831.

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

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

If $\cos \left(2\mathrm{x}\right) = \left(\sqrt{2} + 1\right)\left(\cos \left(\mathrm{x}\right) - \frac{1}{\sqrt{2}}\right), \cos \left(\mathrm{x}\right) \ne \frac{1}{2} \mathrm{then} \mathrm{x} \in$

• $\left\{2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{3} : \mathrm{n} \in \mathrm{Z}\right\}$

• $\left\{2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{6} : \mathrm{n} \in \mathrm{Z}\right\}$

• $\left\{2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{2} : \mathrm{n} \in \mathrm{Z}\right\}$

• $\left\{2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{4} : \mathrm{n} \in \mathrm{Z}\right\}$

In a $∆\mathrm{ABC}$, $\mathrm{a}\left({\cos }^2\left(\mathrm{B}\right) + {\cos }^2\left(\mathrm{C}\right)\right) + \cos \left(\mathrm{A}\right)\left(\mathrm{ccos}\left(\mathrm{C}\right) + \mathrm{bcos}\left(\mathrm{B}\right)\right)$ is equal to

• a

• b

• c

• a + b + c

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

• a

• b

• c

• 0

$\mathrm{If} \cos \left(\mathrm{\theta }\right) - 4\sin \left(\mathrm{\theta }\right) = 1, \mathrm{then} \sin \left(\mathrm{\theta }\right) + 4\cos \left(\mathrm{\theta }\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\pm 1$

• 0

• $\pm 2$

• $\pm 4$

$\mathrm{The} \mathrm{extreme} \mathrm{values} \mathrm{of} 4\cos \left({\mathrm{x}}^2\right)\cos \left(\frac{\mathrm{\pi }}{3} + {\mathrm{x}}^2\right)\cos \left(\frac{\mathrm{\pi }}{3} - {\mathrm{x}}^2\right) \mathrm{over} \mathrm{R}, \mathrm{are}$

• - 1, 1

• - 2, 2

• -3, 3

• - 4, 4

Two sides of a triangle are given by the roots of the equation x² - 5x + 6 = 0 and the angle between the sides is $\frac{\mathrm{\pi }}{3}$.Then, the perimeter of

• $5 +\sqrt{2}$

• $5 +\sqrt{3}$

• $5 +\sqrt{5}$

• $5 +\sqrt{7}$

A tower, of x metres high, has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant y metres from the foot of the tower. Then, the length of the flagstaff(in metres), is

• $\frac{\mathrm{y}\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)}{\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)}$

• $\frac{\mathrm{x}\left({\mathrm{y}}^2 + {\mathrm{x}}^2\right)}{\left({\mathrm{y}}^2 - {\mathrm{x}}^2\right)}$

• $\frac{\mathrm{x}\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)}{\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)}$

• $\frac{\mathrm{x}\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)}{\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)}$