Class 10 - Chapter 4: Quadratic Equations

Section A (1 Mark each)

Q1. The graph of $y = ax^2 + bx + c$ is a parabola opening downwards if:

(a) $a > 0$ (b) $a < 0$ (c) $a = 0$ (d) $a > 0, c < 0$

Q2. If one root of the quadratic equation $2x^2 + kx + 1 = 0$ is $-\frac{1}{2}$, then the value of $k$ is:

(a) 5 (b) -3 (c) 3 (d) -5

Q3. Which of the following equations has 2 as a root?

(a) $x^2 - 4x + 5 = 0$ (b) $x^2 + 3x - 12 = 0$ (c) $2x^2 - 7x + 6 = 0$ (d) $3x^2 - 6x - 2 = 0$

Q4. If the roots of the equation $x^2 - kx + 1 = 0$ are not real, then:

(a) $-2 < k < 2$ (b) $k > 2$ (c) $k < -2$ (d) $k = 2$

Q5. Assertion (A): The equation $(x+1)^2 - x^2 = 0$ has two real roots.
Reason (R): A quadratic equation involving real coefficients has at most two real roots.

(a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.

Section B (2 Marks each)

Q6. Find the value of $p$ so that the quadratic equation $px(x-3) + 9 = 0$ has two equal roots.

Solve on white paper.

Q7. Solve for $x$: $\sqrt{2x + 9} + x = 13$.

Solve on white paper.

Q8. The product of two consecutive odd integers is 483. Find the integers.

Solve on white paper.

Section C (3 Marks each)

Q9. 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.

Solve on white paper.

Q10. Two pipes running together can fill a tank in $11 \frac{1}{9}$ minutes. If one pipe takes 5 minutes more than the other to fill the tank separately, find the time in which each pipe would fill the tank separately.

Solve on white paper.

Section D (Case Study 1 - 4 Marks)

Q11. Case Study: Basketball Trajectory

The path of a basketball thrown by a player is given by the equation $h(t) = -t^2 + 2t + 3$, where $h$ is the height in meters and $t$ is the time in seconds.

(i) What is the shape of the path represented by the equation?

(ii) At what time does the ball hit the ground?

(iii) What is the maximum height achieved by the ball?

Solve on white paper.

Section E (Case Study 2 - 4 Marks)

Q12. Case Study: Room Decor

A rug is placed in a room of dimensions $12 \text{ m} \times 8 \text{ m}$ such that a uniform strip of floor is left uncovered around the rug. The area of the rug is 60 sq. m.

(i) Form a quadratic equation representing the width of the strip ($x$).

(ii) Find the width of the strip left uncovered.

(iii) Find the dimensions (length and breadth) of the rug.

Solve on white paper.