$\mathrm{For} \mathrm{x} \in \mathrm{IR}, 3\cos \left(4\mathrm{x} - 5\right) + 4 \mathrm{lies} \mathrm{in} \mathrm{the} \mathrm{interval}$

• [1, 7]

• [4, 7]

• [0, 7]

• [2, 7]

If $\mathrm{x} = \log \left[\cot \left(\frac{\mathrm{\pi }}{4} + \mathrm{\theta }\right)\right]$, then the value of $\sin \left(\mathrm{hx}\right)$ is

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

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

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

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

If in a $∆$ABC, r₃ = r₁ + r₂ + r, then $\angle \mathrm{A} + \angle \mathrm{B}$ is equal to

• 120°

• 100°

• 90°

• 80°

In a $∆\mathrm{ABC}$${\left(\mathrm{a} - \mathrm{b}\right)}^2{\cos }^2\left(\frac{\mathrm{C}}{2}\right) + {\left(\mathrm{a} +\hspace{0.17em}\mathrm{b}\right)}^2{\sin }^2\left(\frac{\mathrm{C}}{2}\right)$ is equal to

• a²

• c²

• b²

• a² + b²

In a $∆\mathrm{ABC}$, the correct formulae among the following are

$$\begin{gathered} \mathrm{I}. \mathrm{r} = 4\mathrm{Rsin}\left(\frac{\mathrm{A}}{2}\right)\sin \left(\frac{\mathrm{B}}{2}\right)\sin \left(\frac{\mathrm{C}}{2}\right) \\ \mathrm{II}. {\mathrm{r}}_1 = \left(\mathrm{s} - \mathrm{a}\right)\tan \left(\frac{\mathrm{A}}{2}\right) \\ \mathrm{III}. {\mathrm{r}}_3 = \frac{∆}{\mathrm{s} - \mathrm{c}} \end{gathered}$$

• only I, II

• only II, III

• only I, III

• I, II, III

An aeroplane flying with uniform speed horizontally one km above the ground is observed at an elevation of 60°. After 10 s if the elevation is observed to be 30°, then the speed of the plane (in km/h) is

• $\frac{240}{\sqrt{3}}$

• $200\sqrt{3}$

• $240\sqrt{3}$

• $\frac{120}{\sqrt{3}}$

If the distance between the points (acos$\mathrm{\theta }$, asin$\mathrm{\theta }$) and  (acos$\mathrm{\varphi }$, asin$\mathrm{\varphi }$) is 2a, then $\mathrm{\theta }$ is equal to

• $2\mathrm{n\pi } \pm \mathrm{\pi } + \mathrm{\varphi }, \mathrm{n} \in \mathrm{Z}$

• $\mathrm{n\pi } + \frac{\mathrm{\pi }}{2} + \mathrm{\varphi }, \mathrm{n} \in \mathrm{Z}$

• $n\mathrm{\pi } - \mathrm{\varphi }, \mathrm{n} \in \mathrm{Z}$

• $2\mathrm{n\pi } + \mathrm{\varphi }, \mathrm{n} \in \mathrm{Z}$

E₁ : a + b + c = 0, if 1 is a root of ax² + bx + c = 0, E₂ : b² - a² = 2ac, if $\sin \left(\mathrm{\theta }\right), \cos \left(\mathrm{\theta }\right)$ are the roots of ax² + bx + c = 0 Which of the following is true ?

• E₁ is true, E₂ is true

• E₁ is true, E₂ is false

• E₁ is false, E₂ is true

• E₁ is false, E₂ is false

219.

If A + C = 2B, then $\frac{\cos \left(\mathrm{C}\right) - \cos \left(\mathrm{A}\right)}{\sin \left(\mathrm{A}\right) - \sin \left(\mathrm{C}\right)}$ is equal to

• $\cot \left(\mathrm{B}\right)$

• $\cot \left(2\mathrm{B}\right)$

• $\tan \left(2\mathrm{B}\right)$

• $\tan \left(\mathrm{B}\right)$

A + B = C $\Rightarrow$ ${\cos }^2\left(\mathrm{A}\right) + {\cos }^2\left(\mathrm{B}\right) + {\cos }^2\left(\mathrm{C}\right) - 2\cos \left(\mathrm{A}\right)\cos \left(\mathrm{B}\right)\cos \left(\mathrm{C}\right)$ is equal to

• 1

• 2

• 0

• 3