If a particle moves such that the displacement is proportional to the square of the velocity acquired, then its acceleration is :

• proportional to s²

• proportional to $\frac{1}{{\mathrm{s}}^2}$

• proportional to s

• a constant

The maximum value of xy when x + 2y = 8 is :

• 20

• 16

• 24

• 8

The function f(x) = tan^-1(sin(x) + cos(x)), x > 0 is always an increasing function on the interval :

• $\left(0, \mathrm{\pi }\right)$

• $\left(0, \frac{\mathrm{\pi }}{2}\right)$

• $\left(0, \frac{\mathrm{\pi }}{4}\right)$

• $\left(0, \frac{3\mathrm{\pi }}{4}\right)$

The radius of a cylinder is increasing at the rate of 3 m/s and its altitude is decreasing at the rate of 4 m/s. The rate of change of volume when radius is 4 m and altitude is 6 m, is :

• $80 \mathrm{\pi } \mathrm{cu} \mathrm{m}/\mathrm{s}$

• $144 \mathrm{\pi } \mathrm{cu} \mathrm{m}/\mathrm{s}$

• $- 80 \mathrm{\pi } \mathrm{cu} \mathrm{m}/\mathrm{s}$

• $64 \mathrm{\pi } \mathrm{cu} \mathrm{m}/\mathrm{s}$

A ladder 10 m long rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate of 3 emfs. The height of the upper end while it is descending at the rate of 4 emfs, is :

• 4$\sqrt{3}$

• 5$\sqrt{3}$

• 5$\sqrt{2}$

• 6 m

The equation of the tangent to the curve $\mathrm{y} = {\left(1 + \mathrm{x}\right)}^{\mathrm{y}} + {\sin }^{-1}\left({\sin }^2\left(\mathrm{x}\right)\right)$

• x - y + 1 = 0

• x + y + 1 = 0

• 2x - y + 1 = 0

• x + 2y + 2 = 0

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)