In how many ways, words can be made from the letters of the word BHARAT in which B and H are never together ?

• 360

• 300

• 240

• 120

iF a,b and c are in HP, then for any n $\in$ N, which one of the following is true ?

• aⁿ + cⁿ < 2bⁿ

• aⁿ + cⁿ > 2bⁿ

• aⁿ + cⁿ = 2bⁿ

• None of the above

The radius of a cylinder is increasing at the rate of 2 m/s and its height is decreasing at the rate of 3 m/s. When the radius is 3 m and height is 5 m, then the volume of the cylinder would change at the rate of

• $87\mathrm{\pi } {\mathrm{m}}^3/\mathrm{s}$

• $33\mathrm{\pi } {\mathrm{m}}^3/\mathrm{s}$

• $27\mathrm{\pi } {\mathrm{m}}^3/\mathrm{s}$

• $15\mathrm{\pi } {\mathrm{m}}^3/\mathrm{s}$

The values of a, if f(x) =$2{\mathrm{e}}^{\mathrm{x}} - {\mathrm{ae}}^{-\mathrm{x}} + \left(2\mathrm{a} +\hspace{0.17em}1\right)\mathrm{x} - 3$  increases x, are in

• $[0, \infty )$

• $(- \infty , 0]$

• $\left(- \infty , \infty \right)$

• $\left(1, \infty \right)$

By Newton-Raphson method, the positive root of the equation x⁴ - x - 10 = 0 is

• 1.871

• 1.868

• 1.856

• None of these

If ${\sin }^{-1}\left(\mathrm{x}\right) + {\cot }^{-1}\left(\frac{1}{2}\right) = \frac{\mathrm{\pi }}{2}$, then value of x will be

• 0

• $\frac{1}{\sqrt{5}}$

• $\frac{2}{\sqrt{5}}$

• $\frac{\sqrt{3}}{2}$

If 2x - $\left[\begin{array}{cc}1& 2\\ 7& 4\end{array}\right] = \left[\begin{array}{cc}3& 2\\ 0& - 2\end{array}\right]$, then the value of x will be

• $\left[\begin{array}{cc}2& 2\\ 7& 4\end{array}\right]$

• $\left[\begin{array}{cc}1& 2\\ 7/2& 2\end{array}\right]$

• $\left[\begin{array}{cc}2& 2\\ 7/2& 1\end{array}\right]$

• None of the above

$\underset{\mathrm{x}\to 2}{\lim }\frac{2 - \sqrt{2 + \mathrm{x}}}{2^{1/3} - (4 - \mathrm{x}{)}^{1/3} } \mathrm{is} \mathrm{equal} \mathrm{to}$

• $2 . 3^{-1/2}$

• $3 . 2^{-4/3}$

• $- 3 . 2^{4/3}$

• None of the above

29.

If 2a + 3b + 6c = 0, then the equation ax² + bx + c = 0 has atleast one real root in

• (0, 1)

• $\left(0, \frac{1}{2}\right)$

• $\left(\frac{1}{4}, \frac{1}{2}\right)$

• None of the above

If f(x) = $$\begin{gathered} \left\{\sin \left(\frac{\mathrm{\pi x}}{2}\right), \mathrm{if} \mathrm{x} < 1 \\ 3 - 2\mathrm{x}, \mathrm{if} \mathrm{x} \ge 1\right\} \end{gathered}$$, then f(x) has

• local minimum at x = 1

• local maximum at x = 1

• Both local maximum and local minimum at x = 1

• None of the above