The number of integral values of K, for which the equation $7\cos \left(\mathrm{x}\right) + 5\sin \left(\mathrm{x}\right) = 2\mathrm{K} + 1$ has a solution

• 4

• 8

• 10

• 12

If the complex numbers z₁, z₂ and z₃ are in AP, then they lie on a

• circle

• parabola

• line

• ellipse

The number of real solutions of the equation

$\left(\frac{9}{10}\right) = - 3 + \mathrm{x} - {\mathrm{x}}^2$ is

• 0

• 1

• 2

• None of these

If one root is square of the other root of the equation x² + px + q = 0 , then the relations between p and q is

• p³ - (3p - 1)q + q² = 0

• p³ - q(3p + 1) + q² = 0

• p³ + (3p - 1)q + q² = 0

• p³ + (3p + 1)q + q² = 0

The coefficient of x⁵³ in the following expansions

$\sum _{\mathrm{m} = 0}^{100}{}^{100}\mathrm{C}_{\mathrm{m}}{\left(\mathrm{x} - 3\right)}^{100 - \mathrm{m}} . 2^{\mathrm{m}}$ is

• ${}^{100}\mathrm{C}_{47}$

• ${}^{100}\mathrm{C}_{53}$

• - ${}^{100}\mathrm{C}_{53}$

• - ${}^{100}\mathrm{C}_{100}$

216.

If x + iy = ${\left(1 - \mathrm{i}\sqrt{3}\right)}^{100}$, then find (x, y).

• $\left(2^{99}, 2^{99}\sqrt{3}\right)$

• $\left(2^{99}, - 2^{99}\sqrt{3}\right)$

• $\left(- 2^{99}, 2^{99}\sqrt{3}\right)$

• None of these

If $\frac{{\mathrm{x}}^2 + 2\mathrm{x} + 7}{2\mathrm{x} + 3} < 6, \mathrm{x} \in \mathrm{R}$, then

• $\mathrm{x} > 11 \mathrm{or} \mathrm{x} = - \frac{3}{2}$

• x > 11 or x < - 1

• $- \frac{3}{2} < \mathrm{x} < - 1$

• $- 1 < \mathrm{x} < 11 \mathrm{or} \mathrm{x} < - \frac{3}{2}$

If D is the set of all real x such that 1 - e^{(1/x) - 1} is positive, then D is equal to

• $(- \infty , 1]$

• $\left(- \infty , 0\right)$

• $\left(1, \infty \right)$

• $\left(- \infty , 0\right) \cup \left(1, \infty \right)$

If one AM 'A' and two GM p and q are inserted between two given numbers, then. find the value of $\frac{{\mathrm{p}}^2}{\mathrm{q}} + \frac{{\mathrm{q}}^2}{\mathrm{p}}$.

• A

• 2A

• 3A

• 4A

If the roots of the equation x² + ax + b = 0 are c and d, then one of the roots of the equation x² + (2c + a)x + c² + ac + b = 0

• c

• d - c

• 2d

• 2c