Number of integral terms in the expansion of \( \left( \sqrt{7} z + \frac{1}{6 \sqrt{z}} \right)^{824} \) is equal to ______.
Correct Answer: 138
Solution and Explanation
Solution: The general term in the expansion is:
\[ t_{r+1} = \binom{824}{r} \left( \sqrt{7} z \right)^{824 - r} \left( \frac{1}{6 \sqrt{z}} \right)^r \]
which simplifies to \( z^{\frac{824 - r}{7} - \frac{r}{6}} \).
For an integral power, \( r \) must be a multiple of 6.
Thus, \( r = 0, 6, 12, \dots, 822 \), giving 138 integral terms.
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
- 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 \( a_n \) 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
- Find the term independent of $ x $ in $ \left( 2x - \frac{5}{x^2} \right)^6 $
- KEAM - 2025
- Mathematics
- binomial expansion formula
- Binomial Expansion Series $ \left( \frac{(1 + x)}{(n + 1)} \right)' = n_0 x + n_1 \frac{x^2}{2} + n_2 \frac{x^3}{3} + \cdots + n_n \frac{x^n}{n+1} $
- MHT CET - 2025
- Mathematics
- binomial expansion formula
Questions Asked in JEE Main exam
Total number of nucleophiles from the following is: \(\text{NH}_3, PhSH, (H_3C_2S)_2, H_2C = CH_2, OH−, H_3O+, (CH_3)_2CO, NCH_3\)
- JEE Main - 2025
- Nucleophilic and electrophilic substitution reactions (both aromatic and aliphatic)
- The elemental composition of a compound is 54.2%C, 9.2%H, and 36.6%O. If the molar mass of the compound is 132 g/mol, the molecular formula of the compound is:
- JEE Main - 2025
- Thermodynamics
- Let \( a_1, a_2, a_3, \dots \) be a G.P. of increasing positive terms. If \( a_1 a_5 = 28 \) and \( a_2 + a_4 = 29 \), then the value of \( a_6 \) is equal to:
- JEE Main - 2025
- Sequences and Series
- Let the triangle PQR be the image of the triangle with vertices \( (1, 3), (3, 1) \) and \( (2, 4) \) in the line \( x + 2y = 2 \). If the centroid of \( \Delta PQR \) is the point \( (\alpha, \beta) \), then \( 15(\alpha - \beta) \) is equal to :
- JEE Main - 2025
- Coordinate Geometry
- A force \( \mathbf{F} = 2\hat{i} + b\hat{j} + \hat{k} \) is applied on a particle and it undergoes a displacement \( \mathbf{r} = \hat{i} - 2\hat{j} - \hat{k} \). What will be the value of \( b \), if the work done on the particle is zero?
- JEE Main - 2025
- Work and Energy
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 .