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JEE Mathematics Solved Question Paper 2005

Multiple Choice Questions

The value of $\sin (\sin^{- 1} (\frac{1}{3}) + \sec^{- 1} (3)) + \cos (\tan^{- 1} (\frac{1}{2}) + \tan^{- 1} (2))$ is :

If f(x) = x $[\sqrt{x} - \sqrt{x + 1}]$ , then :

f(x) is continuous but not differentiable at x = 0
f(x) is not differentiable at x = 0
f(x) is differentiable at x = 0
None of the above

Let A, B and C be n x n matrices. Which one of the following is a correct statement?

If AB = AC, then B = C
If A³ + 2A² + 3A + 51 = 0, then A is invertible
If A² = 0,then A = 0
None of these

The value of $\cot^{- 1} (9) + \csc^{- 1} (\frac{\sqrt{41}}{4})$ is given by :

0
$\frac{π}{4}$
$\frac{π}{2}$
$\tan^{- 1} (2)$

If the line ax + by + c = 0 is a normal to the curve y =1, then :

a > 0, b > 0
a > 0, b < 0
a < 0, b < 0
Data is insufficient

If f(x) = cos(log(x)), then f(x)f(y) $- \frac{1}{2} [f (\frac{x}{y}) + f (xy)]$ has the value :

- 1
$\frac{1}{2}$
- 2
zero

If y = 3^{x - 1} + 3^{-x - 1} (x real), then the least value ofy is

2
6
2/3
None of these

$\int_{0}^{\frac{π}{2}} \frac{\sqrt{\cot (x)}}{\sqrt{\cot (x)} + \sqrt{\tan (x)}} dx$ is equal to :

1
- 1
$\frac{π}{2}$
$\frac{π}{4}$

Area bounded by the curve y = log_e(x), x = 0, y $\leq$ 0 and x - axis is :

1 sq unit
$\frac{1}{2}$ sq unit
2 sq unit
None of these

10.
If ${|\vec{a} \times \vec{b}|}^{2} + {|\vec{a} . \vec{b}|}^{2} = 144 and |\vec{a}| = 4, then |\vec{b}|$ is equal to

12

3

8

4

$Given, {|\vec{a} \times \vec{b}|}^{2} + {|\vec{a} . \vec{b}|}^{2} = 144 and |\vec{a}| = 4 {|\vec{a} \times \vec{b}|}^{2} + {|\vec{a} . \vec{b}|}^{2} = {|\vec{a}|}^{2} {|\vec{b}|}^{2} \Rightarrow 144 = 16 . {|\vec{b}|}^{2} \Rightarrow {|\vec{b}|}^{2} = 9 \Rightarrow |\vec{b}| = 3$