If the coefficients of x and x² in (1+x)ᵖ(1-x)^q are 4 and -5 respectively, then 2 p+3 q is equal to :

Solution

Maths sir · 4 steps

Write each coefficient in terms of p and q, then spot that the x² one is built from p − q and p + q.

1. Coefficient of x. (1+x)ᵖ = 1 + px + C(p, 2)x² + … and (1-x)^q = 1 - qx + C(q, 2)x² - …. An x comes from one factor's linear term times the other's constant, so the coefficient is p - q = 4.
2. Coefficient of x squared. An x² comes from three products: x² times 1, 1 times x², and x times x. That gives C(p, 2) + C(q, 2) - pq = -5.

   p(p-1)/2 + q(q-1)/2 - pq = ((p-q)² - (p+q))/2 = -5

3. Find p and q. Put p - q = 4: (16 - (p+q))/2 = -5, so p + q = 26. Together with p - q = 4, this gives p = 15 and q = 11.
4. So. 2p + 3q = 30 + 33 = 63, which is option (d).

The idea to keep. For coefficients in a product of two binomials, collect every pair of powers that adds up to the power you want. Then rewrite the result using p+q and p-q; the algebra becomes short.

Answer from the NTA final answer key. Solution by PrepWiser.