The point on the curve y = 2x² - 6x - 4 at which the tangent is parallel to the x-axis, is :

• $\left(\frac{3}{2}, \frac{13}{2}\right)$

• $\left(- \frac{5}{2}, - \frac{17}{2}\right)$

• $\left(\frac{3}{2}, \frac{17}{2}\right)$

• $\left(\frac{3}{2}, - \frac{17}{2}\right)$

The minimum value of $2\cos \left(\mathrm{\theta }\right) + \frac{1}{\sin \left(\mathrm{\theta }\right)} + \sqrt{2}\tan \left(\mathrm{\theta }\right)$ in the interval $\left(0, \frac{\mathrm{\pi }}{2}\right)$ is :

• $2 + \sqrt{2}$

• $3\sqrt{2}$

• $2\sqrt{3}$

• $3 + \sqrt{2}$

If S₁ and S₂ are respectively the sets of local minimum and local maximum points of the function, f(x) = 9x⁴ + 12x³ - 36x² - 25, x $\in$ R, then:

• S₁ = { - 1}; S₂ = {0, 2}

• S₁ = { - 2, 1}, S₂ = {0}

• S₁ = { - 2}; S₂ = {0, 1}

• S₁ = { - 2, 0}; S₂ = {1}

Let f : [0, 2] $\to$ R be a twice differentiable function such that f’’(x) > 0, for all  $\mathrm{x} \in \left(0, 2\right)$. If $\mathrm{\varphi }\left(\mathrm{x}\right)$ = f(x) + f(2 - x), then $\mathrm{\varphi }$ is :

• Increasing on (0, 1) and decreasing on (1, 2)

• Decreasing on (0, 1) and increasing on (1, 2)

• Decreasing on (0, 2)

• Increasing on (0, 2)

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

• $\frac{2}{3}\sqrt{3}$

• $\sqrt{3}$

• $2\sqrt{3}$

• $\sqrt{6}$

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 :

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

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

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

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

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is ${\tan }^{-1}\left(\frac{1}{2}\right)$. Water is poured into it at a constant rate of 5 cubic meter per minute. Then the rate (in m/min), at which the level of water is rising at the instant when the depth of water in the tank is 10m; is :

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$10\mathrm{\pi }$}\right.$

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$15\mathrm{\pi }$}\right.$

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5\mathrm{\pi }$}\right.$

• $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\mathrm{\pi }$}\right.$

If the tangent to the curve, y = x³ + ax - b at the point (- 1, - 5) is perpendicular to the line, - x + y + 4 = 0, then which one of the following points lies on the curve ?

• (2, - 1)

• (- 2, 2)

• (2, - 2)

• (- 2, 1)

149.

Let f(x) = e^x - x and g(x) = x² - x, $\forall \mathrm{x} \in \mathrm{R}$. Then the set of all x $\in$ R, where the function h(x) = (fog)(x) is increasing, is :

• $[- \frac{1}{2}, 0] \cup [1, \infty )$

• $[- 1, - \frac{1}{2}] \cup [\frac{1}{2}, \infty )$

• $[0, \infty )$

• $[0, \frac{1}{2}] \cup [1, \infty )$

The tangent and normal to the ellipse 3x² + 5y² = 32 at the point P(2, 2) meet the x-axis at Q and R, respectively .Then the area (in sq. units) of the triangle PQR is :

• $\frac{68}{15}$

• $\frac{16}{3}$

• $\frac{14}{3}$

• $\frac{34}{15}$