The transformed equation of x² + 6xy + 8y² = 10 when the axes are rotated through an angle $\frac{\mathrm{\pi }}{4}$is :

• 15x² - 14xy + 3y² = 20

• 15x² + 14xy - 3y² = 20

• 15x² + 14xy + 3y² = 20

• 15x² - 14xy - 3y² = 20

$\mathrm{In} ∆ \mathrm{ABC}, \left(\mathrm{a} +\hspace{0.17em}\mathrm{b}\hspace{0.17em}+ \mathrm{c}\right)\left(\tan \left(\frac{\mathrm{A}}{2}\right) + \tan \left(\frac{\mathrm{B}}{2}\right)\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $2\mathrm{ccot}\left(\frac{\mathrm{A}}{2}\right)$

• $2\mathrm{acot}\left(\frac{\mathrm{A}}{2}\right)$

• $2\mathrm{bcot}\left(\frac{\mathrm{B}}{2}\right)$

• $\tan \left(\frac{\mathrm{C}}{2}\right)$

In $∆$ABC, with usual notation, observe the two statements given below :

$$\begin{gathered} \left(\mathrm{I}\right) {\mathrm{rr}}_1{\mathrm{r}}_2{\mathrm{r}}_3 = {∆}^2 \\ \left(\mathrm{II}\right) {\mathrm{r}}_1{\mathrm{r}}_{2 } + {\mathrm{r}}_2{\mathrm{r}}_3 + {\mathrm{r}}_3{\mathrm{r}}_1 = {\mathrm{s}}^2 \\ \mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ? \end{gathered}$$

• Both I and II are true

• I is true, II is false

• I is false, II is true

• Both I and II are false

$\sqrt{3}\csc \left(20°\right) - \sec \left(20°\right)$ is equal to

• 2

• $2\sin \left(20°\right) - \csc \left(40°\right)$

• 4

• $4\sin \left(20°\right) . \csc \left(40°\right)$

If A = 35°, B = 15° and C = 40°, then tan(A) · tan(B) + tan(B) . tan(C) + tan(C) . tan(A) is equal to

• 0

• 1

• 2

• 3

If $\tan \left(\mathrm{\theta }\right) + \tan \left(\mathrm{\theta } + \frac{\mathrm{\pi }}{3}\right) + \tan \left(\mathrm{\theta } + \frac{2\mathrm{\pi }}{3}\right) = 3$, then which of the following is equal to 1 ?

• $\tan \left(2\mathrm{\theta }\right)$

• $\tan \left(3\mathrm{\theta }\right)$

• ${\tan }^2\left(\mathrm{\theta }\right)$

• ${\tan }^3\left(\mathrm{\theta }\right)$

$\frac{1 + \tanh \left({\displaystyle \frac{\mathrm{x}}{2}}\right)}{1 - \tanh \left({\displaystyle \frac{\mathrm{x}}{2}}\right)} = ?$

• ${\mathrm{e}}^{ - \mathrm{x}}$

• ${\mathrm{e}}^{\mathrm{x}}$

• $2{\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

• $2{\mathrm{e}}^{ - \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

$\mathrm{In} ∆\mathrm{ABC}, \mathrm{if} \frac{1}{\mathrm{b} + \mathrm{c}} + \frac{1}{\mathrm{c} +\hspace{0.17em}\mathrm{a}} = \frac{3}{\mathrm{a} +\hspace{0.17em}\mathrm{b} + \mathrm{c}}, \mathrm{then} \mathrm{C} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $90°$

• $60°$

• $45°$

• $30°$

$$\begin{gathered} \mathrm{Observe} \mathrm{the} \mathrm{following} \mathrm{statements} \\ \left(\mathrm{I}\right) \mathrm{In} ∆\mathrm{ABC}, {\mathrm{bcos}}^2\left(\frac{\mathrm{C}}{2}\right) + {\mathrm{ccos}}^2\left(\frac{\mathrm{B}}{2}\right) = \mathrm{s} \\ \left(\mathrm{II}\right) \mathrm{In} ∆\mathrm{ABC}, \cot \left(\frac{\mathrm{A}}{2}\right) = \frac{\mathrm{b} + \mathrm{c}}{2} \Rightarrow \mathrm{B} = 90° \\ \mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ? \end{gathered}$$

• Both I and II are true

• I is true, II is false

• I is false, II is true

• Both I and II are false

220.

$\mathrm{In} \mathrm{a} \mathrm{triangle}, \mathrm{if} {\mathrm{r}}_1 = 2{\mathrm{r}}_2 = 3{\mathrm{r}}_3, \mathrm{then} \frac{\mathrm{a}}{\mathrm{b}} + \frac{\mathrm{b}}{\mathrm{c}} + \frac{\mathrm{c}}{\mathrm{a} } = ?$

• $\frac{75}{60}$

• $\frac{155}{60}$

• $\frac{176}{60}$

• $\frac{191}{60}$