Mathematics · Sample Paper 1

General instructions

Section A · Objective

1. Question 11 mark

   Which of the following is an irrational number?

   1. A.$\sqrt{16}$
   2. B.$0.\overline{45}$
   3. C.$\sqrt{5}$
   4. D.$\frac{22}{7}$
2. Question 21 mark

   The value of $16^{\frac{3}{4}}$ is

   1. A.4
   2. B.6
   3. C.8
   4. D.12
3. Question 31 mark

   $\dfrac{1}{\sqrt{7} - \sqrt{6}}$ is equal to

   1. A.$\sqrt{7} + \sqrt{6}$
   2. B.$\sqrt{7} - \sqrt{6}$
   3. C.$\frac{1}{\sqrt{7} + \sqrt{6}}$
   4. D.$13$
4. Question 41 mark

   The degree of the polynomial $4x^4 + 0x^3 + 0x^5 + 5x + 7$ is

   1. A.5
   2. B.4
   3. C.1
   4. D.0
5. Question 51 mark

   The zero of the polynomial $p(x) = 2x + 5$ is

   1. A.$\frac{5}{2}$
   2. B.$-\frac{5}{2}$
   3. C.$\frac{2}{5}$
   4. D.$-\frac{2}{5}$
6. Question 61 mark

   If $x + 1$ is a factor of $x^3 + kx^2 - x - 2$, then the value of $k$ is

   1. A.1
   2. B.2
   3. C.$-2$
   4. D.3
7. Question 71 mark

   The point $(-3, 5)$ lies in the

   1. A.first quadrant
   2. B.second quadrant
   3. C.third quadrant
   4. D.fourth quadrant
8. Question 81 mark

   The abscissa of every point on the $y$-axis is

   1. A.0
   2. B.1
   3. C.$-1$
   4. D.any number
9. Question 91 mark

   $x = 2,\ y = 1$ is a solution of the linear equation

   1. A.$2x + y = 5$
   2. B.$x + 2y = 5$
   3. C.$x - y = 3$
   4. D.$3x + y = 6$
10. Question 101 mark

    The graph of the linear equation $y = 3$ is a line

    1. A.parallel to the $x$-axis
    2. B.parallel to the $y$-axis
    3. C.passing through the origin
    4. D.which is the $y$-axis itself
11. Question 111 mark

    "A terminated line can be produced indefinitely" is Euclid's

    1. A.Postulate 1
    2. B.Postulate 2
    3. C.Postulate 3
    4. D.Postulate 5
12. Question 121 mark

    Two supplementary angles are in the ratio $4 : 5$. The smaller angle is

    1. A.$80^\circ$
    2. B.$100^\circ$
    3. C.$40^\circ$
    4. D.$72^\circ$
13. Question 131 mark

    In $\triangle ABC$, $AB = AC$ and $\angle B = 50^\circ$. Then $\angle A$ is

    1. A.$50^\circ$
    2. B.$65^\circ$
    3. C.$80^\circ$
    4. D.$130^\circ$
14. Question 141 mark

    In parallelogram $ABCD$, $\angle A = 70^\circ$. Then $\angle B$ is

    1. A.$70^\circ$
    2. B.$110^\circ$
    3. C.$90^\circ$
    4. D.$20^\circ$
15. Question 151 mark

    The angle subtended by a diameter at any point on the circle (other than its end points) is

    1. A.$60^\circ$
    2. B.$90^\circ$
    3. C.$120^\circ$
    4. D.$180^\circ$
16. Question 161 mark

    The area of an equilateral triangle of side 4 cm, found using Heron's formula, is

    1. A.$4\sqrt{3}$ cm²
    2. B.$8\sqrt{3}$ cm²
    3. C.16 cm²
    4. D.12 cm²
17. Question 171 mark

    The volume of a sphere of radius 3 cm is

    1. A.$36\pi$ cm³
    2. B.$12\pi$ cm³
    3. C.$27\pi$ cm³
    4. D.$108\pi$ cm³
18. Question 181 mark

    In the class interval $10 - 20$, the upper class limit is

    1. A.10
    2. B.15
    3. C.20
    4. D.30
19. Question 191 mark

    Assertion (A): $\sqrt{2} + \sqrt{3}$ is an irrational number.

    Reason (R): The sum of two irrational numbers is always irrational.

    1. A.Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
    2. B.Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
    3. C.Assertion (A) is true, but Reason (R) is false.
    4. D.Assertion (A) is false, but Reason (R) is true.
20. Question 201 mark

    Assertion (A): If two angles of a triangle are $50^\circ$ and $60^\circ$, its third angle is $70^\circ$.

    Reason (R): The sum of the angles of a triangle is $180^\circ$.

    1. A.Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
    2. B.Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
    3. C.Assertion (A) is true, but Reason (R) is false.
    4. D.Assertion (A) is false, but Reason (R) is true.

Section B · Very short answer

1. Question 212 marks

   Express $0.\overline{47}$ in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

2. OR

   Question 212 marks

   Simplify: $(3 + \sqrt{3})(2 + \sqrt{2})$

3. Question 222 marks

   Find the remainder when $x^3 + 3x^2 + 3x + 1$ is divided by $x + 1$.

4. Question 232 marks

   If $x = 2,\ y = 1$ is a solution of the equation $2x + 3y = k$, find the value of $k$. Also write one more solution of the equation.

5. OR

   Question 232 marks

   Express $2x = -5y + 7$ in the form $ax + by + c = 0$ and write the values of $a$, $b$ and $c$.

6. Question 242 marks

   Two parallel lines are cut by a transversal. The two interior angles on the same side of the transversal are $(3x + 10)^\circ$ and $(2x - 5)^\circ$. Find $x$ and both angles.

7. Question 252 marks

   Find the area of a triangle whose sides are 13 cm, 14 cm and 15 cm.

Section C · Short answer

1. Question 263 marks

   If $x = 3 + 2\sqrt{2}$, find the value of $x + \dfrac{1}{x}$.

2. Question 273 marks

   Factorise: $x^3 - 2x^2 - x + 2$

3. OR

   Question 273 marks

   Using suitable identities, evaluate: (i) $104 \times 96$ (ii) $99^2$

4. Question 283 marks

   Without plotting, state the quadrant or the axis on which each point lies: (i) $(-2, 4)$ (ii) $(3, -1)$ (iii) $(-1, 0)$ (iv) $(1, 2)$ (v) $(-3, -5)$ (vi) $(0, 6)$

5. Question 293 marks

   In $\triangle ABC$, $AD$ is the perpendicular bisector of $BC$ ($D$ on $BC$). Show that $\triangle ABC$ is an isosceles triangle in which $AB = AC$.

6. OR

   Question 293 marks

   $ABC$ is an isosceles triangle with $AB = AC$. $BE$ and $CF$ are the altitudes to $AC$ and $AB$ respectively. Prove that $BE = CF$.

7. Question 303 marks

   The class marks of a frequency distribution are 15, 25, 35 and 45. Find the class size and write all the class intervals.

8. Question 313 marks

   $ABCD$ is a parallelogram. $AP$ and $CQ$ are perpendiculars from $A$ and $C$ to the diagonal $BD$. Show that (i) $\triangle APB \cong \triangle CQD$ (ii) $AP = CQ$.

Section D · Long answer

1. Question 325 marks

   If $a = \dfrac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}$ and $b = \dfrac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}}$, find the value of $a^2 + b^2 + ab$.

2. OR

   Question 325 marks

   (i) Find rational numbers $a$ and $b$ such that $\dfrac{3 + \sqrt{2}}{3 - \sqrt{2}} = a + b\sqrt{2}$.

   (ii) Express $1.\overline{27}$ in the form $\frac{p}{q}$.

3. Question 335 marks

   (i) If $a + b + c = 9$ and $ab + bc + ca = 26$, find $a^2 + b^2 + c^2$.

   (ii) Factorise: $27x^3 + y^3 + z^3 - 9xyz$

4. Question 345 marks

   (i) Prove that equal chords of a circle subtend equal angles at the centre.

   (ii) $AB$ and $CD$ are equal chords of a circle with centre $O$. If $\angle AOB = (2x + 10)^\circ$ and $\angle COD = (3x - 20)^\circ$, find $x$.

5. OR

   Question 345 marks

   (i) Prove that the sum of the angles of a triangle is $180^\circ$.

   (ii) The angles of a triangle are in the ratio $2 : 3 : 4$. Find the angles.

6. Question 355 marks

   A conical tent is 10 m high and the radius of its base is 24 m. Find (i) the slant height of the tent, (ii) the cost of the canvas required to make the tent, if the cost of 1 m² of canvas is ₹70. (Use $\pi = \frac{22}{7}$)

Section E · Case study

1. Question 364 marks

   Case study — Classroom seating plan. A teacher draws the seating plan of her class on a coordinate grid, with her own desk at the origin. Ravi sits at $(3, 2)$, Meena at $(-2, 4)$, Joseph at $(-3, -1)$ and Ali at $(0, -3)$.

   (i) In which quadrant does Meena sit? (1 mark)

   (ii) Which student sits on an axis, and which axis is it? (1 mark)

   (iii) Find the sum of Joseph's abscissa and Ravi's ordinate. (2 marks)

   OR

   (iii) Write the coordinates of a point on the $x$-axis 4 units to the left of the origin, and of a point on the $y$-axis 3 units above the origin. (2 marks)

2. Question 374 marks

   Case study — Triangular park. A town council is developing a triangular park whose sides measure 50 m, 120 m and 130 m. It will be fenced all round, leaving a 3 m wide gate, and the whole park will be grassed.

   (i) Find the semi-perimeter of the park. (1 mark)

   (ii) Find the length of fencing needed. (1 mark)

   (iii) Using Heron's formula, find the area of the park. (2 marks)

   OR

   (iii) Show that the park is a right-angled triangle and find its area using $\frac{1}{2} \times \text{base} \times \text{height}$. (2 marks)

3. Question 384 marks

   Case study — Decorations for the annual day. For the annual day, students make solid spherical decorative balls of radius 7 cm and solid hemispherical paperweights of radius 3.5 cm. (Use $\pi = \frac{22}{7}$)

   (i) Find the surface area of one ball. (1 mark)

   (ii) Find the volume of one ball. (1 mark)

   (iii) Find the total surface area of one hemispherical paperweight. (2 marks)

   OR

   (iii) A hemispherical bowl of radius 21 cm is used to hold glue. How many litres of glue can it hold? (2 marks)