JEE Mains · Maths · STD 11 - 7. binomial theoram
If \((1 - x^3)^{10} = \sum\limits_{r=0}^{10} a_r x^r (1-x)^{30-2r}\), then \(\dfrac{9a_9}{a_{10}}\) is equal to __________.
- A 30
- B 40
- C 50
- D 60
Answer & Solution
Correct Answer
(A) 30
Step-by-step Solution
Detailed explanation
The given equation is: \((1 - x^3)^{10} = \sum\limits_{r=0}^{10} a_r x^r (1-x)^{30-2r}\) Using the algebraic identity \(1 - x^3 = (1-x)(1+x+x^2)\), the left hand side can be written as: \((1-x)^{10}(1+x+x^2)^{10} = \sum\limits_{r=0}^{10} a_r x^r (1-x)^{30-2r}\) Dividing both…
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 \(f(x)=\lim _{\theta \rightarrow 0}\left(\frac{\cos \pi x-x^{\left(\frac{2}{\theta}\right)} \sin (x-1)}{1+x^{\left(\frac{2}{\theta}\right)}(x-1)}\right), x \in R\).
Consider the following two statements :
(I) \(f ( x )\) is discontinous at \(x =1\).
(II) \(f ( x )\) is continous at \(x =-1\). Then,JEE Mains 2026 Easy - Let \(ABCD\) be a quadrilateral. If \(E\) and \(F\) are the mid points of the diagonals \(AC\) and \(BD\) respectively and \((\overrightarrow{ AB }-\overline{ BC })+(\overrightarrow{ AD }-\overrightarrow{ DC })= k \overline{ FE }\), then \(k\) is equal toJEE Mains 2023 Hard
- The number of matrices \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\), where a \(, b, c, d \in\{-1,0,1,2,3, \ldots \ldots, 10\}\), such that \(A=A^{-1}\), isJEE Mains 2022 Hard
- Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|\vec{a}|=1,|\vec{b}|=4\) and \(\vec{a} \cdot \vec{b}=2\). If \(\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}\) and the angle between \(\vec{b}\) and \(\vec{c}\) is \(\alpha\), then \(192 \sin ^2 \alpha\) is equal toJEE Mains 2024 Medium
- If \(y =\sum \limits_{ k =1}^{6} k \cos ^{-1}\left\{\frac{3}{5} \cos k x -\frac{4}{5} \sin k x \right\}\) then \(\frac{ dy }{ dx }\) at \(x =0\) isJEE Mains 2020 Medium
- Let \(f:\left[ { - 2,3} \right] \to \left[ {0,\infty } \right)\) be a continuous function such that \(f(1-x) = f(x)\) for all \(x \in \left[ { - 2,3} \right]\) . If \(R_1\) is the numerical value of the area of the region bounded by \(y =f (x), x = -2, x = 3\) and the axis of \(x\) and \({R_2} = \int\limits_{ - 2}^3 {x\,f\left( x \right)} dx\) , thenJEE Mains 2013 Hard
More PYQs from JEE Mains
- If the value of \(\lim _{x \rightarrow 0}(2-\cos x \sqrt{\cos 2 x})^{\left(\frac{x+2}{x^{2}}\right)}\) is equal to \(e^{a}\), then \(a\) is equal to \(.....\)JEE Mains 2021 Hard
- The sum \(1+3+11+25+45+71+.\). upto 20 terms, is equal toJEE Mains 2025 Medium
- Let \(A\) be the point of intersection of the lines \(L_1: \frac{x-7}{1}=\frac{y-5}{0}=\frac{z-3}{-1}\) and \(L_2: \frac{x-1}{3}=\frac{y+3}{4}=\frac{z+7}{5}\). Let \(B\) and \(C\) be the point on the lines \(L_1\) and \(L_2\) respectively such that \(\mathrm{AB}=\mathrm{AC}=\sqrt{15}\). Then the square of the area of the triangle ABC is :JEE Mains 2025 Medium
- Let \({ }^{n} C_{r}\) denote the binomial coefficient of \(x^{r}\) in the expansion of \((1+ x )^{ n }.\) If \(\sum_{ k =0}^{10}\left(2^{2}+3 k \right){ }^{ n } C _{ k }=\alpha .3^{10}+\beta \cdot 2^{10}, \alpha, \beta \in R\) then \(\alpha+\beta\) is equal to ....... .JEE Mains 2022 Hard
- Let \(a , b , c\) and \(d\) be positive real numbers such that \(a+b+c+d=11\). If the maximum value of \(a^5 b^3 c^2 d\) is \(3750 \beta\), then the value of \(\beta\) isJEE Mains 2023 Hard
- The set of all \(\alpha \in R\), for which \(w = \frac{{1 + \left( {1 - 8\alpha } \right)z}}{{1 - z}}\) is a purely imaginary number, for all \(z \in C\) satisfying \(\left| z \right| = 1\) and \({\mathop{\rm Re}\nolimits} \,z \ne 1\), isJEE Mains 2018 Hard