APEEJAY SCHOOL SAKET

CLASS XI

SESSION 2018-19

FIRST TERMINAL EXAMINATION

SUBJECT MATHEMATICS

TIME 3 HRS

MAX MARKS 100

GENERAL INSTRUCTIONS

Q1-Q4 are of 1 mark each.

Q5-Q12 are of 2 marks each.

Q13-Q23 are of 4 marks each.

Q24-Q29 are of 6 marks each.

SECTION A

1. Evaluate <math>(1+i)^6+(1-i)^3</math>.
2. Let A={1,2} and B={3,4}. Find the number of relations from A to B.
3. Find the modulus of .

1. There are four bus routes between A and B; and three bus routes between B and C. A man can travel round -trip in number of ways by bus from A to C via
2. If he does not want to use a bus route more than once, in how many ways can he make round trip?

SECTION B

1. What is the middle term in the expansion of <math>(x^2+)^{11}</math>?
2. Show that the triangle formed by the points 1, and i are vertices of an isosceles triangle in the Argand plane.
3. Determine the domain and range of the relation R defined by <math>R=\{(x,x+5):x \in \{0,1,2,3,4,5\}</math>
4. Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements do the vowels never occur together?
5. Show using properties of sets that for any sets A and B, (i)A=(A<math>\cap</math>B)<math>\cup</math>(A-B) (ii)A<math>\cup</math>(B-A) =(A<math>\cup</math>B).
6. Rayi obtained 70 and 75 marks in first two unit test. Find the minimum marks

he should get in the third test to have an average of at least 60 marks.

Find the term independent of x in the expansion of (x^1/3 + )¹⁸,x>0.

What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these four cards belong to four different suits?

SECTION C

The sum of the coefficients of the first three terms in the expansion of

1. m, x≠0, m being a natural number, is 559. Find the term of the expansion containing x3.

1. Prove that <math>\cot x \cot 2x - \cot 2x \cot 3x - \cot 3x \cot x = 1</math>.

Prove that cos²x + cos²(x+) +cos²(x -) =

1. In any triangle ABC, prove that <math>(b^2-c^2)\cot A + (c^2-a^2)\cot B + (a^2-b^2)\cot C = 0</math>.
2. Prove that <math>\cos 20^{\circ} \cos 100^{\circ} + \cos 100^{\circ} \cos 140^{\circ} - \cos 140^{\circ} \cos 200^{\circ} = -3/4</math>
3. Find p if the 17^th and 18^th terms of the expansion (2+a)⁵⁰are equal.
4. For any sets A and B show that <math>P(A \cap B) = P(A) \cap P(B)</math>

Find the number of words with or without meaning which can be made using all the letters of the word MOTHER? If the words are written as in a dictionary, what will the rank of word MOTHER?

1. How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25%but less than 30% acid content?

Show that the co-efficient of the middle term in the expansion of (1+x)²ⁿ is equal to the sum of the co-efficients of two middle terms in the expansion of (1+x)^2n-1.

Prove that 1+2 +3+....+n< (2n+1)².

SECTION D

1. Let R be a relation from Q to Q defined by <math>R = \{(a, b): a, b \in Q \text{ and } a-b \in Z\}.</math> Show that (i)(a,a)∈R for all a∈Q (ii)(a,b)∈R implies that (b,a)∈R (iii)(a,b)<math>\in</math>R and (b,c)<math>\in</math>R implies that (a,c)<math>\in</math>R.

1. (a) Find the square root of 3-4i.
2. Solve <math>\sqrt{5} x^2 + x + \sqrt{5} = 0</math>

(a)Find (a+b)⁶-(a-b)⁶.

(b)A college awarded 38 medals in football, 15 in basketball and 20 in cricket.