If a > 1, b > 1 and c > 1 are in geometric progression, then $\frac{1}{1 + {\log }_{\mathrm{e}}\left(\mathrm{a}\right)}, \frac{{\displaystyle 1}}{{\displaystyle 1 + {\log }_{\mathrm{e}}\left(\mathrm{b}\right)}}, \frac{{\displaystyle 1}}{{\displaystyle 1 + {\log }_{\mathrm{e}}\left(\mathrm{c}\right)}}$ will be in

• arithmetic proression

• geometric progression

• harmonic progresssion

• None of the above

If z = x + iy is a complex numberwhere x and y are integers. Then, the area of the rectangle whose vertices are the roots of the equation $\mathrm{z}{\overline{\mathrm{z}}}^3 + \overline{\mathrm{z}}{\mathrm{z}}^3 = 350$, is

• 48

• 32

• 40

• 90

How many real roots does the quadratic equation f(x) = x ² + 3$\left|\mathrm{x}\right|$ + 2 = 0 have ?

• One

• Two

• Four

• No real root

20 persons are invited for a party. The number of ways in which they and the host can be seated at a circular table, if two particular persons be seated on eitherside of the host is equal to

• 2 . (18)!

• 18! . 3!

• 19! . 2!

• None of these

The number of solution of $\tan \left(5\mathrm{\pi cos}\left(\mathrm{\theta }\right)\right) = \cot \left(5\mathrm{\pi sin}\left(\mathrm{\theta }\right)\right) \mathrm{for} \mathrm{\theta } \mathrm{in} \left(0, 2\mathrm{\pi }\right)$ will be

• 28

• 14

• 4

• 2

The equation $\sin \left(\mathrm{x}\right) + 2\sin \left(2\mathrm{x}\right) + 3\sin \left(3\mathrm{x}\right) = \frac{8}{\mathrm{\pi }}$ has atleast one root in

• $\left[\mathrm{\pi }, \frac{3\mathrm{\pi }}{2}\right]$

• $\left[0, \frac{\mathrm{\pi }}{2}\right]$

• $\left[\frac{\mathrm{\pi }}{2}, \mathrm{\pi }\right]$

• $\left[\frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{4}\right]$

AB is a vertical pole. The end A is on the ground level. C is the middle point of AB and P is a point on the ground level. The portion BC subtends an angle $\mathrm{\beta }$ at P. If AP = nAB, then $\tan \left(\mathrm{\beta }\right)$ is equal to

• $\frac{\mathrm{n}}{2{\mathrm{n}}^2 + 1}$

• $\frac{\mathrm{n}}{{\mathrm{n}}^2 + 1}$

• $\frac{\mathrm{n}}{\mathrm{n} + 1}$

• None of these

18.

If the line x + 2by + 7 = 0 is a diameter of the circle x² + y² - 6x + zy = 0, then b is equal to

• - 5

• - 3

• 2

• 5

If the line y = 2x+ c is tangent to the parabola y² = 4x, then c is equal to

• $- \frac{1}{2}$

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{1}{4}$

The length of the latus rectum of the ellipse 9x² + 4y² = 1 is

• $\frac{3}{2}$

• $\frac{4}{9}$

• $\frac{8}{3}$

• $\frac{8}{9}$