Mathematics · Sample Paper 3

General instructions

Section A · Objective

1. Question 11 mark

   The HCF of 26 and 91 is

   1. A.13
   2. B.7
   3. C.26
   4. D.1
2. Question 21 mark

   The product of a non-zero rational number and an irrational number is always

   1. A.rational
   2. B.irrational
   3. C.an integer
   4. D.zero
3. Question 31 mark

   The number of zeroes of a polynomial whose graph cuts the x-axis at three points is

   1. A.1
   2. B.2
   3. C.3
   4. D.0
4. Question 41 mark

   If one zero of $x^2-4x+k$ is 1, then k is

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

   The pair $x+y=5$, $2x+2y=12$ has

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

   Which of these is a quadratic equation?

   1. A.$x^3-x=0$
   2. B.$(x+1{)}^2=x^2+3$
   3. C.$x(x+2)=5$
   4. D.$2x+3=7$
7. Question 71 mark

   The roots of $x^2+6x+9=0$ are

   1. A.real and distinct
   2. B.real and equal
   3. C.not real
   4. D.zero
8. Question 81 mark

   The 15th term of the AP 7, 4, 1, … is

   1. A.−35
   2. B.−32
   3. C.−38
   4. D.49
9. Question 91 mark

   The sum of the first 20 odd natural numbers is

   1. A.400
   2. B.200
   3. C.420
   4. D.380
10. Question 101 mark

    In $△ABC$, $DE\parallel BC$ with $AD=3$, $DB=2$, $AE=4.5$. Then $EC$ is

    1. A.3
    2. B.2
    3. C.4.5
    4. D.6.75
11. Question 111 mark

    The distance between $(-1,3)$ and $(2,-1)$ is

    1. A.5
    2. B.4
    3. C.7
    4. D.3
12. Question 121 mark

    The point dividing the segment joining (0, 0) and (6, 9) in the ratio 1 : 2 is

    1. A.(2, 3)
    2. B.(3, 4.5)
    3. C.(4, 6)
    4. D.(1, 2)
13. Question 131 mark

    $\tan 60^{\circ }$ equals

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

    $(1+{\tan }^2\theta ){\cos }^2\theta$ equals

    1. A.1
    2. B.0
    3. C.${\sin }^2\theta$
    4. D.${\sec }^2\theta$
15. Question 151 mark

    The shadow of a pole is equal to its height. The angle of elevation of the sun is

    1. A.$30^{\circ }$
    2. B.$45^{\circ }$
    3. C.$60^{\circ }$
    4. D.$90^{\circ }$
16. Question 161 mark

    From a point 13 cm from the centre of a circle of radius 5 cm, the length of the tangent is

    1. A.12 cm
    2. B.8 cm
    3. C.18 cm
    4. D.10 cm
17. Question 171 mark

    The area of a sector of angle $60^{\circ }$ in a circle of radius 6 cm is

    1. A.$6\pi {\text{cm}}^2$
    2. B.$12\pi {\text{cm}}^2$
    3. C.$3\pi {\text{cm}}^2$
    4. D.$36\pi {\text{cm}}^2$
18. Question 181 mark

    A cone and a cylinder have the same base and height. The ratio of their volumes is

    1. A.1 : 3
    2. B.3 : 1
    3. C.1 : 2
    4. D.2 : 3
19. Question 191 mark

    Assertion (A): For a symmetric distribution, mean, median and mode are equal.

    Reason (R): In a symmetric distribution, the data are evenly spread around the centre.

    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): The probability of an event can be 1.2.

    Reason (R): The probability of any event lies between 0 and 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 LCM of 12, 15 and 21 by prime factorisation.

2. Question 222 marks

   Find a quadratic polynomial whose sum and product of zeroes are $-3$ and $2$.

3. OR

   Question 222 marks

   Which term of the AP 21, 18, 15, … is −81?

4. Question 232 marks

   Find the value of k if the points $(2,3)$, $(4,k)$ and $(6,-3)$ are collinear.

5. Question 242 marks

   If $\sec \theta =\frac{13}{12}$, find $\sin \theta$ and $\tan \theta$.

6. OR

   Question 242 marks

   Evaluate $\frac{\sin 30^{\circ }+\tan 45^{\circ }-\csc 60^{\circ }}{\sec 30^{\circ }+\cos 60^{\circ }+\cot 45^{\circ }}$.

7. Question 252 marks

   Two tangents TP and TQ are drawn to a circle with centre O from an external point T. If $\angle PTQ=70^{\circ }$, find $\angle POQ$.

Section C · Short answer

1. Question 263 marks

   Prove that $\sqrt{5}$ is irrational.

2. Question 273 marks

   The cost of 2 pencils and 3 erasers is ₹9, and of 4 pencils and 6 erasers is ₹18. Represent the situation algebraically and say whether the solution is unique.

3. OR

   Question 273 marks

   Solve by elimination: $3x+4y=10$ and $2x-2y=2$.

4. Question 283 marks

   Find two consecutive positive integers whose squares add up to 365.

5. Question 293 marks

   Prove that if a line divides two sides of a triangle in the same ratio, then it is parallel to the third side (converse of BPT).

6. Question 303 marks

   Prove that the angle between two tangents drawn from an external point is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

7. OR

   Question 303 marks

   A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the minor segment. (Use $\pi =3.14$)

8. Question 313 marks

   A bag has 3 red, 5 black and 4 white balls. One ball is drawn at random. Find the probability that it is (i) white (ii) not black (iii) red or white.

Section D · Long answer

1. Question 325 marks

   A train travels 480 km at a uniform speed. If the speed had been 8 km/h less, it would have taken 3 hours more. Find the speed of the train.

2. OR

   Question 325 marks

   The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages was 124. Find their present ages.

3. Question 335 marks

   From a point on the ground, the angles of elevation of the bottom and top of a transmission tower fixed at the top of a 20 m high building are $45^{\circ }$ and $60^{\circ }$. Find the height of the tower. (Use $\sqrt{3}=1.732$)

4. Question 345 marks

   A wooden article is made by scooping out a hemisphere from each end of a solid cylinder of height 10 cm and radius 3.5 cm. Find the total surface area of the article. (Use $\pi =\frac{22}{7}$)

5. OR

   Question 345 marks

   A medicine capsule is shaped like a cylinder with two hemispheres stuck to its ends. Its diameter is 0.5 cm and its total length is 2 cm. Find the volume and surface area of the capsule. (Use $\pi =\frac{22}{7}$)

6. Question 355 marks

   Find the median of the following distribution:

   | Marks | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 | |---|---|---|---|---|---|---| | Students | 5 | 8 | 20 | 15 | 7 | 5 |

Section E · Case study

1. Question 364 marks

   Case study — School fair tickets. At a school fair, tickets for adults cost ₹x and for children ₹y. One family paid ₹250 for 2 adults and 3 children; another paid ₹350 for 3 adults and 4 children.

   (i) Write the pair of linear equations. (1 mark)

   (ii) Are the lines intersecting, parallel or coincident? (1 mark)

   (iii) Find the cost of one adult ticket and one child ticket. (2 marks)

   OR

   (iii) How much would a group of 4 adults and 5 children pay? (2 marks)

2. Question 374 marks

   Case study — Planting saplings. In a rectangular school garden drawn on a grid (1 unit = 1 m), Riya plants a sapling at A(2, 3) and Aman plants one at B(8, 11). A water tap is to be placed exactly midway between them.

   (i) Find the distance between the two saplings. (1 mark)

   (ii) Find the position of the tap. (1 mark)

   (iii) A third sapling C is to be planted on AB so that AC : CB = 1 : 3. Find C. (2 marks)

   OR

   (iii) Is the point (5, 7) equidistant from A and B? Justify. (2 marks)

3. Question 384 marks

   Case study — Pizza slices. A pizza of diameter 28 cm is cut into 8 equal slices. (Use $\pi =\frac{22}{7}$)

   (i) What is the angle of each slice at the centre? (1 mark)

   (ii) Find the area of the whole pizza. (1 mark)

   (iii) Find the area of one slice and the length of its curved crust. (2 marks)

   OR

   (iii) Rahul eats 3 slices. What area of pizza is left? (2 marks)