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
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
3. The zeroes of the polynomial x2 – 3x – m(m + 3) are
4. The value of k for which the system of linear equations x + 2y = 3, 5x + ky + 7 = 0 is inconsistent is
5. The roots of the quadratic equation x2 – 0.04 = 0 are
6. The common difference of the A.P. is
7. The nth term of the A.P. a, 3a, 5a, …… is
8. The point P on x-axis equidistant from the points A(–1, 0) and B(5, 0) is
9. The co-ordinates of the point which is reflection of point (–3, 5) in x-axis are
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
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 = _________.
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 O1 and O2 respectively of the object A are _________, _________.
Q. Nos. 16 to 20 are short answer type questions of 1 mark each.
16. If sin A + sin2 A = 1, then find the value of the expression (cos2 A + cos4 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 x2 < 4.
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 :
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
In figure, if AD ⊥ BC, then prove that AB2 + CD2 = BD2 + AC2.
23. Prove that
Show that tan4 θ + tan2 θ = sec4 θ – sec2 θ
24. The volume of a right circular cylinder with its height equal to the radius is 25 1/7 cm3. 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).
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)
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.
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 cm2. 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.
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.
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.
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.