Chapter 9

If \(C\) stands for \({ }^{\mathrm{n}} C_{r}\), then the sum of the series \(\frac{2\left(\frac{n}{2}\right) !\left(\frac{n}{2}\right)}{n !}\left[C_{0}^{2}-2 C_{1}^{2}+3 C_{2}^{2}-\right.\) \(\left.\ldots+(-1)^{n}(n+1) C_{n}^{2}\right]\), where \(n\) is an even positive integer, is (A) 0 (B) \((-1)^{w 2}(n+1)\) (C) \((-1)^{n / 2}(n+2)\) (D) \((-1)^{\mathrm{n}} n\)

If coefficient of \(x^{n}\) in \((1+x)^{101}\left(1-x+x^{2}\right)^{100}\) is nonzero, then \(n\) cannot be of the form (A) \(3 t+1\) (B) \(3 t\) (C) \(3 t+2\) (D) \(4 t+1\)

The digit at unit's place in the number \(17^{1995}+11^{1995}\) \(-7^{1995}\) is (A) 0 (B) 1 (C) 2 (D) 3

\(\left({ }^{\mathrm{m}} C_{0}+{ }^{\mathrm{m}} C_{1}-{ }^{\mathrm{m}} C_{2}-{ }^{\mathrm{m}} C_{3}\right)+\left({ }^{\mathrm{m}} C_{4}+{ }^{\mathrm{m}} C_{5}-{ }^{\mathrm{m}} C_{6}-{ }^{\text {" }} C_{7}\right)+\ldots\) \(=0\) if and only if for some positive integer \(k, m=\) (A) \(4 k\) (B) \(4 k+1\) (C) \(4 k-1\) (D) \(4 k+2\)

Let \(n(>1)\) be a positive integer. Then, largest integer \(m\) such that \(\left(n^{\mathrm{m}}+1\right)\) divides \(1+n+n^{2}+\ldots+n^{255}\) is (A) 128 (B) 63 (C) 64 (D) 32

The value of \(2\left({ }^{n} C_{0}\right)+\frac{3}{2}\left({ }^{-} C_{1}\right)+\frac{4}{3}\left({ }^{-} C_{2}\right)+\frac{5}{4}\left({ }^{-} C_{3}\right) \ldots\) is (A) \(\frac{2^{n}(1-n)-1}{n+1}\) (B) \(\frac{2^{n}(n+3)-1}{n+1}\) (C) \(\frac{2^{n}-1}{n+1}\) (D) \(\frac{2^{n}+2}{n-1}\)

If \(A={ }^{2 \mathrm{n}} \mathrm{C}_{0}{\underline{\phantom{xx}}}^{2 n} \mathrm{C}_{1}+{ }^{2 \mathrm{n}} \mathrm{C}_{1}{\underline{\phantom{xx}}}^{2 \mathrm{n}-1} \mathrm{C}_{1}+{ }^{2 \mathrm{n}} \mathrm{C}_{2}{\underline{\phantom{xx}}}^{2 \mathrm{n}-2} \mathrm{C}_{1}+\ldots\), then \(A\) is (A) 0 (B) \(2^{\mathrm{n}}\) (C) \(n 2^{2 n}\) (D) 1

The sum to \((n+1)\) terms of the series \(\frac{C_{0}}{2}-\frac{C_{1}}{3}+\frac{C_{2}}{4}-\frac{C_{3}}{5}+\ldots\) is (A) \(\frac{1}{n(n+1)}\) (B) \(\frac{1}{n+2}\) (C) \(\frac{1}{n+1}\) (D) none of these

Let \(R=(5 \sqrt{5}+11)^{2 \mathrm{n}+1}\) and \(f=R-[R]\) where [ ] denotes the greatest integer function. Then \(R f=\) (A) \(2^{2 n+1}\) (B) \(\mathrm{W} 2^{4 n+1}\) (C) \(4^{2 n+1}\) (D) none of these

Let \(n\) and \(k\) be positive integers such that \(n \geq \frac{k(k+1)}{2}\) The number of solutions \(\left(x_{1}, x_{2}, \ldots, x_{k}\right), x_{1} \geq 1, x_{2} \geq 2\), \(\ldots, x_{\mathrm{k}} \geq k\), all integers, satisfying \(x_{1}+x_{2}+\ldots+x_{\mathrm{k}}=n\), is (A) \({ }^{\mathrm{m}} \mathrm{C}_{\mathrm{k}-1}\) (B) \({ }^{\mathrm{m}} \mathrm{C}_{\mathrm{k}}\) (C) \({ }^{\mathrm{m}} \mathrm{C}_{\mathrm{k}+1}\) (D) none of these where \(m=\frac{1}{2}\left(2 n-k^{2}+k-2\right)\)

\({ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}+{ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{n}}+{ }^{\mathrm{n}+2} \mathrm{C}_{\mathrm{n}}+\ldots+{ }^{\mathrm{n}+\mathrm{k}} \mathrm{C}_{\mathrm{n}}=\) \((\text { A })^{n+k-1} C\) (B) \({ }^{n+k} C_{n+1}\) (C) \({ }^{\mathrm{n}+\mathrm{k}+{ }^{1}} C_{\mathrm{n}+1}\) (D) none of these

If \((1+x)^{15}=C_{0}+C_{1} x+C_{2} x^{2}+\ldots+C_{15} x^{15}\), then the value of \(C_{2}+2 C_{3}+3 C_{4}+\ldots+14 C_{15}\) is (A) 219923 (B) 16789 (C) 219982 (D) none of these

The coefficient of \(x^{30}\) in the expression \((1+x)^{1000}+2 x(1+x)^{999}+3 x^{2}(1+x)^{998}+\ldots+1001 x^{1000}\) is (A) \({ }^{1000} \mathrm{C}_{50}\) (B) \({ }^{1001} C_{s e}\) (C) \({ }^{1002} C_{50}\) (D) none of these

If \((1+x)^{\mathrm{m}}=C_{0}+C_{1} x+C_{2} x^{2}+\ldots+C_{\mathrm{n}} x^{\mathrm{n}}\), then for \(n\) even, \(C_{0}^{2}-C_{1}^{2}+C_{2}^{2}-\ldots+(-1)^{n} C_{n}^{2}\) is equal to (A) 0 (B) \((-1)^{n / 2 \cdot n} C_{n / 2}\) (C) \({ }^{n} C_{n / 2}\) (D) none of these

\(\sum_{k=0}^{n} \frac{{ }^{n} C_{k}}{(k+1)(k+2)}=\) (A) \(\frac{2^{n+1}-n-3}{(n+1)(n+2)}\) (B) \(\frac{2^{n+2}-n-3}{(n+1)(n+2)}\) (C) \(\frac{2^{n+2}-n+3}{(n+1)(n+2)}\) (D) none of these

For all \(n \in N\), the integer just above \((\sqrt{3}+1)^{2 n}\) is divisible by (A) \(2^{n+1}\) (B) \(2^{\mathrm{n}}+1\) (C) \(2^{\mathrm{n}+1}+1\) (D) none of these

If \(C_{0}, C_{1}, C_{2}, \ldots, C_{n}\) be the coefficients in the expansion of \((1+x)^{\text {n, then }}\) \(\frac{2^{2} \cdot C_{0}}{1 \cdot 2}+\frac{2^{3} \cdot C_{1}}{2 \cdot 3}+\ldots+\frac{2^{n+2} \cdot C_{n}}{(n+1)(n+2)}\) is equal to (A) \(\frac{3^{n+1}-2 n-5}{(n+1)(n+2)}\) (B) \(\frac{3^{n+2}-2 n-5}{(n+1)(n+2)}\) (C) \(\frac{3^{n+2}+2 n-5}{(n+1)(n+2)}\) (D) none of these

If \(a, b, c\) and \(d\) are any four consecutive coefficients of any binomial expansion, then \(\frac{a+b}{a}, \frac{b+c}{b}, \frac{c+d}{c}\) are (A) A.P. (B) G.P. (C) H.P. (D) none of these

The last two digits of the number \(3^{400}\) are (A) 38 (B) 27 (C) 01 (D) none of these

If \(n\) is an even positive integer and \(k=\frac{3 n}{2}\), then \(\sum_{r=1}^{k}(-3)^{r-1}{\underline{\phantom{xx}}}^{3 n} C_{2 r-1}=\) (A) \(\underline{1}\) (B) \(-1\) (C) 0 (D) none of these

The coefficient of \(x^{301}\) in the expansion of \((1+x)^{500}+x(1+x)^{494}+x^{2}(1+x)^{498}+\ldots .+x^{500}\) is \(\begin{array}{lll}\text { (A) }{\underline{\phantom{xx}}}^{501} C_{301} & \text { (B) } & { }^{500} C_{301}\end{array}\) (C) \({ }^{501} C_{300}\) (D) none of these

The fractional part of \(\frac{(\sqrt{6})^{2 n}}{5}, n \in N\) is equal to (A) \(\frac{1}{3}\) (B) \(\frac{1}{5}\) (C) \(\frac{1}{6}\) (D) none of these

Let \(n\) be an odd natural number greater than \(1 .\) Then, the number of zeros at the end of the sum \(99^{\mathrm{n}}+1\) is (A) 2 (B) 3 (C) 4 (D) none of these

\(\sum_{r=0}^{n} \frac{1}{(2 r) !(2 n-2 r) !}=\) (A) \(\frac{2^{2 n}}{(2 n) !}\) (B) \(\frac{2^{2 n-1}}{(2 n) !}\) (C) \(\frac{2^{2 n+1}}{(2 n) !}\) (D) none of these

The coefficient of \(x^{\mathrm{n}}\) in polynomial \(\left(x+{ }^{2 \mathrm{n}+1} C_{0}\right)\left(x+{ }^{2 \mathrm{n}+1} C_{1}\right)\left(x+{ }^{2 \mathrm{n}+1} C_{2}\right) \ldots .\left(x+{ }^{2 \mathrm{n}+1} C_{\mathrm{n}}\right)\) is (A) \(2^{2 n+1}\) (B) \(2^{2 n}\) (C) \(2^{2 n-1}\) (D) none of these

If 7 divides \(32^{3211}\), the remainder is (A) 2 (B) 4 (C) 8 (D) none of these

If the 4 th term in the expansion of \(\left(2+\frac{3}{8} x\right)^{10}\) has the maximum numerical value, then the range of values of \(x\) is (A) \(-2 \leq x \leq 2\) (B) \(-\frac{64}{21} \leq x \leq-2\) (C) \(2 \leq x \leq \frac{64}{21}\) (D) none of these

Three consecutive binomial coefficients can never be in (A) G.P. (B) H.P. (C) A.P. (D) A.G.P.

The greatest term in the expansion of \((1+x)^{10}\), when \(x=\frac{2}{3}\) is (A) \(210\left(\frac{2}{3}\right)^{4}\) (B) \(6300\left(\frac{2}{3}\right)^{3}\) (C) \(\left(\frac{2}{3}\right)^{5}\) (D) none of these

The numerically greatest term in the expansion of \((3-5 x)^{15}\), when \(x=\frac{1}{5}\) is (A) 4 th term (B) 5 th term (C) 6 th term (D) none of these

The greatest term in the expansion of \(\sqrt{3}\left(1+\frac{1}{\sqrt{3}}\right)^{20}\) is (A) \(\frac{25840}{9}\) (B) \(\frac{24840}{9}\) (C) \(\frac{26840}{9}\) (D) none of these

If 4th term in the expansion of \(\left(2+\frac{3 x}{8}\right)^{10}\) has the greatest numerical value, then \(x\) belongs to (A) \((-\infty,-2] \cup[2, \infty)\) (B) \(\left(-\frac{64}{21}, \frac{64}{21}\right)\) (C) \(\left(-\frac{64}{21},-2\right) \cup\left(2, \frac{64}{21}\right)\) (D) none of these

Let \(R=(5+2 \sqrt{6})^{\mathrm{n}}\) and \(f=\) fractional part of \(R\), then \(R(1-f)=\) (A) 1 (B) \(-1\) (C) 0 (D) none of these

\(\left[(3+\sqrt{5})^{2 n}\right]+1\), where \([x]\) denotes the integral part of 1 \(x\), is divisible by (A) \(2^{\mathrm{w}-1}\) (B) \(2^{\mathrm{n}}\) (C) \(2^{\mathrm{n}+1}\) (D) none of these

If \(n \in N\) such that \((7+4 \sqrt{3})^{\mathrm{n}}=I+f\), where \(I \in N\) and \(0

\begin{tabular}{ll} \hline Column-I & Column-II \\ \hline (A) The number of integral terms in the & 1\. 210 \\ expansion of \(\left(5^{1 / 2}+7^{18}\right)^{1028}\) is & \\ (B) The coefficient of the term & 2\. 2520 \\ independent of \(x\) in the expansion of \\ \(\qquad\left(\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right)^{10}\) is \\ (C) The coefficient of \(x^{2} y^{3} z^{5}\) in the \\ expansion \((x+y+z)^{10}\) is \\ (D) The least remainder when \(17^{30}\) is & 4\. 4 \\ divided by 5 is \end{tabular}

Assertion: If \(P_{n}\) denotes the product of the binomial coefficients in the expansion of $$ (1+x)^{n}, \text { then } \frac{P_{n+1}}{P_{n}} \text { equals } \frac{(n+1)^{n}}{n !} $$ Reason: \({ }^{n+1} C_{r+1}=\frac{n+1}{r+1}{\underline{\phantom{xx}}}^{n} C_{r}\)

Assertion: The value of \(\frac{{ }^{11} C_{0}}{1}+\frac{{ }^{11} C_{1}}{2}+\frac{{ }^{11} C_{2}}{3}+\) \(\ldots+\frac{{ }^{11} C_{11}}{12}\) is \(\frac{1}{12}\left(2^{12}-1\right)\) Reason: For \(0 \leq k \leq n,{ }^{n} C_{k}=\frac{n}{k} \cdot{ }^{n-1} C_{k-1}\)

The coefficient of \(x^{5}\) in \(\left(1+2 x+3 x^{2}+\ldots\right)^{-3 / 2}\) is: (A) 21 (B) 25 [2002] (C) 26 (D) none of these

The number of integral terms in the expansion of \((\sqrt{3}+\sqrt[8]{5})^{256}\) is (A) 32 (B) 33 (C) 34 (D) 35

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 [2004] (A) \(-\frac{5}{3}\) (B) \(\frac{3}{5}\) (C) \(-\frac{3}{10}\) (D) \(\frac{10}{3}\)

If the coefficients of \(\mathrm{rth},(\mathrm{r}+1)\) th and \((\mathrm{r}+2)\) th terms in the binomial expansion of \((1+y) m\) are in A. P., then \(m\) and \(\mathrm{r}\) satisfy the equation \([2005]\) (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) \({ }^{56} \mathrm{C}_{3}\) (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) \(a-b=1\) (B) \(a+b=1\) (B) \(\frac{a}{b}=1\) (D) \(a b=1\)

If \(x\) is so small that \(x^{3}\) and higher powers of \(x\) may be \(\begin{aligned}&\text { neglected, then } \\\&\text { mated as } & (1+x)^{3 / 2}-\left(1+\frac{1}{2} x\right)^{3} \\\&\text { may be approxi- } \\\&(1-x)^{1 / 2} & {[2005]}\end{aligned}\) (A) \(1-\frac{3}{8} x^{2}\) (B) \(3 x+\frac{3}{8} x^{2}\) (C) \(-\frac{3}{8} x^{2}\) (D) \(\frac{x}{2}-\frac{3}{8} x^{2}\)

If the expansion in powers of \(x\) of the function \(\frac{1}{(1-a x)(1-b x)}\) is \(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\ldots\), then \(\mathrm{a}_{n}\) is (A) \(\frac{b^{n}-a^{n}}{b-a}\) (B) \(\frac{a^{n}-b^{n}}{b-a}\) [2006] (a) \(\frac{a^{n+1}-b^{n+1}}{b-a}\) (D) \(\frac{b^{n+1}-a^{n+1}}{b-a}\)

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) \((35,20)\) \([2006]\) (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 [2007] (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) \(^{2}{\underline{\phantom{xx}}}^{0} \mathrm{C}_{10}\)

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 \([2008]\) (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) \(\frac{4}{\log _{10}^{4}-\log _{10}^{3}}\)