$$\begin{gathered} \mathrm{Let} \mathrm{f} : \left(0, \infty \right) \to \left(0, \infty \right) \mathrm{be} \mathrm{a} \mathrm{differentiable} \mathrm{function} \mathrm{such} \mathrm{that} \mathrm{f}\left(1\right) = \mathrm{e} \mathrm{and} \\ \underset{\mathrm{t}\to \mathrm{x}}{\lim } \frac{{\mathrm{t}}^2{\mathrm{f}}^2\left(\mathrm{x}\right) - {\mathrm{x}}^2{\mathrm{f}}^2\left(\mathrm{t}\right)}{\mathrm{t} - \mathrm{x}}. \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = 1, \mathrm{then} \mathrm{x} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $\frac{1}{\mathrm{e}}$

• 2e

• $\frac{1}{2\mathrm{e}}$

• e

Contrapositive o the statement : 'If a function f is differentiable at a, then it is also continuous at a', is :

• If a function f is not continuous at a, then it is not differentiable at a.

• If a function f is continuous at a, then it is differentiable at a.

• If a function f is not continuous at a. then it is differentiable at a.

• If a function f is continuous at a, then it is not differentiable at a.

If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of

(1 + x)^{n + 5} are in the ratio 5 : 10 : 14, then the largest coefficient in the expansion is :

• 330

• 252

• 792

• 462

The circle passing through the intersection of the circles, x² + y² – 6x = 0 and x² + y² – 4y = 0, having its centre on the line, 2x – 3y + 12 = 0, also passes through the point :

• $\left(1, - 3\right)$

• $\left(- 1, 3\right)$

• ( - 3, 6)

• ( - 3, 1)

The angle of elevation of a cloud C from a point P, 200 m above a still take is 30°. If the angle of depression of the image of C in the lake from the point P is 60°, then PC (in m) is equal to

• 100

• $400\sqrt{3}$

• $200\sqrt{3}$

• 400

6.

$$\begin{gathered} \mathrm{Let} {\mathrm{a}}_1, {\mathrm{a}}_2, .... {\mathrm{a}}_{\mathrm{n}} \mathrm{be} \mathrm{a} \mathrm{given} \mathrm{A}.\mathrm{P}. \mathrm{whose} \mathrm{common} \mathrm{difference} \mathrm{is} \mathrm{an} \mathrm{integer} \mathrm{and} {\mathrm{S}}_{\mathrm{n}}= {\mathrm{a}}_1 + {\mathrm{a}}_2 + ... + \mathrm{an}. \\ \mathrm{If} {\mathrm{a}}_1 = 1, {\mathrm{a}}_1 = 300 \mathrm{and} 15 \le \mathrm{n} \le 50, \mathrm{then} \mathrm{the} \mathrm{ordered} \mathrm{pair} ({\mathrm{S}}_{\mathrm{n} – 4}, {\mathrm{a}}_{\mathrm{n} – 4}) \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• (2490, 248)

• (2490, 249)

• (2480, 249)

• (2480, 248)

$$\begin{gathered} \mathrm{Let} \underset{\mathrm{i} = 1}{\overset{50}{\cup }}\mathrm{X} = \underset{\mathrm{i} = 1}{\overset{\mathrm{n}}{\cup }}{\mathrm{y}}_{\mathrm{i}} = \mathrm{T}, \mathrm{where} \mathrm{each} {\mathrm{X}}_{\mathrm{i}} \mathrm{contains} 10 \mathrm{elements} \mathrm{and} \mathrm{each} {\mathrm{Y}}_{\mathrm{i}} \mathrm{contains} 5 \mathrm{elements}. \mathrm{If} \mathrm{each} \\ \mathrm{element} \mathrm{of} \mathrm{the} \mathrm{set} \mathrm{T} \mathrm{is} \mathrm{an} \mathrm{element} \mathrm{of} \mathrm{exactly} 20 \mathrm{of} \mathrm{sets} {\mathrm{X}}_{\mathrm{i}}\text{'}\mathrm{s} \mathrm{and} \mathrm{exactly} 6 \mathrm{of} \mathrm{sets} {\mathrm{Y}}_{\mathrm{i}}\text{'}\mathrm{s} \mathrm{then} \mathrm{n} \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• 45

• 15

• 30

• 50

$$\begin{gathered} \mathrm{Let} \mathrm{x} = 4 \mathrm{be} \mathrm{a} \mathrm{directrix} \mathrm{to} \mathrm{an} \mathrm{ellipse} \mathrm{whose} \mathrm{centre} \mathrm{is} \mathrm{at} \mathrm{the} \mathrm{origin} \mathrm{and} \mathrm{its} \mathrm{eccentricity} \mathrm{is} 21. \\ \mathrm{If} \mathrm{P}(1, \mathrm{\beta }), \mathrm{\beta } > 0 \mathrm{is} \mathrm{a} \mathrm{point} \mathrm{on} \mathrm{this} \mathrm{ellipse}, \mathrm{then} \mathrm{the} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{normal} \mathrm{to} \mathrm{it} \mathrm{at} \mathrm{P} \mathrm{is} \end{gathered}$$

• 7x - 4y = 1

• 4x - 2y = 1

• 8x - 2y = 5

• 4x - 3y = 2

$$\begin{gathered} \mathrm{The} \mathrm{distance} \mathrm{of} \mathrm{the} \mathrm{point} (1, – 2, 3) \mathrm{from} \mathrm{the} \mathrm{plane} \mathrm{x} – \mathrm{y} + \mathrm{z} = 5 \\ \mathrm{measured} \mathrm{parallel} \mathrm{to} \mathrm{the} \mathrm{line} \frac{\mathrm{x}}{2} = \frac{\mathrm{y}}{3} = \frac{\mathrm{z}}{ - 6} \mathrm{is} : \end{gathered}$$

• $\frac{1}{7}$

• 7

• 1

• $\frac{7}{5}$

If the perpendicular bisector of the line segment joining the points P(1, 4) and Q(k, 3) has y-intercept equal to – 4, then a value of k is ;

• $\sqrt{15}$

•  4

•  - 2

• $\sqrt{14}$