The minimum value of 2x² + x - 1 is :

• $- \frac{1}{4}$

• $\frac{3}{2}$

• $- \frac{9}{8}$

• $\frac{9}{8}$

A point is moving on y = 4 - 2x². The x-coordinate of the point is decreasing at the rate of 5 units/second. Then, the rate at which y coordinate of the point is changing when the point is at (1, 2) is

• 5 units/s

• 10 units/s

• 15 units/s

• 20 units/s

Match the points on the curve 2y² = x + 1 with the slopes of normals at those points and choose the correct answer.

      Point           Slope of the normal

A    (7, 2)         1 - 4$\sqrt{2}$

B    (0, $\frac{1}{\sqrt{2}}$)     2  - 8

C    (1, - 1)       3, 4

D    (3, $\sqrt{2}$)      4, 0

$\mathrm{f}\left(\mathrm{x}, \mathrm{y}\right) = 2{\left(\mathrm{x} - \mathrm{y}\right)}^2 - {\mathrm{x}}^4 - {\mathrm{y}}^4$ ${\left|\left({\mathrm{f}}_{\mathrm{xx}}{\mathrm{f}}_{\mathrm{yy}} - {\mathrm{f}}_{\mathrm{xy}}^2\right)\right|}_{\left(0, 0\right)}$

• 32

• 16

• 0

• - 1

If x is real, then the minimum value of $\frac{{\mathrm{x}}^2 - \mathrm{x} + 1}{{\mathrm{x}}^2 + \mathrm{x} + 1}$, is

• $\frac{1}{3}$

• 3

• $\frac{1}{2}$

• 1

A stone thrown upwards, has its equation of motion s = 490t - 4.9t². Then the maximum height reached by it, is

• 24500

• 12500

• 12250

• 25400

757.

The radius of a circular plate is increasing at the rate of 0.01 cm/s when the radius is 12 cm. Then, the rate at which the area increases, is

• 0.24 $\mathrm{\pi } \mathrm{cm}/\mathrm{s}$

• $60 \mathrm{\pi } \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $24\mathrm{\pi } \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $1.2\mathrm{\pi } \mathrm{sq} \mathrm{cm}/\mathrm{s}$

Observe the following statements

A: f(x) = 2x³ - 9x² + 12x - 3 is increasing outside the interval (1, 2)

R : f'(x) < 0 for x $\in$ (1, 2).

Then, which of the following is true ?

• Both A and R are true, and R is not the correct reason for A

• Both A and R are true, and R is the correct reason for A

• A is true but R is false

• A is false but R is true

If $\mathrm{\theta }$ is the angle between the curves xy = 2 and x² + 4y = 0 and x² + 4y = 0, then tan$\mathrm{\theta }$ is equal to :

• 1

• - 1

• 2

• 3

In the interval(- 3, 3) the function f(x) = $\frac{\mathrm{x}}{3} + \frac{3}{\mathrm{x}}, \mathrm{x} \ne 0$ is

• increasing

• decreasing

• neither increasing nor decreasing

• partly increasing and partly decreasing