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JEE Mathematics Solved Question Paper 2005

Multiple Choice Questions

21.
In a , is equal to

a

b

c

a + b + c

$a (\cos^{2} (B) + \cos^{2} (C)) + \cos (A) (ccos (C) + bcos (B)) = a [(\frac{a^{2} + c^{2} - b^{2}}{2 ac}) + (\frac{a^{2} + b^{2} - c^{2}}{2 ab})] + \frac{b^{2} + c^{2} - a^{2}}{2 bc} [c (\frac{b^{2} + a^{2} - c^{2}}{2 ab}) + b (\frac{a^{2} + c^{2} - b^{2}}{2 ac})] = (\frac{a^{2} + c^{2} - b^{2}}{2 ac}) + \frac{{(a^{2} + b^{2} - c^{2})}^{2}}{4 {ab}^{2}} + \frac{(b^{2} + c^{2} - a^{2}) (b^{2} + a^{2} - c^{2})}{4 {ab}^{2}} + \frac{(b^{2} + c^{2} - a^{2}) (a^{2} + c^{2} - b^{2})}{4 {ac}^{2}} = \frac{(a^{2} + c^{2} - b^{2}) (a^{2} + c^{2} - b^{2} + b^{2} + c^{2} - a^{2})}{4 {ac}^{2}} + \frac{(a^{2} + b^{2} - c^{2}) (a^{2} + b^{2} - c^{2} + b^{2} + c^{2} - a^{2})}{4 {ab}^{2}} = \frac{a^{2} + c^{2} - b^{2}}{2 a} + \frac{a^{2} + b^{2} - c^{2}}{2 a} = \frac{2 a^{2}}{2 a} = a$

22.

$In a ∆ ABC, \sum (b + c) \tan (\frac{A}{2}) \tan (\frac{B - C}{2})$ is equal to

23.

If $α_{1}, α_{2}, α_{3}$ respectively denote the moduli of the complex number - i, $\frac{1}{3}$ (1 + i) and - 1 + i, 3 then their increasing order is

$α_{1}, α_{2}, α_{3}$
$α_{3}, α_{2}, α_{1}$
$α_{2}, α_{1}, α_{3}$
$α_{3}, α_{1}, α_{2}$

24.

$If α is a non - real root of x^{6} = 1, then \frac{α^{5} + α^{3} + α + 1}{α^{2} + 1} is equal to,$

$α^{2}$
0
$- α^{2}$
$α$

25.

$If \cos (θ) - 4 \sin (θ) = 1, then \sin (θ) + 4 \cos (θ) is equal to$

$\pm 1$
0
$\pm 2$
$\pm 4$

26.

$The extreme values of 4 \cos (x^{2}) \cos (\frac{π}{3} + x^{2}) \cos (\frac{π}{3} - x^{2}) over R, are$

- 1, 1
- 2, 2
-3, 3
- 4, 4

27.

Two sides of a triangle are given by the roots of the equation x² - 5x + 6 = 0 and the angle between the sides is $\frac{π}{3}$ .Then, the perimeter of

$5 + \sqrt{2}$
$5 + \sqrt{3}$
$5 + \sqrt{5}$
$5 + \sqrt{7}$

28.

A tower, of x metres high, has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant y metres from the foot of the tower. Then, the length of the flagstaff(in metres), is

$\frac{y (x^{2} - y^{2})}{(x^{2} + y^{2})}$
$\frac{x (y^{2} + x^{2})}{(y^{2} - x^{2})}$
$\frac{x (x^{2} + y^{2})}{(x^{2} - y^{2})}$
$\frac{x (x^{2} - y^{2})}{(x^{2} + y^{2})}$

29.

A coin and six faced die, both unbiassed, are thrown simultaneously. The probability of getting a head on the coin and an odd number on the die, is

$\frac{1}{2}$
$\frac{3}{4}$
$\frac{1}{4}$
$\frac{2}{3}$

30.

A number n is chosen at random from S = {1, 2, 3, ... , 50}. LetA = {n ∈ S:n + 50/n > 27}, B={n ∈ S : n is a prime) and C = {n ∈ S : n is a square). Then,correct order of their probabilities is