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JEE Mathematics Solved Question Paper 2010

Multiple Choice Questions

21.

If A is a non-singular matrix of order 3, then adj (adj A) is equal to

A
A^{- 1}
$\frac{1}{|A|} A$
$|A| A$

22.

If $[\begin{matrix} x - y - z \\ - y + z \\ z \end{matrix}] = [\begin{matrix} 0 \\ 5 \\ 3 \end{matrix}]$ , then the value of x, y and z are respectively

5, 2, 2
1, - 2, 3
0, - 3, 3
11, 8, 3

23.

Which one of the following is true always for any two non-singular matrices A and B of same order ?

AB = BA
(AB)^t = A^tB^t
(A + B)(A - B) = A² - B²
(AB)^{- 1} = B^{- 1}A^{- 1}

24.

If $\tan^{- 1} (x + 2) + \tan^{- 1} (x - 2) - \tan^{- 1} (\frac{1}{2}) = 0$ , then one of the values of x is equal to

- 1
5
$\frac{1}{2}$
1

25.

The value of $\sin^{- 1} (\frac{4}{5}) + 2 \tan^{- 1} (\frac{1}{3})$ is equal to

$\frac{π}{3}$
$\frac{π}{4}$
$\frac{π}{2}$
$\frac{π}{6}$

26.

The value of $\sin^{- 1} (\cos (4095 °))$ is equal to

$- \frac{π}{3}$
$\frac{π}{6}$
$- \frac{π}{4}$
$\frac{π}{4}$

27.

If $|\vec{a}| = 5, |\vec{b}| = 6 and \vec{a} . \vec{b} = - 25, then |\vec{a} \times \vec{b}| is equal to$

25
$6 \sqrt{11}$
11 $\sqrt{5}$
$5 \sqrt{11}$

28.
If $\vec{p}, \vec{q} and \vec{r}$ are perpendicular to $\vec{q} + \vec{r},$ $\vec{r} + \vec{p},$ $\vec{p} + \vec{q}$ respectively and if $|\vec{p} + \vec{q}| = 6,$ $|\vec{q} + \vec{r}| = 4 \sqrt{3}$ and $|\vec{r} + \vec{p}| = 4,$ then $|\vec{p} + \vec{q} + \vec{r}|$ is

5 $\sqrt{2}$

10

15

5

5 $\sqrt{2}$

$\vec{p} ⊥ (\vec{q} + \vec{r}) \Rightarrow \vec{p} . (\vec{q} + \vec{r}) = 0 \Rightarrow \vec{p} . \vec{q} + \vec{p} . \vec{r} = 0 . . . (ii) \vec{q} ⊥ (\vec{r} + \vec{p}) \Rightarrow \vec{q} . (\vec{r} + \vec{p}) = 0 \Rightarrow \vec{q} . \vec{r} + \vec{q} . \vec{p} = 0 . . . (ii) \vec{r} ⊥ (\vec{p} + \vec{q}) \Rightarrow \vec{r} . (\vec{p} + \vec{q}) = 0 \Rightarrow \vec{r} . \vec{p} + \vec{r} . \vec{q} = 0 . . . (ii)$

Adding Eqs. (i), (ii) and (iii), we get

$\vec{p} . \vec{q} + \vec{q} . \vec{r} + \vec{r} . \vec{p} = 0 . . . (iv)$

$Now, |\vec{p} + \vec{q}| = 6 \Rightarrow (\vec{p} + \vec{q}) . (\vec{p} + \vec{q}) = 36 \Rightarrow {|\vec{p}|}^{2} + \vec{p} . \vec{q} + \vec{q} . \vec{p} + {|\vec{q}|}^{2} = 36 . . . (v) Similarly, |\vec{q} + \vec{r}| = 4 \sqrt{3} \Rightarrow {|\vec{q}|}^{2} + \vec{q} . \vec{r} + \vec{r} . \vec{p} + {|\vec{r}|}^{2} = 48 . . . (vi) and |\vec{r} + \vec{p}| = 4 \Rightarrow {|\vec{r}|}^{2} + \vec{r} . \vec{p} + \vec{p} . \vec{r} + {|\vec{p}|}^{2} = 16 . . . (vii) Adding Eqs . (v), (vi) and (vii), we get 2 {|\vec{p}|}^{2} + 2 {|\vec{q}|}^{2} + 2 {|\vec{r}|}^{2} + 2 (\vec{p} . \vec{q} + \vec{q} . \vec{r} + \vec{r} . \vec{p}) = 100 \Rightarrow {|\vec{p}|}^{2} + {|\vec{q}|}^{2} + {|\vec{r}|}^{2} = \frac{100}{2} = 50 . . . (viii) [∵ using Eq . (iv)] Now, {(\vec{p} + \vec{q} + \vec{r})}^{2} = {|\vec{p}|}^{2} + {|\vec{q}|}^{2} + {|\vec{r}|}^{2} + 2 (\vec{p} . \vec{q} + \vec{q} . \vec{r} + \vec{r} . \vec{p}) = 50 [∵ using Eqs . (iv) and (viii)] \Rightarrow |\vec{p} + \vec{q} + \vec{r}| = 5 \sqrt{2}$

29.

The vectors of magnitude a, 2a, 3a meet at a point and their directions are along the diagonals of three adjacent faces of a cube. Then, the magnitude of their resultant is

30.

Which one of the following vectors is of magnitude 6 and perpendicular to both $\vec{a} = 2 \hat{i} + 2 \hat{j} + \hat{k}$ $and \vec{b} = \hat{i} - 2 \hat{j} + 2 \hat{k}$ ?