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The total number of terms in the expansion of (x + y)¹⁰⁰ + (x – y)¹⁰⁰ after simplification is

1) 51 2) 202 3) 100 4) 50 Solution: (1) 51 If x and y are real numbers, then for... View Article

Sum of infinite series 1 + (2 / 3) (1 / 2) + (2 / 3) (5 / 6) (1 / 2²) + …… ∞ is

1) 21/3 2) 41/3 3) 81/3 4) 21/5 Solution: (2) 41/3 \(\begin{array}{l}(1+x)^{n}=1+n x+\frac{n(n-1)}{2 !} x^{2}+\frac{n(n-1)(n-2)}{3 !} x^{3} \\ +\cdots \frac{n(n-1)(n-2)... View Article

7⁹ + 9⁷ is divided by

1) 128 2) 24 3) 64 4) 72 Solution: (3) 64 \(\begin{array}{l}\begin{array}{l} 7^{9}+9^{7}=(8-1)^{9}+(8+1)^{7}\\ =\left[{ }^{9} \mathbf{C}_{0} \mathbf{8}^{9}-{ }^{9} \mathbf{C}_{1} \mathbf{8}^{8}+{... View Article

The remainder left out when 8²ⁿ – (62)²ⁿ⁺¹ is divided by 9, is

1) 0 2) 2 3) 7 4) 8 Solution: (2) 2 \(\begin{array}{l}\begin{array}{l} \text { Given expresssion is written as }\left(8^{2}\right)^{n}-(62)^{2... View Article

If in the expansion of (a – 2b)ⁿ, the same of 5th and 6th terms is zero, then the value of a/b is

1) (n – 4)/ 5 2) 2(n – 4)/5 3) 5/(n – 4) 4) 5/2(n – 4) Solution: (2) 2(n... View Article

The coefficient of pⁿqⁿ in the expansion of [(1 + p) (1 + q) (p + q)]ⁿ is

1) ∑k=0n [C (n, k)]2 2) ∑k=0n [C (n, k + 2)]2 3) ∑k=0n [C (n, k + 3)]2 4)... View Article

5th term from the end in the expansion of [(x³ / 2) – (2 / x²)]¹² is

1) – 7920 x– 4 2) 7920 x4 3) 7920 x– 4 4) – 7920 x4 Solution: (3) 7920 x–... View Article

In the expansion of (x³ – 1 / x²)ⁿ, n ∈ N, if the sum of the coefficients of x⁵ and x¹⁰ is 0, then n is equal to

1) 25 2) 20 3) 15 4) None of these Solution: (3) 15 \(\begin{array}{l}\left(x^{3}-\frac{1}{x^{2}}\right)^{n} \text \ is \ T_{r+1}=n_{c_{r}}\left(x^{3}\right)^{n-r}\left(-\frac{1}{x^{2}}\right)^{r}\\ \Rightarrow... View Article

If A and B are coefficients of xⁿ in the expansion of (1 + x)²ⁿ and (1 + x)^{2n – 1} respectively, then A/B is equal

1) 4 2) 2 3) 9 4) 6 Solution: (2) 2 \(\begin{array}{l}\begin{array}{l} \text { Since } A=2 n_{c_{n}} \text {... View Article

If (1 + ax)ⁿ = 1 + 6x + (27 / 2) x² + aⁿxⁿ, then the values of a and n are respectively

1) 2 , 3 2) 3 , 2 3) 3/2 , 4 4) 1 , 6 5) 3/2 , 6... View Article

If rth and (r + 1)th terms in the expansion of (p + q)ⁿ are equal, then the value of (n + 1)q / r(p+q) is

1) 0 2) 1 3) 1/4 4) 1/2 Solution: (2) 1 \(\begin{array}{l}\begin{array}{l} \text { Given } \quad T_{r}=T_{r-1} \\ \text... View Article

The expression [x + (x³ – 1)^½]⁵ + [x – (x³ – 1)^½]⁵ is a polynomial of degree

1) 5 2) 6 3) 7 4) 8 Solution: (3) 7 \(\begin{array}{l}(a+b)^{5}+(a-b)^{5}\\ ={ }^{5} C_{0} a^{5}+{ }^{5} C_{1} a^{4} b+{... View Article

If e^x / 1 -x = B₀ + B₁x + B₂x² +….+ B_nxⁿ + …, then B_n – B_n-1 equals

1) 1/n! 2) 1/(n -1)! 3) 1/n! – (1/(n -1)!) 4) 1 Solution: (1) 1/n! \(\begin{array}{l}e^{x}=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\cdots \infty\\ (1-x)^{-1}=1+x+x^{2}+\cdots... View Article

The coefficient of x⁴ in [(x / 2) – (3 / x²)]¹⁰ is

1) 405 / 256 2) 504 / 259 3) 450 / 263 4) None of these Solution: (1) 405 /... View Article

If the 4th term in the expansion of [ax + (1 / x)]ⁿ is 5 / 2, for all x belongs to R, then the values of a and n are

1) 1 / 2, 6 2) 6, 1/2 3) 2, 6 4) None of these Solution: (1) 1 / 2,... View Article

If the 6^th term in the expansion of the binomial [latex]left[sqrt{2^{log left(10-3^{x}right)}}+sqrt[5]{2^{(x-2) log 3}}right]^{m}[/latex] = 21 and known that the binomial coefficient of 2nd, 3rd and 4th terms in the expansion represent respectively the 1st, 3rd and 5th of an AP, then x =

1) 0 2) 1 3) – 2 4) 3 Solution: (1) 0 \(\begin{array}{l}2 M_{c_{2}}=M_{c_{1}}+M_{c_{3}}\\ \Rightarrow 2 \frac{\mathrm{M}(\mathrm{M}-1)}{1.2}=M+\frac{M(M-1)(M-2)}{1.2 .3}\\ \Rightarrow \mathrm{M}(\mathrm{M}-1)-\frac{\mathrm{M}(\mathrm{M}-1)(\mathrm{M}-2)}{6}=M\\... View Article

The coefficient of three consecutive of (1 + x)ⁿ⁺⁵ are in the ratio 5 : 10 : 14. Then, n is equal to (answer lies between 0 to 9 )

1) 1 2) 3 3) 4 4) 6 Solution: (4) 6 \(\begin{array}{l}T_{r-1}: T_{r}: T_{r+1}=5: 10: 14\\ \frac{T_{r+1}}{T_{r}}=\frac{n+5-r+1}{r}, \frac{T_{r}}{T_{r-1}}=\frac{n+5-r+2}{r-1}\\ \Rightarrow \frac{T_{r+1}}{T_{r}}=\frac{14}{10},... View Article

If (a + bx)^-3 = 1 / 27 + (1 / 3) x + …, then the ordered pair (a, b) equals

1) (3, – 27) 2) (1, 1/3) 3) (3, 9) 4) (3, – 9) Solution: (4) (3, – 9) \(\begin{array}{l}(a+b... View Article

The coefficient of x³ in the infinite series expansion of 2/(1- x) (2 – x), for |x| < 1 is

1) – (1/16) 2) 15/8 3) – (1/8) 4) 15/16 Solution: (2) 15/8 \(\begin{array}{l}2(1-x)^{-1}(2-x)^{-1} =\frac{2}{2}(1-x)^{-1}\left(1-\frac{x}{2}\right)^{-1} \\ =(1-x)^{-1}\left(1-\frac{x}{2}\right)^{-1} \\ \Rightarrow(1-x)^{-1}\left(1-\frac{x}{2}\right)^{-1} =\left[1+x+x^{2}+\cdots\right]\left[1+\frac{x}{2}+\frac{x^{2}}{4}+\frac{x^{3}}{8}+\cdots\right]... View Article

If the coefficient of x⁸ in (ax² + 1 / bx)¹³ is equal to the coefficient of x^-8 in (ax – 1 / bx²)¹³, then a and b will satisfy the relation

1) ab + 1 = 0 2) ab = 1 3) a = 1 – b 4) a + b... View Article

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