Chapter 9

The coefficient of \(x^{17}\) in the expansion of \((x-1)(x-2)(x-3) \ldots(x-18)\) is (A) \(\frac{171}{2}\) (B) 342 (C) \(-171\) (D) 684

The fractional part of \(\frac{2^{4 n}}{15}\) is (A) \(\frac{2}{15}\) (B) \(\frac{1}{15}\) (C) \(\frac{4}{15}\) (D) none of these

The sum of the coefficients of all the integral powers of \(\mathrm{x}\) in the expansion of \((1+2 \sqrt{x})^{80}\) is (A) \(\frac{1}{2}\left(3^{80}+1\right)\) (B) \(\frac{1}{2}\left(3^{80}-1\right)\) (C) \(\left(3^{80}+1\right)\) (D) \(\left(3^{80}-1\right)\)

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

The two consecutive terms in the expansion of \((3 x+2)^{74}\), whose coefficients are equal, are (A) 20 th and \(21 \mathrm{st}\) (B) 30 th and 31 st (C) 40 th and \(41 \mathrm{st}\) (D) none of these

If in the expansion of \(\left(2^{x}+\frac{1}{4^{x}}\right)^{n}, \frac{T_{3}}{T_{2}}=7\) and the sum of the coefficients of 2 nd and 3 rd terms is 36 , then the value of \(x\) is (A) \(-\frac{1}{3}\) (B) \(-\frac{1}{2}\) (C) \(\frac{1}{3}\) (D) \(\frac{1}{2}\)

The interval in which \(x\) must lie so that the numerically greatest term in the expansion of \((1-x)^{21}\) has the greatest coefficient is, \((x>0)\). (A) \(\left[\frac{5}{6}, \frac{6}{5}\right]\) (B) \(\left(\frac{5}{6}, \frac{6}{5}\right)\) (C) \(\left(\frac{4}{5}, \frac{5}{4}\right)\) (D) \(\left[\frac{4}{5}, \frac{5}{4}\right]\)

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

If \(\frac{1}{\sqrt{4 x+1}}\left\\{\left(\frac{1+\sqrt{4 x+1}}{2}\right)^{n}-\left(\frac{1-\sqrt{4 x+1}}{2}\right)^{n}\right\\}\) \(=a_{0}+a_{1} x+\ldots+a_{5} x^{5}\), then \(n\) equals (A) 11 (B) 9 (C) 10 (D) none of these

The sum \(\sum_{i=0}^{m}\left(\begin{array}{c}10 \\\ i\end{array}\right)\left(\begin{array}{c}20 \\ m-i\end{array}\right)\), (where \(\left(\begin{array}{l}p \\ q\end{array}\right)=0\) if \(p

The number of distinct terms in the expansion of \(\left(x^{3}+1+\frac{1}{x^{3}}\right)^{n} ; x \in R^{+}\)and \(n \in N\) is (A) \(2 n\) (B) \(3 n\) (C) \(2 n+1\) (C) \(3 n+1\)

The number of terms with integral coefficients in the expansion of \(\left(17^{1 / 3}+35^{1 / 2} x\right)^{600}\) is (A) 100 (B) 50 (C) 150 (D) 101

If \(z=\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)^{5}+\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)^{5}\), then (A) \(\operatorname{Re}(z)=0\) (B) \(I_{m}(z)=0\) (C) \(\operatorname{Re}(z)>0, I_{m}(z)>0\) (D) \(\operatorname{Re}(z)>0, I_{m}(z)<0\)

The greatest value of the term independent of \(x\) in the expansion of \(\left(x \sin \alpha+x^{-1} \cos \alpha\right)^{10}, \alpha \in R\), is (A) \(\frac{10 !}{2^{5}}\) (B) \(\frac{10 !}{(5 !)^{2}}\) (C) \(\frac{1}{2^{5}} \frac{10 !}{(5 !)^{2}}\) (D) none of these

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

The sum of the last ten coefficients in the expansion of \((1+x)^{19}\) when expanded in ascending powers of \(x\) is (A) \(2^{18}\) (B) \(2^{19}\) (C) \(2^{18}-{ }^{19} C_{10}\) (D) \(\frac{1}{2}\left(2^{19}-1\right)\)

The number of integral terms in the expansion of \((2 \sqrt{5}+\sqrt[6]{7})^{642}\) is (A) 105 (B) 107 (C) 321 (D) 108

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 positive integer which is just greater than \((1+0.0001)^{1000}\) is (A) 3 (B) 4 (C) 5 (D) 2

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

The interval in which \(x(>0)\) must be so that the greatest term in the expansion of \((1+x)^{2 n}\) has the greatest coefficient is (A) \(\left(\frac{n-1}{n}, \frac{n}{n-1}\right)\) (B) \(\left(\frac{n}{n+1}, \frac{n+1}{n}\right)\) (C) \(\left(\frac{n}{n+2}, \frac{n+2}{n}\right)\) (D) none of these

If \(n\) is positive integer and \(k\) is a positive integer not exceeding \(n\), then \(\sum_{k=1}^{n} k^{3}\left(\frac{C_{k}}{C_{k-1}}\right)^{2}\), where \(C_{k}={ }^{n} C_{k}\), is (A) \(\frac{n(n+1)(n+2)}{12}\) (B) \(\frac{n(n+1)^{2}(n+2)}{12}\) (C) \(\frac{n(n+1)^{2}(n+2)}{6}\) (D) none of these

If the fourth term in the expansion of \(\left(\sqrt{\frac{1}{x^{\log x+1}}}+x^{1 / 12}\right)^{6}\) is equal to 200 and \(x>1\), then \(x\) is equal to (A) \(10^{\sqrt{2}}\) (B) 10 (C) \(10^{4}\) (D) none of these

The coefficient of \(\lambda^{n} \mu^{n}\) in the expansion of \([(1+\lambda)(1+\mu)\) \((\lambda+\mu)]^{n}\) is (A) \(\sum_{r=0}^{n} C_{r}^{2}\) (B) \(\sum_{r=0}^{n} C_{r+2}^{2}\) (C) \(\sum_{r=0}^{n} C_{r+3}^{2}\) (D) \(\sum_{r=0}^{n} C_{r}^{3}\)

If \(\alpha=18^{3}+7^{3}+3.18 .7 .25\), and \(\beta=3^{6}+6.243 .2+15.81 .4+20.27 .8\) \(+15.9 .16+6.3 .32+64\) then the value of \(\alpha \beta^{-1}\) is (A) 1 (B) 5 (C) 25 (D) 100

If there is a term containing \(x^{2 r}\) in \(\left(x+\frac{1}{x^{2}}\right)^{n-3}\), then (A) \(n-2 r\) is a positive integral multiple of 3 (B) \(n-2 r\) is even (C) \(n-2 r\) is odd (D) none of these

If \(P_{n}\) denotes the product of the bipomial coefficients in the expansion of \((1+x)^{n}\), then \(\frac{P_{n+1}}{P_{n}}\) equals \(\begin{array}{ll}\text { (A) } \frac{(n+1)^{n}}{n !} & \text { (B) } \frac{n^{n}}{n !}\end{array}\) (C) \(\frac{(n+1)^{n}}{(n+1) !}\) (C) \(\frac{(n+1)^{n+1}}{(n+1) !}\)

The value of \(\frac{1}{81^{n}}-\frac{10}{81^{n}}{\underline{\phantom{xx}}}^{2 n} C_{2}+\frac{10^{2}}{81^{n}}{\underline{\phantom{xx}}}^{2 n} C_{2}\) \(-\frac{10^{3}}{81^{n}}{\underline{\phantom{xx}}}^{2 n} C_{3}+\ldots+\frac{10^{2 n}}{81^{n}}\) is (A) 2 (B) 0 (C) \(1 / 2\) (D) 1

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

Coefficient of \(t^{24}\) in \(\left(1+t^{2}\right)^{12}\left(1+t^{12}\right)\left(1+t^{24}\right)\) is (A) \({ }^{12} C_{6}+3\) (B) \({ }^{12} C_{6}+1\) (C) \({ }^{12} C_{6}\) (D) \({ }^{12} C_{6}+2\)

If the sum of the coefficients in the expansions of \((1+2 x)^{m}\) and \((2+x)^{n}\) are respectively 6561 and 243 , then the position of the point \((m, n)\) with respect to the circle \(x^{2}+y^{2}-4 x-6 y-32=0\) (A) is inside the circle (B) is outside the circle (C) is on the circle (D) can not be fixed

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

The coefficient of \(x^{n}\) in the expansion \((2 x+3)^{n}-\) \((2 x+3)^{n-1}(5-2 x)+(2 x+3)^{n-2}(5-2 x)^{2}+\ldots+(-1)^{n}\) \((5-2 x)^{n}\) is (A) \(\frac{1}{8} 2^{n}\) (B) \((n+1) 2^{n}\) (C) \((n+1) 2^{n-3}\) (D) \(-(n+1) 2^{n-2}\)

The value of \(2\left({ }^{n} C_{0}\right)+\frac{3}{2}\left({ }^{n} C_{1}\right)+\frac{4}{3}\left({ }^{n} C_{2}\right)+\frac{5}{4}\left({ }^{n} C_{3}\right) \ldots\). (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) \(\frac{2^{n}+2}{n-1}\)

Which of the following expansions will have term containing \(x^{3} ?\) (A) \(\left(x^{-\frac{1}{5}}+2 x^{\frac{3}{5}}\right)^{25}\) (B) \(\left(x^{\frac{3}{5}}+2 x^{-\frac{1}{5}}\right)^{24}\) (C) \(\left(x^{\frac{3}{5}}-2 x^{-\frac{1}{5}}\right)^{23}\) (D) \(\left(x^{\frac{3}{5}}+2 x^{-\frac{1}{5}}\right)^{22}\)

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

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

If \(a_{i}(i=0,1,2, \ldots, 16)\) be real constants such that for every real value of \(x,\left(1+x+x^{2}\right)^{8}=a_{0}+a_{1} x_{2}+a_{2} x^{2} \ldots\) \(+a_{16} x^{16}\), then \(a_{5}\) is equal to (A) 502 (B) 504 (C) 506 (D) 508

Statement-1: \(\sum_{r=0}^{n}(r+1){ }^{n} C_{r}=(n+2) 2^{n-1}\) Statement-2: \(\sum_{r=0}^{n}(r+1)^{n} C_{r} x^{r}=(1+x)^{n}+\) \(+n x(1+x)^{n-1}\) (A) Statement- 1 is false, Statement- 2 is true (B) Statement-1 is true, Statement- 2 is true, Statement-2 is a correct explanation for Statement-1 (C) Statement- 1 is true, Statement- 2 is true; Statement-2 is not a correct explanation for Statement-1 (D) Statement- 1 is true, Statement-2 is false

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

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

If \(C_{0}, C_{1}, C_{2}, \ldots, C_{\mathrm{n}}\) denote the binomial coefficients in the expansion of \((1+x)^{\mathrm{n}}\), then \(\sum_{r=0}^{n}(-1)^{r} \cdot{ }^{n} C_{r} \cdot \frac{1+r \log _{e} 10}{\left(1+\log _{e} 10^{n}\right)^{r}}=\) (A) 2 (B) 1 (C) 0 (D) none of these

If \(C_{0}, C_{1}, C_{2}, \ldots, C_{\mathrm{n}}\) are the coefficients of the expansion of \((1+x)^{n}\), then the value of \(\sum_{0}^{n} \frac{C_{k}}{k+1}\) is (A) 0 (B) \(\frac{2^{n}-1}{n}\) (C) \(\frac{2^{n+1}-1}{n+1}\) (D) none of these

Larger of \(99^{50}+100^{50}\) and \(101^{50}\) is (A) \(101^{50}\) (B) \(99^{50}+100^{50}\) (C) both are equal (D) none of these

The greatest coefficient in the expansion of \((x+y+z+w)^{15}\) is (A) \(\frac{15 !}{3 !(4 !)^{3}}\) (B) \(\frac{15 !}{(3 !)^{3} 4 !}\) (C) \(\frac{15 !}{2 !(4 !)^{2}}\) (D) none of these

The sum of the series \(\sum_{r=0}^{10}{\underline{\phantom{xx}}}^{20} C_{r}\) is (A) \(2^{19}-\frac{1}{2} \cdot{ }^{20} C_{10}\) (B) \(2^{19}+\frac{1}{2} \cdot{ }^{20} C_{10}\) (C) \(2^{19}\) (D) \(2^{20}\)

\({ }^{\mathrm{n}+1} \mathrm{C}_{2}+2\left[{ }^{2} \mathrm{C}_{2}+{ }^{3} \mathrm{C}_{2}+{ }^{4} \mathrm{C}_{2}+\ldots+{ }^{\mathrm{n}} C_{2}\right]=\) (A) \(\frac{n(n+1)(2 n+1)}{6}\) (B) \(\frac{n(n+1)}{2}\) (C) \(\frac{n(n-1)(2 n-1)}{6}\) (D) none of these

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

The greatest integer which divides the number \(101^{100}-1\) is (A) 100 (B) 1,000 (C) 10,000 (D) \(1,00,000\)

Given positive integers \(r>1, n>2\) and the coefficients of \((3 r)\) th term and \((r+2)\) th term in the binomial expansion of \((1+x)^{2 \mathrm{n}}\) are equal, then \(r=\) (A) \(\frac{n}{2}, n\) even (B) \(\frac{n}{2}\) (C) \(n\) (D) 1