In Z, the set of all integers, the inverse of - 7 w.r.t. * defined by ab = a + b + 7 for all a,b $\in$ Z is

• - 14

• 7

• 14

• - 7

$\mathrm{In} ∆\mathrm{ABC} \mathrm{if} \left|\begin{array}{ccc}1& \mathrm{a}& \mathrm{b}\\ 1& \mathrm{c}& \mathrm{a}\\ 1& \mathrm{b}& \mathrm{c}\end{array}\right|$ = 0, if then sin²(A) + sin²(B) + sin²(C) is equal to

• $\frac{4}{9}$

• $\frac{9}{4}$

• $3\sqrt{3}$

• 1

Assuming that the sums and products given below are defined which ofthe following is not true for matrices?

• AB = AC does not imply B = C

• A + B = B + A

• (AB)' = B'A'

• AB = 0 implies A = 0 or B = 0

Inverse of the matrix $\left[\begin{array}{cc}1& - 2\\ 3& 4\end{array}\right]$ is

• $\frac{1}{10}\left[\begin{array}{cc}1& - 2\\ 3& 4\end{array}\right]$

• $\frac{1}{10}\left[\begin{array}{cc}4& 2\\ - 3& 1\end{array}\right]$

• $\left[\begin{array}{cc}4& 2\\ - 3& 1\end{array}\right]$

• $\frac{1}{10}\left[\begin{array}{cc}4& - 2\\ - 3& 1\end{array}\right]$

$\left|\begin{array}{ccc}{\mathrm{b}}^2{\mathrm{c}}^2& \mathrm{bc}& \mathrm{b} +\hspace{0.17em}\mathrm{c}\\ {\mathrm{c}}^2{\mathrm{a}}^2& \mathrm{ca}& \mathrm{c} + \mathrm{a}\\ {\mathrm{a}}^2{\mathrm{b}}^2& \mathrm{a} \mathrm{b}& \mathrm{a} + \mathrm{b}\end{array}\right|$ is equal to

• $\frac{1}{\mathrm{abc}}\left(\mathrm{ab} +\mathrm{bc} + \mathrm{ca}\right)$

• ab  + bc + ca

• 0

• a + b + c

$\sin \left(\frac{1}{2}{\cos }^{-1}\left(\frac{4}{5}\right)\right)$

• $- \frac{1}{\sqrt{10}}$

• $\frac{1}{\sqrt{10}}$

• $- \frac{1}{10}$

• $\frac{1}{10}$

The derivative of ${\cos }^{-1}\left(\frac{1 - {\mathrm{x}}^2}{1 + {\mathrm{x}}^2}\right) \mathrm{w}. \mathrm{r}. \mathrm{t}. {\cot }^{-1}\left(\frac{1 - 3{\mathrm{x}}^2}{3\mathrm{x} - {\mathrm{x}}^3}\right)$ is

• $\frac{3}{2}$

• 1

• $\frac{1}{2}$

• $\frac{2}{3}$

If y = 1 - x + $\frac{{\mathrm{x}}^2}{2!} - \frac{{\mathrm{x}}^3}{3!} + \frac{{\mathrm{x}}^4}{4!} ..., \mathrm{then} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is equal to

• - x

• x

• y

• - y

The slope of the tangent to the curve x = 3t² + 1, y = t³ - 1 at x = 1 is

• $\frac{1}{2}$

• 0

• - 2

• $\infty$

The rate ofchange of the surface area ofthe sphere of radius r when the radius is increasing at the rate of 2 cm/sec is proportional to

• $\frac{1}{{\mathrm{r}}^2}$

• $\frac{1}{\mathrm{r}}$

• r²

• r