Let f(x)+2 f(1/x)=x²+5 and 2 g(x)-3 g(1/2)=x, x>0. If α=∫₁² f(x) d x, and β=∫₁² g(x) d x, then the value of 9 α+β is :

Solution

Maths sir · 5 steps

Swap x with 1/x to get a second equation for f, and put x = 1/2 to pin down the constant in g.

1. Find f(x). Replace x by 1/x to get f(1/x) + 2f(x) = 1/x² + 5. Double this and subtract the original equation, so f(1/x) cancels.

   3f(x) = 2/x² - x² + 5 ⇒ f(x) = 1/3(2/x² - x² + 5)

2. Compute alpha. α = 1/3[-2/x - x³/3 + 5x]₁² = 1/3(19/3 - 8/3) = 11/9, so 9α = 11.
3. Find g(x). Here g(1/2) is just a constant. Put x = 1/2: 2g(1/2) - 3g(1/2) = 1/2, so g(1/2) = -1/2 and g(x) = 1/2(x - 3/2).
4. Compute beta. β = 1/2[x²/2 - 3x/2]₁² = 1/2((2 - 3) - (1/2 - 3/2)) = 1/2(-1 + 1) = 0.
5. So. 9α + β = 11 + 0 = 11, which is option (d).

The idea to keep. When f(x) and f(1/x) appear together, substitute x → 1/x and solve the two equations like simultaneous equations. A term like g(1/2) is a constant: find it by putting x = 1/2.

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