First Fundamental Theorem of Calculus

Q.

Evaluate $\underset{x\to 0}{lim}\frac{sin5x}{tan3x}$

Q. The general values of x for which $\cos 2x,\frac{1}{2}$ and $\sin 2x$ are in AP, are given by

1. 
2. 
3. 
4. 

Q.

The minimum value of $f\left(x\right)=si{n}^{4}x+co{s}^{4}x,0\le x\le \frac{\pi }{2}$ is

1. 

2. 

3. 

4. 

Q. ${\int }_0^{\frac{\pi }{2}}\frac{cosx}{1+cosx+sinx}dx=$

1. $\frac{\pi }{4}+\frac{1}{2}log2$
2. $\frac{\pi }{4}+log2$
3. $\frac{\pi }{4}-\frac{1}{2}log2$
4. $\frac{\pi }{4}-log2$

Q. $\int \frac{e^x}{4+e^{2x}}dx$ is equal to
(where $C$ is constant of integration)

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

Q. $\underset{x\to a}{\lim }\frac{x^{2/3}-a^{2/3}}{x^{3/4}-a^{3/4}}$

Q. If x=(-∞, -1) find value of 4 tan inverse x + sin inverse(2x/1+x²)+ Cos inverse (1-x²/1+x²) .

Q. a₀ xⁿ + a₁ xⁿ⁻¹ + a₂ xⁿ⁻² + ... + a_n−1 x + a_n.

Q.

${\int }_1^{a}x.a^{-\left[lo{g}_{a}x\right]}dx=\frac{e-1}{2},\text{where}a>1\text{and}[.]$ denotes the greatest integer function, then the value of $a^2$is

1. $e-1$

2. $e$

3. $e+1$

4. $e^2-1$

Q. Evaluate the following:
(i) $2x^3+2x^2-7x+72,\mathrm{when}x=\frac{3-5i}{2}$
(ii) $x^4-4x^3+4x^2+8x+44,\mathrm{when}x=3+2i$
(iii) $x^4+4x^3+6x^2+4x+9,\mathrm{when}x=-1+i\sqrt{2}$
(iv) $x^6+x^4+x^2+1,\mathrm{when}x=\frac{1+i}{\sqrt{2}}.$

Q. If w=-1+ root 3i/2, then arg(w^2) is

Q.

The most general valuie that satisfies the equation cosecθ = 2 and cotθ = -√3 is

1. 2nπ + π/6, n ∈ Z

2. 2nπ + 5π/ 6, n ∈ Z

3. 2nπ ± 5π/ 6, n ∈ Z

4. 2nπ - 5π/6, n ∈ Z

Q. Let f(x) be a function satisfying f ’(x) = f(x) with f(0) = 1 and g(x) be a function that satisfies f(x) + g(x) = $x^2$. Then, the value of the integral ${\int }_0^{1}f\left(x\right)g\left(x\right)dx,$ is

1. $e-\frac{e^2}{2}-\frac{5}{2}$
2. $e+\frac{e^2}{2}-\frac{3}{2}$
3. $e-\frac{e^2}{2}-\frac{3}{2}$
4. $e+\frac{e^2}{2}+\frac{5}{2}$

Q. If ${\int }_0^{a^2}\frac{1}{1+16x^2}dx=\frac{\pi }{16}$, then the value of a is

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

Q.

${\int }_0^{\frac{\pi }{2}}(cosx-sinx)e^{x}dx=$

1. 0

2. 1

3. -1

4. 2

Q. ${\int }_0^{10\pi }|sinx|dxis$

1. 20
2. 8
3. 10
4. 18

Q. The maximum value of the function
$f\left(x\right)=\underset{0}{\overset{1}{\int }}t\sin (x+\pi t)dt,x\in R$ is

1. $\frac{1}{\pi }\sqrt{{\pi }^2+4}$
2. $\frac{1}{{\pi }^2}\sqrt{{\pi }^2+4}$
3. $\sqrt{{\pi }^2+4}$
4. $\frac{1}{2{\pi }^2}\sqrt{{\pi }^2+4}$

Q. If $l(m,n)={\int }_0^1{t}^m(1+t{)}^{n}dt$, then the expression for $l(m,n)$ in terms of $l(m+1,n-1)$ is

1. $\frac{2^n}{m+1}-\frac{n}{m+1}l(m+1,n-1)$
2. $\frac{n}{m+1}l(m+1,n-1)$
3. $\frac{2^n}{m+1}+\frac{n}{m+1}l(m+1,n-1)$
4. $\frac{m}{m+1}l(m+1,n-1)$

Q. Write the following in the simplest form : tan^-1{2(x)^1/2/(1-x)}

Q. ${\int }_1^e\frac{e^x}{x}(1+xlogx)dx=$

1. $e^e$
2. $e^e-e$
3. $e^e+e$
4. $Noneofthese$

Q. The value of the integral ${\int }_a^{a+\frac{\pi }{2}}(|sinx|+|cosx|)dxis$

1. $a\pi$
2. $2a\pi$
3. $\frac{a\pi }{2}$
4. independent of a

Q. $\mathrm{If}{\int }_1^a\left(3x^2+2x+1\right)dx=36,\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}a\mathrm{is}$

Q. ${\int }_0^2\left[x^2\right]dx$ is (where [.] is greastest integral function

1. $2-\sqrt{2}$
2. $2+\sqrt{2}$
3. $\sqrt{2}-1$
4. $-\sqrt{2}-\sqrt{3}+5$

Q. ${\int }_0^{\frac{\pi }{8}}\frac{se{c}^{2}2x}{2}dx=$

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