11.

Let P(h, k) be a point on the curve y = x² + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is :

• x + 3y - 62 = 0

• x + 3y + 26 = 0 

• x - 3y - 11 = 0

• x - 3y + 22 = 0

Let S be the set of all $\mathrm{\lambda } \in \mathrm{R}$ for which the system of linear equations

2x – y + 2z = 2

x – 2y + $\mathrm{\lambda }$z = – 4

x + $\lambda$y + z = 4

has no solution. Then, the set S

• is a singleton

• contains exactly two elements

• contains more than two elements

• is an empty set

If a function f(x) defined by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{{\mathrm{ae}}^{\mathrm{x}} + {\mathrm{be}}^{ - \mathrm{x}}, - 1 \le \mathrm{x} < 1 \\ {\mathrm{cx}}^2, 1 \le \mathrm{x} \le 3 \\ {\mathrm{ax}}^2 2\mathrm{cx}, 3 < \mathrm{x} \le 4\right\} \\ \mathrm{be} \mathrm{continuous} \mathrm{for} \mathrm{some} \mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{R} \mathrm{and} \mathrm{f},\left(0\right) + \mathrm{f}\text{'}\left(2\right) = \mathrm{e}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{a} \mathrm{is} : \end{gathered}$$

• $\frac{1}{{\mathrm{e}}^2 - 3\mathrm{e} + 13}$

• $\frac{\mathrm{e}}{{\mathrm{e}}^2 - 3\mathrm{e} + 13}$

• $\frac{\mathrm{e}}{{\mathrm{e}}^2 - 3\mathrm{e} - 13}$

• $\frac{\mathrm{e}}{{\mathrm{e}}^2 + 3\mathrm{e} + 13}$

$$\begin{gathered} \mathrm{Let} \mathrm{\alpha } \mathrm{and} \mathrm{\beta } \mathrm{be} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation}, 5{\mathrm{x}}^2 + 6\mathrm{x} – 2 = 0. \\ \mathrm{If} \mathrm{Sn} = {\mathrm{\alpha }}^{\mathrm{n}}+ {\mathrm{\beta }}^{\mathrm{n}}, \mathrm{n} = 1, 2, 3,..., \mathrm{then} : \end{gathered}$$

• $6{\mathrm{S}}_6 +\hspace{0.17em}5{\mathrm{S}}_5 = 2{\mathrm{S}}_4$

• $5{\mathrm{S}}_6 +\hspace{0.17em}6{\mathrm{S}}_5 = 2{\mathrm{S}}_4$

• $5{\mathrm{S}}_6 +\hspace{0.17em}6{\mathrm{S}}_5 +\hspace{0.17em}2{\mathrm{S}}_4 = 0$

• $6{\mathrm{S}}_6 + 5{\mathrm{S}}_5 +\hspace{0.17em}2{\mathrm{S}}_4 = 0$

Let X = $\left\{\mathrm{x} \in \mathrm{N} : 1 \le 1 \le 17\right\}$ and Y = {ax + b : x $\in$ X and a, b $\in$ R, a > 0}. If mean and variance of elements of Y are 17 and 216 respectively then a + b is equal to :

• 7

• 9

•  - 7

•  - 27

Area (in sq. units) of the region outside $\frac{\left|\mathrm{x}\right|}{2} + \frac{\left|\mathrm{y}\right|}{3} = 1$ and inside the ellipse $\frac{{\mathrm{x}}^2}{4} + \frac{{\mathrm{y}}^2}{9} = 1$

• $3\left(\mathrm{\pi } - 2\right)$

• $3\left(4 - \mathrm{\pi }\right)$

• $6\left(\mathrm{\pi } - 2\right)$

• $6\left(4 - \mathrm{\pi }\right)$

Let A be a 2 × 2 real matrix with entries from {0, 1} and |A| $\ne$ 0. Consider the following two statements;

(P)If A $\ne$ I₂, then |A| = – 1

(Q)If |A| = 1, then tr(A) = 2

where I₂ denotes 2 × 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then :

• (P) is true and (Q) are false

• Both (P) and (Q) are true

• Both (P) and (Q) are false

• (P) is false and (Q) is true

If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining $\left(0, \frac{3}{2}\right) \mathrm{and} \left(\frac{1}{2}, 2\right)$ then

• $\left|\mathrm{b} - \mathrm{a}\right| = 1$

• $\mathrm{b} = \frac{\mathrm{\pi }}{2} + \mathrm{a}$

• $\left|\mathrm{a} +\mathrm{b}\right| = 1$

• b = a

Let y = y(x) be the solution of the differential equation,

$$\begin{gathered} \frac{2 + \sin \left(\mathrm{x}\right)}{\mathrm{y} + 1} . \frac{\mathrm{dy}}{\mathrm{dx}} = - \cos \left(\mathrm{x}\right), \mathrm{y} > 0, \mathrm{y}\left(0\right) = 1. \mathrm{If} \mathrm{y}\left(\mathrm{\pi }\right) = \mathrm{a} \\ \mathrm{and} \frac{\mathrm{dy}}{\mathrm{dx}} \mathrm{at} \mathrm{x} = \mathrm{\pi } \mathrm{is} \mathrm{b}, \mathrm{then} \mathrm{the} \mathrm{ordered} \mathrm{pair} \left(\mathrm{a}, \mathrm{b}\right) = ? \end{gathered}$$

• $\left(2, \frac{3}{2}\right)$

• (1, - 1)

• (2, 1)

• (1, 1)

The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y, z = 1 also passes through the point :

• (2, 0, - 1)

• (0, 6, - 2)

• (0,  - 6, 2)

• (- 2, 0 , 1)