Chapter 9

If \(\\{x\\}\) denotes the fractional part of \(x\), then \(\left\\{\frac{2^{2003}}{17}\right\\}\) is (A) \(\frac{2}{17}\) (B) \(\frac{4}{17}\) (C) \(\frac{8}{17}\) (D) \(\frac{16}{17}\)

If \([x]\) denotes the greatest integer less than or equal to \(x\), then \(\left[(6 \sqrt{6}+14)^{2 n+1}\right]\) (A) is an even integer (B) is an odd integer (C) depends on \(n\) (D) none of these

If \(C_{r}\) stands for \({ }^{n} C_{r}\), 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) \((-1)^{n / 2}(n+1)\) (C) \((-1)^{n / 2}(n+2)\) (D) \((-1)^{\mathrm{n}} n\)

The sum of the series \(1+\frac{1}{3^{2}}+\frac{1.4 .1}{1.2 .3^{4}}+\frac{1.4 .7}{1.2 .3} \frac{1}{3^{6}}+\ldots\) is (A) \(\sqrt{\frac{3}{2}}\) (B) \(\left(\frac{3}{2}\right)^{\frac{1}{3}}\) (C) \(\sqrt{\frac{1}{3}}\) (D) \(\left(\frac{1}{3}\right)^{\frac{1}{3}}\)

If coefficient of \(x^{\mathrm{n}}\) 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

The coefficient of \(x^{\mathrm{n}}\) in the polynomial \(\left(x+{ }^{\mathrm{n}} C_{0}\right)(x+3\) \(\left.{ }^{\mathrm{n}} C_{1}\right)\left(x+5{ }^{\mathrm{n}} C_{2}\right) \ldots\left(x+(2 n+1){ }^{\mathrm{n}} C_{\mathrm{n}}\right)\) is (A) \(n .2^{\mathrm{n}}\) (B) \(n \cdot 2^{n+1}\) (C) \((n+1) \cdot 2^{\mathrm{n}}\) (D) \(n \cdot 2^{\mathrm{n}-1}\)

If \(n\) is an even integer and \(a, b, c\) are distinct, the number of distinct terms in the expansion of \((a+b+c)^{\mathrm{n}}\) \(+(a+b-c)^{\mathrm{n}}\) is (A) \(\left(\frac{n}{2}\right)^{2}\) (B) \(\left(\frac{n+1}{2}\right)^{2}\) (C) \(\left(\frac{n+2}{2}\right)^{2}\) (D) \(\left(\frac{n+3}{2}\right)^{2}\)

\(\left({ }^{(\mathrm{m}} C_{0}+{ }^{m} C_{1}-{ }^{m} C_{2}-{ }^{m} C_{3}\right)+\left({ }^{\mathrm{m}} C_{4}+{ }^{m} C_{5}-{ }^{m} C_{6}-{ }^{m} C_{7}\right)+\ldots\) \(=0\) 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

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

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) \(\mathrm{W} 2^{4 \mathrm{n}+1}\) (C) \(4^{2 \mathrm{n}+1}\) (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 \(m=\frac{1}{2}\left(2 n-k^{2}+k-2\right)\)

\(\sum_{r=0}^{n}{\underline{\phantom{xx}}}^{n} \mathrm{C}_{r} \sin r x \cos (n-r) x=\) (A) \(2^{n-1} \sin (n-1) x\) (B) \(2^{\mathrm{n}} \sin n x\) (C) \(2^{\mathrm{n}-1} \sin n x\) (D) none of these

If \(S_{\mathrm{n}}=1+q+q^{2}+q^{3}+\ldots+q^{\mathrm{n}}\) and \(S_{\mathrm{n}}^{\prime}=1+\left(\frac{q+1}{2}\right)+\left(\frac{q+1}{2}\right)^{2}+\ldots+\left(\frac{q+1}{2}\right)^{n}, q \neq 1\), then \({ }^{\mathrm{n}+1} C_{1}+{ }^{\mathrm{n}+1} C_{2} \cdot S_{1}+{ }^{\mathrm{n}+1} C_{3} \cdot S_{2}+\ldots+{ }^{\mathrm{n}+1} C_{\mathrm{n}+1} \cdot S_{\mathrm{n}}=\) (A) \(2^{n-1} \cdot S_{n}^{\prime}\) (B) \(2^{\mathrm{n}} \cdot S_{\mathrm{n}}\) (C) \(2^{\mathrm{n}+1} \cdot S_{\mathrm{n}}^{\mathrm{n}}\) (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

If \(a_{0}, a_{1}, a_{2}, \ldots, a_{2 \mathrm{n}}\) be the coefficients in the expansion of \(\left(1+x+x^{2}\right)^{\mathrm{n}}\) in ascending powers of \(x\), then \(a_{0}^{2}-a_{1}^{2}+a_{2}^{2}-a_{3}^{2}+\ldots-a_{2 n-1}^{2}+a_{2 n}^{2}=\) (A) \(a_{2 \mathrm{n}}\) (B) \(a_{\mathrm{n}}\) (C) \(a_{0}\) (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) \({ }^{1001} C_{50}\) (C) \({ }^{1002} C_{50}\) (D) none of these

If \((1+x)^{n}=C_{0}+C_{1} x+C_{2} x^{2}+\ldots+C_{n} x^{n}\), 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) \((-1)^{n / 2 n} C_{n / 2}\) (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) \(2^{\mathrm{n}+1}\) (B) \(2^{\mathrm{n}}+1\) (C) \(2^{\mathrm{n}+1}+1\) (D) none of these

If \(C_{0}, C_{1}, C_{2}, \ldots, C_{\mathrm{n}}\) be the coefficients in the expansion of \((1+x)^{\mathrm{n}}\), 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

\({ }^{\mathrm{m}} C_{\mathrm{r}}+{ }^{\mathrm{m}} C_{\mathrm{r}-1} \cdot{ }^{\mathrm{n}} C_{1}+{ }^{\mathrm{m}} C_{\mathrm{r}-2} \cdot{ }^{\mathrm{n}} C_{2}+\ldots+{ }^{\mathrm{m}} C_{1} \cdot{ }^{\mathrm{n}} C_{\mathrm{r}-1}+{ }^{\mathrm{n}} C_{\mathrm{r}}=\) (A) \({ }^{\mathrm{m}+\mathrm{n}} C_{r-1}\) (B) \({ }^{\mathrm{m}+\mathrm{n}} \mathrm{C}_{\mathrm{r}}\) (C) \({ }^{\mathrm{m}+\mathrm{n}} C_{\mathrm{r}+1}\) (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) 1 (B) \(-1\) (C) 0 (D) none of these

The coefficient of \(x^{301}\) in the expansion of \((1+x)^{500}+x(1+x)^{499}+x^{2}(1+x)^{498}+\ldots .+x^{300}\) is \((1+x)^{500}+x(1+x)^{499}+x^{2}(1+x)^{498}+\ldots .+x^{500}\) is (A) \({ }^{501} C_{301}\) (B) \({ }^{500} C_{301}\) (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

The number of irrational terms in the expansion of \((\sqrt[8]{5}+\sqrt[6]{2})^{100}\) is (A) 96 (B) 97 (C) 98 (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 \(\left(x+{ }^{2 n+1} C_{0}\right)\left(x+{ }^{2 n+1} C_{1}\right)\left(x+{ }^{2 n+1} C_{2}\right) \ldots .\left(x+{ }^{2 n+1} C_{n}\right)\) is (A) \(2^{2 n+1}\) (B) \(2^{2 \mathrm{n}}\) (C) \(2^{2 n-1}\) (D) none of these

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

If the 4th term in the expansion of \(\left(2+\frac{3}{8} x\right)^{10}\) has the maximum numerical value, then the range of values of 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

If the 4th term in the expansion of \(\left(2+\frac{3}{8} x\right)^{10}\) has the maximum numerical value, then the range of values of 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 value of \(x\), for which the 6 th term in the expansion of the binomial \(\left[\sqrt{ \left.2^{\log \left(10-3^{x}\right.}\right)}+\sqrt[5]{2^{(x-2) \log 3}}\right]^{m}\) is equal to 21 and it is known that the binomial coefficient of the 2nd, 3rd and 4th terms in the expansion represent respectively the first, third and fifth terms of an A.P. (the symbol log stands for logarithm to the base 10), is (A) 1 (B) 0 (C) 2 (D) none of these

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 4 th 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

\(\left[(3+\sqrt{5})^{2 n}\right]+1\), where \([x]\) denotes the integral part of \(x\), is divisible by (A) \(2^{n-1}\) (B) \(2^{\text {n }}\) (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

Assertion: If \(n\) is a positive integer and \(k\) is a positive integer not exceeding \(n\), then $$ \begin{gathered} \sum_{k=1}^{n} k^{3}\left(\frac{C_{k}}{C_{k-1}}\right)^{2} \text {, where } C_{k}={ }^{n} C_{k}, \text { is } \\ \frac{n(n+1)^{2}(n+2)}{12} \\ \text { Reason: } \frac{C_{k}}{C_{k-1}}=\frac{{ }^{n} C_{k}}{{ }^{n} C_{k-1}}=\frac{n-k+1}{k} \end{gathered} $$

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 interval in which \(x(x>0)\) must lie so that the numerically greatest term in the expansion of \((1-x)^{21}\) has the greatest coefficient is, \(\left(\frac{5}{6}, \frac{6}{5}\right)\). Reason: If \(n\) is odd, then numerically greatest coefficient in the expansion of \((1-x)^{n}\) is \(\frac{{ }^{n} C_{n-1}}{2}\) or \(\frac{{ }^{n} C_{n+1}}{2}\).

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 (C) 26 (D) none of these

If \(|x|<1\), then the coefficient of \(x^{n}\) in expansion of \((1\) \(\left.+x+x^{2}+x^{3}+\ldots\right)^{2}\) is: (A) \(n\) (B) \(n-1\) (C) \(n+2\) (D) \(n+1\)

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