Quadratic Equations: Class 10 Important Questions with Answers

HOTS (higher-order) practice

H1. Solve $\frac{1}{x+4}-\frac{1}{x-7}=\frac{11}{30}$, $x\ne -4,7$.

H2. If $-5$ is a root of $2x^2+px-15=0$ and $p(x^2+x)+k=0$ has equal roots, find $k$.

Case-based practice

20 MCQs (1 mark each)

1. Which is a quadratic equation?
   (a) $x^2+\frac{1}{x}=2$ (b) $(x+1{)}^2=x^2+3$ (c) $x(x+2)=5$ (d) $x^3+x=2$
2. The roots of $x^2-7x+12=0$ are
   (a) 3, 4 (b) −3, −4 (c) 2, 6 (d) −3, 4
3. The discriminant of $2x^2-5x+3=0$ is
   (a) 1 (b) 49 (c) −1 (d) 5
4. The roots of $x^2-4x+4=0$ are
   (a) real, distinct (b) real, equal (c) not real (d) irrational
5. If $x=2$ is a root of $x^2+kx-10=0$, then $k$ is
   (a) 2 (b) −3 (c) 3 (d) 5
6. $x^2+kx+9=0$ has equal roots when $k$ is
   (a) 3 (b) ±3 (c) ±6 (d) 9
7. $3x^2+2x+1=0$ has
   (a) 2 distinct real roots (b) equal roots (c) no real roots (d) one root
8. The sum of the roots of $3x^2-6x+1=0$ is
   (a) −2 (b) 2 (c) $\frac{1}{3}$ (d) 6
9. The product of the roots of $2x^2+7x-4=0$ is
   (a) 2 (b) −2 (c) $-\frac{7}{2}$ (d) $\frac{7}{2}$
10. The quadratic equation with roots 3 and −5 is
    (a) $x^2-2x-15=0$ (b) $x^2+2x-15=0$ (c) $x^2+2x+15=0$ (d) $x^2-8x+15=0$
11. The roots of $a{x}^2+bx+c=0$ are real and distinct if
    (a) $b^2-4ac=0$ (b) $b^2-4ac>0$ (c) $b^2-4ac<0$ (d) $b^2=4a$
12. The roots of $x^2-2=0$ are
    (a) ±2 (b) $\pm \sqrt{2}$ (c) 2, 0 (d) $\sqrt{2},\sqrt{2}$
13. Two consecutive positive integers have product 72. The equation is
    (a) $x^2+x-72=0$ (b) $x^2-x-72=0$ (c) $x^2+x+72=0$ (d) $x^2-72=0$
14. $k{x}^2-4x+1=0$ has equal roots when $k$ is
    (a) 2 (b) 4 (c) −4 (d) 16
15. The roots of $2x^2-x-6=0$ are
    (a) 2, $\frac{3}{2}$ (b) −2, $\frac{3}{2}$ (c) 2, $-\frac{3}{2}$ (d) −2, $-\frac{3}{2}$
16. If $D=0$, each root of $a{x}^2+bx+c=0$ is
    (a) $-\frac{b}{a}$ (b) $-\frac{b}{2a}$ (c) $\frac{b}{2a}$ (d) $\frac{c}{a}$
17. Which equation has two distinct real roots?
    (a) $x^2+x+1=0$ (b) $x^2-2x+1=0$ (c) $2x^2-3x-1=0$ (d) $x^2+4=0$
18. A rectangle's length is 3 m more than its breadth and its area is 40 m². The breadth is
    (a) 8 m (b) 5 m (c) 4 m (d) 10 m
19. The roots of $x^2+6x+5=0$ are
    (a) 1, 5 (b) −1, −5 (c) −1, 5 (d) 1, −5
20. One root of $2x^2+kx+4=0$ is 2. The other root is
    (a) 1 (b) −1 (c) 2 (d) 4

Answer key

1 (c) · 2 (a) · 3 (a) · 4 (b) · 5 (c) · 6 (c) · 7 (c) · 8 (b) · 9 (b) · 10 (b) · 11 (b) · 12 (b) · 13 (a) · 14 (b) · 15 (c) · 16 (b) · 17 (c) · 18 (b) · 19 (b) · 20 (a)

Hints: Q1 (b) simplifies to $2x-2=0$. Q5 $4+2k-10=0$. Q7 $4-12<0$. Q14 $16-4k=0$. Q17 $D=9+8=17$. Q18 $x(x+3)=40\Rightarrow x=5$. Q20 product $=\frac{4}{2}=2=2\times 1$.

Assertion–Reason (10)

(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.

1. A: $x^2+2x+3=0$ has no real roots. R: If $b^2-4ac<0$, a quadratic equation has no real roots.
2. A: $(x-1{)}^2=x^2+5$ is a quadratic equation. R: An equation of degree 2 is a quadratic equation.
3. A: $3x^2-4\sqrt{3}x+4=0$ has two equal roots. R: When $D=0$, the two roots are equal.
4. A: The roots of $x^2-5x+6=0$ are 2 and 3. R: Every quadratic equation has two distinct real roots.
5. A: If $x=1$ is a root of $x^2+kx-3=0$, then $k=2$. R: A root of an equation satisfies the equation.
6. A: The sum of the roots of $2x^2-8x+3=0$ is 4. R: The sum of the roots of $a{x}^2+bx+c=0$ is $-\frac{b}{a}$.
7. A: The product of the roots of $x^2-9=0$ is −9. R: The roots of $x^2-9=0$ are equal in size and opposite in sign.
8. A: $x^2+4=0$ has roots ±2. R: $x^2=4$ has roots ±2.
9. A: $4x^2-12x+9=0$ has the equal roots $\frac{3}{2},\frac{3}{2}$. R: A quadratic equation has at most two roots.
10. A: $2x^2+3x+5=0$ has two real roots. R: The product of the roots of $2x^2+3x+5=0$ is $\frac{5}{2}$.

Key: 1 (a) · 2 (d) · 3 (a) · 4 (c) · 5 (a) · 6 (a) · 7 (b) · 8 (d) · 9 (b) · 10 (d)

Q2: expanding gives $-2x-4=0$, which is linear. Q8: $x^2=-4$ has no real root. Q10: $D=9-40<0$.

Very short answer (1 mark each)

1. Find the discriminant of $x^2-3x-4=0$. Ans. $9+16=25$
2. Is $x=-2$ a root of $x^2+5x+6=0$? Ans. $4-10+6=0$, yes
3. Find $k$ if $x^2-6x+k=0$ has equal roots. Ans. $36-4k=0\Rightarrow k=9$
4. Find the roots of $x^2-16=0$. Ans. ±4
5. Write the quadratic equation whose roots are 0 and −4. Ans. $x^2+4x=0$
6. State the nature of the roots of $2x^2+x+1=0$. Ans. $D=-7$: no real roots
7. Find the sum of the roots of $x^2-5x+2=0$. Ans. 5
8. If the product of the roots of $k{x}^2-3x+6=0$ is 2, find $k$. Ans. $\frac{6}{k}=2\Rightarrow k=3$

Short answer practice (2 marks each)

1. Solve by factorisation: $4\sqrt{3}{x}^2+5x-2\sqrt{3}=0$.

2. For what values of $k$ does $k{x}^2+2x+1=0$ have two distinct real roots?

3. Solve $\frac{1}{x}-\frac{1}{x-2}=3$, $x\ne 0,2$.

Short answer practice (3 marks each)

4. The sum of a number and its reciprocal is $\frac{10}{3}$. Find the number.

5. A motorboat whose speed in still water is 18 km/h takes 1 hour more to go 24 km upstream than to return downstream. Find the speed of the stream.

6. Find $k$ for which $(k-12)x^2+2(k-12)x+2=0$ has equal roots.

7. The diagonal of a rectangular field is 60 m more than the shorter side. If the longer side is 30 m more than the shorter side, find the sides.

Long answer (5 marks each)

8. 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.

9. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.