Match the following. List-I List-II1) 1.31+2.41.2+3.51.2.3+4.61.2.3.4+……+∞ a) 232) 1+12!+14!+16!+…2-11!+13!+15!+…2 b) 4e3) coefficient x2 in e2x+3 c) 14) coefficient of x4 in cosh(2x) d) 2e3The correct match is

(1) ∑1∞ n(n+2)n-1 =∑1∞ (n+2)n-1 =∑1∞ (n+1)+3n-1 =∑1∞1n-1+3n-1 =e+3e=4e (2) 1+12+14+16+....2-1+11+13+15+....2 =e+e-122-e-e-122 =144e e-1=1 (3)e2x+3=e3·e2x =e31+2x+(2x)22+(2x)33+...x2 coef=e3 222 =2e3 (4) cos h 2x =e2x+e-2x2 =1+(2x)22+(2x)44+....x4 coff=244=23

Match the following. List-I List-II1) 1.31+2.41.2+3.51.2.3+4.61.2.3.4+……+∞ a) 232) 1+12!+14!+16!+…2-11!+13!+15!+…2 b) 4e3) coefficient x2 in e2x+3 c) 14) coefficient of x4 in cosh(2x) d) 2e3The correct match is

(1) ∑1∞ n(n+2)n-1 =∑1∞ (n+2)n-1 =∑1∞ (n+1)+3n-1 =∑1∞1n-1+3n-1 =e+3e=4e (2) 1+12+14+16+....2-1+11+13+15+....2 =e+e-122-e-e-122 =144e e-1=1 (3)e2x+3=e3·e2x =e31+2x+(2x)22+(2x)33+...x2 coef=e3 222 =2e3 (4) cos h 2x =e2x+e-2x2 =1+(2x)22+(2x)44+....x4 coff=244=23

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Match the following.
List-I List-II
1) $\frac{1.3}{1} + \frac{2.4}{1.2} + \frac{3.5}{1.2.3} + \frac{4.6}{1.2.3.4} + \dots \dots + \infty$ a) $\frac{2}{3}$
2) ${(1 + \frac{1}{2!} + \frac{1}{4!} + \frac{1}{6!} + \dots)}^{2} - {(\frac{1}{1!} + \frac{1}{3!} + \frac{1}{5!} + \dots)}^{2}$ b) 4e
3) coefficient $x^{2}$ in $e^{2 x + 3}$ c) 1
4) coefficient of $x^{4}$ in cosh(2x) d) $2 e^{3}$
The correct match is

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1 2 3 4

b c a d

1 2 3 4

b d c a

1 2 3 4

c b d a

1 2 3 4

b c d a

answer is B.

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Detailed Solution

(1) $\begin{array}{rcl} \sum_{1}^{\infty} \frac{n (n + 2)}{n - 1} \\ = & \sum_{1}^{\infty} \frac{(n + 2)}{n - 1} \\ {= \sum}_{1}^{\infty} \frac{(n + 1) + 3}{n - 1} \\ = & \sum_{1}^{\infty} (\frac{1}{n - 1} + \frac{3}{n - 1}) \\ = & e + 3 e = 4 e \end{array}$

(2) $\begin{array}{rcl} {(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + . . . .)}^{2} - {(1 + \frac{1}{1} + \frac{1}{3} + \frac{1}{5} + . . . .)}^{2} \\ = & {(\frac{e + e^{- 1}}{2})}^{2} - {(\frac{e - e^{- 1}}{2})}^{2} \\ = & \frac{1}{4} [4 e e^{- 1}] = 1 \end{array}$

(3) $\begin{array}{rcl} e^{2 x + 3} & = & e^{3} \cdot e^{2 x} \\ = & e^{3} [1 + 2 x + \frac{{(2 x)}^{2}}{2} + \frac{{(2 x)}^{3}}{3} + . . .] \\ x^{2} coef & = & e^{3} \frac{2^{2}}{2} \\ = & 2 e^{3} \end{array}$

(4) $\begin{array}{rcl} \cos h 2 x \\ = & \frac{e^{2 x} + e^{- 2 x}}{2} \\ = & [1 + \frac{{(2 x)}^{2}}{2} + \frac{{(2 x)}^{4}}{4} + . . . .] \\ x^{4} coff & = & \frac{2^{4}}{4} = \frac{2}{3} \end{array}$