Let a and b be two nonzero real numbers. If the coefficient of x5 in the expansion of (\(ax^2+(\frac{70}{27bx})^4\) is equal to the coefficient of x-5 in the expansion of \((ax-\frac{1}{bx^2})^7\). then the value of 2b is
Solution and Explanation
We are given the expression:
\[ T_{r+1} = \ ^4C_r (ax^2)^{(4-r)} \times \left(\frac{70}{27bx}\right)^r \]
For the coefficient of \( x^5 \), we solve for \( r \):
\[ 8 - 2r - r = 5 \quad \Rightarrow \quad r = 1 \]
Thus, the coefficient of \( x^5 \) is:
\[ \ ^4C_1 a^3 \left( \frac{70}{27b} \right) \]
Next, for the expression:
\[ t_{r+1} = \ ^7C_r (ax)^{7-r} \left( -\frac{1}{bx^2} \right)^r \]
For the coefficient of \( x^{-5} \), we solve for \( r \):
\[ 7 - r - 2r = -5 \quad \Rightarrow \quad r = 4 \]
The coefficient of \( x^{-5} \) is:
\[ \ ^7C_4 a^3 \frac{1}{b^4} \]
Thus, we have the equation:
\[ 2b = 3 \]
Final Answer:
The value of \( 2b \) is \( \boxed{3} \).
Top Questions on Binomial theorem
- If the number of terms in the binomial expansion of \((2x + 3)^n\) is 22, then the value of \(n\) is:
- KCET - 2025
- Mathematics
- Binomial theorem
- A radar system can detect an enemy plane in one out of ten consecutive scans. The probability that it can detect an enemy plane at least twice in four consecutive scans is:
- AP EAPCET - 2025
- Mathematics
- Binomial theorem
- A string of letters is to be formed by using 4 letters from all the letters of the word “MATHEMATICS”. The number of ways this can be done such that two letters are of same kind and the other two are of different kind is
- AP EAPCET - 2025
- Mathematics
- Binomial theorem
- The number of trials conducted in a binomial distribution is 6. If the difference between the mean and variance of this variate is \(\frac{27}{8}\), then the probability of getting at most 2 successes is:
- AP EAPCET - 2025
- Mathematics
- Binomial theorem
- The number of ways of dividing 15 persons into 3 groups containing 3, 5 and 7 persons so that two particular persons are not included into the 5 persons group is
- AP EAPCET - 2025
- Mathematics
- Binomial theorem
Questions Asked in JEE Advanced exam
- Let $$ \alpha = \frac{1}{\sin 60^\circ \sin 61^\circ} + \frac{1}{\sin 62^\circ \sin 63^\circ} + \cdots + \frac{1}{\sin 118^\circ \sin 119^\circ}. $$ Then the value of $$ \left( \frac{\csc 1^\circ}{\alpha} \right)^2 $$ is \rule{1cm}{0.15mm}.
- JEE Advanced - 2025
- Some Applications of Trigonometry
Figure 1 shows the configuration of main scale and Vernier scale before measurement. Fig. 2 shows the configuration corresponding to the measurement of diameter $ D $ of a tube. The measured value of $ D $ is:
- JEE Advanced - 2025
- Units and measurement
- Let $ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $ and $ P = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} $. Let $ Q = \begin{pmatrix} x & y \\ z & 4 \end{pmatrix} $ for some non-zero real numbers $ x, y, z $, for which there is a $ 2 \times 2 $ matrix $ R $ with all entries being non-zero real numbers, such that $$ QR = RP $$ Then which of the following statements is (are) TRUE?
- JEE Advanced - 2025
- Matrices
- Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.
The center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity):
- JEE Advanced - 2025
- Oscillations
Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.