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
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
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
C.
6
Given, f(x) = ax2 + bx + c, g(x) = px2 + qx + r
Since f(1) = g(1)
...(i)
f(2) = g(2)
...(ii)
Subtracting Eq. (ii) from Eq. (i), we get
3a + b = 3p + q ...(iii)
f(3) - g(3) = 2
From Eq. (i),
(a - p) + (b - q) + (c - r) = 0
...(v)
From Eq. (ii),
4(a - p) + 2(b - q) + c - r = 0
...(vi)
Subtracting Eq.(v) from Eq. (vi), we get
(a - p) - 1 = 0
a - p = 1
From Eq. (v),
b - q = - 3
Now,
f(4) - g(4) = (16a + 4b + c) - (16p + 4q + r)
= 16(a - p) + 4(b - q) + (c - r) ...(vii)
Substituting the values of (a - p), (b - q), (c - r) from above in Eq. (vii), we get
f(4) - g(4) =
= 16 - 12 + 2 = 6
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 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