Sufficient Condition for an Extrema

Q.

Find two positive numbers x and y such that x+y = 60 and $x{y}^3$ is maximum.

Q.

The differential coefficient of $\sec \left({\tan }^{-1}\left(x\right)\right)$.

Q. The area of the region bounded by the curves $y=|x-1|$ and $y=3-|x|$ is

1. $6$ sq. units
2. $2$ sq. units
3. $3$ sq. units
4. $4$ sq. units

Q.

Is Modulus always positive?

Q. Let $f\left(x\right)=3x^2-7x+c$ where ${}^{\prime }{c}^{\prime }$ is a parameter and $x\ge \frac{7}{6}$. Then the value of $\left[c\right]$ such that $f\left(x\right)$ touches $f^{-1}\left(x\right)$ is :
([.] represents the greatest integer function)

Q.

The real-valued function $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{cosec}}^{-1}\left(\frac{\mathrm{x}}{\sqrt{\mathrm{x}-\left[\mathrm{x}\right]}}\right)$ where $\left[x\right]$ denotes the greatest integer less than or equal to $x$, is defined for all $x$ belonging to:

1. All non- integers except the interval $[–1,1]$

2. All integers except $0,–1,1$

3. All reals except integers

4. All reals except the interval $[–1,1]$

Q.

Find the sum of all odd numbers between $100$ and $200.$

Q. $20$ is divided in two parts so that product of cube of one quantity and square of other quantity is maximum.The parts are-

1. $16,4$
2. $10,10$
3. $6,14$
4. $12,8$

Q.

What is the value of $i^2$ in complex number?

Q. If ${\cos }^{-1}x={\tan }^{-1}x,$ then

1. $x^2=\frac{\sqrt{5}-1}{2}$
2. there are two such values of $x$ in $[-1,1]$
3. $\sin \left({\cos }^{-1}x\right)=\frac{\sqrt{5}-1}{2}$
4. $\tan \left({\cos }^{-1}x\right)=\frac{\sqrt{5}-1}{2}$

Q.

If $\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}1,\mathrm{when}\mathrm{x}\mathrm{is}\mathrm{rational}\\ 0,\mathrm{when}\mathrm{x}\mathrm{is}\mathrm{irrational}\end{array}\right.$, then $\underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)=?$

1. $0$

2. $1$

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

4. None of these

Q.

Sum of series $\frac{1}{1.2.3}+\frac{5}{3.4.5}+\frac{9}{5.6.7}+\dots$ is equal to

1. $\frac{3}{2}-3{\log }_e\left(2\right)$

2. $\frac{5}{2}-3{\log }_e\left(2\right)$

3. $1-4{\log }_e\left(2\right)$

4. None of these

Q.

If ${\log }_k\left(x\right)·{\log }_5\left(k\right)={\log }_x\left(5\right),k\ne 1,k>0$, then $x$ is equal to:

1. $k$

2. $0$

3. $5$

4. None of these

Q. $f\left(x\right)=x^2-4|x|andg\left(x\right)=\{\begin{array}{c}min\left\{f\right(t):-6\le t\le x\},xϵ[-6,0]\\ max\left\{f\right(t):0\le t\le x\},xϵ[0,6]\end{array},thang\left(x\right)$

1. Exactly one point of local minima
2. Exactly one point of local maxima
3. No point of local maxima but exactly one point of local minima
4. Neither a point of local maxima nor minima

Q. Which among the following function is not differentiable at $x=0$

1. $f\left(x\right)=x^2$
2. $f\left(x\right)=x+\left|x\right|$
3. $f\left(x\right)=2x$
4. $\sin x$

Q. The domain of the function $f\left(x\right)={\log }_2\left({\log }_3\right({\log }_4(x^2-3x+6)\left)\right)$ is

1. $[2,\infty )$
2. $[0,\infty )$
3. $(-\infty ,2)$
4. $(-\infty ,1)\cup (2,\infty )$

Q.

$\frac{2}{3!}+\frac{4}{5!}+\frac{6}{7!}+.......\mathrm{to}\infty$is

1. $e$

2. $2e$

3. $3e$

4. none of these

Q.

$\left(\frac{1}{1.3}\right)+\left(\frac{1}{2.5}\right)+\left(\frac{1}{3.7}\right)+\left(\frac{1}{4.9}\right)+...is$

1. $2{\log }_{e}2-2$

2. $2-{\log }_{e}2$

3. $2{\log }_{e}4$

4. ${\log }_{e}4$

Q.

The equation ${\sin }^{-1}x-{\cos }^{-1}x={\cos }^{-1}\left(\raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$ has

1. no solution

2. unique solution

3. infinite number of solutions

4. None of the above

Q.

Which of the following numbers is/are rational?

1. $\sin \left({15}^{°}\right)$

2. $\cos \left({15}^{°}\right)$

3. $\sin \left({15}^{°}\right)\cos \left({15}^{°}\right)$

4. $\sin \left({15}^{°}\right)\cos \left({75}^{°}\right)$

Q.

The sequence $\log a,\log \left(\frac{a^2}{b}\right),\log \left(\frac{a^3}{b^2}\right),....$ is

1. GP

2. AP

3. HP

4. GP and HP

Q.

The equation $y^2-8y-x+19=0$ represents

1. A parabola whose focus is $\left(\frac{1}{4},0\right)$ and directrix is $x=-\frac{1}{4}$

2. A parabola whose vertex is $(3,4)$ and directrix is $x=\frac{11}{4}$

3. A parabola whose focus is $\left(\frac{13}{4},4\right)$ and vertex is $(0,0)$

4. A curve which is not a parabola

Q.

$\frac{d}{dx}\left[{\sin }^{-1}\frac{x}{a}\right]=$ , $a<0,\left|\frac{x}{a}\right|<1$

1. $\frac{1}{\sqrt{a^2-x^2}}$

2. $\frac{1}{\sqrt{x^2-a^2}}$

3. $-\frac{1}{\sqrt{a^2-x^2}}$

4. $-\frac{1}{\sqrt{x^2-a^2}}$

Q.

How do you find the stationary point of a function?

Q.

$f\left(x\right)=si{n}^{-1}{\left[4-(x-7{)}^3\right]}^{\frac{1}{5}}$ then $f^{-1}\left(x\right)=$ ...

1. $(7-si{n}^{5}x{)}^{⅓}$

2. $7+(4-si{n}^{5}x{)}^{⅓}$

3. $7+(4+si{n}^{5}x{)}^{⅓}$

4. $(7+si{n}^{5}x{)}^{⅓}$

Q.

If $f\left(x\right)$ is continuous function such that the area bounded by the curve $y=f\left(x\right),$ the $x$-axis and the lines $x=a$and $x=0$ is $\frac{\left(a^2\right)}{2}+\frac{a}{2}\sin a+\left(\frac{\pi }{2}\right)\cos a.$

Then the value of $f\left(\frac{\mathrm{\pi }}{2}\right)$ is.

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

2. $\frac{a}{2}$

3. $\frac{a^2}{2}$

4. $\frac{\mathrm{\pi }}{2}$

Q.

An infinite G.P has the first term $x$ and sum $5$ , then $x$ belongs to :

1. $\left|x\right|<10$

2. $-10<x<0$

3. $0<x<10$

4. $x>10$

Q. Let f(x) and g(x) be two functions having finite non-zero third order derivatives f'''(x) and g'''(x) for all $xϵR$ . If $f\left(x\right)g\left(x\right)=1$ for all $xϵR$ then $\frac{f^{\prime \prime \prime }}{f^{\prime }}-\frac{g^{\prime \prime \prime }}{g^{\prime }}$ is equal to :

1. $3(\frac{f^{\prime \prime }}{g}-\frac{g^{\prime \prime }}{f})$
2. $3(\frac{g^{\prime \prime }}{g}-\frac{f^{\prime \prime }}{g})$
3. $3(\frac{f^{\prime \prime }}{f}-\frac{g^{\prime \prime }}{g})$
4. $3(\frac{f^{\prime \prime }}{g}-\frac{g^{\prime \prime }}{f})$

Q. Let $a_i,i=1,2,3,\dots ,n$ denote the integers in the domain of function $f\left(x\right)=\sqrt{{\log }_{1/2}\left(\frac{4x-25}{x-21}\right)}$ where $a_i<a_{i+1}\forall i\in N.$ A line $L:\frac{2x-a_1}{4}=\frac{y+a_1}{a_2}=\frac{z-a_3}{a_5}$ meets the $xy,yz$ and $zx$ planes at $A,B$ and $C$ respectively. If volume of the tetrahedron $OABD$ is $V$ cubic units where $O$ is origin and $D$ is the image of $C$ with respect to $x-$axis, then the value of $90V$ is

Q.

Find the values of x for which y=$\left[x\right(x-2){]}^2$ is an increasing function.