Let the numbers 2, b, c be in an A.P. and A = $\left|\begin{array}{ccc}1& 1& 1\\ 2& \mathrm{b}& \mathrm{c}\\ 4& {\mathrm{b}}^2& {\mathrm{c}}^2\end{array}\right|$. If det(A) $\in$ [2, 16], then c lies in the interval :

• [4, 6]

• (2 + 2^3/4, 4)

• [2, 3)

• [3, 2 + 2^3/4]

Let f : $\mathrm{R} \to \mathrm{R}$ be a diiferentiable function Satisfyingf'(3) + f'(2) = 0. Then $\underset{\mathrm{x}\to 0}{\lim }{\left(\frac{1 + \mathrm{f}\left(3 + \mathrm{x}\right) - \mathrm{f}\left(3\right)}{1 + \mathrm{f}\left(2 - \mathrm{x}\right) - \mathrm{f}\left(2\right)}\right)}^{\frac{1}{\mathrm{x}}}$ is equal to :

• 1

• e

• e^{- 1}

• e²

Let f : [- 1, 3] $\to$ R be defined as f(x) = $$\begin{gathered} \left\{\left|\mathrm{x}\right| + \left[\mathrm{x}\right], - 1 \le \mathrm{x} < 1 \\ \mathrm{x} + \left|\mathrm{x}\right|, 1 \le \mathrm{x} < 2 \\ \mathrm{x} +\hspace{0.17em}\left[\mathrm{x}\right], 2 \le \mathrm{x} \le 3\right\} \end{gathered}$$ where [t] denotes the greatest integer less than or equal to t. Then, f is discontinuous at :

• only three points

• only one point

• only two points

• four or more points

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is :

• $\frac{2}{3}\sqrt{3}$

• $\sqrt{3}$

• $2\sqrt{3}$

• $\sqrt{6}$

5.

Let f(x) = a^x(a > 0) be written as f(x) = f₁(x) + f₂(x), where f₁(x) is an even function and f₂(x) is an odd function. Then f₁(x + y) + f₁(x – y) equals :

• 2f₁(x)f₁(y)

• 2f₁(x + y)f₁(x - y)

• 2f₁(x + y)f₂(x - y)

• 2f₁(x)f₂(y)

Given that the slope of the tangent to a curve y = y(x) at any point (x, y) is $\frac{2\mathrm{y}}{{\mathrm{x}}^2}$. If the curve passes through the centre of the circle x² + y² - 2x - 2y = 0, then its equation is :

• ${\mathrm{xlog}}_{\mathrm{e}}\left|\mathrm{y}\right| = 2\left(\mathrm{x} - 1\right)$

• ${\mathrm{xlog}}_{\mathrm{e}}\left|\mathrm{y}\right| = - 2\left(\mathrm{x} - 1\right)$

• ${\mathrm{x}}^2{\log }_{\mathrm{e}}\left|\mathrm{y}\right| = - 2\left(\mathrm{x} - 1\right)$

• ${\mathrm{xlog}}_{\mathrm{e}}\left|\mathrm{y}\right| = \left(\mathrm{x} - 1\right)$

Let $\overrightarrow{\mathrm{a}} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \mathrm{x}\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$ , for some real x. Then $\left|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right| = \mathrm{r}$ is possible if :

• $0 < \mathrm{r} \le \sqrt{\frac{3}{2}}$

• $\sqrt{\frac{3}{2}} < \mathrm{r} \le 3\sqrt{\frac{3}{2}}$

• $3\sqrt{\frac{3}{2}} < \mathrm{r} \le 5\sqrt{\frac{3}{2}}$

• $\mathrm{r} \ge 5\sqrt{\frac{3}{2}}$

If $\int \frac{\mathrm{dx}}{{\mathrm{x}}^3{\left(1 + {\mathrm{x}}^6\right)}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}} = \mathrm{xf}\left(\mathrm{x}\right)(1 +\hspace{0.17em}{\mathrm{x}}^6{)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} + \mathrm{C}$ where C is a constant of integration, then the function f(x) is equal to :

• - $\frac{1}{2{\mathrm{x}}^2}$

• $- \frac{1}{6{\mathrm{x}}^3}$

• $\frac{3}{{\mathrm{x}}^2}$

• $- \frac{1}{2{\mathrm{x}}^3}$

Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the Lines joining the top of each pole to the foot of the other, from this horizontal plane is :

• 12

• 18

• 15

• 16

Let f(x) = ${\int }_0^{\mathrm{x}}\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt}$, where g is a non–zero even function. If f(x + 5) = g(x), then ${\int }_0^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$ equals :

• ${\int }_{\mathrm{x} + 5}^5\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt}$

• $2{\int }_5^{\mathrm{x} + 5}\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt}$

• ${\int }_5^{\mathrm{x} +\hspace{0.17em}5}\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt}$

• $5{\int }_{\mathrm{x} + 5}^5\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt}$