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 ∪ (A ∩ B).

 

Question 2:

Find the value of lim_x->0  

 

Question 3:

If (a - 1, b + 2) = (3, 1), find the values of a and b.

 

Question 4:

The minute hand of a watch is 1.5 cm long. How far does its tip move in 40 minutes? (Use π = 3.14).

 

                                                            SECTION – B                                      (2 * 8 = 16)

Question 5:

If the 3^rd 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 6:

In a ΔABC, find value of (b + c) cos A + (c + a) cos B + (a + b) cos C.

 

Question 7:

Find the value of cos (–1710°).

 

Question 8:

If f(x) = x³ - , then find the value of f(x) + f .

 

Question 9:

Consider the experiment of rolling a die. Let A be the event ‘getting a prime number’, B be the event ‘getting an odd number’. Write the sets representing the events (i) A or B              (ii) A and B           (iii) A but not B            (iv) ‘not A’

 

Question 10:

Find the component statements of the following and check whether they are true or not.

(i) A square is a quadrilateral and its four sides equal.

(ii) All prime numbers are either even or odd.

 

Question 11:

Evaluate: lim_x->1

 

Question 12:

It is needed to seat 5 boys and 4 girls in a row so that the girl gets the even places. How many are such arrangements possible?

 

                                                     SECTION – C                                     (4 * 11 = 44)

Question 13:

A survey shows that 73% of the Indians like apples, whereas 65% like oranges. What % Indians like both apples and oranges?

 

Question 14:

Find the domain and the range of the real function f defined by

f(x) = √(x − 1)

 

Question 15:

Prove that cos 4x = 1 – 8 * sin² x * cos² x

 

Question 16:

Use principle of mathematical induction to prove that x²ⁿ – y²ⁿ is divisible by x + y.

 

Question 17:

Solve:  ≥  –

 

Question 18:

Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements

(i) do the words start with P

(ii) do all the vowels always occur together

OR

Find n, if  and  are in the ratio 2 : 1.

 

Question 19:

The ratio of the sums of m and n terms of an A.P. is m² : n². Show that the ratio of m^th and n^th term is (2m – 1) : (2n – 1).

 

Question 20:

Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x + y = 16.

 

Question 21:

Find the coordinates of a point on y-axis which are at a distance of 5√2 from the point P(3, –2, 5).

 

Question 22:

Find the derivative of  for some constant a.

 

Question 23:

If E and F are events such that P(E) = , P(F) =  and P(E and F) = , find

(i) P(E or F)                      (ii) P(not E and not F)

 

                                                     SECTION – D                                         (6 * 6 = 36)

Question 24:

Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 * cos x * cos 2x * sin 4x

 

Question 25:

Convert the complex number √3 + i in polar form.

 

Question 26:

Find the value of {a² + √(a² - 1)}⁴ + {a² + √(a² - 1)}⁴

 

Question 27:

The sum of first three terms of a G.P. is  and their product is 1. Find the common ratio and the terms.

 

Question 28:

The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12 and 14. Find the remaining two observations.

OR

The mean and standard deviation of marks obtained by 50 students of a class in three subjects, Mathematics, Physics and Chemistry are given below:

Subject	Mathematics	Physics	Chemistry
Mean	42	32	40.9
Standard Deviation	12	15	20

Which of the three subjects shows the highest variability in marks and which shows the lowest?

 

Question 29:

Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (–4, 0, 0) is equal to 10.

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