The angle between the curves y = a^x and y = b^x is equal to

• ${\tan }^{-1}\left(\left|\frac{\mathrm{a} - \mathrm{b}}{1 + \mathrm{ab}}\right|\right)$

• ${\tan }^{-1}\left(\left|\frac{\mathrm{a} + \mathrm{b}}{1 - \mathrm{ab}}\right|\right)$

• ${\tan }^{-1}\left(\left|\frac{\log \left(\mathrm{b}\right) - \log \left(\mathrm{a}\right)}{1 + \log \left(\mathrm{a}\right)\log \left(\mathrm{b}\right)}\right|\right)$

• ${\tan }^{-1}\left(\left|\frac{\log \left(\mathrm{a}\right) - \log \left(\mathrm{b}\right)}{1 + \log \left(\mathrm{a}\right)\log \left(\mathrm{b}\right)}\right|\right)$

Let f(x) = (x - 7)²(x - 2)⁷, x $\in$ [2, 7].  The value of $\mathrm{\theta } \in \left(2, 7\right)$ such that $\mathrm{f}\text{'}\left(\mathrm{\theta }\right)$ = 0 is equal to

• $\frac{49}{4}$

• $\frac{53}{9}$

• $\frac{53}{7}$

• $\frac{49}{9}$

The distance between the origin and the normal to the curve y = e^2x + x² at x = 0 is

• 2

• $\frac{2}{\sqrt{3}}$

• $\frac{2}{\sqrt{5}}$

• $\frac{1}{2}$

The point on the curve x² + y² = a², y $\ge$ 0 at which the tangent is parallel to x-axis is

• (a, 0)

• (- a, 0)

• $\left(\frac{\mathrm{a}}{2}, \frac{\sqrt{3}}{2}\mathrm{a}\right)$

• (0, a)

An edge of a variable cube is increasing at the rate of 10 cm/s. How fast the volume of the cube will increase when the edge is 5 cm long?

• 750 cm³/ s

• 75 cm³/ s

• 300 cm³/ s

• 150 cm³/ s

The minimum value of f(x) = $\left|3 - \mathrm{x}\right| + 7$ is

• 0

• 6

• 7

• 8

The total revenue in rupees received from the sale of x units of a product is given by R(x) = 13x² + 26x + 15. Then, the marginal revenue in rupees, when x = 15 is

• 116

• 126

• 136

• 416

If a tangent of the curve y = 2 + $\sqrt{4\mathrm{x} + 1}$ has slope $\frac{2}{5}$ at a point, then the point is

• (0, 2)

• $\left(\frac{3}{4}, 4\right)$

• (2, 5)

• (6, 7)

The points on the graph y = x³ - 3x at which the tangent is parallel to x-axis are

• (2, 2) and (1, - 2)

• (- 1, 2) and (- 2, - 2)

• (2, 2) and (- 1, 2)

• (1, - 2) and (- 1, 2)

The slope of the normal to the curve y² - xy - 8 = 0 at the point (0, 2) is equal to

• - 3

• - 6

• 3

• 6