In a AABC, if $\angle$C = 90°, r and R are the inradius and circumradius of the $∆$ABC respectively, then 2(r + R) is equal to

• b + c

• c + a

• a + b

• a + b + c

Let $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be two distinct roots of $\mathrm{acos}\left(\mathrm{\theta }\right) + \mathrm{bsin}\left(\mathrm{\theta }\right) = \mathrm{c}$  where a, b, c are three real constants and $\mathrm{\theta } \in \left[0, 2\mathrm{\pi }\right]$. Then, $\mathrm{\alpha } + \mathrm{\beta }$ is also a root of the same equation, if

• a + b = c

• b + c = a

• c + a = b

• c = a

If $\cos \left(\mathrm{x}\right) \mathrm{and} \sin \left(\mathrm{x}\right)$ are solutions of the differential equation

${\mathrm{a}}_0\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + {\mathrm{a}}_1\frac{d\mathrm{y}}{d\mathrm{x}} + {\mathrm{a}}_2\mathrm{y} = 0$

where a₀, a₁ and a₂ are real constants, then which of the following is/are always true?

• $\mathrm{Acos}\left(\mathrm{x}\right) + \mathrm{Bsin}\left(\mathrm{x}\right)$ is a solution, where A and B are real constants 

• $\mathrm{Acos}\left(\mathrm{x} + \frac{\mathrm{\pi }}{4}\right)$ is a solution, where A is a real constant

• $\mathrm{Acos}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)$ is a solution, where A is a real constant

• $\mathrm{Acos}\left(\mathrm{x} + \frac{\mathrm{\pi }}{4}\right) + \mathrm{Bsin}\left(\mathrm{x} - \frac{\mathrm{\pi }}{4}\right)$ is a souton, where A and B are real constants 

Let 16x² - 3y - 32x - 12y = 44 represents a hyperbola. Then,

• length of the transverse axis is $2\sqrt{3}$

• length of each latusrectum is $32/\sqrt{3}$

• eccentricity is $\sqrt{19/3}$

• equation of a directrix is x = $\frac{\sqrt{19}}{3}$

Which of the following statements is /are correct for $0 < \mathrm{\theta } < \frac{\mathrm{\pi }}{2}$

• ${\left(\cos \left(\mathrm{\theta }\right)\right)}^{1/2} \le \cos \left(\frac{\mathrm{\theta }}{2}\right)$

• ${\left(\cos \left(\mathrm{\theta }\right)\right)}^{3/4} \ge \cos \left(\frac{3\mathrm{\theta }}{4}\right)$

• $\cos \left(\frac{5\mathrm{\theta }}{6}\right) \ge {\left(\cos \left(\mathrm{\theta }\right)\right)}^{5/6}$

• $\cos \left(\frac{7\mathrm{\theta }}{8}\right) \le {\left(\cos \left(\mathrm{\theta }\right)\right)}^{7/8}$

For the function f(x) = $\left[\frac{1}{\left[\mathrm{x}\right]}\right]$. where [x] denotes the greatest integer less than or equal to x, which of the following statements are true?

• The domain is $\left(- \infty , \infty \right)$

• The range is $\left\{0\right\} \cup \left\{- 1\right\} \cup \left\{1\right\}$

• The domain is $\left(- \infty , 0\right) \cup [1, \infty )$

• The range is $\left\{0\right\} \cup \left\{1\right\}$

Which of the following is /are always false?

• A quadratic equation with rational coefficients has zero or two irrational roots

• A quadratic equation with real coefficients has zero or two non-real roots

• A quadratic equation with irrational coefficients has zero or two irrational roots

• A quadratic equation with integer coefficients has zero or two irrational roots

48.

If the straight line (a - 1)x - by + 4 = 0 is normal to the hyperbola xy = 1, then which of the following does not hold?

• a > 1, b > 0

• a > 1, b < 0

• a < 1, b < 0

• a < 1, b > 0

Let f: $\mathrm{R} \to \mathrm{R}$ be a continuous function which satisfies f(x) = ${\int }_0^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)d\mathrm{t}$. Then, the value of $\mathrm{f}\left({\log }_{\mathrm{e}}\left(5\right)\right)$ is

• 0

• 2

• 5

• 3

Let f: [2, 2] $\to$ R  be a continuous function such that f(x) assumes only irrational values. If $\mathrm{f}\left(\sqrt{2}\right) = \sqrt{2}$, then

• f(0) = 0

• $\mathrm{f}(\sqrt{2} - 1) = \sqrt{2} - 1$

• $\mathrm{f}(\sqrt{2} - 1) = \sqrt{2} + 1$

• $\mathrm{f}(\sqrt{2} - 1) = \sqrt{2}$