This question paper comprises four sections - A, B, C and D. This question paper carries 40 questions. All questions are compulsory. Section A from Question no. 1 to 20 comprises of 20 questions of one mark each.

Section B from Question no. 21 to 26 comprises of 6 questions of two marks each. Section C from Question no. 27 to 34 comprises of 8 questions of three marks each. Section D from Question no. 35 to 40 comprises of 6 questions of four marks each.

Question numbers 1 to 10 are multiple choice questions of 1 mark each. Select the correct option.

1. The sum of exponents of prime factors in the prime-factorisation of 196 is

- 3
- 4
- 5
- 2

2. Euclid’s division Lemma states that for two positive integers a and b, there exists unique integer q and r satisfying a = bq + r, and

- 0 < r < b
- 0 < r ≤ b
- 0 ≤ r < b
- 0 ≤ r ≤ b

3. The zeroes of the polynomial x^{2} – 3x – m(m + 3) are

- m, m + 3
- –m, m + 3
- m, –(m + 3)
- –m, –(m + 3)

4. The value of k for which the system of linear equations x + 2y = 3, 5x + ky + 7 = 0 is inconsistent is

- –14/3
- 2/5
- 5
- 10

5. The roots of the quadratic equation x^{2} – 0.04 = 0 are

- ± 0.2
- ± 0.02
- 0.4
- 2

6. The common difference of the A.P. is

- 1
- 1/p
- –1
- –1/p

7. The n^{th} term of the A.P. a, 3a, 5a, …… is

- na
- (2n – 1)a
- (2n + 1)a
- 2na

8. The point P on x-axis equidistant from the points A(–1, 0) and B(5, 0) is

- (2, 0)
- (0, 2)
- (3, 0)
- (2, 2)

9. The co-ordinates of the point which is reflection of point (–3, 5) in x-axis are

- (3, 5)
- (3, –5)
- (–3, –5)
- (–3, 5)

10. If the point P (6, 2) divides the line segment joining A(6, 5) and B(4, y) in the ratio 3 : 1, then the value of y is

- 4
- 3
- 2
- 1

In Q. Nos. 11 to 15, fill in the blanks. Each question is of 1 mark.

11. In figure, MN || BC and AM : MB = 1 : 2, then ar(Δ AMN)/ar(Δ ABC) = _________.

12. In given Figure, the length PB = _________ cm.

13. In ΔABC, AB = 6√3 cm, AC = 12 cm and BC = 6 cm, then ∠B = _________.

OR

Two triangles are similar if their corresponding sides are _________.

14. The value of (tan 1º tan 2º …… tan 89º) is equal to _________.

15. In figure, the angles of depressions from the observing positions O_{1} and O_{2} respectively of the object A are _________, _________.

Q. Nos. 16 to 20 are short answer type questions of 1 mark each.

16. If sin A + sin^{2} A = 1, then find the value of the expression (cos^{2} A + cos^{4} A).

17. In figure is a sector of circle of radius 10.5 cm. Find the perimeter of the sector. (Take π = 22/7)

18. If a number x is chosen at random from the numbers –3, –2, –1, 0, 1, 2, 3, then find the probability of x^{2} < 4.

OR

What is the probability that a randomly taken leap year has 52 Sundays ?

19. Find the class-marks of the classes 10-25 and 35-55.

20. A die is thrown once. What is the probability of getting a prime number.

21. A teacher asked 10 of his students to write a polynomial in one variable on a paper and then to handover the paper. The following were the answers given by the students :

Answer the following questions :

- How many of the above ten, are not polynomials ?
- How many of the above ten, are quadratic polynomials ?

22. In figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that

ar (Δ ABC)/ar(Δ DBC) = AO/DO

OR

In figure, if AD **⊥** BC, then prove that AB^{2} + CD^{2} = BD^{2} + AC^{2}.

23. Prove that

OR

Show that tan^{4} θ + tan^{2} θ = sec^{4} θ – sec^{2} θ

24. The volume of a right circular cylinder with its height equal to the radius is 25 1/7 cm^{3}. Find the height of the cylinder. (Use π = 22/7)

25. A child has a die whose six faces show the letters as shown below :

A B C D E A

The die is thrown once. What is the probability of getting (i) A, (ii) D ?

26. Compute the mode for the following frequency distribution :

27. If 2x + y = 23 and 4x – y = 19, find the value of (5y – 2x) and (y/x - 2).

OR

Solve for x :

28. Show that the sum of all terms of an A.P. whose first term is a, the second term is b and the last term is c is equal to (a + c)(b + c – 2a)/2(b – a)

OR

Solve the equation :

1 + 4 + 7 + 10 + … + x = 287.

29. In a flight of 600 km, an aircraft was slowed down due to bad weather. The average speed of the trip was reduced by 200 km/hr and the time of flight increased by 30 minutes. Find the duration of flight.

30. If the mid-point of the line segment joining the points A(3, 4) and B(k, 6) is P (x, y) and x + y – 10 = 0, find the value of k.

OR

Find the area of triangle ABC with A (1, –4) and the mid-points of sides through A being (2, –1) and (0, –1).

31. In Figure, if Δ ABC ~ Δ DEF and their sides of lengths (in cm) are marked along them, then find the lengths of sides of each triangle.

32. If a circle touches the side BC of a triangle ABC at P and extended sides AB and AC at Q and R, respectively, prove that

AQ = 1/2 (BC + CA + AB)

33. If sin θ + cos θ = √2, prove that tan θ + cot θ = 2.

34. The area of a circular play ground is 22176 cm^{2}. Find the cost of fencing this ground at the rate of Rs. 50 per metre.

35. Prove that √5 is an irrational number.

36. It can take 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for four hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled. How long would it take for each pipe to fill the pool separately ?

37. Draw a circle of radius 2 cm with centre O and take a point P outside the circle such that OP = 6.5 cm. From P, draw two tangents to the circle.

OR

Construct a triangle with sides 5 cm, 6 cm and 7 cm and then construct another triangle whose sides are 3/4 times the corresponding sides of the first triangle.

38. From a point on the ground, the angles of elevation of the bottom and the top of a tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower.

39. Find the area of the shaded region in figure, if PQ = 24 cm, PR = 7 cm and O is the centre of the circle.

OR

Find the curved surface area of the frustum of a cone, the diameters of whose circular ends are 20 m and 6 m and its height is 24 m.

40. The mean of the following frequency distribution is 18. The frequency f in the class interval 19 - 21 is missing. Determine f.

OR

The following table gives production yield per hectare of wheat of 100 farms of a village :

Change the distribution to a ‘more than’ type distribution and draw its ogive.