Mathematics · Sample Paper 1

General instructions

Section A · Objective

1. Question 11 mark

   The HCF of 96 and 404 is

   1. A.2
   2. B.4
   3. C.8
   4. D.12
2. Question 21 mark

   The HCF of two numbers is 6 and their LCM is 180. If one number is 36, the other is

   1. A.30
   2. B.36
   3. C.18
   4. D.60
3. Question 31 mark

   The zeroes of the polynomial $x^2 - 5x + 6$ are

   1. A.2 and 3
   2. B.$-2$ and $-3$
   3. C.1 and 6
   4. D.$-1$ and $-6$
4. Question 41 mark

   If $\alpha$ and $\beta$ are the zeroes of $x^2 + 7x + 10$, then $\alpha + \beta$ is

   1. A.7
   2. B.$-7$
   3. C.10
   4. D.$-10$
5. Question 51 mark

   The pair of equations $2x + 3y = 5$ and $4x + 6y = 10$ has

   1. A.a unique solution
   2. B.no solution
   3. C.infinitely many solutions
   4. D.exactly two solutions
6. Question 61 mark

   The discriminant of the quadratic equation $2x^2 - 4x + 3 = 0$ is

   1. A.8
   2. B.$-8$
   3. C.40
   4. D.$-40$
7. Question 71 mark

   The 10th term of the AP $2, 7, 12, \ldots$ is

   1. A.45
   2. B.47
   3. C.50
   4. D.52
8. Question 81 mark

   The common difference of the AP $3, 1, -1, -3, \ldots$ is

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

   In $\triangle ABC$, $DE \parallel BC$ with $D$ on $AB$ and $E$ on $AC$. If $AD = 2$ cm, $DB = 3$ cm and $AE = 4$ cm, then $EC$ is

   1. A.4 cm
   2. B.5 cm
   3. C.6 cm
   4. D.8 cm
10. Question 101 mark

    The distance of the point $(3, 4)$ from the origin is

    1. A.4
    2. B.5
    3. C.7
    4. D.25
11. Question 111 mark

    The mid-point of the line segment joining $(2, -4)$ and $(-6, 8)$ is

    1. A.$(-2, 2)$
    2. B.$(2, -2)$
    3. C.$(-4, 4)$
    4. D.$(4, 2)$
12. Question 121 mark

    If $\sin A = \frac{3}{5}$ and $A$ is acute, then $\cos A$ is

    1. A.$\frac{3}{4}$
    2. B.$\frac{4}{5}$
    3. C.$\frac{5}{4}$
    4. D.$\frac{4}{3}$
13. Question 131 mark

    The value of $\dfrac{\tan 45^\circ}{\sin 30^\circ + \cos 60^\circ}$ is

    1. A.0
    2. B.$\frac{1}{2}$
    3. C.1
    4. D.2
14. Question 141 mark

    The angle of elevation of the top of a tower from a point on the ground 50 m from its foot is $45^\circ$. The height of the tower is

    1. A.25 m
    2. B.50 m
    3. C.$50\sqrt{3}$ m
    4. D.$\frac{50}{\sqrt{3}}$ m
15. Question 151 mark

    The number of tangents that can be drawn to a circle from a point outside it is

    1. A.0
    2. B.1
    3. C.2
    4. D.infinitely many
16. Question 161 mark

    The tangent at any point of a circle is ______ to the radius through the point of contact.

    1. A.parallel
    2. B.perpendicular
    3. C.equal
    4. D.inclined at $45^\circ$
17. Question 171 mark

    The area of a sector of a circle of radius 14 cm with central angle $90^\circ$ is (use $\pi = \frac{22}{7}$)

    1. A.154 cm²
    2. B.77 cm²
    3. C.308 cm²
    4. D.44 cm²
18. Question 181 mark

    A die is thrown once. The probability of getting a prime number is

    1. A.$\frac{1}{3}$
    2. B.$\frac{1}{2}$
    3. C.$\frac{2}{3}$
    4. D.$\frac{1}{6}$
19. Question 191 mark

    Assertion (A): The HCF of any two consecutive natural numbers is 1.

    Reason (R): Any two consecutive natural numbers are co-prime.

    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 the probability of an event is 0.3, the probability of the event not happening is 0.7.

    Reason (R): For any event $E$, $P(E) + P(\text{not } E) = 1$.

    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

   Find the HCF and LCM of 6 and 20 by the prime factorisation method.

2. OR

   Question 212 marks

   Check whether $6^n$ can end with the digit 0 for any natural number $n$.

3. Question 222 marks

   Solve the pair of linear equations: $2x + y = 7$ and $x - y = 2$.

4. Question 232 marks

   Find the ratio in which the point $(-4, 6)$ divides the line segment joining $A(-6, 10)$ and $B(3, -8)$.

5. OR

   Question 232 marks

   Find the point on the $x$-axis which is equidistant from $(2, -5)$ and $(-2, 9)$.

6. Question 242 marks

   Evaluate: $2\tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ$

7. Question 252 marks

   From a point $Q$, the length of the tangent to a circle is 24 cm and the distance of $Q$ from the centre is 25 cm. Find the radius of the circle.

Section C · Short answer

1. Question 263 marks

   Prove that $\sqrt{3}$ is irrational.

2. Question 273 marks

   Find the zeroes of the quadratic polynomial $6x^2 - 3 - 7x$ and verify the relationship between the zeroes and the coefficients.

3. OR

   Question 273 marks

   If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2 - 6x + k$ such that $3\alpha + 2\beta = 20$, find the value of $k$.

4. Question 283 marks

   A fraction becomes $\frac{9}{11}$ if 2 is added to both its numerator and denominator. If 3 is added to both, it becomes $\frac{5}{6}$. Find the fraction.

5. Question 293 marks

   Prove that $\dfrac{1 + \tan^2 A}{1 + \cot^2 A} = \tan^2 A$.

6. OR

   Question 293 marks

   Prove that $\sqrt{\dfrac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A$.

7. Question 303 marks

   Prove that the lengths of the tangents drawn from an external point to a circle are equal.

8. Question 313 marks

   A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out at random. Find the probability that it is (a) red (b) white (c) not green.

Section D · Long answer

1. Question 325 marks

   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.

2. OR

   Question 325 marks

   The sum of the reciprocals of Rehman's ages (in years) 3 years ago and 5 years from now is $\frac{1}{3}$. Find his present age.

3. Question 335 marks

   State and prove the Basic Proportionality Theorem.

4. Question 345 marks

   From the top of a 7 m high building, the angle of elevation of the top of a cable tower is $60^\circ$ and the angle of depression of its foot is $45^\circ$. Find the height of the tower.

5. OR

   Question 345 marks

   The angles of elevation of the top of a tower from two points at distances of 4 m and 9 m from its base, on the same straight line, are complementary. Prove that the height of the tower is 6 m.

6. Question 355 marks

   Find the mean and the median of the following data:

   | Class | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | |---|---|---|---|---|---| | Frequency | 5 | 8 | 15 | 16 | 6 |

Section E · Case study

1. Question 364 marks

   Case study — TV production. A manufacturer of TV sets produced 600 sets in the third year and 700 sets in the seventh year. The production increases uniformly by a fixed number every year.

   (i) Find the increase in production every year. (1 mark)

   (ii) Find the production in the first year. (1 mark)

   (iii) Find the total production in the first 7 years. (2 marks)

   OR

   (iii) In which year will the production be 800 sets? (2 marks)

2. Question 374 marks

   Case study — Town map. On a town map drawn on a coordinate grid (1 unit = 1 km), the school is at $A(1, 3)$, the library at $B(4, 7)$ and the park at $C(7, 3)$.

   (i) Find the distance between the school and the library. (1 mark)

   (ii) Find the mid-point of the school and the park. (1 mark)

   (iii) Show that the triangle formed by the school, the library and the park is isosceles. (2 marks)

   OR

   (iii) A bus stop lies on $AB$ and divides it in the ratio $1 : 2$. Find its coordinates. (2 marks)

3. Question 384 marks

   Case study — A wooden toy. A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. (Use $\pi = \frac{22}{7}$)

   (i) Find the height of the conical part. (1 mark)

   (ii) Find the slant height of the cone. (1 mark)

   (iii) Find the total surface area of the toy. (2 marks)

   OR

   (iii) Find the volume of the toy. (2 marks)