Maximum Marks : 100                                                              Time Taken : 3 Hours

General Instructions:

(i) All the questions are compulsory.

(ii) This question paper consists of 4 sections A, B, C and D.

(iii) Questions from Section A carry 1 mark each.

(iv) Questions from Section B carry 2 marks each.

(v) Questions from Section C carry 4 marks each.

(vi) Questions from Section D carry 6 marks each.

(vii) Use of calculator is not permitted.

 

                                                            SECTION – A                                      (1 * 4 = 4)

Question 1:

Prove that if A ∪ B = C and A ∩ B = φ then A = C – B

  

Question 2:

If f(x) = , where a > 0 and n ∈ N, the find f(f(x)).

 

Question 3:

The large hand of a clock Is 42 cm long. How many centimetres does its extremity move in 20 minutes?

 

Question 4:

Everyone in a room shakes hands with everybody else. The total number of handshakes is 66. Find the total number of persons in the room.

 

                                                            SECTION – B                                      (2 * 8 = 16)

Question 5:

In a survey, 85 % of Indians like mangoes, whereas 55 % like guavas. What percentage of the Indians like both mangoes and guavas?

 

Question 6:

Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by

{(a, b): a, b ∈ A, b is exactly divisible by a}.

(i) Write R in roster form

(ii) Find the domain of R

(iii) Find the range of R.

 

Question 7:

If sin x + sin² x = 1 then find the value of cos² x + cos⁴ x

 

Question 8:

Given, 3 * tan (θ - 15⁰) = tan (θ + 15⁰), 0 < θ < π, then what is the value of θ?

 

Question 9:

If 9 times the 9^th term of an A.P. is equal to 13 times the 13^th term, then find the 22^nd term of the A.P.

 

Question 10:

Find the positive integer n so that lim_x->3  = 108.

 

Question 11:

For the given if-then statements, write the contra-positive statement.

If a triangle is equilateral, then all of its angles are 60°

If a number is multiple of 9, then it is multiple of 3.

 

Question 12:

Find the probability of getting 53 Mondays in a leap year.

 

                                                     SECTION – C                                     (4 * 11 = 44)

Question 13:

For A, B and C three sets under the universal set U, ifn(U) = 600, n(A) = 250, n(B) = 200 and n(A ∩ B) = 50. Find n(A’ ∩ B’).

 

Question 14:

Find the domain and range of the function f(x) = √(9 – x²).

 

Question 15:

If tan x = , then find the value of x. 

 

Question 16:

Prove that 10^2n-1 + 1 is divisible by 11 for all n ∈ N.

 

Question 17:

Ravi scored 70 and 75 marks in the first two-unit test. Calculate the minimum marks he should get in the third test to have an average of at least 60 marks.

 

Question 18:

Find the value of r if ⁵P_r = 2 * ⁶P_r-1                                 

 

Question 19:

In an AP, if m^th term is n and the n^th term is m, where m ≠ n, find the p^th term.

OR

A man saved Rs. 66000 in 20 years. In each succeeding year after the first year, he saved Rs. 200 more than what he saved in the previous year. How much did he save in the first year?

 

Question 20:

If the eccentricity of an ellipse is  and the distance between its foci is 10, then find latus rectum of the ellipse.

 

Question 21:

Find the ratio in which the YZ-plane divides the line segment formed by joining the points (–2, 4, 7) and (3, –5, 8).

 

Question 22:

Find derivative of 1 + x + x² + x³ + ……….+ x⁵⁰ at x = 1

 

Question 23:

Six boys and six girls sit in a row at random. Find the probability that the boys and girls sit alternatively.

 

                                                     SECTION – D                                       (6 * 6 = 36)

Question 24:

In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.

OR

If tan² θ = 1 - e², then what is the value of sec θ + tan³ θ * cosec θ?

 

Question 25:

If  then find the least positive integral value of m.

 

Question 26:

Find the middle terms in the expansions of (  + 9y)¹⁰

 

Question 27:

Find the sum of the series (3³ – 2³) + (5³ – 4³) + (7³ – 6³) + … to n terms.

 

Question 28:

(i) Find the co ordinate of the point which divides the join of P(2, -1, 4) and Q(4, 3, 2) in the ratio 2 : 5 (a) internally (b) externally.

(ii) If the distance between the points (a, 0, 1) and (0, 1, 2) is √27, then what is the value of a?

 

Question 29:

Find the mean deviation about the median for the data.

xi	15	21	27	30	35
yi	3	5	6	7	8

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