Chapter 9

The coefficient of the middle term in the binomial expansion in powers of \(x\) of \((1+\alpha x)^{4}\) and of \((1-\alpha x)\) is the same if \(\alpha\) equals (A) \(-\frac{5}{3}\) (B) \(\frac{3}{5}\) (C) \(-\frac{3}{10}\) (D) \(\frac{10}{3}\)

The coefficient of \(x^{\mathrm{n}}\) in expansion of \((1+x)(1-x)^{\mathrm{n}}\) is (A) \(-\frac{5}{3}\) (B) \(\frac{3}{5}\)(C) \((-1)^{n-1}(n-1)^{2}\) (D) \((-1)^{n-1} n\)

If the coefficients of rth, (r+1)th and (r +2)th terms in the binomial expansion of \((1+y) m\) are in \(A . P .\) then \(m\) and \(r\) satisfy the equation (A) \(m^{2}-m(4 r-1)+4 r^{2}-2=0\) (B) \(m^{2}-m(4 r+1)+4 r^{2}+2=0\) (C) \(m^{2}-m(4 r+1)+4 r^{2}-2=0\) (D) \(m^{2}-m(4 r-1)+4 r^{2}+2=0\)

The value of \({ }^{50} C_{4}+\sum_{r-1}^{6}{\underline{\phantom{xx}}}^{56-r} C_{3}\) is (A) \({ }^{55} C_{4}\) (B) \({ }^{55} C_{3}\) (C) \({ }^{56} C_{3}\) (D) \({ }^{56} C_{4}\)

If the coefficient of \(x^{7}\) in \(\left[a x^{2}+\left(\frac{1}{b x}\right)\right]^{11}\) equals the coefficient of \(x^{-7}\) in \(\left[a x^{2}-\left(\frac{1}{b x}\right)\right]^{11}\), then a and \(b\) satisfy the relation (A) \(\mathrm{a}-\mathrm{b}=1\) (B) \(\mathrm{a}+\mathrm{b}=1\) (B) \(\frac{a}{b}=1\) (D) \(\mathrm{ab}=1\)

If \(x\) is so small that \(x^{3}\) and higher powers of \(x\) may be neglected, then \(\frac{(1+x)^{3 / 2}-\left(1+\frac{1}{2} x\right)^{3}}{(1-x)^{1 / 2}}\) may be approximated as (A) \(1-\frac{3}{8} x^{2}\) (B) \(3 x+\frac{3}{8} x^{2}\) (C) \(-\frac{3}{8} x^{2}\) (D) \(\frac{x}{2}-\frac{3}{8} x^{2}\)

For natural numbers \(m, n\) if \((1-y)^{m}(1+y)^{n}=1+a_{1} y\) \(+a_{2} y^{2}+\ldots\), and \(a_{1}=a_{2}=10\), then \((m, n)\) is (A) \((20,45)\) (B) \((35,20)\) (C) \((45,35)\) (D) \((35,45)\)

In the binomial expansion of \((a-b)^{n}, n \geq 5\), the sum of \(5^{\text {th }}\) and \(6^{\text {th }}\) terms is zero, then \(\frac{a}{b}\) equals (A) \(\frac{5}{n-4}\) (B) \(\frac{6}{n-5}\) (C) \(\frac{n-5}{6}\) (D) \(\frac{n-4}{5}\)

The sum of the series \({ }^{20} \mathrm{C}_{0}-{ }^{20} \mathrm{C}_{1}+{ }^{20} \mathrm{C}_{2}-{ }^{20} \mathrm{C}_{3}+\ldots-\ldots+{ }^{20} \mathrm{C}_{10}\) is (A) \(-{ }^{20} \mathrm{C}_{10}\) (B) \(\frac{1}{2}{\underline{\phantom{xx}}}^{20} C_{10}\) (C) 0 (D) \({ }^{2}{\underline{\phantom{xx}}}^{\circ} \mathrm{C}_{10}\)

In a binomial distribution \(\mathrm{B}\left(n, p=\frac{1}{4}\right)\), if the probability of at least one success is greater than or equal to \(\frac{9}{10}\), then \(\mathrm{n}\) is greater than (A) \(\frac{1}{\log _{10}^{4}-\log _{10}^{3}}\) (B) \(\frac{1}{\log _{10}^{4}+\log _{10}^{3}}\) (C) \(\frac{9}{\log _{10}^{4}-\log _{10}^{3}}\) (D) \(\frac{4}{\log _{10}^{4}-\log _{10}^{3}}\)

The remainder left out when \(8^{2 n}-(62)^{2 n+1}\) is divided by 9 is (A) 0 (B) 2 (C) 7 (D) 8

The coefficient of \(x^{7}\) in the expansion of the expression \(\left(1-x-x^{2}+x^{3}\right)^{6}\) is (A) \(-132\) (B) \(-144\) (c) 132 (D) 144

If \(n\) is a natural number, then \((\sqrt{3}+1)^{2 n}-(\sqrt{3}-1)^{2 n}\) is (1) an irrational number (2) an odd positive integer (3) an even positive integer (4) a rational number other than positive integers

If \(x=-1\) and \(x=2\) are extreme points of \(f(x)=\alpha \log |x|+\beta x^{2}+x\), then (A) \(\alpha=-6, \beta=\frac{1}{2}\) (B) \(\alpha=-6, \beta=-\frac{1}{2}\) (C) \(\alpha=2, \beta=-\frac{1}{2}\) (D) \(\alpha=2, \beta=\frac{1}{2}\)

If the coefficients of \(x^{3}\) and \(x^{4}\) in the expansion of \(\left(1+a x+b x^{2}\right)(1-2 x)^{18}\), in powers of \(x\), are both zero, then \((a, b)\) is equal to (A) \(\left(16, \frac{251}{3}\right)\) (B) \(\left(14, \frac{251}{3}\right)\) (C) \(\left(14, \frac{272}{3}\right)\) (D) \(\left(16, \frac{272}{3}\right)\)

If X \(=\left\\{4^{n}-3 n-1: n \in N\right\\}\) and \(Y=\\{9(n-1): n \in N\\}\), where \(\mathrm{N}\) is the set of natural numbers, then the set \(X \cup Y\) is equal to (A) \(N\) (B) \(Y-X\) (C) \(X\) (D) \(Y\)

The sum of the coefficients of integral powers of \(x\) in the binomial expansion of \((1-2 \sqrt{x})^{50}\) is: (A) \(\frac{1}{2}\left(3^{50}\right)\) (B) \(\frac{1}{2}\left(3^{50}-1\right)\) (C) \(\frac{1}{2}\left(2^{50}+1\right)\) (D) \(\frac{1}{2}\left(3^{50}+1\right)\)