Question:
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 :
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 :
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Write the coefficients of \(x^2\) and \(x^3\) using \(\binom{15}{r}(-3)^r\) and solve the two linear equations in \(a, b\).
Updated On: Nov 5, 2026
- \((28, 861)\)
- \((28, 315)\)
- \((-21, 714)\)
- \((-54, 315)\)
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The Correct Option is B
Solution and Explanation
\((1-3x)^{15} = 1 - 45x + 945x^2 - 12285x^3 + \dots\) since \(\binom{15}{2}\cdot 9 = 105\cdot 9 = 945\) and \(\binom{15}{3}(-27) = -455\cdot 27 = -12285\).
Coefficient of \(x^2\): \(945 - 45a + b = 0 \Rightarrow b = 45a - 945\) ... (i)
Coefficient of \(x^3\): \(-12285 + 945a - 45b = 0 \Rightarrow 21a - b = 273\) ... (ii)
Substitute (i) in (ii): \(21a - 45a + 945 = 273 \Rightarrow -24a = -672 \Rightarrow a = 28\).
\(b = 45\cdot 28 - 945 = 1260 - 945 = 315\).
Answer: (2) \((28, 315)\)
Coefficient of \(x^2\): \(945 - 45a + b = 0 \Rightarrow b = 45a - 945\) ... (i)
Coefficient of \(x^3\): \(-12285 + 945a - 45b = 0 \Rightarrow 21a - b = 273\) ... (ii)
Substitute (i) in (ii): \(21a - 45a + 945 = 273 \Rightarrow -24a = -672 \Rightarrow a = 28\).
\(b = 45\cdot 28 - 945 = 1260 - 945 = 315\).
Answer: (2) \((28, 315)\)
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