 5/4 = 0$, then value of k is:

1. 2
2. -2
3. 1/4
4. 1/2

6. The quadratic equation $2x^2 - \sqrt{5}x + 1 = 0$ has:

1. Two distinct real roots
2. Two equal real roots
3. No real roots
4. More than 2 roots

7. Which constant must be added and subtracted to solve the quadratic equation $9x^2 + \frac{3}{4}x - \sqrt{2} = 0$ by the method of completing the square?

1. $1/8$
2. $1/64$
3. $1/4$
4. $9/64$

8. If equation $9x^2 + 6kx + 4 = 0$ has equal roots, then value of k is:

1. $\pm 2$
2. $\pm 3/2$
3. 0
4. $\pm 3$

9. The roots of $x^2 + 7x + 10 = 0$ are:

1. 2, 5
2. -2, -5
3. -2, 5
4. 2, -5

10. Assertion (A): The equation $x^2 + 3x + 1 = (x-2)^2$ is a linear equation.
Reason (R): The highest power of x in the equation is 1.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is not the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

11. Find the roots of the quadratic equation by factorization: $3x^2 - 5x + 2 = 0$.

12. Find the value of p so that the quadratic equation $px(x-3) + 9 = 0$ has two equal roots.

13. Solve for x: $4x^2 - 4a^2x + (a^4 - b^4) = 0$.

14. The product of two consecutive positive integers is 306. Form the quadratic equation to find the integers.

15. A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

16. Find the roots of the quadratic equation $5x^2 - 6x - 2 = 0$ by the method of completing the square (or quadratic formula).

17. Two water taps together can fill a tank in $9\frac{3}{8}$ hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

18. Is it possible to design a rectangular park of perimeter 80 m and area 400 $m^2$? If so, find its length and breadth.

19. If the equation $(1+m^2)x^2 + 2mcx + c^2 - a^2 = 0$ has equal roots, then show that $c^2 = a^2(1+m^2)$.

20. Solve for x: $\frac{1}{a+b+x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x}$, $a+b \neq 0, x \neq 0, x \neq -(a+b)$.

21. A plane left 30 minutes later than the scheduled time and in order to reach its destination 1500 km away in time, it has to increase its speed by 250 km/h from its usual speed. Find its usual speed.

22. Rs 6500 were divided equally among a certain number of persons. Had there been 15 more persons, each would have got Rs 30 less. Find the original number of persons.

23. Case Study: Basketball Trajectory
A basketball is thrown into the hoop. The height h of the ball (in feet) at t seconds is given by $h = -16t^2 + 20t + 6$.

(i) What is the initial height of the ball (at t=0)? (1 Mark)
(ii) Find the height of the ball after 0.5 seconds. (1 Mark)
(iii) When will the ball hit the ground (h=0)? (2 Marks)