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:

If A and B are two sets, then find the value of (A ∩ B) ∪ (A - B).

  

Question 2:

If f(x) = x² – 3x + 1, find x such that f(2x) = f(x).

 

Question 3:

Find the angle between the minute hand and hour hand of a clock when the time is 7: 20.

 

Question 4:

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has at least one boy and one girl?

                                                            SECTION – B                                      (2 * 8 = 16)

Question 5:

How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits is not allowed?

 

Question 6:

Find the domain and range of the function f(x) = .

 

Question 7:

Find the value of the trigonometric function tan  + cot( ).

 

Question 8:

Find the principal and general solutions of the equation cot x = -√3

 

Question 9:

The 17^th term of an AP exceeds its 10^th term by 7. Find the common difference.

 

Question 10:

Evaluate lim_x->0

 

Question 11:

Write the contra positive and converse of the following statements.

(i) If x is a prime number, then x is odd.

(ii) It two lines are parallel, then they do not intersect in the same plane.

 

Question 12:

A letter is chosen at random from the word ‘ASSASSINATION’. Find the probability that letter is (i) a vowel (ii) a consonant

 

                                                     SECTION – C                                     (4 * 11 = 44)

Question 13:

Are the following pair of sets equal? Give reasons.

(i) A = {2, 3}, B = {x : x is solution of x² + 5x + 6 = 0}

(ii) A = { x : x is a letter in the word FOLLOW}

      B = { y : y is a letter in the word WOLF}

 

Question 14:

Find the domain and range of the function f(x) = 2 - |x - 5|.

 

Question 15:

Prove that sin 3x = 3 sin x – 4 sin³ x

 

Question 16:

Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.2² + 3.2² + ... + n.2ⁿ = (n – 1)2ⁿ⁺¹ + 2

 

Question 17:

If ²ⁿC₃ : ⁿC₃ = 11 : 1, the what is the value of n?

 

Question 18:

Solve the following system of inequalities graphically:

3x + 2y ≤ 150, x + 4y ≤ 80, x ≤ 15, y ≥ 0, x ≥ 0

 

Question 19:

If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.

OR

If the third term of an A.P. is 7 and its 7^th term is 2 more than three times of its third term, then find the sum of its first 20 terms. 

 

Question 20:

Find the centre and radius of the circle x² + y² – 8x + 10y – 12 = 0

 

Question 21:

Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) and C (–1, 1, 2). Find the coordinates of the fourth vertex.

 

Question 22:

If y =  +  then prove that 2x *  + y = 2

 

Question 23:

In a town of 6000 people, 1200 are over 50 years old and 2000 are females. It is known that 30% of the females are over 50 years. What is the probability that a randomly chosen individual from the town is either female of over 30 years.

 

                                                     SECTION – D                                       (6 * 6 = 36)

Question 24:

Find the value of:

(i) sin 75°                                                  (ii) tan 15°

 

Question 25:

Solve the equation 3x² – 4x + 203  = 0

 

Question 26:

Show that 9ⁿ⁺¹ - 8n - 9 is divisible by 64, whenever n is a positive integer.

 

Question 27:

If S₁, S₂, S₃ are the sum of first n natural numbers, their squares and their cubes, respectively, show that 9S₂² = S₃(1 + 8S₁)

 

Question 28:

Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and (6, 0, 0).

 

Question 29:

Find the mean and variance for the first n natural numbers.

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