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

• $\frac{234}{7}$ units

• $\frac{288}{21}$ units

• $\frac{221}{3}$ units

• $\frac{234}{21}$ units

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

• a minimum

• a discontinuity

• a point of inflexion

• a maximum

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

• $\log \left(1 + \mathrm{x}\right) < \frac{\mathrm{x}}{1 + \mathrm{x}} < \mathrm{x}$

• $\frac{\mathrm{x}}{1 + \mathrm{x}} < \mathrm{x} < \log \left(1 + \mathrm{x}\right)$

• $\mathrm{x} < \log \left(1 + \mathrm{x}\right) < \frac{\mathrm{x}}{1 + \mathrm{x}}$

• $\frac{\mathrm{x}}{1 + \mathrm{x}} < \log \left(1 +\hspace{0.17em}\mathrm{x}\right) < \mathrm{x}$

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

• 4

• 1

• 3

• 2

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

• c = $\frac{2}{3}$

• $\sqrt{\frac{73}{5}}$

• $\mathrm{c} = \frac{14}{\sqrt{73}}$

• $\sqrt{\frac{5}{7}}$

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

• 13

• 12

• 9

• 14

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.

• $\left(~ \mathrm{p} \wedge ~ \mathrm{q}\right) \Rightarrow \left(\mathrm{r} \wedge \mathrm{s}\right)$

• $~ \left(\mathrm{p} \wedge \mathrm{q}\right) \Rightarrow \left(\mathrm{r} \wedge \mathrm{s}\right)$

• $~ \left(\mathrm{p} \vee \mathrm{q}\right) \Rightarrow \left(\mathrm{r} \wedge \mathrm{s}\right)$

• None of these

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

• $\left[\begin{array}{ccc}12& 36& 12\\ - 12& 24& - 36\\ 0& 12& 24\end{array}\right]$

• $\left[\begin{array}{ccc}12& 26& - 12\\ 24& 36& - 36\\ 0& 12& - 24\end{array}\right]$

• $\left[\begin{array}{ccc}12& - 12& 36\\ 24& - 24& - 36\\ 0& 12& 24\end{array}\right]$

• None of the above

If N is a set of natural numbers, then under binary operation a · b = a + b, (N, ·) is

• quasi-group

• semi-group

• monoid

• group

On the interval [0, 1], the function x²⁵(1- x)⁷⁵ takes its maximum value at the point

• 0

• 1/4

• 1/2

• 1/3