CBSE Class 9 Mathematics · Exam Practice

Practice Paper — Set 1

CBSE Class 9 Mathematics (Ganita Prakash), covering number systems, polynomials, algebraic identities, coordinate geometry, mensuration, probability, and arithmetic progressions, with a full answer key and worked solutions.

🎯 Full Marks: 63 ⏱️ Time: 2½ Hours 📝 Questions: 42
SectionTopicQsMarks
ANumber Systems & Polynomials1515
BAlgebraic Identities & Coordinate Geometry1520
CMensuration, Probability & Progressions1228
Total4263
Before you start
  • Attempt the paper first, without looking at the answer key at the bottom of the page.
  • Show your working for Sections B and C — method marks are given even if the final answer is wrong.
15 MARKS

Section A — Number Systems & Polynomials

1. Multiple Choice Questions (MCQs) (1 mark each)

1. Which of the following is an irrational number?

A. √4B. √9C. √2D. √16

2. If p(x) = 2x + 3, then p(2) =

A. 5B. 7C. 8D. 9

3. The zero of the polynomial p(x) = x − 5 is

A. −5B. 0C. 5D. 1

4. Is 0 a rational number?

A. YesB. NoC. Only sometimesD. Cannot be determined

5. The decimal expansion of 1/3 is

A. TerminatingB. Non-terminating, repeatingC. Non-terminating, non-repeatingD. Exactly 0.3

6. The degree of the polynomial 3x² + 2x + 1 is

A. 0B. 1C. 2D. 3

7. √25 is

A. RationalB. IrrationalC. NeitherD. Undefined

8. Which of these is a linear polynomial?

A. x²B. 2x + 1C. x³ + 1D. 1/x

9. If p(x) = x + 1, then p(0) =

A. 0B. 1C. −1D. 2

10. Between any two rational numbers, how many rational numbers can be found?

A. NoneB. Exactly oneC. Exactly tenD. Infinitely many

2. Fill in the Blanks (1 mark each)

1. π is a/an ______ number (rational/irrational).

2. The zero of the polynomial p(x) = 2x − 6 is ______.

3. Every integer is a ______ number.

4. The coefficient of x in 5x + 7 is ______.

5. √16 = ______.

20 MARKS

Section B — Algebraic Identities & Coordinate Geometry

3. True or False (1 mark each)

1. (a + b)² = a² + 2ab + b². (   )

2. The point (0, 0) is called the origin. (   )

3. (x + 2)(x − 2) = x² − 4. (   )

4. The x-coordinate of a point is also called its ordinate. (   )

5. The point (3, 4) lies in the first quadrant. (   )

4. Match the Following (1 mark each)

Column A
1.(a + b)²
2.(a − b)²
3.a² − b²
4.(x + 3)(x + 5)
5.(x + 1)² at x = 4
Column B
a)a² − 2ab + b²
b)x² + 8x + 15
c)a² + 2ab + b²
d)25
e)(a + b)(a − b)

5. Very Short Answer Type Questions (2 marks each)

1. Expand: (2x + 3)².

2. Find the value of (x + y)² if x = 3, y = 4.

3. Factorise: x² − 9.

4. The point (2, −3) lies in which quadrant?

5. Find p(2) if p(x) = x² − 4x + 3.

28 MARKS

Section C — Mensuration, Probability & Progressions

6. Solve the Following (2 marks each)

1. Find the area of a triangle with sides 3 cm, 4 cm, and 5 cm.

2. A die is thrown once. Find the probability of getting a number greater than 4.

3. Find the 10th term of the AP: 2, 5, 8, 11, ...

4. Find the sum of the first 10 terms of the AP: 1, 3, 5, 7, ...

5. Find the area of a rectangle with length 15 m and width 8 m.

7. Do As Directed (2 marks each)

1. A coin is tossed twice. Find the probability of getting exactly one head.

2. Find the 5th term of an AP with first term 7 and common difference 4.

3. Find the area of a circle with radius 14 cm (use π = 22/7).

4. A bag has 3 red and 2 blue balls. Find the probability of drawing a red ball.

5. Find the perimeter of a rectangle 20 cm long and 12 cm wide.

8. Think & Solve (Word Problems) (4 marks each)

1. The sum of the first n terms of an AP is given by Sₙ = 3n² + 2n. Find (a) the first term, and (b) the common difference.

2. A rectangular park is 40 m long and 25 m wide, with a path of width 2.5 m running all around the inside edge. Find (a) the area of the park, and (b) the area of the path.

CBSE Class 9 Mathematics · Practice Paper — Set 1 · Answer Key

Answers & Worked Solutions

Mark your own paper, then read the solution for every question you got wrong. For the multiple-choice questions, the note in pink shows the answer most students pick by mistake, and why.

Section A — Number Systems & Polynomials (15 marks)
1. Multiple Choice Questions
Q1. C. √2

√4=2, √9=3, and √16=4 are all whole numbers (rational). √2 cannot be written as a simple fraction — it's irrational.

Q2. B. 7

p(2) = 2(2) + 3 = 4 + 3 = 7.

Q3. C. 5

A zero of a polynomial is the value of x that makes p(x) = 0: x − 5 = 0 → x = 5.

Q4. A. Yes

0 can be written as 0/1, which fits the definition of a rational number (a fraction of two integers).

Q5. B. Non-terminating, repeating

1/3 = 0.3333... — the digit 3 repeats forever.

Tempting wrong answer: Exactly 0.3 — this rounds the decimal instead of recognising it repeats infinitely.

Q6. C. 2

The degree is the highest power of x present — here it's x² (power 2).

Q7. A. Rational

√25 = 5, a whole number, which is rational.

Q8. B. 2x + 1

A linear polynomial has degree exactly 1 (highest power of x is 1). x² has degree 2, x³+1 has degree 3, and 1/x is not a polynomial at all.

Q9. B. 1

p(0) = 0 + 1 = 1.

Q10. D. Infinitely many

Between any two distinct rational numbers, there are always infinitely many more rational numbers.

2. Fill in the Blanks
Q1. Irrational

π's decimal expansion never terminates or repeats, so it cannot be written as a simple fraction.

Q2. 3

2x − 6 = 0 → 2x = 6 → x = 3.

Q3. Rational

Every integer n can be written as n/1, a ratio of two integers.

Q4. 5

In 5x + 7, the number multiplying x is 5.

Q5. 4

4 × 4 = 16.

Section B — Algebraic Identities & Coordinate Geometry (20 marks)
3. True or False
Q1. True

This is a standard algebraic identity.

Q2. True

(0, 0) is indeed called the origin — the point where both axes meet.

Q3. True

This is the difference-of-squares identity: (x+2)(x−2) = x² − 4.

Q4. False

The x-coordinate is called the abscissa. The y-coordinate is called the ordinate.

Q5. True

Both coordinates of (3, 4) are positive, which places it in the first quadrant.

4. Match the Following
1 → c) a² + 2ab + b²

Standard identity for (a + b)².

2 → a) a² − 2ab + b²

Standard identity for (a − b)².

3 → e) (a + b)(a − b)

Standard identity for a² − b².

4 → b) x² + 8x + 15

(x+3)(x+5) = x² + 5x + 3x + 15 = x² + 8x + 15.

5 → d) 25

(4 + 1)² = 5² = 25.

5. Very Short Answer Type Questions
Q1. 4x² + 12x + 9

(2x + 3)² = (2x)² + 2(2x)(3) + 3² = 4x² + 12x + 9.

Q2. 49

(3 + 4)² = 7² = 49.

Q3. (x − 3)(x + 3)

x² − 9 = x² − 3² — a difference of squares.

Q4. Fourth quadrant

x is positive (2) and y is negative (−3): positive-x, negative-y is the fourth quadrant.

Q5. −1

p(2) = 2² − 4(2) + 3 = 4 − 8 + 3 = −1.

Section C — Mensuration, Probability & Progressions (28 marks)
6. Solve the Following
Q1. 6 sq cm

Since 3² + 4² = 9 + 16 = 25 = 5², this is a right triangle with legs 3 and 4. Area = ½ × 3 × 4 = 6 sq cm.

Q2. 1/3

Numbers greater than 4 on a die: 5 and 6 — that's 2 out of 6 outcomes. P = 2/6 = 1/3.

Q3. 29

a = 2, d = 3. 10th term = a + 9d = 2 + 27 = 29.

Q4. 100

a = 1, d = 2, n = 10. Sum = (n/2)[2a + (n−1)d] = 5 × [2 + 18] = 5 × 20 = 100.

Q5. 120 sq m

Area = length × width = 15 × 8 = 120 sq m.

7. Do As Directed
Q1. 1/2

Outcomes: HH, HT, TH, TT. Exactly one head occurs in HT and TH — 2 out of 4. P = 2/4 = 1/2.

Q2. 23

5th term = a + 4d = 7 + 4(4) = 7 + 16 = 23.

Q3. 616 sq cm

Area = πr² = (22/7) × 14 × 14 = 22 × 28 = 616 sq cm.

Q4. 3/5

3 red out of 5 total balls: P(red) = 3/5.

Q5. 64 cm

Perimeter = 2 × (20 + 12) = 2 × 32 = 64 cm.

8. Think & Solve (Word Problems)
Q1. (a) first term = 5 (b) common difference = 6
  1. S₁ = 3(1)² + 2(1) = 5 — this is the first term, a₁ = 5.
  2. S₂ = 3(2)² + 2(2) = 12 + 4 = 16.
  3. Second term a₂ = S₂ − S₁ = 16 − 5 = 11.
  4. Common difference d = a₂ − a₁ = 11 − 5 = 6.
Q2. (a) 1,000 sq m (b) 300 sq m
  1. Area of park = 40 × 25 = 1,000 sq m.
  2. Since the path is 2.5 m wide all around the inside, the inner rectangle (excluding the path) has length 40 − 2(2.5) = 35 m and width 25 − 2(2.5) = 20 m.
  3. Inner area = 35 × 20 = 700 sq m.
  4. Area of path = 1,000 − 700 = 300 sq m.
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