AP EAMCET · Maths · Binomial Theorem
The coefficient of \(x^{15}\) in the product \((1-x)(1-2 x)\left(1-2^2 x\right)\left(1-2^3 x\right) \ldots\left(1-2^{15} x\right)\) is
- A \(2^{121}-2^{104}\)
- B \(2^{105}+2^{124}\)
- C \(2^{105}-2^{121}\)
- D \(2^{120}-2^{104}\)
Answer & Solution
Correct Answer
(C) \(2^{105}-2^{121}\)
Step-by-step Solution
Detailed explanation
Let the product be \(P(x) = \prod_{k=0}^{15} (1 - 2^k x)\). The coefficient of \(x^{15}\) is \( (-1)^{15} \sum_{0 \le i_1 This sum can be expressed as \( \left(\prod_{k=0}^{15} 2^k\right) \left(\sum_{m=0}^{15} \frac{1}{2^m}\right) \).…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(f: N \rightarrow N\) be a function such that \(f(x+y)=f(x)+f(y)+x y\) for every \(x, y \in N\). If \(f(1)=2\), then \(\sum_{k=0}^{10} f(k)=\)AP EAMCET 2025 Medium
- If the \(\operatorname{Arg} z_1\) and \(\operatorname{Arg} \overline{z_2}\) are \(\frac{\pi}{3}\) and \(\frac{\pi}{5}\) respectively then the value of \(\operatorname{Arg} z_1+\operatorname{Arg} z_2\) isAP EAMCET 2022 Easy
- The value of \(k\), if \((1,2),(k,-1)\) are conjugate points with respect to the ellipse \(2 x^2+3 y^2=6\) isAP EAMCET 2007 Hard
- Suppose is an isosceles triangle with and . Then the centroid of the triangle isAP EAMCET 2022 Medium
- If \([a, b]\) is the range of the function \(\frac{x+2}{2 x^2+3 x+6}\) for \(x \in \mathbf{R}\), thenAP EAMCET 2020 Easy
- If \(x+i y=\frac{3}{2+\cos (\theta)+i \sin (\theta)}\), then \(x^2+y^2=\)AP EAMCET 2020 Medium
More PYQs from AP EAMCET
- A bomb at rest explodes into three pieces of equal masses. If two pieces move perpendicular to each other, each with a speed \(v\) then the speed of the third piece isAP EAMCET 2023 Easy
- \(\int \frac{d x}{x^3+3 x^2+2 x}=\)AP EAMCET 2018 Hard
- The line \(x+y=1\) meets the lines represented by the equation \(y^3-6 x y^2+11 x^2 y-6 x^3=0\) at the points \(P, Q, R\). If \(O\) is the origin, then \((O P)^2+(O Q)^2+(O R)^2=\)AP EAMCET 2020 Medium
- A purple colored compound of Manganese decomposes on heating to liberate oxygen and forms compounds of Manganese ( and ) compound reacts with in presence of potassium nitrate to give compound . Compounds and respectively areAP EAMCET 2021 Medium
- If the petrol burnt in driving a motor boat varies as the cube of the velocity, then the speed (in \(\mathrm{km} /\) hour) of the boat going against a water flow of \(C \mathrm{kms} /\) hour so that the quantity of petrol burnt is minimum isAP EAMCET 2018 Medium
- According to Cahn-Ingold-Prelog sequence rules, the correct order of priority for the given groups isAP EAMCET 2008 Easy