Let f : R $\to$ R be a function which satisfies f(x + y) = f(x) + f(y) $\forall$ x, y $\in$ R. If f(1) = 2 and g(n) = $\sum _{\mathrm{k} = 1}^{\left(\mathrm{n} - 1\right)}\mathrm{f}\left(\mathrm{k}\right) \mathrm{n} \in \mathrm{N}$, then the value of g(n) = 20, is

• 9

• 20

• 5

• 4

12.

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{first} 11 \mathrm{terms} \mathrm{of} \mathrm{an} \mathrm{A}.\mathrm{P}. , {\mathrm{a}}_1, {\mathrm{a}}_2, {\mathrm{a}}_3 ..... \mathrm{is} 0({\mathrm{a}}_1 \ne 0), \\ \mathrm{then} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{A}.\mathrm{P}., {\mathrm{a}}_1, {\mathrm{a}}_3, {\mathrm{a}}_5, ..... {\mathrm{a}}_{23} \mathrm{is} {\mathrm{ka}}_1, \mathrm{where} \mathrm{k} \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• $\frac{121}{10}$

• $- \frac{121}{10}$

• $- \frac{72}{5}$

• $\frac{72}{5}$

Let n > 2 be an integer. Suppose that there are n Metro stations in a city located around a circular path. Each pair of nearest stations is connected by a straight track only. Further, each pair of nearest station is connected by blue line, whereas all remaining pairs of stations are connected by red line. If number of red lines is 99 times the number of blue lines, then the value of n is

• 199

• 101

• 201

• 200

$$\begin{gathered} \mathrm{Let} \mathrm{f} : \left( - 1, \infty \right) \to \mathrm{R} \mathrm{be} \mathrm{defined} \mathrm{by} \mathrm{f}\left(0\right) = 1 \\ \mathrm{and} \mathrm{f}\left(\mathrm{x}\right) = \frac{1}{\mathrm{x}}{\log }_{\mathrm{e}}\left(1 + \mathrm{x}\right), \mathrm{x} \ne 0. \mathrm{Then} \mathrm{the} \mathrm{function} \mathrm{f} : \end{gathered}$$

• increases in ( – 1, 0) and decreases in (0, $\infty$).

• $\mathrm{decreases} \mathrm{in} ( – 1, \infty )$

• decreases in ( – 1, 0) and increases in (0, $\infty$).

• increases in ( – 1, $\infty$)

Let A = {x = (x, y, z)^T : PX = 0 and x² + y² + z² = 1}, where $\left[\begin{array}{ccc}1& 2& 1\\ - 2& 3& - 4\\ 1& 9& - 1\end{array}\right]$ P then the set A

• is a singleton

• contains more than two elements

• contains exactly two elements

• is an empty set.

Let a, b, c $\in$ R be all non-zero satisfy a³ + b³ + c³ = 2.If the matrix A = $\left[\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{b}& \mathrm{c}& \mathrm{a}\\ \mathrm{c}& \mathrm{a}& \mathrm{b}\end{array}\right]$ satifies A^TA = I, then a value of abc can be :

• $\frac{1}{3}$

• $- \frac{1}{3}$

• 3

• $\frac{2}{3}$

A plane passing through the point (3, 1, 1) contains two lines whose direction ratios are 1, – 2, 2 and 2, 3, – 1 respectively. If this plane also passes through the point($\mathrm{\alpha }$,–3, 5), then $\mathrm{\alpha }$ is equal to

• 5

• 10

•  - 5

•  - 10

The equation of the normal to the curve y = (1 + x)^2y + cos²(sin ^{– 1}(x)) at x = 0 is :

• y = 4x + 2

• y + 4x = 2

• x + 4y = 8

• 2y + x = 4

If a curve y = f(x), passing through the point (1,2), is the solution of the differential equation, $2{\mathrm{x}}^2\mathrm{dy} = \left(2\mathrm{xy} + {\mathrm{y}}^2\right)\mathrm{dx}, \mathrm{then} \mathrm{f}\left(\frac{1}{2}\right)$ is equal to

• $\frac{1}{1 - {\log }_{\mathrm{e}}\left(2\right)}$

• $1 + \log {\left(\mathrm{e}\right)}^2$

• $\frac{1}{1 + {\log }_{\mathrm{e}}\left(2\right)}$

• $\frac{ - 1}{1 + {\log }_{\mathrm{e}}\left(2\right)}$

Consider a region R = {(x, y) $\in$ R² : x^{2 $\le \mathrm{y} \le 2\mathrm{x}$}}. If a line y = $\mathrm{\alpha }$ divides the area of region R into two equal parts, then which of the following is true ?

• $3{\mathrm{\alpha }}^2 - 8\mathrm{\alpha } + 8 = 0$

• ${\mathrm{\alpha }}^3 - 6{\mathrm{\alpha }}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} - 16 = 0$

• ${\mathrm{\alpha }}^3 - 6{\mathrm{\alpha }}^2 + 16 = 0$

• ${\text{3α}}^2 - 8{\mathrm{\alpha }}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} + 8 = 0$