CBSE Class 10 · Chapter 4 · Algebra · 2026-27

Quadratic Equations Class 10: MCQ and case study questions

15 multiple-choice questions and 2 case-based questions on the current (rationalised) syllabus, in the style of the board paper's Sections A and E. Try each one first; the answer, a one-line reason and a worked solution open on a tap.

15 MCQs (1 mark each)
2 case studies (4 marks each)
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Multiple-choice questions

Tap an option to check it; the answer and a worked solution open. Section A of the board paper has 18 MCQs of 1 mark each, spread over all chapters.

Q1

·1 mark·Multiple choiceStandard form

Which of the following is not a quadratic equation?

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Answer: (b)

(x - 3)^{2} = x^{2} + 1

Why: (x − 3)² = x² + 1 simplifies to −6x + 8 = 0, which is linear.

$(x - 3)^{2} = x^{2} + 1 \Rightarrow x^{2} - 6 x + 9 = x^{2} + 1 \Rightarrow - 6 x + 8 = 0$ : degree $1$ . The others reduce to degree $2$ .

Q2

·1 mark·Multiple choiceSolution by factorisation / quadratic formula

The roots of $x^{2} - 11 x + 28 = 0$ are

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Answer: (a)

4

and

7

Why: 28 = 4 × 7 and 4 + 7 = 11: (x − 4)(x − 7) = 0.

$x^{2} - 11 x + 28 = (x - 4) (x - 7) = 0 \Rightarrow x = 4$ or $x = 7$ .

Q3

·1 mark·Multiple choiceDiscriminant and nature of roots

The discriminant of $2 x^{2} - 5 x + 4 = 0$ is

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Answer: (b)

- 7

Why: D = b² − 4ac = 25 − 32.

$D = (- 5)^{2} - 4 (2) (4) = 25 - 32 = - 7$ .

Q4

·1 mark·Multiple choiceDiscriminant and nature of roots

The equation $x^{2} + 5 x + 7 = 0$ has

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Answer: (c) no real roots

Why: D = 25 − 28 = −3 < 0.

$D = 5^{2} - 4 (1) (7) = - 3 < 0$ , so there are no real roots.

Q5

·1 mark·Multiple choiceSolution by factorisation / quadratic formula

Using the quadratic formula, the roots of $x^{2} - 6 x + 4 = 0$ are

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Answer: (b)

3 \pm \sqrt{5}

Why: x = (6 ± √(36 − 16))/2 = (6 ± 2√5)/2.

$x = \frac{6 \pm \sqrt{36 - 16}}{2} = \frac{6 \pm 2 \sqrt{5}}{2} = 3 \pm \sqrt{5}$ .

Q6

·1 mark·Multiple choiceSolution by factorisation / quadratic formula

The roots of $x^{2} + p x + q = 0$ are $3$ and $- 5$ . Then $p + q$ equals

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Answer: (d)

- 13

Why: The equation is (x − 3)(x + 5) = x² + 2x − 15 = 0, so p = 2, q = −15.

A quadratic with roots $3$ and $- 5$ is $(x - 3) (x + 5) = x^{2} + 2 x - 15$ . So $p = 2$ , $q = - 15$ and $p + q = - 13$ .

Q7

·1 mark·Multiple choiceSolution by factorisation / quadratic formula

The roots of $2 x^{2} - 3 x - 5 = 0$ are

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Answer: (c)

\frac{5}{2}

and

- 1

Why: Split −3x as −5x + 2x: (2x − 5)(x + 1) = 0.

$2 x^{2} - 5 x + 2 x - 5 = x (2 x - 5) + 1 (2 x - 5) = (2 x - 5) (x + 1) = 0$ , so $x = \frac{5}{2}$ or $x = - 1$ .

Q8

·1 mark·Multiple choiceDiscriminant and nature of roots

The equation $2 x^{2} + 4 x + k = 0$ has no real roots exactly when

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Answer: (a)

k > 2

Why: No real roots needs D = 16 − 8k < 0.

$D = 16 - 8 k < 0 \Rightarrow k > 2$ . At $k = 2$ , $D = 0$ (equal roots).

Q9

·1 mark·Multiple choiceSituational problems

The product of two consecutive positive integers is $132$ . The integers are

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Answer: (a)

11

and

12

Why: n(n + 1) = 132 gives n² + n − 132 = 0 = (n + 12)(n − 11).

$n (n + 1) = 132 \Rightarrow n^{2} + n - 132 = 0 \Rightarrow (n + 12) (n - 11) = 0$ . As $n > 0$ , $n = 11$ : the integers are $11$ and $12$ .

Q10

·1 mark·Multiple choiceSituational problems

The length of a rectangular plot is $3$ m more than its breadth and its area is $70 m^{2}$ . The breadth is

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Answer: (c)

7

Why: b(b + 3) = 70 gives b² + 3b − 70 = 0 = (b + 10)(b − 7).

$b (b + 3) = 70 \Rightarrow b^{2} + 3 b - 70 = 0 \Rightarrow (b + 10) (b - 7) = 0$ . Breadth is positive, so $b = 7$ m.

Q11

·1 mark·Multiple choiceSolution by factorisation / quadratic formula

The roots of $(x - 3) (x + 2) = 14$ are

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Answer: (b)

5

and

- 4

Why: Expand first: x² − x − 20 = 0 = (x − 5)(x + 4).

$x^{2} - x - 6 = 14 \Rightarrow x^{2} - x - 20 = 0 \Rightarrow (x - 5) (x + 4) = 0$ , so $x = 5$ or $- 4$ . (Setting each bracket to $0$ is wrong: the right side is not $0$ .)

Q12

·1 mark·Multiple choiceDiscriminant and nature of roots

The values of $k$ for which $x^{2} - k x + 25 = 0$ has equal roots are

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Answer: (a)

\pm 10

Why: Equal roots need D = k² − 100 = 0.

$D = (- k)^{2} - 4 (1) (25) = k^{2} - 100 = 0 \Rightarrow k = \pm 10$ .

Q13

·1 mark·Multiple choiceDiscriminant and nature of roots

Which of the following equations has two equal roots?

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Answer: (c)

x^{2} - 10 x + 25 = 0

Why: x² − 10x + 25 = (x − 5)², so D = 100 − 100 = 0.

Discriminants: $16$ , $- 3$ , $0$ , $8$ . Only $x^{2} - 10 x + 25 = 0$ has $D = 0$ (root $5$ twice).

Q14

·1 mark·Multiple choiceSituational problems

The sum of a positive number and its reciprocal is $\frac{17}{4}$ . The number is

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Answer: (d)

4

\frac{1}{4}

Why: x + 1/x = 17/4 gives 4x² − 17x + 4 = 0 = (4x − 1)(x − 4).

$4 x^{2} + 4 = 17 x \Rightarrow 4 x^{2} - 17 x + 4 = 0 \Rightarrow (4 x - 1) (x - 4) = 0$ , so $x = 4$ or $\frac{1}{4}$ (both work).

Q15

·1 mark·Multiple choiceSituational problems

The product of Tara's age $5$ years ago and her age $5$ years from now is $144$ . Her present age is

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Answer: (d)

13

years

Why: (x − 5)(x + 5) = 144 gives x² = 169.

$(x - 5) (x + 5) = 144 \Rightarrow x^{2} - 25 = 144 \Rightarrow x^{2} = 169 \Rightarrow x = 13$ (age is positive).

Case study questions

Section E of the board paper has three case-based questions of 4 marks: a real-life passage, then parts of 1, 1 and 2 marks, with a choice (OR) on the 2-mark part. All Class 10 case study questions by chapter.

Case study 1: Cycling to the village (4 marks)

Aarav cycles $36$ km from his town to his grandmother's village at a steady speed of $x$ km/h. He notices that if he rode $3$ km/h faster, the trip would take $1$ hour less.

(i) Show that

x

satisfies

x^{2} + 3 x - 108 = 0

. [1 mark]

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Answer:

x^{2} + 3 x - 108 = 0

\frac{36}{x} - \frac{36}{x + 3} = 1 \Rightarrow 108 = x (x + 3) \Rightarrow x^{2} + 3 x - 108 = 0

. A1

(ii) Find the discriminant of this equation and state the nature of its roots. [1 mark]

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Answer:

441

; two distinct real roots

D = 9 + 432 = 441 > 0

: two distinct real roots. A1

(iii) Find Aarav's usual speed and the time his trip takes. [2 marks]

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Answer:

9

km/h;

4

hours

(x + 12) (x - 9) = 0

M1; speed is positive, so

x = 9

km/h and time

= 36 \div 9 = 4

hours. A1

OR Solve

x^{2} + 3 x - 108 = 0

by the quadratic formula and explain which root is rejected. [2 marks]

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Answer:

x = 9

- 12

; reject

- 12

x = \frac{- 3 \pm \sqrt{441}}{2} = \frac{- 3 \pm 21}{2}

M1, so

x = 9

- 12

; a speed cannot be negative, so

- 12

is rejected. A1

Case study 2: Buying school bags (4 marks)

A shopkeeper spends ₹ $1200$ on $x$ identical school bags. He notices that if each bag had cost ₹ $20$ less, he could have bought $5$ more bags for the same ₹ $1200$ .

(i) Show that

x^{2} + 5 x - 300 = 0

. [1 mark]

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Answer:

x^{2} + 5 x - 300 = 0

\frac{1200}{x} - \frac{1200}{x + 5} = 20 \Rightarrow 6000 = 20 x (x + 5) \Rightarrow x^{2} + 5 x - 300 = 0

. A1

(ii) Find the discriminant of

x^{2} + 5 x - 300 = 0

. [1 mark]

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Answer:

1225

D = 25 + 1200 = 1225 (= 35^{2})

. A1

(iii) How many bags did he buy, and what did each bag cost? [2 marks]

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Answer:

15

bags at ₹

80

each

(x + 20) (x - 15) = 0 \Rightarrow x = 15

(positive) M1; cost of each bag

= 1200 \div 15 = ₹ 80

. A1

OR Find the roots of

x^{2} + 5 x - 300 = 0

using the quadratic formula. [2 marks]

Show answer

Answer:

15

and

- 20

x = \frac{- 5 \pm 35}{2}

M1, so

x = 15

x = - 20

. A1

Next steps for Quadratic Equations

This free set is separate from the chapter's question bank. On the Quadratic Equations chapter page the revision notes and the first step of the Route to 95 are free for everyone. CBSE Pro adds all 49 questions in the chapter bank (short and long answers, assertion–reason and more case studies) with their full step-marking schemes, the Route to 95 checkpoints with your progress saved, Skill Builders, the full common-mistakes library and AI marking of your written working.

Quadratic Equations: important questions What Pro includes

Original questions written by CBSE Math Revision for the CBSE 2026-27 syllabus; not taken from NCERT, NCERT Exemplar or CBSE papers. Every answer was checked twice, the second time by an independent reviewer. CBSE Math Revision is independent and not affiliated with CBSE or NCERT. Spotted a slip? Tell us and it goes in the corrections log.