Property 7

Q.

How do you find the exact value of${\sin }^{-1}(-1)$?

Q.

If$u={\sin }^{-1}\left(\frac{x}{y}\right)+{\tan }^{-1}\left(\frac{y}{x}\right),$ then the value of$x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}$ is

1. $0$

2. $1$

3. $2$

4. none of these

Q.

What is the area bounded by the parabola $y^2=8x$ and its latus rectum.

1. $\frac{16}{3}$sq units

2. $\frac{32}{3}$sq units

3. $\frac{8}{3}$sq units

4. $\frac{64}{3}$sq units

5. $\frac{4}{3}$sq units

Q. Write the derivative of sinx with respect to cosx

Q. The area bounded by the parabolas $y^2=4x$ and $x^2=4y$ is

1. $\frac{8}{3}$
2. $\frac{16}{3}$
3. $\frac{20}{3}$
4. $\frac{32}{3}$

Q.

Let $\theta =\frac{\pi }{5}$ and $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$. If $B=A+A^4$, then $det\left(A\right)$:

1. is one

2. lies in $\left(1,2\right)$

3. lies in $\left(2,3\right)$

4. is zero

Q.

Differential coefficient of ${\sin }^{-1}\left(\frac{1-x}{1+x}\right)$ w.r.t. $\sqrt{x}$ is

1. $\frac{1}{2\sqrt{x}}$

2. $\frac{\sqrt{x}}{\sqrt{\left(1-x\right)}}$

3. $1$

4. $-\frac{2}{\left(1+x\right)}$

Q.

If ${\sin }^{-1}x-{\cos }^{-1}x=\frac{\pi }{6}$, then $x=$

1. $\frac{1}{2}$

2. $\frac{\sqrt{3}}{2}$

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

4. None of these

Q. The value of $\underset{x\to 2}{lim}\frac{x^3-6x^2+11x-6}{x^2-6x+8}$ is

1. $2$
2. $-2$
3. $\frac{1}{2}$
4. Limit doesn't exists.

Q.

If $x=a\sin 2\theta (1+\cos 2\theta ),y=b\cos 2\theta (1-\cos 2\theta )$, then $dy/dx=$?

1. $(b\tan \theta )/a$

2. $(a\tan \theta )/b$

3. $a/(b\tan \theta )$

4. $b/a\tan \theta$

Q.

The area enclosed by the pair of lines $xy=0$, the line $x-4=0$ and $y+5=0$ is

1. $20\mathrm{sq}.\mathrm{units}$

2. $10\mathrm{sq}.\mathrm{units}$

3. $\frac{5}{4}\mathrm{sq}.\mathrm{units}$

4. $0\mathrm{sq}.\mathrm{units}$

Q. Let $y=y\left(x\right)$ satisfies the equation $\frac{dy}{dx}-|A|=0,$ for all $x>0,$ where $A=\left[\begin{array}{ccc}y& \sin x& 1\\ 0& -1& 1\\ 2& 0& \frac{1}{x}\end{array}\right]$.
If $y\left(\pi \right)=\pi +2,$ then the value of $y\left(\frac{\pi }{2}\right)$ is:

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

Q.

The area of the feasible region for the following constraints $3y+x\ge 3$, $x\ge 0$ and $y\ge 0$will be

1. bounded

2. unbounded

3. convex

4. concave

Q.

If $f\left(x\right)=\frac{\sin x}{e^x}$ in $\left[0,\mathrm{\pi }\right]$, then $f\left(x\right)$

1. satisfies Rolles theorem and $c=\frac{\mathrm{\pi }}{4}$, so that $f\left(\frac{\mathrm{\pi }}{4}\right)=4$

2. does not satisfy Rolles theorem but $f\left(\frac{\mathrm{\pi }}{4}\right)>0$

3. satisfies Rolles theorem but $f\left(\frac{\mathrm{\pi }}{4}\right)=0$

4. satisfies LaGranges Mean value theorem but $f\left(\frac{\mathrm{\pi }}{4}\right)\ne 0$

Q.

The area enclosed within the curve $\left|x\right|+\left|y\right|=1$ is

1. $\sqrt{2}$

2. $1$

3. $\sqrt{3}$

4. $2$

Q. The value of ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{\cos x}{1+e^x}\mathrm{dx}$ is equal to

Q. The option(s) with the value of $a$ and $L$ that satisfy the following equation is(are)
$\frac{\underset{0}{\overset{4\pi }{\int }}{e}^t({\sin }^{6}at+{\cos }^{4}at)dt}{\underset{0}{\overset{\pi }{\int }}{e}^t({\sin }^{6}at+{\cos }^{4}at)dt}=L?$

1. $a=2,L=\frac{e^{4\pi }-1}{e^{\pi }-1}$
2. $a=2,L=\frac{e^{4\pi }+1}{e^{\pi }+1}$
3. $a=4,L=\frac{e^{4\pi }-1}{e^{\pi }-1}$
4. $a=4,L=\frac{e^{4\pi }+1}{e^{\pi }+1}$

Q.

If $f\left(x\right)=x+\frac{1}{x}$ then the value of $f\left(x\right)$ is

1. $x^2+\log x+c$

2. $\frac{x^2}{2}+\log x+c$

3. $\frac{x}{2}-\log x+c$

4. None of these

Q.

The perimeter of the right angled triangle is five times the length of the shortest side. The numerical value of the area of the triangle is 15 times the numerical value of the length of the shortest side. Find the lengths of the three sides of the triangle.

Q.

Area under the curve $y=\sqrt{3x+4}$between x=0 and x=4, is

1. 
   

2. 
   

3. None of these

4. 8 sq. unit

Q. Area between the curve y=cos x and x-axis when $0\le x\le 2\pi$ is

1. 2
2. 4
3. 0
4. 3

Q.

The area of the region bounded by the curves $\mathrm{y}=|\mathrm{x}-2|,\mathrm{x}=1,\mathrm{x}=3$ and the x-axis is:

1. $1$

2. $2$

3. $3$

4. $4$

Q. (Sinx + cosx) ÷ cos^3x = tan^3x + tan^2x + tanx + 1 ; prove LHS = RHS

Q.

If $0<p<q,$ then $\underset{n\to \infty }{\lim }{\left(q^n+p^n\right)}^{\frac{1}{n}}=?$

1. $e$

2. $p$

3. $q$

4. $0$

Q.

Log6+ 2log5+log4-log3-log2=2

Prove it

Q.

Consider a region$R=\left\{(x,y)\in R^2:x^2\le y\le 2x\right\}$. If a line $y=\alpha$divides the area of region R into two equal parts, then which of the following is true?

1. ${\alpha }^3-6{\alpha }^2+16=0$

2. $3{\alpha }^2-8{\alpha }^{3/2}+8=0$

3. ${\alpha }^3-6{\alpha }^{3/2}-16=0$

4. $3{\alpha }^2-8\alpha +8=0$

Q.

If(1+x)^n =C0+C1x+C2x^2+......Cnx^n, prove that

1) 3C0-8C1+13C2-18C3+.....upto (n+1) terms

Q. 1.Find the derivatives of the following 1. (x3-3x2+4)(4x5+x2-1)

Q.

Integrate the following functions.
$\int \frac{x+3}{x^2-2x-5}dx.$

Q. The area contained between the curve $xy=a^2$, the vertical line x = a, x = 4a(a > 0) and x-axis is

1. $a^{2}log2$
2. $2a^{2}log2$
3. $alog2$
4. $2alog2$