If ${\sin }^2\left(\mathrm{\theta }\right) + 3\cos \left(\mathrm{\theta }\right) = 2 \mathrm{then} {\cos }^3\left(\mathrm{\theta }\right) + {\sec }^3\left(\mathrm{\theta }\right)$ is equal to

• 1

• 4

• 9

• 18

If $\mathrm{x} = 1 + \frac{1}{2 \times 1!} + \frac{1}{4 \times 2!} + \frac{1}{8 \times 3!} + ...$ and y = $1 + \frac{{\mathrm{x}}^2}{1!} + \frac{{\mathrm{x}}^4}{2!} + \frac{{\mathrm{x}}^6}{3!} +\hspace{0.17em}...$ Then, the value of ${\log }_{\mathrm{e}}\left(\mathrm{y}\right)$ is

• e

• e²

• 1

• $\frac{1}{\mathrm{e}}$

The value of the infinite series

$\frac{1^2 + 2^2}{3!} + \frac{1^2 + 2^2 + 3^2}{4!} + \frac{1^2 + 2^2 + 3^2 +\hspace{0.17em}{4}^2}{5!} + ...$ is

• e

• 5e

• $\frac{5\mathrm{e}}{6} - \frac{1}{2}$

• $\frac{5\mathrm{e}}{6}$

If P be a point on the parabola y = 4ax with focus F. Let Q denote the foot of the perpendicular from P onto the directrix. Then, $\frac{\tan \left(\angle \mathrm{PQF}\right)}{\tan \left(\angle \mathrm{PFQ}\right)}$ is

• 1

• $\frac{1}{2}$

• 2

• $\frac{1}{4}$

The equations of the circles, which touch both the axes and the line 4x + 3y = 12 and have centres in the first quadrant, are

• x² + y² + x - y + 1 = 0

• x² + y² - 2x - 2y + 1 = 0

• x² + y² - 12x - 12y + 36 = 0

• x² + y² - 6x - 6y + 36 = 0

If $\sin \left(\mathrm{\alpha }\right), \cos \left(\mathrm{\alpha }\right)$  be the roots of the equation x² - bx + c = 0, Then, which of the following statements is/are correct?

• $\mathrm{c} \le \frac{1}{2}$

• $\mathrm{b} \le \sqrt{2}$

• $\mathrm{c} > \frac{1}{2}$

• b $> \sqrt{2}$

57.

Which of the following real valued functions is/are not even functions?

• $\mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^3\sin \left(\mathrm{x}\right)$

• $\mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^2 \cos \left(\mathrm{x}\right)$

• $\mathrm{f}\left(\mathrm{x}\right) = {\mathrm{e}}^{\mathrm{x}}{\mathrm{x}}^3\sin \left(\mathrm{x}\right)$

• f(x) = x - [x], where [x] denotes the greatest integer less than or equal to x.

If I = $\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right) \mathrm{and} \mathrm{P} = \left(\begin{array}{ccc}1& 0& 0\\ 0& - 1& 0\\ 0& 0& - 2\end{array}\right)$. Then, the matrix P³ + 2P² is equal to

• P

• I - P

• 2I + P

• 2I - P

The value of determinant

$\left|\begin{array}{ccc}1 + {\mathrm{a}}^2 - {\mathrm{b}}^2& 2\mathrm{ab}& - 2\mathrm{b}\\ 2\mathrm{ab}& 1 - {\mathrm{a}}^2 + {\mathrm{b}}^2& 2\mathrm{a}\\ 2\mathrm{b}& - 2\mathrm{a}& 1 - {\mathrm{a}}^2 - {\mathrm{b}}^2\end{array}\right|$ is

• 0

• (1 + a² + b²)

• (1 + a² + b²)²

• (1 + a² + b²)³

In the set of all 3 x 3 real matrices a relation is defined as follows. A matrix A is related to a matrix B, if and only if there is a non-singular 3 x 3 matrix P, such that B = P^-1AP. This relation is

• reflexive, symmetric but not transitive

• reflexive, transitive but not symmetric

• symmetric, transitive but not reflexive

• an equivalence relation