Chapter 8

A ratio of the \(5^{\text {th }}\) term from the begining to the 5 th term from the end in the binomial expansion of \(\left(2^{\frac{1}{3}}+\frac{1}{2(3)^{\frac{1}{3}}}\right)^{10}\) is: (a) \(1: 2(6)^{\frac{1}{3}} \quad\) (b) \(1: 4(16)^{\frac{1}{3}}\) (c) \(4(36)^{\frac{1}{3}}: 1\) (d) \(2(36)^{\frac{1}{3}}: 1\)

The term independent of \(\mathrm{x}\) in the binomial expansion of \(\left(1-\frac{1}{x}+3 x^{5}\right)\left(2 x^{2}-\frac{1}{x}\right)^{8}\) is : (a) 496 (b) \(-496\) (c) 400 (d) \(-400\)

The ratio of the coefficient of \(x^{15}\) to the term independent of \(x\) in the expansion of \(\left(x^{2}+\frac{2}{x}\right)^{15}\) is: (a) \(7: 16\) (b) \(7: 64\) (c) \(1: 4\) (d) \(1: 32\)

The coefficient of the middle term in the binomial expansion in powers of \(x\) of \((1+\alpha x)^{4}\) and of \((1-\alpha x)^{6}\) is the same if \(\alpha\) equals (a) \(\frac{3}{5}\) (b) \(\frac{10}{3}\) (c) \(\frac{-3}{10}\) (d) \(\frac{-5}{3}\)

The value of \(\sum_{r=0}^{20}{\underline{\phantom{xx}}}^{50-r} C_{6}\) is equal to: (a) \({ }^{51} C_{7}-{ }^{30} C_{7}\) (b) \({ }^{50} C_{7}-{ }^{30} C_{7}\) (c) \({ }^{50} C_{6}-{ }^{30} C_{6}\) (d) \({ }^{51} C_{7}^{\prime}+{ }^{30} C_{7}\)

The coefficient of \(x^{4}\) in the expansion of \(\left(1+x+x^{2}\right)^{10}\) is

If the sum of the coefficients of all even powers of \(x\) in the product \(\left(1+x+x^{2}+\ldots+x^{20}\right)\left(1-x+x^{2}-x^{3}+\ldots+x^{2 n}\right)\) is 61, then \(n\) is equal to

The term independent of \(x\) in the expansion of \(\left(\frac{1}{60}-\frac{x^{8}}{81}\right) \cdot\left(2 x^{2}-\frac{3}{x^{2}}\right)^{6}\) is equal to: (a) \(-72\) (b) 36 (c) \(-36\) (d) \(-108\)

If \({ }^{29} \mathrm{C}_{1}+\left(2^{2}\right)^{27} \mathrm{C}_{2}+\left(3^{2}\right)^{29} \mathrm{C}_{3}+\ldots \ldots \ldots .+\left(20^{2}\right)^{2 \mathrm{C}} \mathrm{C}_{20}=\mathrm{A}\left(2^{\mathrm{e}}\right)\), then the ordered pair \((\mathrm{A}, \beta)\) is equal to: \(\quad\) (a) \((420,19)\) (b) \((420,18)\) (c) \((380,18)\) (d) \((380,19)\)

The coefficient of \(x^{18}\) in the product \((1+x)(1-x)^{10}\) \(\left(1+x+x^{2}\right)^{9}\) is : \(\quad\) (a) 84 (b) \(-126\) (c) \(-84\) (d) 126

If the coefficients of \(\mathrm{x}^{2}\) and \(\mathrm{x}^{3}\) are both zero, in the expansion of the expression \(\left(1+a x+b x^{2}\right)(1-3 x)^{15}\) in powers of \(x\), then the ordered pair (a, b) is equal to: \(\quad\) (a) \((28,861)\) (b) \((-54,315)\) (c) \((28,315)\) (d) \((-21,714)\)

The sum of the real values of \(x\) for which the middle term in the binomial expansion of \(\left(\frac{x^{3}}{3}+\frac{3}{x}\right)^{8}\) equals 5670 is : (a) 0 (b) 6 (c) 4 (d) 8

The value of \(\mathrm{r}\) for which \({ }^{20} C_{r}{\underline{\phantom{xx}}}^{20} C_{0}+{ }^{20} C_{r-1}{\underline{\phantom{xx}}}^{20} C_{1}+{ }^{20} C_{r-2}{\underline{\phantom{xx}}}^{20} C_{2}+\ldots+{ }^{20} C_{0}{\underline{\phantom{xx}}}^{20} C_{r}\) ismaximum, is: (a) 15 (b) 20 (c) 11 (d) 10

If \(\sum_{r=0}^{25}\left\\{{ }^{50} \mathrm{C}_{\mathrm{r}}{\underline{\phantom{xx}}}^{50-\mathrm{r}} \mathrm{C}_{25-\mathrm{r}}\right\\}=\mathrm{K}\left({ }^{50} \mathrm{C}_{25}\right)\), then \(\mathrm{K}\) is equal to: (a) \((25)^{2}\) (c) \(2^{24}\) (b) \(2^{25}-1\) (d) \(2^{25}\)

The coefficient of \(\mathrm{t}^{4}\) in the expansion of \(\left(\frac{1-t^{6}}{1-t}\right)^{3}\) (a) 14 (c) 10 (b) 15 (d) 12

The value of \(\left({ }^{21} \mathrm{C}_{1}-{ }^{10} \mathrm{C}_{1}\right)+\left({ }^{21} \mathrm{C}_{2}-{ }^{10} \mathrm{C}_{2}\right)+\left({ }^{21} \mathrm{C}_{3}-{ }^{10} \mathrm{C}_{3}\right)+\left({ }^{21} \mathrm{C}_{4}-{ }^{10} \mathrm{C}_{4}\right)\) \(+\ldots+\left({ }^{21} \mathrm{C}_{10}-{ }^{10} \mathrm{C}_{10}\right)\) is: (a) \(2^{20}-2^{10}\) (b) \(2^{21}-2^{11}\) (c) \(2^{21}-2^{10}\) (d) \(2^{20}-2^{9}\)

If the number of terms in the expansion of \(\left(1-\frac{2}{x}+\frac{4}{x^{2}}\right)^{n}\), \(\mathrm{x} \neq 0\), is 28 , then the sum of the coefficients of all the terms in this expansion, is: (a) 243 (b) 729 (c) 64 (d) 2187

The sum of coefficients of integral power of \(x\) in the binomial expansion \((1-2 \sqrt{x})^{50}\) is : (a) \(\frac{1}{2}\left(3^{50}-1\right)\) (b) \(\frac{1}{2}\left(2^{50}+1\right)\) (c) \(\frac{1}{2}\left(3^{50}+1\right)\) (d) \(\frac{1}{2}\left(3^{50}\right)\)

The coefficient of \(\mathrm{x}^{1012}\) in the expansion of \(\left(1+x^{n}+x^{253}\right)^{10}\), (where \(n \leq 22\) is any positive integer), is (a) 1 (b) \({ }^{10} \mathrm{C}_{4}\) (c) \(4 \mathrm{n}\) (d) \({ }^{253} \mathrm{C}_{4}\)

In a shop there are five types of ice-creams available. A child buys six ice- creams. Statement-1: The number of different ways the child can buy the six ice-creams is \({ }^{10} C_{5}\). Statement \(-2\) : The number of different ways the child can buy the six ice- creams is equal to the number of different ways of arranging \(6 \mathrm{~A}\) 's and \(4 \mathrm{~B}\) 's in a row. (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

The sum of the series \({ }^{20} C_{0}-{ }^{20} C_{1}+{ }^{20} C_{2}-{ }^{20} C_{3}+\ldots . .-\ldots . .+{ }^{20} C_{10}\) is (a) 0 (b) \({ }^{20} C_{10}\) (c) \(-{ }^{20} C_{10}\) (d) \(\frac{1}{2}{\underline{\phantom{xx}}}^{20} C_{10}\)

If \(x\) is so small that \(x^{3}\) and higher powers of \(x\) may be neglected, then \(\frac{(1+x)^{\frac{3}{2}}-\left(1+\frac{1}{2} x\right)^{3}}{(1-x)^{\frac{1}{2}}}\) may be approximated as (a) \(1-\frac{3}{8} x^{2}\) (b) \(3 x+\frac{3}{8} x^{2}\) (c) \(-\frac{3}{8} x^{2}\) (d) \(\frac{x}{2}-\frac{3}{8} x^{2}\)

The positive integer just greater than \((1+0.0001)^{10000}\) is (a) 4 (b) 5 (c) 2 (d) 3