The term independent of x in the expansion of
\((1−x^2+3x^3)(\frac{5}{2}x^3−\frac{1}{5x^2})11,x≠0 \)
is:
The term independent of x in the expansion of
\((1−x^2+3x^3)(\frac{5}{2}x^3−\frac{1}{5x^2})11,x≠0 \)
is:
\(\frac{7}{40}\)
\(\frac{33}{200}\)
\(\frac{39}{200}\)
\(\frac{11}{50}\)
The Correct Option is B
Solution and Explanation
The correct answer is (B) : \(\frac{33}{200}\)
\((1−x^2+3x^3)(\frac{5}{2}x^3−\frac{1}{5x^2})^{11},x≠0\)
General term of \((\frac{5}{2}x^3−\frac{1}{5x^2})^{11}\) is
\(T_{r+1}=^{11}C_r(\frac{5}{2}x^3)^{11−r}(\frac{−1}{5x^2})^r\)
\(=^{11}C_r(\frac{5}{2})^{11−r}(\frac{−1}{5})^r x^{33−5r}\)
So, term independent from x in given expression
\(=^{−11}C_7(\frac{5}{2})^4(\frac{−1}{5})^7=\frac{11×10×9×8}{24}×\frac{1}{16×125 }\)
\(=\frac{33}{200}\)
Top Questions on binomial expansion formula
- Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
- JEE Advanced - 2025
- Mathematics
- binomial expansion formula
- Let the coefficients of three consecutive terms \( T_r \), \( T_{r+1} \), and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{12}\) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( \left( 4\sqrt{3} + 3\sqrt{4} \right)^{12} \). Then \( p + q \) is equal to:
- JEE Main - 2025
- Mathematics
- binomial expansion formula
- For some \( n \neq 10 \), let the coefficients of the 5th, 6th, and 7th terms in the binomial expansion of \( (1 + x)^{n+4} \) be in A.P. Then the largest coefficient in the expansion of \( (1 + x)^{n+4} \) is:
- JEE Main - 2025
- Mathematics
- binomial expansion formula
- Let \( \alpha, \beta, \gamma \) and \( \delta \) be the coefficients of \( x^7, x^5, x^3, x \) respectively in the expansion of \( (x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, \, x > 1 \). If \( \alpha u + \beta v = 18 \), \( \gamma u + \delta v = 20 \), then \( u + v \) equals:
- JEE Main - 2025
- Mathematics
- binomial expansion formula
- Let \( a_n \) denote the term independent of \( x \) in the expansion of \( \left[ x + \frac{\sin(1/n)}{x^2} \right]^{3n} \). Then \( \lim_{n \to \infty} \frac{(a_n) n!}{3^n n^n} \) equals:
- WBJEE - 2025
- Mathematics
- binomial expansion formula
Questions Asked in JEE Main exam
In the following \(p\text{–}V\) diagram, the equation of state along the curved path is given by \[ (V-2)^2 = 4ap, \] where \(a\) is a constant. The total work done in the closed path is:
- JEE Main - 2026
- Thermodynamics
Let \( ABC \) be a triangle. Consider four points \( p_1, p_2, p_3, p_4 \) on the side \( AB \), five points \( p_5, p_6, p_7, p_8, p_9 \) on the side \( BC \), and four points \( p_{10}, p_{11}, p_{12}, p_{13} \) on the side \( AC \). None of these points is a vertex of the triangle \( ABC \). Then the total number of pentagons that can be formed by taking all the vertices from the points \( p_1, p_2, \ldots, p_{13} \) is ___________.
- JEE Main - 2026
- Coordinate Geometry
- A voltage regulating circuit consisting of a Zener diode having breakdown voltage of $10\,\text{V}$ and maximum power dissipation of $0.4\,\text{W}$ is operated at $15\,\text{V}$. The approximate value of protective resistance in this circuit is ___ $\Omega$.
- JEE Main - 2026
- Semiconductor electronics: materials, devices and simple circuits
- Two point charges of \(1\,\text{nC}\) and \(2\,\text{nC}\) are placed at two corners of an equilateral triangle of side \(3\) cm. The work done in bringing a charge of \(3\,\text{nC}\) from infinity to the third corner of the triangle is ________ \(\mu\text{J}\). \[ \left(\frac{1}{4\pi\varepsilon_0}=9\times10^9\,\text{N m}^2\text{C}^{-2}\right) \]
- JEE Main - 2026
- Electrostatics
Consider the following two reactions A and B:
The numerical value of [molar mass of $x$ + molar mass of $y$] is ___.
- JEE Main - 2026
- Organic Chemistry
Concepts Used:
Binomial Expansion Formula
The binomial expansion formula involves binomial coefficients which are of the form
(n/k)(or) nCk and it is calculated using the formula, nCk =n! / [(n - k)! k!]. The binomial expansion formula is also known as the binomial theorem. Here are the binomial expansion formulas.
This binomial expansion formula gives the expansion of (x + y)n where 'n' is a natural number. The expansion of (x + y)n has (n + 1) terms. This formula says:
We have (x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn
General Term = Tr+1 = nCr xn-r . yr
- General Term in (1 + x)n is nCr xr
- In the binomial expansion of (x + y)n , the rth term from end is (n – r + 2)th .