Let x, y, z be positive real numbers such that x, y, z are in GP and ${\tan }^{-1}\left(\mathrm{x}\right), {\tan }^{-1}\left(\mathrm{y}\right), {\tan }^{-1}\left(\mathrm{z}\right)$ are in AP. Then which one of the following is correct ?

• x = y = z

• xz = 1

• $\mathrm{x}\ne \mathrm{y}$ and y = z

• $\mathrm{x} = \mathrm{y} \mathrm{and} \mathrm{y} \ne \mathrm{z}$

$$\begin{gathered} \mathrm{If} {\mathrm{S}}_{\mathrm{n}} = \mathrm{nP} + \frac{\mathrm{n}\left(\mathrm{n} - 1\right)\mathrm{Q}}{2}, \mathrm{where} {\mathrm{S}}_{\mathrm{n}} \mathrm{denotes} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{first} \mathrm{n} \mathrm{terms} \mathrm{of} \mathrm{an} \mathrm{AP}, \\ \mathrm{then} \mathrm{the} \mathrm{common} \mathrm{difference} \mathrm{is} : \end{gathered}$$

•  P + Q

• 2P + 3Q

• 2Q`

• Q

The value of the product $6^{\frac{1}{2}} \times 6^{\frac{1}{4}} \times 6^{\frac{1}{8}} \times 6^{\frac{1}{16}} \times ...$ Upto infinite terms is

• 6

• 36

• 216

• 512

The number of terms in the expansion of (x + a)¹⁰⁰ + (x - a)¹⁰⁰ after simpiification is :

• 202

• 101

• 51

• 50

45.

A person is to count 4500 notes. Let a_n denote the number of notes he counts in the nth minute. If a₁ = a₂ = a₃ = ... = a₁₀ = 150, and a₁₀, a₁₁, a₁₂, ... are in AP with the common difference - 2, then the time taken by him to count all the notes is

• 24minutes

• 34minutes

• 125minutes

• 135minutes

If y = x + x² + x³ + ... up to infinite terms, where x $<$1, then which one of the following is correct ?

• $\mathrm{x} = \frac{\mathrm{y}}{1 + \mathrm{y}}$

• $\mathrm{x} = \frac{\mathrm{y}}{1 - \mathrm{y}}$

• $\mathrm{x} = \frac{1 + \mathrm{y}}{\mathrm{y}}$

• $\mathrm{x} = \frac{1 - \mathrm{y}}{\mathrm{y}}$

The value of $\frac{1}{{\log }_3\left(\mathrm{e}\right)} + \frac{{\displaystyle 1}}{{\displaystyle {\log }_3{\left(\mathrm{e}\right)}^2}} + \frac{{\displaystyle 1}}{{\displaystyle {\log }_3{\left(\mathrm{e}\right)}^4}} +\hspace{0.17em}... \mathrm{up} \mathrm{to} \mathrm{infinite} \mathrm{is} :$

• ${\log }_{\mathrm{e}}\left(9\right)$

• 0

• 1

• ${\log }_{\mathrm{e}}\left(3\right)$

A tea party is arranged for 16 people along two sides of a long table with eight chairs on each side. Four particular men wish to sit on one particular side and two particular men on the other side. The number of ways they can be seated is

• $24 \times 8! \times 8!$

• ${\left(8!\right)}^3$

• $210 \times 8! \times 8!$

• 16!

If. 1 . 3 + 2 . 3² + 3 . 3³ + ... + n . 3ⁿ = $\frac{\left(2\mathrm{n} - 1\right)3^{\mathrm{a}} + \mathrm{b}}{4}$ then a and b are respectively

• n, 2

• n, 3

• n + 1, 2

• n + 1, 3

$$\begin{gathered} \mathrm{In} ∆\mathrm{PQR}, <\mathrm{R} = \frac{\mathrm{\pi }}{2}. \mathrm{If} \tan \left(\frac{\mathrm{P}}{2}\right) \mathrm{and} \tan \left(\frac{\mathrm{Q}}{2}\right) \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{ax}}^2 + \mathrm{bx} +\hspace{0.17em}\mathrm{c} = 0, \\ \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ? \end{gathered}$$

• a = b + c

• b = c + a

• c = a + b

• b = c