Theory question 111
Integration(a) If f(x) = ∫(4x − x2) dx and f(3) = 21, find f(x).
(b) The second, fourth and eight terms of an Arithmetic Progression (A.P.) form the first three consecutive terms of a Geometric Progression (G.P.). The sum of the third and fifth terms of the A.P. is 20. Find the: (i) first four terms of the A.P., (ii) sum of the first ten terms of the A.P.
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Model answer
(a) f(x) = 2x² − x³/3 + c; f(3) = 18 − 9 + c = 21, c = 12. f(x) = 2x² − x³/3 + 12.
(b) (a + 3d)² = (a + d)(a + 7d) gives 2d² = 2ad, so a = d.
(a + 2d) + (a + 4d) = 8d = 20, d = 2.5 = a.
(i) 2.5, 5, 7.5, 10.
(ii) S₁₀ = 5(5 + 22.5) = 137.5.
Explanation
For G.P. terms, (middle)² = first × third.