The distance of the point on the curve x² = 2y, which is nearest to the pont (0, 5) is

• 3

• 4

• $2\sqrt{2}$

• None of these

The minimum value of $\left(\mathrm{x} - \mathrm{\alpha }\right)\left(\mathrm{x} - \mathrm{\beta }\right)$ is

• 0

• $\mathrm{\alpha \beta }$

• $\frac{1}{4}{\left(\mathrm{\alpha } - \mathrm{\beta }\right)}^2$

• $- \frac{1}{4}{\left(\mathrm{\alpha } - \mathrm{\beta }\right)}^2$

The equation of tangent to the curve 6y = 7 - x³ at (1, 1) is

• 2x + y = 3

• x + 2y = 3

• x + y = 1

• x + y + 2 = 0

The maximum value of xy subject to x + y = 7 is

• 10

• 12

• $\frac{49}{4}$

• $\frac{55}{4}$

The family of curves in which the sub-tangent at any point to any curve is double the abscissa is given by

• x = Cy²

• y = Cx²

• x² = Cy²

• y² = Cx²

If log (1 + x) -  $\frac{2\mathrm{x}}{2 + \mathrm{x}}$ is increasing, then

• $0 < \mathrm{x} < \infty$

• $- \infty < \mathrm{x} < 0$

• $- \infty < \mathrm{x} < \infty$

• $- 1 < \mathrm{x} < 2$

The functions f(x) =xe ^{- X}, $\forall \left(\mathrm{x} \in \mathrm{R}\right)$ attains a maximum value at x is equal

• 1

• 2

• $\frac{1}{\mathrm{e}}$

• 3

The approximate value of ${\left(1.0002\right)}^{3000} \mathrm{is}$

• $1·2$

• $1·4$

• $1·6$

• $1·8$

The sum of two numbers is 20. If the product of the square of one number and cube of the other is maximum, then the numbers are :

• 12, 8

• 3, 4

• 9, 12

• 15, 18

Gas is being pumped into a spherical balloon at the rate of 30 ft³/min. The rate at which the radius increase when it reaches the value 15ft, is :

• $\frac{1}{30\mathrm{\pi }} \mathrm{ft}/\min$

• $\frac{1}{15\mathrm{\pi }} \mathrm{ft}/\min$

• $\frac{1}{20\mathrm{\pi }} \mathrm{ft}/\min$

• $\frac{1}{25\mathrm{\pi }} \mathrm{ft}/\min$