Let y = t¹⁰ + 1 and x = t⁸ + 1, then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is equal to :

• $\frac{5}{2}\mathrm{t}$

• 20t⁸

• $\frac{5}{16{\mathrm{t}}^6}$

• None of these

2.

The vectors $\overrightarrow{\mathrm{AB}} = 3\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + 4\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{AC}} = 5\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ are the sides of a triangle ABC. The length ofthe median through A is :

• $\sqrt{13} \mathrm{unit}$

• 2$\sqrt{5}$ unit

• 5 unit

• 10 unit

A function f on R into itself is continuous at a point ain R, iff for each $\in$ > 0, there exists, $\mathrm{\delta }$ > 0 such that :

• $\left|\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(\mathrm{a}\right)\right| < \in \Rightarrow \left|\mathrm{x} - \mathrm{a}\right| < \mathrm{\delta }$

• $\left|\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(\mathrm{a}\right)\right| > \in \Rightarrow \left|\mathrm{x} - \mathrm{a}\right| > \mathrm{\delta }$

• $\left|\mathrm{x} - \mathrm{a}\right| > \mathrm{\delta } \Rightarrow \left|\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(\mathrm{a}\right)\right| > \in$

• $\left|\mathrm{x} - \mathrm{a}\right| < \mathrm{\delta } \Rightarrow \left|\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(\mathrm{a}\right)\right| < \in$

If A = $\left[\begin{array}{ccc}2& 4& 5\\ 4& 8& 10\\ - 6& - 12& - 15\end{array}\right]$, then rank of A is equal to :

• 0

• 1

• 2

• 3

The minimum value of $3\sin \left(\mathrm{\theta }\right) + 4\cos \left(\mathrm{\theta }\right)$ is :

• 5

• 1

• 3

• - 5

${\cos }^{-1}\left(\frac{1}{2}\right) + 2{\sin }^{-1}\left(\frac{1}{2}\right)$ is equal to :

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{2\mathrm{\pi }}{3}$

If f(x) = (a - xⁿ)^1/n where a > 0 and n $\in$ N, then fof(x) is equal to :

• a

• x

• xⁿ

• aⁿ

Maximum value of sin(x) - cos(x) is equal to :

• $\sqrt{2}$

• 1

• 0

• none of these

A particle possess two velocities simultaneously at an angle of ${\tan }^{-1}\left(\frac{12}{5}\right)$; to each other. Their resultant is 15 m/s. If one velocity is 13 m/s, then the other will be :

• 5 m/s

• 4 m/s

• 12 m/s

• 13 m/s

Define f on R into itself by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\mathrm{xsin}\left(\frac{1}{\mathrm{x}}\right), \mathrm{when} \mathrm{x} \ne 0 \\ 0, \mathrm{when} \mathrm{x} = 0\right\} \end{gathered}$$, then :

• f is continuous at 0 but not differentiable at 0

• f is both continuous and differentiable at 0

• f is differentiable but not continuous at 0

• None of these