Reduction Formulae

Q.

$$\begin{gathered} \mathrm{Given}; \\ {\mathrm{C}}_{\left(\mathrm{graphite}\right)}+{\mathrm{O}}_2\left(\mathrm{g}\right)\to {\mathrm{CO}}_2\left(\mathrm{g}\right);{∆}_{\mathrm{r}}{\mathrm{H}}^{°}=393.5{\mathrm{kJmol}}^{-1} \\ {\mathrm{H}}_2\left(\mathrm{g}\right)+[1/2]{\mathrm{O}}_2\left(\mathrm{g}\right)\to {\mathrm{H}}_2\mathrm{O}\left(\mathrm{I}\right) \\ {\mathrm{CO}}_2\left(\mathrm{g}\right)+2{\mathrm{H}}_2\mathrm{O}\left(\mathrm{I}\right)\to {\mathrm{CH}}_4\left(\mathrm{g}\right)+2{\mathrm{O}}_2\left(\mathrm{g}\right) \\ {∆}_{\mathrm{r}}{\mathrm{H}}^{°}=+890.3{\mathrm{kJmol}}^{-1} \end{gathered}$$

Based on the above thermochemical equations, the value of ${∆}_{\mathrm{r}}{\mathrm{H}}^{\circ }$ at $298\mathrm{K}$ for the reaction ${\mathrm{C}}_{\left(\mathrm{graphite}\right)}+2{\mathrm{H}}_2\left(\mathrm{g}\right)\to {\mathrm{CH}}_4\left(\mathrm{g}\right)$ will be:

1. ${}^{-}74.8{\mathrm{kJmol}}^{-1}$

2. ${}^{-}144{\mathrm{kJmol}}^{-1}$

3. ${}^{+}74.8\mathrm{kJ}$

4. ${}^{+}144{\mathrm{kJmol}}^{-1}$

Q.

Find the second order derivative of the given functions.

$e^{x}sin5x$

Q. Prove that : $\frac{1}{1+x^{b-a}+x^{c-a}}+\frac{1}{1+x^{a-b}+x^{c-b}}+\frac{1}{1+x^{b-c}+x^{a-c}}=1$

Q. Assertion(A): ${\cos }^{-1}x$ and ${\tan }^{-1}x$ are positive for all positive real values of $x$ in their domain.
Reason(R): The domain of $f\left(x\right)={\cos }^{-1}x+{\tan }^{-1}x$ is $[-1,1].$

1. Both A and R are true and R is the correct explanation of A
2. A is false but R is true
3. Both A and R are true but R is not correct explanation of A
4. A is true but R is false

Q. Evaluate the given definite integrals as limit of sums:
${\int }_2^3{x}^{2}dx$

Q. Find the value of ${\left(\frac{x^a}{x^b}\right)}^{\frac{1}{ab}}\times {\left(\frac{x^b}{x^c}\right)}^{\frac{1}{bc}}\times {\left(\frac{x^c}{x^a}\right)}^{\frac{1}{ca}}$

Q. If $I_n=\int ta{n}^{n}xdx$ then which of the following relation is correct -

1. $I_n=\frac{ta{n}^{n-1}\left(x\right)}{n-1}-I_{n-2}$
2. $I_n=\frac{ta{n}^{n-2}\left(x\right)}{n-2}-I_{n-2}$
3. $I_n=\frac{ta{n}^{n-3}\left(x\right)}{n-3}-I_{n-2}$
4. $I_n=\frac{ta{n}^{n-4}\left(x\right)}{n-4}-I_{n-2}$

Q. If the reduction formula for $I_n=\int \frac{\sin \mathrm{nx}}{\cos x}\mathrm{dx}$ is given by
$I_n+I_{n-2}=-\frac{2\cos \left\{\right(n-1\left)x\right\}}{n-1}$, then $\int \frac{\sin 3x}{\cos x}\mathrm{dx}$ is.

1. 
   $\cos 2x+\log \left|\sec x\right|+C$
2. 
   $\cos 2x-\log \left|\sec x\right|+C$
3. 
   $-\cos 2x+\log \left|\sec x\right|+C$
4. 
   $-\cos 2x-\log \left|\sec x\right|+C$

Q. If $I_n=\int ta{n}^{n}xdx,then{I}_0+I_1+2(I_2+...I_8)+I_9+I_{10}$, is equal to

1. ${\sum }_{n=1}^9\frac{ta{n}^{n}x}{n}$
2. $1+{\sum }_{n=1}^8\frac{ta{n}^{n}x}{n}$
3. ${\sum }_{n=1}^9\frac{ta{n}^{n}x}{n+1}$
4. ${\sum }_{n=2}^{10}\frac{ta{n}^{n}x}{n+1}$

Q. Evaluate the following definite integrals as limit of sums.
${\int }_1^4(x^2-x)dx$.

1. $\frac{81}{2}$
2. $\frac{63}{3}$
3. $\frac{35}{3}$
4. $\frac{27}{2}$

Q. Prove that ${\int }_0^x\left[t\right]dt=\frac{\left[x\right]\left(\right[x]-1)}{2}+\left[x\right](x-[x\left]\right)$, where $[\cdot ]$ denotes the greatest integer function.

Q. Prove that $C_0+\frac{C_1}{2}+\frac{C_2}{3}+\frac{C_3}{4}+....+\frac{C_n}{n+1}=\frac{2^{n+1}-1}{(n+1)}$

Q.

Find the integrals of the functions.
$\int ta{n}^{4}xdx.$

Q. $\underset{n\to \infty }{lim}(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}......\frac{1}{5n})$

1. $\pi /4$
2. ${\tan }^{-1}$
3. $\pi /2$
4. ${\tan }^{-1}\left(2\right)$

Q. If $a+3b+2c=12$, then the maximum value of $6bc(1+a)+a(3b+2c)$ is

1. $92$
2. $112$
3. $102$
4. $115$

Q. If the reduction formula for $I_n=\int {\tan }^n\mathrm{xdx}$ is given by
$I_n=\frac{1}{n-1}{\tan }^{n-1}x-I_{n-2}$, then $\int {\tan }^{3}x\mathrm{dx}\mathrm{is}$

1. 
   $\frac{1}{2}{\tan }^{2}x-\frac{1}{2}\ln \left|1+{\tan }^{2}x\right|+C$
2. 
   $-\frac{1}{2}{\tan }^{2}x-\frac{1}{2}\ln \left|1+{\tan }^{2}x\right|+C$
3. 
   $\frac{1}{2}{\tan }^{2}x+\frac{1}{2}\ln \left|1+{\tan }^{2}x\right|+C$
4. 
   $-\frac{1}{2}{\tan }^{2}x+\frac{1}{2}\ln \left|1+{\tan }^{2}x\right|+C$

Q. If $\frac{9^{{}^{\frac{1}{3}}}\times {27}^{-\frac{1}{2}}}{3^{\frac{1}{6}}\times 3^{-\frac{2}{3}}}$ is $m^{-\frac{1}{3}}$, then $m$ is equal to:

Q. Let $u=\stackrel{\infty }{\underset{0}{\int }}\frac{dx}{x^4+7x^2+1}\&v=\stackrel{\infty }{\underset{0}{\int }}\frac{x^{2}dx}{x^4+7x^2+1}$ then:

1. v > u
2. $3u+2v=5\pi /6$
3. 6v = $\pi$
4. $u+v=\pi /3$

Q. If $2x+\frac{1}{3x}=5$ then the value of $\frac{5x}{6x^2+20x+1}$ is

1. $\frac{3}{7}$
2. $\frac{2}{7}$
3. $\frac{1}{7}$
4. $\frac{4}{7}$

Q. Let $I_n=\int {\tan }^{n}xdx,(n>1).$ If $I_4+I_6=a{\tan }^{5}x+b{x}^5+C,$ where $C$ is a constant of integration, then the ordered pair $(a,b)$ is equal to:

1. $(-\frac{1}{5},1)$
2. $(\frac{1}{5},0)$
3. $(\frac{1}{5},-1)$
4. $(-\frac{1}{5},0)$

Q. If $\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+....+\infty =\frac{{\pi }^4}{90}$ then $\frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+....+\infty =$

1. $\frac{{\pi }^4}{90}$
2. $\frac{{\pi }^4}{96}$
3. $\frac{{\pi }^4}{92}$
4. $\frac{{\pi }^4}{108}$

Q. Integrate the function $\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}$

Q. The integral ${\int }_0^{1}ta{n}^{-1}\left(\frac{2x-1}{1+x-x^2}\right)dx$ simlifies to a value:

Q. Evaluate the given definite integrals as limit of sums:
${\int }_0^5(x+1)dx$

Q. If ${\sin }^{-1}(1-x)-2{\sin }^{-1}x=\pi /2$, then $x$ equals-

1. $0,-1/2$
2. $0,\frac{1}{2}$
3. $0$
4. None of these

Q. The value of integral $\stackrel{3}{\underset{-1}{\int }}({\tan }^{-1}\left(\frac{x}{1+x^2}\right)+{\tan }^{-1}\left(\frac{x^2+1}{x}\right))dx$

1. $\frac{3\pi }{2}$
2. $\pi$
3. $2\pi$
4. Can't be found

Q. If ${}^{56}{P}_{r+6}:{}^{54}{P}_{r+3}=30800:1$, then $r$ $=$

1. $40$
2. $41$
3. $42$
4. $39$

Q. Assertion :Let $f:R\to R$ be any function. Define $g:R\to R$ by $g\left(x\right)=\left|f\left(x\right)\right|$ for all $x$. Then, $g$ is continuous is $f$ is continuous. Reason: Composition of two continuous functions is a continuous function

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. State true or false:

1. True
2. False

Q. Integrate $\int \frac{x(x^2+3)+3\left({\tan }^{-1}x\right){(1+x^2)}^2}{(1+x^2)}dx$ (where $C$ is an integration constant)

1. $\frac{(x^2+3)}{(1+x^2)}{\tan }^{-1}x+C$
2. $\frac{{\tan }^{-1}x}{1+x^2}+C$
3. $x(x^2+3){\tan }^{-1}x+C$
4. none