Class 9 – Mathematics Question Paper

Time: 3 Hours Maximum Marks: 80

Section A – MCQs (1 × 20 = 20 Marks)

1. Rationalize: 1 / √5

   a) √5

   b) √5 / 5

   c) 5√5

   d) 1/5

2. (2³)² =

   a) 2⁵

   b) 2⁶

   c) 4⁶

   d) 8²

3. Degree of polynomial 5x³ + 2x² − x + 7 is:

   a) 1

   b) 2

   c) 3

   d) 4

4. If p(x) = x² − 4, then p(2) =

   a) 0

   b) 2

   c) 4

   d) −4

5. If (x − 3) is a factor of a polynomial, then remainder on division by (x − 3) is:

   a) 1

   b) 0

   c) 3

   d) −3

6. (a + b)² =

   a) a² + b²

   b) a² + 2ab + b²

   c) a² − 2ab + b²

   d) 2a² + 2b²

7. Two equal chords of a circle are:

   a) Parallel

   b) Equal in length

   c) Perpendicular

   d) Diameter

8. Radius of a circle is 7 cm. Diameter is:

   a) 7 cm

   b) 14 cm

   c) 21 cm

   d) 49 cm

9. Heron’s formula is used to find:

   a) Perimeter

   b) Area of triangle

   c) Volume

   d) Circumference

10. Curved surface area of a cone =

    a) πr²

    b) 2πrh

    c) πrl

    d) 4πr²

11. Volume of sphere =

    a) 4/3 πr³

    b) πr²h

    c) 2πr²

    d) πr³

12. If frequency of a class is zero, it means:

    a) No observation

    b) Negative observation

    c) Infinite observation

    d) One observation

13. Histogram is used for:

    a) Ungrouped data

    b) Grouped continuous data

    c) Pie chart

    d) Probability

14. If mean = 10 and number of observations = 5, total sum =

    a) 2

    b) 15

    c) 50

    d) 5

15. (x³ − 1) is divisible by:

    a) x − 1

    b) x + 1

    c) x − 3

    d) x + 3

16. √18 can be simplified as:

    a) 3√2

    b) 2√3

    c) 9√2

    d) 6√3

17. If radius doubles, volume of sphere becomes:

    a) Double

    b) 4 times

    c) 6 times

    d) 8 times

18. Sum of frequencies in a distribution is called:

    a) Mean

    b) Total frequency

    c) Class interval

    d) Mode

19. In a circle, perpendicular from centre to chord:

    a) Bisects the chord

    b) Doubles the chord

    c) Is tangent

    d) Is radius

20. (a − b)(a + b) =

    a) a² + b²

    b) a² − b²

    c) a² − 2ab + b²

    d) a² + 2ab + b²

Section B – Very Short Answer (2 × 5 = 10 Marks)

1. Rationalize: 5 / √3

2. Find degree of polynomial 7x⁴ − 3x + 9.

3. If p(x) = x³ − 2x² + 1, find p(1).

4. Find curved surface area of cone of radius 7 cm and slant height 10 cm.

5. Find mean of 5, 7, 9.

Section C – Short Answer Questions (3 × 6 = 18 Marks)

1. Factorize: x³ − 27.

2. Using Remainder Theorem, find remainder when x³ − 3x² + 2 is divided by (x − 2).

3. Prove: Equal chords of a circle subtend equal angles at the centre.

4. Find area of triangle with sides 7 cm, 8 cm, 9 cm using Heron’s formula.

5. Find volume of sphere of radius 7 cm.

6. Construct a frequency polygon from given grouped data.

Section D – Long Answer Questions (5 × 4 = 20 Marks)

1. Simplify using identities:

   (2x + 5)² − (2x − 5)²

2. Prove that perpendicular from centre to chord bisects the chord.

3. Find total surface area and volume of a cone of radius 7 cm and height 24 cm.

4. The following data shows marks of students:

Class Interval:

0–10, 10–20, 20–30, 30–40, 40–50

Frequency:

3, 5, 9, 7, 6

(a) Construct histogram

(b) Find total frequency

Section E – Case Study Based Questions (4 × 3 = 12 Marks)

Case Study 1:

A tent is in the shape of a cone placed on a circular base. Radius = 7 m, height = 24 m.

(i) Find slant height.

(ii) Find curved surface area.

(iii) Find volume of tent.

Case Study 2:

A survey of 40 students’ marks is grouped into class intervals.

(i) Why is histogram suitable here?

(ii) What is total frequency?

(iii) How is frequency polygon drawn?

Case Study 3:

A triangle has sides 13 cm, 14 cm, 15 cm.

(i) Find semi-perimeter.

(ii) Find area using Heron’s formula.

(iii) Verify area using identity method if possible.