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JEE Mathematics Solved Question Paper 2004

Multiple Choice Questions

11.

If , then 16A + 9B is equal to

12.

The value of the series + $\frac{x^{5}}{5!} {(\log_{e} (a))}^{5}$ + ... is

$\cosh ({xlog}_{e} (a))$
$\coth ({xlog}_{e} (a))$
$\sinh ({xlog}_{e} (a))$
$\tanh ({xlog}_{e} (a))$

13.

Coefficient of x¹⁰ in the expansion of (2 + 3x)e^{- x} is

$\frac{- 26}{(10)!}$
$\frac{- 28}{(10)!}$
$\frac{- 30}{(10)!}$
$\frac{- 32}{(10)!}$

14.

The set of all solutions of the inequation $x^{2} - 2 x + 5 \leq 0$ in R is

$R - (- \infty, - 5)$
$R - (5, \infty)$
$ϕ$
$R - (- \infty, - 4)$

15.

If (x - 2) is a common factor of the expressions x² + ax + b and x² + cx + d, then, $\frac{b - d}{c - a}$ is equal to

16.

If the roots of the equation 4x³ - 12x² + 11x + k = 0 are in arithmetic progression, then k is equal to

17.

If f(x) is apolynomial ofdegreen with rational coefficients and 1 + 2i , $2 - \sqrt{3}$ and 5 are three roots of f(x) = 0 , then the least value of n is

18.
If z₁, z₂ are two complex numbers satisfying $|\frac{z_{1} - 3 z_{2}}{3 - z_{1} z_{2}}| = 1, |z_{1}| \neq 3$ , then $|z_{2}|$ is equal to

1

2

3

4

$Given that, |\frac{z_{1} - 3 z_{2}}{3 - z_{1} z_{2}}| = 1, |z_{1}| \neq 3 \Rightarrow |z_{1} - 3 z_{2}| = |3 - z_{1} z_{2}| [∵ |\frac{z_{1}}{z_{2}}| = \frac{|z_{1}|}{|z_{2}|}] \Rightarrow {|z_{1} - 3 z_{2}|}^{2} = {|3 - z_{1} z_{2}|}^{2} \Rightarrow (z_{1} - 3 z_{2}) (z_{1} - 3 z_{2}) = (3 - z_{1} z_{2}) (3 - z_{1} z_{2}) \Rightarrow {|z_{1}|}^{2} - 3 z_{1} z_{2} - 3 z_{1} z_{2} + 9 {|z_{2}|}^{2} = 9 - 3 z_{1} z_{2} - 3 z_{1} z_{2} + {|z_{1}|}^{2} {|z_{2}|}^{2} \Rightarrow (9 - {|z_{1}|}^{2}) (1 - {|z_{2}|}^{2}) = 0 \Rightarrow {|z_{1}|}^{2} = 9 or {|z_{2}|}^{2} = 1 \Rightarrow |z_{1}| = 3 or |z_{2}| = 1 But, |z_{1}| \neq 3 ∴ |z_{2}| = 1$