A vehicle registration number consists of 2 letters of English alphabet followed by 4 digits, where the first digit is not zero. Then, the total number of vehicles with distinct registration numbers is

• ${26}^2 \times {10}^4$

• ${}^{26}\mathrm{P}_2 \times {}^{10}\mathrm{P}_4$

• ${}^{26}\mathrm{P}_2 \times 9 \times {}^{10}\mathrm{P}_3$

• ${}^{26}\mathrm{P}_2 \times 9 \times {10}^3$

The number of words that can be written using all the letters of the word 'IRRATIONAL' is

• $\frac{10!}{{\left(2!\right)}^3}$

• $\frac{10!}{{\left(2!\right)}^2}$

• $\frac{10!}{2!}$

• 10!

23.

Four speakers will address a meeting where speaker Q will always speak P. Then, the number of ways in which the order of speakers can be prepared is

• 256

• 128

• 24

• 12

The number of diagonals in a regular polygon of 100 sides is

• 4950

• 4850

• 4750

• 4650

Let the coefficients of powers of x in the 2nd, 3rd and 4th terms in the expansion of (1 + x)ⁿ, where n is a positive integer, be in arithmetic progression. Then, the sum of the coefficients of odd powers of x in the expansion is

• 32

• 64

• 128

• 256

Let f(x) = ax² + bx + c, g(x) = px² + qx + r such that f(1) = g(1), f(2) = g(2) and f(3) - g(3) = 2. Then, f(4) - g(4) is

• 4

• 5

• 6

• 7

The sum 1 x 1! + 2 x 2! + ... + 50 x 50! equals

• 51!

• 51! + 1

• 51! + 1

• 2 $\times$ 51!

Six numbers are in AP such that their sum is 3. The first term is 4 times the third term. Then, the fifth term is

• - 15

• - 3

• 9

• - 4

The sum of the infinite series $1 +\hspace{0.17em}\frac{1}{3} + \frac{1 . 3}{3 . 6} +\hspace{0.17em}\frac{1 . 3 . 5}{3 . 6 . 9} + \frac{1 . 3 . 5 . 7}{3 . 6 . 9 . 12} + ...$ is equal to

• $\sqrt{2}$

• $\sqrt{3}$

• $\sqrt{\frac{3}{2}}$

• $\sqrt{\frac{1}{3}}$

The equations x² + x + a= 0 and x² + ax + 1 = 0 have a common real root

• for no value of a

• for exactly one value of a

• for exactly two value of a

• for exactly three value of a