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:

Find the radian measures corresponding to the degree measures –47° 30'.

  

Question 2:

If 4x + i(3x – y) = 3 + i (– 6), where x and y are real numbers, then find the values of x and y.

 

Question 3:

Solve the linear inequality: x - 42≥x + 14- 1

 

Question 4:

Two dice are thrown simultaneously. Find the probability of getting a total of 9.

 

                                                            SECTION – B                                      (2 * 8 = 16)

Question 5:

If xⁿ – 1 is divisible by x – k, then find the least positive integral value of k.

 

Question 6:

If X = {(x, y) : y = e^-x; x ∈ W} and Y = {(x, y) : y = e^x, x ∈ W}, then what will be the value of (X∩Y)?

 

Question 7:

For any set N = {1, 2, 3, ..., 50},

(i) Write the subset A of N, whose elements are odd numbers.

(ii) Write the subset B of N, whose elements are represented by x + 2, where x ∈ N.

 

Question 8:

Express (1 + 3i)^-1 in the form of a + ib.

 

Question 9:

Solve the equation x² – x + 2 = 0

 

Question 10:

Find n if ^n-1P₃ : ⁿP₄ = 1 : 9

 

Question 11:

Find the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digit is to be repeated.

 

Question 12:

Find the coefficient of a⁵ b⁷ in (a – 2b)¹²

 

                                                     SECTION – C                                     (4 * 11 = 44)

Question 13:

If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and X ∩ Y has 10 elements, how many elements does Y have?

 

Question 14:

Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true?

(i) f is a relation from A to B

(ii) f is a function from A to B.

Justify your answer in each case.

 

Question 15:

Find the values of other five trigonometric functions if sin x = 3/5, x lies in second quadrant.

 

Question 16:

Prove that: cos  * cos  – sin  * sin  = sin(x + y)

 

Question 17:

If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that:

(a² + b²)(c² + d²)(e² + f²)(g² + h²) = A² + B²

 

Question 18:

While drilling hole it was found that temperature (T) changes at (x) km below the surface. It was given by T = 30 + 25(x - 3) and 3 < x < 15. Determine the depth at which temperature will be between 400°C and 700°C.

 

Question 19:

Solve the given inequality and show the graph of the solution on number line:

 

OR

If the third term in the binomial expansion of (1 + x)^m is ( )x² then find the

rational value of m.

 

Question 20:

Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their sum is less than 40.

 

Question 21:

How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?

 

Question 22:

A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. Then resulting mixture is to be more than 4% but less than 6% boric acid. If there are 640 litres of the 8% solution, how many litres of 2% solution will have to be added?

 

Question 23:

Prove by the Principle of Mathematical Induction that

1 × 1! + 2 × 2! + 3 × 3! + ... + n × n! = (n + 1)! – 1 for all natural numbers n.

 

                                                     SECTION – D                                       (6 * 6 = 36)

Question 24:

Find the number of different 8-letter arrangements that can be made

from the letters of the word DAUGHTER so that

(i) all vowels occur together (ii) all vowels do not occur together.

 

Question 25:

Find the modulus and the argument of the complex number z = -√3 + i

 

Question 26:

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

 

Question 27:

There are 100 multiple choice questions in an examination. For every correct answer the student gets 5 marks and for every wrong answer 2 marks are deducted from the total score of correct answers. Suppose the student attempts all the questions and his total score is 220, how many questions did he attempt correctly?

 

Question 28:

Find the general solution of the equation sin 2x + cos x = 0

 

Question 29:

Prove that For all n ≥ 1, 8ⁿ – 3ⁿ is divisible by 5.

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