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(C₁ / C₀) + 2 (C₂ / C₁) + 3 (C₃ / C₂) + …….. + n (C_n / C_n-1) =

1) [n (n – 1)] / 2 2) [n (n + 1)] / 2 3) [(n + 1) (n +... View Article

If n is an integer with 0 ≤ n ≤ 11, then the minimum value of n!(11 – 1) ! is attained, when a value of n equals

1) 11 2) 5 3) 7 4) 9 Solution: (2) 5 \(\begin{array}{l}\begin{array}{l} \text { Given, } n !(11-n) !\\ \text... View Article

^mC_r+1 + ∑_k=mⁿ ^kC_r =

1) nCr+1 2) n+1Cr+1 3) nCr 4) None of these Solution: (2) n+1Cr+1 \(\begin{array}{l}\begin{array}{l} \text { Given, }\\ M_{C_{r+1}}+\sum_{k=M}^{n} k_{C_{r}}\\... View Article

If S = given expression, then S =

1) [2n+1 – 1] / (n + 1) 2) [3n+1 – 1] / (n + 1) 3) [3n – 1]... View Article

The value of (ⁿC₁)² + (ⁿC₂)² + (ⁿC₃)² +… + (ⁿC_n)² is

1) (2nCn)2 2) 2nCn 3) 2nCn + 1 4) 2nCn – 1 Solution: (4) 2nCn – 1 \(\begin{array}{l}(1+x)^{n}=n_{C_{0}}+n_{C_{1}} x+n_{C_{2}} x^{2}+\cdots+n_{C_{n}}... View Article

The term independent of x in the expansion of [x³ – (2 / x²)]¹⁵ is

1) T7 2) T8 3) T9 4) T10 Solution: (4) T10 \(\begin{array}{l}\begin{array}{l} \begin{array}{l} \text { General term } T_{r+1} =15_{c_{r}}\left(x^{3}\right)^{15-r}\left(\frac{2}{x^{2}}\right)^{r}... View Article

The middle term in the expansion of [x – (1 / x)]¹⁸ is

1) 18C9 2) –18C9 3) 18C10 4) –18C10 Solution: (2) –18C9 The middle term of the expansion [x – (1... View Article

The sum of coefficients in the expansion of [(1 / x) + 2x]ⁿ is 6561. The coefficient of term independent of x is

1) 16 8C4 2) 8C4 3) 8C5 4) None of these Solution: (1) 16 8C4 \(\begin{array}{l}\text \ Given \ sum... View Article

The term independent of x in the expansion of [(2√x / 5) – (1 / 2x √x)]¹¹ is

1) 5th term 2) 6th term 3) 11th term 4) No term Solution: (4) No term \(\begin{array}{l}\begin{array}{l} \text { General... View Article

The largest coefficient in the expansion of (1 + x)²⁴ is

1) 24C24 2) 24C13 3) 24C12 4) 24C11 Solution: (3) 24C12 The binomial coefficient of the middle term is the... View Article

The term independent of x in the expansion of [x – (1 / x)]⁴ + [x + (1 / x)]³ is

1) – 3 2) 0 3) 3 4) 1 Solution: (2) 0 Rewrite the given expansion [x – (1 /... View Article

If the expansion of [(3√x / 7) – (5 / 2x√x)]¹³ⁿ contains a term independent of x in 14th term, then n should be

1) 10 2) 5 3) 6 4) 4 5) 11 Solution: (4) 4 \(\begin{array}{l}\\T_{13+1}=13 n_{C_{13}}\left(\frac{3 \sqrt{x}}{7}\right)^{13 n-13}\left(\frac{-5}{2 x \sqrt{x}}\right)^{13} \\... View Article

The coefficient of the middle term in the binomial expansion in powers of x in the expansion of (1 + αx)⁴ and (1 – αx)⁶ is the same if α equals

1) -(5/3) 2) 10/3 3) -(3/10) 4) 3/5 Solution: (3) -(3/10) \(\begin{array}{l}\\\text \ Coefficient \ of \ the \ middle... View Article

The greatest term in the expansion of (1 + 3x)⁵⁴, where x = 1/3 is

1) T28 2) T25 3) T26 4) T24 Solution: (1) T28 \(\begin{array}{l}\\ \text \ We \ know \ \frac{n-r+1}{r}\left|\frac{a}{x}\right| \geq... View Article

The greatest term in the expansion of √3 [1 + (1 / √3)]²⁰ is

1) 26840/9 2) 24840/9 3) 25840/9 4) None of these Solution: (3) 25840/9 \(\begin{array}{l}\text \ We \ know \ greatest... View Article

If in the expansion [(3x) – (2 / x²)]¹⁵, rth term is independent of x, then the value of r is

1) 6 2) 10 3) 9 4) 12 Solution: (1) 6 \(\begin{array}{l}\begin{array}{l} \text { Given } T_{r}=15_{C_{r-1}}\left(3 x 1^{15-r+1}\left(\frac{-2}{x^{2}}\right)^{r-1}\right.\\ =15_{C_{r-1}}... View Article

For |x| < 1, the constant term in the expansion of 1 / [(x - 1)² (x – 2)] is

1) 2 2) 1 3) 0 4) -(1/2) Solution: (4) -(1/2) \(\begin{array}{l}\begin{array}{l} \frac{1}{(x-1)^{2}(x-2)}=\frac{1}{-2(1-x)^{2}(1-x / 2)}\\ =-\frac{1}{2}\left[(1-x)^{-2}\left(1-\frac{x}{2}\right)^{-1}\right]\\ \text { [Note: The... View Article

The coefficient of the term independent of x in the expansion of {[(x + 1) / (x^2/3 – x^1/3 + 1)] – [(x – 1) / (x – x^1/2)]}¹⁰ is

1) 210 2) 105 3) 70 4) 112 Solution: (1) 210 \(\begin{array}{l}\begin{array}{l} \text { Rewrite the given expansionas }\\ \left[\frac{(\sqrt[3]{x})^{3}+1^{3}}{x^{\frac{2}{3}}-x^{\frac{1}{3}+1}}-\frac{(\sqrt{x})^{2}-1^{2}}{\sqrt{x}(\sqrt{x}-1)}\right]^{10}=\frac{\left(\sqrt[3]{x}+1-1-\frac{1}{\sqrt{x}}\right)^{10}}{1}\\... View Article

The coefficient of the middle term in the expansion of (x + 2y)⁶ is

1) 6C3 2) 8(6C3) 3) 8(6C5) 4) 6C4 5) 8(6C4) Solution: (2) 8(6C3) The middle term of the expansion is... View Article

In the expansion (1 – 3x + 3x² – x³)²ⁿ, the middle term is

1) (n + 1)th term 2) (2n + 1)th term 3) (3n + 1)th term 4) None of these Solution:... View Article

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