The total revenue in rupees received from the sale of x units of a product is given by R(x) = 13x² + 26x + 15. Then, the marginal revenue in rupees, when x = 15 is

• 116

• 126

• 136

• 416

If a tangent of the curve y = 2 + $\sqrt{4\mathrm{x} + 1}$ has slope $\frac{2}{5}$ at a point, then the point is

• (0, 2)

• $\left(\frac{3}{4}, 4\right)$

• (2, 5)

• (6, 7)

The value of x in the interval [ 4, 9] at which the function f(x) = $\sqrt{\mathrm{x}}$ satisfies the mean value theorem is

• $\frac{13}{4}$

• $\frac{17}{4}$

• $\frac{21}{4}$

• $\frac{25}{4}$

4.

If the function f : $[1, \infty ) \to [1, \infty )$ is defined by f(x) = 2^{x(x - 1)}, then f^-1(x) is

• ${\left(\frac{1}{2}\right)}^{\mathrm{x}\left(\mathrm{x} - 1\right)}$

• $\frac{1}{2}\left(1 - \sqrt{1 + 4{\log }_2\left(\mathrm{x}\right)}\right)$

• $\frac{1}{2}\sqrt{1 + 4{\log }_2\left(\mathrm{x}\right)}$

• $\frac{1}{2}\left[1 + \sqrt{1 + 4{\log }_2\left(\mathrm{x}\right)}\right]$

If n(A) = 8 and $\mathrm{n}\left(\mathrm{A} \cap \mathrm{B}\right) = 2$, then $\mathrm{n}\left(\left(\mathrm{A} \cap \mathrm{B}\right)\text{'} \cap \mathrm{A}\right)$ is equal to

• 2

• 4

• 6

• 8

If f(x) = $\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)$, $\mathrm{x} \in \left(- \infty , \infty \right)$ and g(x) = x², $\mathrm{x} \in \left(- \infty , \infty \right)$, then (fog)(x) is equal to

• 1

• 0

• ${\sin }^2\left(\mathrm{x}\right) + \cos \left({\mathrm{x}}^2\right)$

• $\sin \left({\mathrm{x}}^2\right) + \cos \left({\mathrm{x}}^2\right)$

If A = $\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& - 2\\ \mathrm{a}& 2& \mathrm{b}\end{array}\right]$ is a matrix satisfying AA^T = 9I₃, then the values of a and b are respectively.

• 1, 2

• - 1, 2

• - 1, - 2

• - 2, - 1

If A = $\left[\begin{array}{cc}2& 1\\ 0& \mathrm{x}\end{array}\right]$ and A^{- 1} = $\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{6}\\ 0& \frac{1}{\mathrm{x}}\end{array}\right]$, then the value of x is equal to

• - 3

• 3

• - 2

• 6

If A = $\left(\begin{array}{cc}\cos \left(\mathrm{\alpha }\right)& \sin \left(\mathrm{\alpha }\right)\\ \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\end{array}\right)$, then A¹⁰ is equal to

• $\left(\begin{array}{cc}{\cos }^{10}\left(\mathrm{\alpha }\right)& {\sin }^{10}\left(\mathrm{\alpha }\right)\\ - {\sin }^{10}\left(\mathrm{\alpha }\right)& {\cos }^{10}\left(\mathrm{\alpha }\right)\end{array}\right)$

• $\left(\begin{array}{cc}{\cos }^{10}\left(\mathrm{\alpha }\right)& - {\sin }^{10}\left(\mathrm{\alpha }\right)\\ {\sin }^{10}\left(\mathrm{\alpha }\right)& {\cos }^{10}\left(\mathrm{\alpha }\right)\end{array}\right)$

• $\left(\begin{array}{cc}{\cos }^{10}\left(\mathrm{\alpha }\right)& {\sin }^{10}\left(\mathrm{\alpha }\right)\\ - {\sin }^{10}\left(\mathrm{\alpha }\right)& - {\cos }^{10}\left(\mathrm{\alpha }\right)\end{array}\right)$

• $\left(\begin{array}{cc}\cos \left(10\mathrm{\alpha }\right)& \sin \left(10\mathrm{\alpha }\right)\\ - \sin \left(10\mathrm{\alpha }\right)& \cos \left(10\mathrm{\alpha }\right)\end{array}\right)$

If A = $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ \mathrm{a}& \mathrm{b}& - 1\end{array}\right]$ and I is the unit matrix of order 3, then A² + 2A⁴ + 4A⁶ is equal to

• 7 A⁸

• 7 A⁷

• 8 I

• 6 I