InterviewSolution
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InterviewSolution
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If the coefficients of `x^(2)` and `x^(3)`are both zero, in the expansion of the expression `(1 + ax + bx^(2)) (1 - 3x)^15` in powers of x, then the ordered pair (a,b) is equal to(A)` (28, 315)`(B)`(-21, 714) `(C) `(28, 861)`(D) `(-54, 315)`A. (28, 315)B. (-21, 714)C. (28, 861)D. (-54, 315) |
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Answer» Given expression is `(1 + ax + bx^(2)) (1 - 3x)^(15)`. In the expanison of binominal `(1 - 3x)^(15)`. The `(r + 1)` th term is `T_(r + 1) = .^(15)C_(r ) - (3x)^(r ) = .^(15)C_(r ) (-3)^(r ) x^(r )` Now, coefficient of `x^(2)` in the expansion of `(1 + ax + bx^(2)) (1 - 3x)^(15)` is `.^(15)C_(2) (-3)^(2) + a .^(15)C_(1) (-3)^(1) + b .^(15)C_(0) (-3)^(0) = 0` (given) `implies (105 xx 9) - 45a + b = 0` `implies 45 a - b = 945` Similarly, the coefficient of `x^(3)`, in the expanison of `(1 + ax + bx^(2)) (1 - 3x)^(15)` is `.^(15)C_(3) (-3)^(3) + a .^(15)C_(2) (-3)^(2) + b .^(15)C_(1) (-3)^(1) = 0` `implies - 12285 + 945 a - 45 b = 0` `implies 63 a - 3b = 819` `implies 21a - b = 273` From Eqs. (i) and (ii) We get `24 a = 672 implies a = 28` So, b = 315 `implies (a, b) = (28, 315)` |
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