On the ellipse 4x² + 9y² = 1, the points at which the tangents are parallel to the line 8x = 9y, are

• $\left(\frac{2}{5}, \frac{1}{5}\right)$

• $\left(- \frac{2}{5}, \frac{1}{5}\right)$

• $\left(- \frac{2}{5}, - \frac{1}{5}\right)$

• $\left(\frac{2}{5}, - \frac{1}{5}\right)$

A container s the shape of an inverted cone. Its height is 6 m and radius is 4m at the top. If it is filled with water at the rate of 3m/min then the rate of change of height of water(in mt/min) when the water level is 3 m is

• $\frac{3}{4\mathrm{\pi }}$

• $\frac{2}{9\mathrm{\pi }}$

• $16\mathrm{\pi }$

• $2\mathrm{\pi }$

$\mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \mathrm{the} \mathrm{lengths} \mathrm{of} \mathrm{the} \mathrm{tangents} \mathrm{from} \mathrm{the} \mathrm{vertices} \mathrm{of} \mathrm{a} \mathrm{triangle} \mathrm{to} \mathrm{its} \mathrm{incircle}. \mathrm{Then}$

• $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } = \frac{1}{{\mathrm{r}}^2}\left(\mathrm{\alpha \beta \gamma }\right)$

• $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } = \frac{1}{\mathrm{r}}\left(\mathrm{\alpha \beta \gamma }\right)$

• $\frac{1}{\mathrm{\alpha }} + \frac{1}{\mathrm{\beta }} + \frac{1}{\mathrm{\gamma }} = \mathrm{r}\left(\mathrm{\alpha \beta \gamma }\right)$

• ${\mathrm{\alpha }}^2 + {\mathrm{\beta }}^2 + {\mathrm{\gamma }}^2 = \frac{2}{\mathrm{r}}\left(\mathrm{\alpha \beta \gamma }\right)$

If a cylindrical vessel of given volume V with no lid on the top is to be made from a sheet of metal, then the radius (r) and height(h) of the vessel so that the metal sheet used is minimum is

• $\mathrm{r} = \sqrt[3]{\frac{\mathrm{\pi }}{\mathrm{V}}}, \mathrm{h} = \sqrt[3]{\frac{\mathrm{\pi }}{\mathrm{V}}}$

• $\mathrm{r} = \sqrt{\mathrm{\pi V}}, \mathrm{h} = \sqrt{\mathrm{\pi V}}$

• $\mathrm{r} = \sqrt[3]{\frac{\mathrm{V}}{\mathrm{\pi }}}, \mathrm{h} = \sqrt[3]{\frac{\mathrm{V}}{\mathrm{\pi }}}$

• $\mathrm{r} = \sqrt{\frac{\mathrm{V}}{\mathrm{\pi }}}, \mathrm{h} = \sqrt{\frac{\mathrm{V}}{\mathrm{\pi }}}$

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side=x units and a circle of radius=r units. If the sum of the areas of the square and the circle so formed is minimum, then:

• 2x=(π+4)r

• (4−π)x=πr

• x=2r

• x=2r

Let f (x) be a polynomial of degree four having extreme values at x =1 an x =2. If  then f(2) is equal to 

• -8

• -4

• 0

• 0

A spherical balloon is filled with 4500π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is

• 9/7

• 7/9

• 2/9

• 2/9

8.

Let a, b ∈ R be such that the function f given by f(x) = ln |x| + bx
2+ ax, x ≠ 0 has extreme values at x = –1 and x = 2.
Statement 1: f has local maximum at x = –1 and at x = 2.
Statement 2: 

• Statement 1 is false, statement 2 is true

• Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

• Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

• Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

If dy/dx = y + 3 > 0 and y(0) = 2, then y(ln2) is equal to:

• 7

• 5

• 13

• 13

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point (-3, 1) and has eccentricity  is

• 3x² + 5y² -32 = 0

• 5x² + 3y² - 48 = 0

• 3x² + 5y² - 15 = 0 

• 3x² + 5y² - 15 = 0