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JEE Mathematics Solved Question Paper 2015

Multiple Choice Questions

11.
The shortest distance between the lines $\frac{x - 7}{3} = \frac{y + 4}{- 16} = \frac{z - 6}{7}$ and $\frac{x - 10}{3} = \frac{y - 30}{8} = \frac{4 - z}{5}$ is

$\frac{234}{7}$ units

$\frac{288}{21}$ units

$\frac{221}{3}$ units

$\frac{234}{21}$ units

$\frac{288}{21}$ units

Given, lines are $\frac{x - 7}{3} = \frac{y + 4}{- 16} = \frac{z - 6}{7}$ and $\frac{x - 10}{3} = \frac{y - 30}{8} = \frac{4 - z}{5}$

The vector form of given equations are

$r = 7 \hat{i} - 4 \hat{j} + 6 \hat{k} + λ (3 \hat{i} - 16 \hat{j} + 7 \hat{k})$

and $r = 10 \hat{i} + 30 \hat{j} + 4 \hat{k} + μ (3 \hat{i} + 8 \hat{j} - 5 \hat{k})$

On comparing these equations with

$r = a_{1} + {λb}_{1} and r = a_{2} + {μb}_{2}$ , we get

$a_{1} = 7 \hat{i} - 4 \hat{j} + 6 \hat{k} a_{2} = 10 \hat{i} + 30 \hat{j} + 4 \hat{k} b_{1} = 3 \hat{i} - 16 \hat{j} + 7 \hat{k} b_{2} = 3 \hat{i} + 8 \hat{j} - 5 \hat{k}$

Now, (a₂ - a₁)

= $(10 \hat{i} + 30 \hat{j} + 4 \hat{k}) - (7 \hat{i} - 4 \hat{j} + 6 \hat{k})$

= $3 \hat{i} + 34 \hat{j} - 2 \hat{k}$

$b_{1} \times b_{2} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & - 16 & 7 \\ 3 & 8 & - 5 \end{matrix}| = (80 - 56) \hat{i} + (21 + 15) \hat{j} + (24 + 48) \hat{k} = 24 \hat{i} + 36 \hat{j} - 2 \hat{k}$

Then, $|b_{1} \times b_{2}| = \sqrt{{(24)}^{2} + {(36)}^{2} + {(72)}^{2}}$

$= \sqrt{576 + 1296 + 5184} = \sqrt{7056} = 84$

$∴ Shortest distance = |\frac{(a_{2} - a_{1}) . (b_{1} \times b_{2})}{|b_{1} \times b_{2}|}| = |\frac{(3 \hat{i} + 34 \hat{j} - 2 \hat{k}) (24 \hat{i} + 36 \hat{j} + 72 \hat{k})}{84}| = |\frac{72 + 1224 - 144}{84}| = |\frac{1152}{84}| = \frac{288}{21} units$

12.

At t = 0, the function f(t) = $\frac{\sin (t)}{t}$ has

a minimum
a discontinuity
a point of inflexion
a maximum

13.

Which of the following inequality is true for x > 0?

$\log (1 + x) < \frac{x}{1 + x} < x$
$\frac{x}{1 + x} < x < \log (1 + x)$
$x < \log (1 + x) < \frac{x}{1 + x}$
$\frac{x}{1 + x} < \log (1 + x) < x$

14.

If the complex number z lies on a circle with centre at the origin and radius = $\frac{1}{4}$ , then the 4 complex number - 1 + 8z lies on a circle with radius

15.

If line y = 2x + c is a normal to the ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{16} = 1$ ,then

c = $\frac{2}{3}$
$\sqrt{\frac{73}{5}}$
$c = \frac{14}{\sqrt{73}}$
$\sqrt{\frac{5}{7}}$

16.

If x² + x + 1 = 0, then the value of $\sum_{n = 1}^{6} {(x^{n} + \frac{1}{x^{n}})}^{2}$ is

17.

If p : It rains today, q : I go to school, r : I shall meet any friends ands : I shall go for a movie, then which of the following is the proportion? If it does not rain or if I do not go to school, then I shall meet my friend and go for a movie.

$(~ p \land ~ q) \Rightarrow (r \land s)$
$~ (p \land q) \Rightarrow (r \land s)$
$~ (p \lor q) \Rightarrow (r \land s)$
None of these

18.

If the matrix A = $[\begin{matrix} 1 & 3 & 1 \\ - 1 & 2 & - 3 \\ 0 & 1 & 2 \end{matrix}]$ , then adj (adj A) is equal to

$[\begin{matrix} 12 & 36 & 12 \\ - 12 & 24 & - 36 \\ 0 & 12 & 24 \end{matrix}]$
$[\begin{matrix} 12 & 26 & - 12 \\ 24 & 36 & - 36 \\ 0 & 12 & - 24 \end{matrix}]$
$[\begin{matrix} 12 & - 12 & 36 \\ 24 & - 24 & - 36 \\ 0 & 12 & 24 \end{matrix}]$
None of the above