What is the minimum value of $\frac{{\mathrm{a}}^2}{{\cos }^2\left(\mathrm{x}\right)} + \frac{{\mathrm{b}}^2}{{\displaystyle {\sin }^2}\left(\mathrm{x}\right)}$, where a > 0 and b > 0 ?

• (a + b)²

• (a - b)²

• a²  + b²

• $\left|{\mathrm{a}}^2 + {\mathrm{b}}^2\right|$

If $\mathrm{f}\left(\mathrm{x}\right) = \frac{{\mathrm{x}}^2}{3} - \frac{5{\mathrm{x}}^2}{2} + 6\mathrm{x} + 7$ increases in the interval T and decreases in the interval S, then which one of the following is correct ?

• $\mathrm{T} = \left(- \infty , 2\right) \cup \left(3, \infty \right) \mathrm{and} \mathrm{S} = \left(2, 3\right)$

• $\mathrm{T} = \mathrm{\phi } \mathrm{and} \mathrm{S} = \left( - \infty , \infty \right)$

• $\mathrm{T} = \left( - \infty , \infty \right) \mathrm{and} \mathrm{S} = \mathrm{\phi }$

• $\mathrm{T} = \left(2, 3\right) \mathrm{and} \mathrm{S} = \left(- \infty , 2\right) \cup \left(3, \infty \right)$

Consider the function

f(x) = 3x⁴ – 20x³ – 12x² + 288x + 1

In which one of the following intervals is the function decreasing ?

• ( - 2, 3)

• (3, 4)

• (4, 6)

• No

Consider the function

f(x) = 3x⁴ – 20x³ – 12x² + 288x + 1

In which one of the following intervals is the function increasing ?

• ( - 2, 3)

• (3, 4)

• (4, 6)

• (6, 9)

25.

Let l be the length and b be the breadth of a rectangle such that l x b = k. What is the maximum area of the rectangle ?

• 2k²

• k²

• $\frac{{\mathrm{k}}^2}{2}$

• $\frac{{\mathrm{k}}^2}{4}$

What is the minimum value of $3\cos \left(\mathrm{A} + \frac{\mathrm{\pi }}{3}\right)$ where A ∈ R ?

• - 3

• - 1

• 0

• 3

The radius of a circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference ?

• 4.4 cm/sec

   

• 8.4 cm/sec

• 8.8 cm/sec

• 15.4 cm/sec