Chapter 8

If \(\\{p\\}\) denotes the fractional part of the number \(p\), then \(\left\\{\frac{3^{200}}{8}\right\\}\), is equal to: (a) \(\frac{5}{8}\) (b) \(\frac{7}{8}\) (c) \(\frac{3}{8}\) (d) \(\frac{1}{8}\)

The natural number \(m\), for which the coefficient of \(x\) in the binomial expansion of \(\left(x^{m}+\frac{1}{x^{2}}\right)^{22}\) is 1540, is

The coefficient of \(x^{4}\) in the expansion of \(\left(1+x+x^{2}+x^{3}\right)^{6}\) in powers of \(x\), is

Let \(\left(2 x^{2}+3 x+4\right)^{10}=\sum_{r=0}^{20} a_{r} x^{r}\). Then \(\frac{a_{7}}{a_{13}}\) is equal to

If \(\alpha\) and \(\beta\) be the coefficients of \(x^{4}\) and \(x^{2}\) respectively in the expansion of \(\left(x+\sqrt{x^{2}-1}\right)^{6}+\left(x-\sqrt{x^{2}-1}\right)^{6}\), then: (a) \(\alpha+\beta=60\) (b) \(\alpha+\beta=-30\) (c) \(\alpha-\beta=60\) (d) \(\alpha-\beta=-132\)

The smallest natural number \(n\), such that the coefficient of \(x\) in the expansion of \(\left(x^{2}+\frac{1}{x^{3}}\right)^{n}\) is \({ }^{n} C_{23}\), is : (a) 38 (b) 58 (c) 23 (d) 35

If the fourth term in the Binomial expansion of \(\left(\frac{2}{x}+x^{\log _{8} x}\right)^{6}(x>0)\) is \(20 \times 8^{7}\), then a value of \(x\) is: (a) \(8^{3}\) (b) \(8^{2}\) (c) 8 (d) \(8^{-2}\)

The sum of the co-efficients of all even degree terms in \(x\) in the expansion of \(\left(x+\sqrt{x^{3}-1}\right)^{6}+\left(x-\sqrt{x^{3}-1}\right)^{6},(x>\) 1) is equal to: (a) 29 (b) 32 (c) 26 (d) 24

If the fourth term in the binomial expansion of \(\left(\sqrt{\frac{1}{x^{1+\log _{w} x}}}+x^{\frac{1}{12}}\right)^{6}\) is equal to 200, and \(x>1\), then the value of \(x\) is: (a) 100 (b) 10 (c) \(10^{3}\) (d) \(10^{4}\)

Let \((x+10)^{50}+(x-10)^{50}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{50} x^{50}\) for all \(x \in \mathbf{R} ;\) then \(\frac{a_{2}}{a_{0}}\) is equal to: (a) \(12.50\) (b) \(12.00\) (c) \(12.25\) (d) \(12.75\)

If the third term in the binomial expansion of \(\left(1+x^{\log _{2} x}\right)^{5}\) equals 2560 , then a possible value of \(x\) is: (a) \(\frac{1}{4}\) (b) \(4 \sqrt{2}\) (c) \(\frac{1}{8}\) (d) \(2 \sqrt{2}\)

The positive value of \(\lambda\) for which the co-efficient of \(x^{2}\) in the expression \(x^{2}\left(\sqrt{x}+\frac{\lambda}{x^{2}}\right)^{10}\) is 720 , is: (a) 4 (b) \(2 \sqrt{2}\) (c) \(\sqrt{5}\) (d) 3

If the fractional part of the number \(\frac{2^{403}}{15}\) is \(\frac{k}{15}\), then \(\mathrm{k}\). is equal to: \(\quad\) (a) 6 (b) 8 (c) 4 (d) 14

The coefficient of \(x^{10}\) in the expansion of \((1+x)^{2}\left(1+x^{2}\right)^{3}\) \(\left(1+x^{3}\right)^{4}\) is equal to [Online April \(\mathbf{1 5}, \mathbf{2 0 1 8}\) ] (a) 52 (b) 44 (c) 50 (d) 56

If \(\mathrm{n}\) is the degree of the polynomial, \(\left[\frac{1}{\sqrt{5 x^{3}+1}-\sqrt{5 x^{3}-1}}\right]^{8}+\left[\frac{1}{\sqrt{5 x^{3}+1}+\sqrt{5 x^{3}-1}}\right]^{8}\) and \(\mathrm{m}\) is the coefficient of \(\mathrm{x}^{\mathrm{n}}\) in it, then the ordered pair \((\mathrm{n}, \mathrm{m})\) is equal to (a) \(\left(12,(20)^{4}\right)\) (b) \(\left(8,5(10)^{4}\right)\) (c) \(\left(24,(10)^{8}\right)\) (d) \(\left(12,8(10)^{4}\right)\)

The coefficient of \(x^{2}\) in the expansion of the product \(\left(2-x^{2}\right) \cdot\left(\left(1+2 x+3 x^{2}\right)^{6}+\left(1-4 x^{2}\right)^{6}\right)\) is (a) 106 (b) 107 (c) 155 (d) 108

The sum of the co-efficients of all odd degree terms in the expansion of \(\left(x+\sqrt{x^{3}-1}\right)^{5}+\left(x-\sqrt{x^{3}-1}\right)^{5},(x>1)\) is : (a) 0 (b) 1 (c) 2 \((\) d) \(-1\)

If \((27)^{999}\) is divided by 7 , then the remainder is : (a) 1 (b) 2 (c) 3 (d) 6

If the coefficients of \(\mathrm{x}^{-2}\) and \(\mathrm{x}^{-4}\) in the expansion of \(\left(x^{\frac{1}{3}}+\frac{1}{2 x^{\frac{1}{3}}}\right)^{18},(x>0)\), are \(m\) and \(n\) respectively, then \(\frac{m}{n}\) is equal to : (a) 27 (b) 182 (c) \(\frac{5}{4}\) (d) \(\frac{4}{5}\)

If the coefficients of the three successive terms in the binomial expansion of \((1+x)^{n}\) are in the ratio \(1: 7: 42\), then the first of these terms in the expansion is: (a) \(8^{\text {th }}\) (b) \(6^{\text {th }}\) (c) \(7^{\text {? }}\) (d) \(9^{\text {th }}\)

If the coefficents 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) \(\left(14, \frac{272}{3}\right)\) (b) \(\left(16, \frac{272}{3}\right)\) (c) \(\left(16, \frac{251}{3}\right)\) (d) \(\left(14, \frac{251}{3}\right)\)

If \(X=\left\\{4^{n}-3 n-1: n \in N\right\\}\) and \(Y=\\{9(n-1): n \in N\\}\), where \(N\) is the set of natural numbers, then \(X \cup Y\) is equal to: (a) \(\bar{X}\) (b) \(Y\) (c) \(N\) (d) \(Y-X\)

If \(1+x^{4}+x^{5}=\sum_{i=0}^{5} a_{i}(1+x)^{i}\), for all \(x\) in \(R\), then \(a_{2}\) is: (a) \(-4\) (b) 6 (c) \(-8\) (d) 10

If \(\left(2+\frac{x}{3}\right)^{55}\) is expanded in the ascending powers of \(x\) and the coefficients of powers of \(x\) in two consecutive terms of the expansion are equal, then these terms are: (a) \(7^{\text {th }}\) and \(8^{\text {th }}\) (b) \(8^{\text {th }}\) and \(9^{\text {th }}\) (c) \(28^{\text {th }}\) and \(29^{\text {th }}\) (d) \(27^{\text {th }}\) and \(28^{\text {th }}\)

The number of terms in the expansion of \((1+x)^{101}\left(1+x^{2}-x\right)^{100}\) in powers of \(x\) is: (a) 302 (b) 301 (c) 202 (d) 101

If for positive integers \(r>1, n>2\), the coefficients of the \((3 r)^{t h}\) and \((r+2)^{\text {th }}\) powers of \(x\) in the expansion of \((1+x)^{2 n}\) are equal, then \(n\) is equal to: (a) \(2 r+1\) (b) \(2 r-1\) (c) \(3 r\) (d) \(r+1\)

The sum of the rational terms in the binomial expansion of \(\left(2^{\frac{1}{2}}+3^{\frac{1}{5}}\right)^{10}\) (a) 25 (b) 32 (c) 9 (d) 41

If the 7 th term in the binomial expansion of \(\left(\frac{3}{\sqrt[3]{84}}+\sqrt{3} \ln x\right)^{9}, x>0\), is equal to 729, then \(x\) can be : (a) \(e^{2}\) (b) \(e\) (c) \(\frac{e}{2}\) (d) \(2 e\)

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

The number of terms in the expansion of \(\left(y^{1 / 5}+x^{1 / 10}\right)^{55}\), in which powers of \(x\) and \(y\) are free from radical signs are (a) six (b) twelve (c) seven (d) five

If \(f(y)=1-(y-1)+(y-1)^{2}-(y-1)^{3}\) then the coefficient of \(y^{2}\) in it is (a) \({ }^{17} \mathrm{C}_{2}\) (b) \({ }^{17} \mathrm{C}\) (c) \({ }^{18} \mathrm{C}_{2}\) (d) \({ }^{18} \mathrm{C}_{3}\)

Statement - \(1:\) For each natural number \(n,(n+1)^{7}-1\) is divisible by 7 Statement - \(2:\) For each natural number \(n, n^{7}-n\) is divisible by \(7 .\) (a) Statement- 1 is true, Statement- 2 is true; Statement- 2 is a correct explanation for Statement-1. (b) Statement- 1 is true, Statement- 2 is true; Statement- 2 is NOT a correct explanation for Statement- 1 (c) Statement- 1 is true, Statement-2 is false (d) Statement- 1 is false, Statement- 2 is true

The coefficient of \(x^{7}\) in the expansion of \(\left(1-x-x^{2}+x^{3}\right)^{6}\) is (a) \(-132\) (b) \(-144\) (c) 132 (d) 144

The remainder left out when \(8^{2 n}-(62)^{2 n+1}\) is divided by 9 is (a) 2 (b) 7 (c) 8 (d) 0

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 the binomial expansion of \((a-b)^{n}, n \geq 5\), the sum of \(5^{\text {th }}\) and \(6^{\text {th }}\) terms is zero, then \(\mathrm{a} / \mathrm{b}\) equals (a) \(\frac{n-5}{6}\) (b) \(\frac{n-4}{5}\) (c) \(\frac{5}{n-4}\) (d) \(\frac{6}{n-5}\).

For natural numbers \(\mathrm{m}, \mathrm{n}\) 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 \(\begin{array}{llll}\text { (a) }(20,45) & \text { (b) }(35,20) & \underline{\phantom{xxx}}\end{array}\) (c) \((45,35)\) (d) \((35,45)\)

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 \(\left[a x-\left(\frac{1}{b x^{2}}\right)\right]^{11}\), then a and bsatisfy the relation (a) \(a-b=1\) (b) \(a+\underline{b}=1\) (c) \(\frac{a}{b}=1\) (d) \(a b=1\)

The coefficient of \(x^{n}\) in expansion of \((1+x)(1-x)^{n}\) is (a) \((-1)^{n-1} n\) (b) \((-1)^{n}(1-n)\) (c) \((-1)^{n-1}(n-1)^{2}\) (d) \((n-1)\)

The number of integral terms in the expansion of \((\sqrt{3}+\sqrt[8]{5})^{256}\) is (a) 35 (b) 32 (c) 33 (d) 34

\(r\) and \(n\) are positive integers \(r>1, n>2\) and coefficient of \((r+2)^{\text {th }}\) term and \(3 r^{\text {th }}\) term in the expansion of \((1+x)^{2 \pi}\) are equal, then \(n\) equals (a) \(3 r\) (b) \(3 r+1\) (c) \(2 r\) (d) \(2 r+1\)

The coefficients of \(x^{p}\) and \(x^{q}\) in the expansion of \((1+x)^{p+q}\) are (a) equal (b) equal with opposite signs (c) reciprocals of each other (d) none of these

If the constant term in the binomial expansion of \(\left(\sqrt{x} \frac{k}{x^{2}}\right)^{10}\) is 405, then \(|k|\) equals: \(\quad\) (a) 9 (b) 1 (c) 3 (d) 2

If for some positive integer \(n\), the coefficients of three consecutive terms in the binomial expansion of \((1+x)^{n+5}\) are in the ratio \(5: 10: 14\), then the largest coefficient in this expansion is: \(\quad\) (a) 462 (b) 330 (c) 792 (d) 252

If the number of integral terms in the expansion of \(\left(3^{1 / 2}+5^{1 / 8}\right)^{n}\) is exactly 33 , then the least value of \(n\) is : (a) 264 (b) 128 (c) 256 (d) 248

If the term independent of \(x\) in the expansion of \(\left(\frac{3}{2} x^{2}-\frac{1}{3 x}\right)^{9}\) is \(k\), then \(18 k\) is equal to : (a) 5 (b) 9 (c) 7 (d) 11

Let \(\alpha>0, \beta>0\) be such that \(\alpha^{3}+\beta^{2}=4\). If the maximum value of the term independent of \(x\) in the binomial expansion of \(\left(\alpha x^{\frac{1}{9}}+\beta x^{-\frac{1}{6}}\right)^{10}\) is \(10 k\), then \(k\) is equal to: (a) 336 (b) 352 (c) 84 (d) 176

For a positive integer \(n,\left(1+\frac{1}{x}\right)^{n}\) is expanded in increasing powers of \(x\). If three consecutive coefficients in this expansion are in the ratio, \(2: 5: 12\), then \(n\) is equal to

In the expansion of \(\left(\frac{x}{\cos \theta}+\frac{1}{x \sin \theta}\right)^{16}\), if \(l_{1}\) is the least value of the term independent of \(x\) when \(\frac{\pi}{8} \leq \theta \leq \frac{\pi}{4}\) and \(l_{2}\) is the least value of the term independent of \(x\) when \(\frac{\pi}{16} \leq \theta \leq \frac{\pi}{8}\), then the ratio \(l_{2}: l_{1}\) is equal to : (a) \(1: 8\) (b) \(16: 1\) (c) \(8: 1\) (d) \(1: 16\)

The total number is irrational terms in the binomial expansion of \(\left(7^{\frac{1}{5}}-3^{\frac{1}{10}}\right)^{60}\) is: \(\quad\) (a) 55 (c) 48 (b) 49 (d) 54