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

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

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

The two consecutive terms in the expansion of \((3 x+2)^{74}\), whose coefficients are equal, are (A) 20 th and 21 st (B) 30 th and 31 st (C) 40 th and 41 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 \(\frac{2\left(\frac{n}{2}\right) !\left(\frac{n}{2}\right) !}{n !}\left[C_{0}^{2}-2 C_{1}^{2}+3 C_{2}^{2}-\quad \cdots+(-1)^{n}(n+1)\right.\) \(\left.C_{n}^{2}\right]\), where \(n\) is an even positive integer, is (A) 0 (B) \((-1)^{w / 2}(n+1)\) (C) \((-1)^{n / 2}(n+2)\) (D) \((-1)^{n} n\)

\(\mathrm{f}\left(1+2 x+x^{2}\right)^{n}=\sum_{r=0}^{2 n} a, x^{r}\), then \(a_{r}=\) (A) \(\left({ }^{n} C_{r}\right)^{2}\) (B) \({ }^{n} C_{r} \cdot{ }^{n} C_{r+1}\) (C) \({ }^{2 n} C_{r}\) (D) \({ }^{2 n} C_{r+1}\)

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) \(\operatorname{Re}(z)>0, I_{s}(z)>0\) (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

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

The coefficient of the term independent of \(x\) in the expansion of \(\left(\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right)^{10}\) is (A) 210 (B) 105 (C) 70 (D) 112

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) \(\frac{1}{8} 2^{x}\) (B) \((n+1) 2^{n}\) (C) \((n+1) 2^{n-3}\) (D) \(-(n+1) 2^{n-2}\)

The value of the sum of the series \(3^{n} C_{0}-8^{n} C_{1}+\) \(13^{\text {" }} C_{2}-18^{n} C_{3}+\ldots\) upto \((n+1)\) terms is (A) 0 (B) \(3^{n}\) (B) \(5^{\text {" }}\) (D) none of these

The value of \(2\left({ }^{"} C_{0}\right)+\frac{3}{2}\left({ }^{n} C_{1}\right)+\frac{4}{3}\left({ }^{n} C_{2}\right)+\frac{5}{4}\left({ }^{n} C_{3}\right) \ldots\). is (A) \(\frac{2^{H}(1-n)-1}{n+1}\) (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 \((\sqrt{3}+1)^{2 u}-(\sqrt{3}-1)^{2 n}\) is (A) an irrational number (B) an odd positive integer (C) an even positive integer (D) a rational number other than positive integers

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) \(\frac{1}{\log _{10}^{4}-\log _{10}^{3}}\) (B) \(\frac{1}{\log _{10}{\underline{\phantom{xx}}}^{4}+\log _{10}{\underline{\phantom{xx}}}^{3}}\) (C) \(\frac{9}{\log _{10}{\underline{\phantom{xx}}}^{4}-\log _{10}^{3}}\) (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 \(C_{0}, C_{1}, C_{2}, \ldots, C_{n}\) are the coefficients of the expansion of \((1+x)^{\mathrm{n}}\), 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 \(\sum_{r=0}^{10}{\underline{\phantom{xx}}}^{21} C_{r}\) 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) \(2^{20}\)

If \(A={ }^{2 \mathrm{n}} \mathrm{C}_{0} \cdot{ }^{2 \mathrm{n}} \mathrm{C}_{1}+{ }^{2 \mathrm{n}} \mathrm{C}_{1}{\underline{\phantom{xx}}}^{2 \mathrm{n}-1} \mathrm{C}_{1}+{ }^{2} C_{2}{\underline{\phantom{xx}}}^{2 \mathrm{n}-2} \mathrm{C}_{1}+\ldots\), then \(A\) is (A) 0 (B) \(2^{\mathrm{a}}\) (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 \((1+x)^{2 n}\) are equal, then \(r=\) (A) \(\frac{n}{2}, n\) even (B) \(\frac{n}{2}\) (C) \(n\) (D) 1

Let \(n\) be a positive integer such that \(\left(1+x+x^{2}\right)^{\mathrm{n}}=a_{0}+a_{1} x+a_{z} x^{2}+\ldots+a_{2 a} x^{2 \mathrm{n}}\), then \(a_{\mathrm{r}}=\) (A) \(a_{\text {n- }-r}, 0 \leq r \leq 2 n\) (B) \(a_{2 n}, 0 \leq r \leq 2 n\) (C) \(a_{2 n-r}, 0 \leq r \leq 2 n\) (D) none of these

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