Critical Point

Q.

If the function: $R\to R$, is defined by $f\left(x\right)=\left|x\right|\left(x-\sin x\right)$, then which of the following statements is TRUE?

1. Function is one-one, but not onto

2. Function is onto, but not one-one

3. Function is both one-one and onto

4. Function is neither one-one nor onto

Q.

If $x=a(1+\cos \theta ),y=a(\theta +\sin \theta )$ , then $\frac{d^{2}y}{d{x}^2}at\theta =\pi /2$ is equal to:

1. $-\frac{1}{a}$

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

3. $-1$

4. $-2$

Q. Let $g\left(x\right)=\frac{1}{2}x+7$ and $g^{-1}\left(x\right)=ax-b$. Then number of solutions of $\sin x=\frac{a}{b}$ in the interval $[-100\pi ,100\pi ]$ is

Q. If the function $f$ given by
$f\left(x\right)=x^3-3(a-2)x^2+3ax+7$, for some $a\in R$ is increasing in $(0,1]$ and decreasing in $[1,5)$, then a root of the equation,
$\frac{f\left(x\right)-14}{(x-1{)}^2}=0(x\ne 1)$ is :

1. $-7$
2. $5$
3. $6$
4. $7$

Q.

What are the minimum and maximum values of the function $x^5-5x^4+5x^3-10$.

Q. The area bounded by the curves $y=x\ln x$ and $y=2x-2x^2$ is

1. $\frac{7}{12}\text{sq. units}$
2. $\frac{7}{4}\text{sq. units}$
3. $\frac{5}{4}\text{sq. units}$
4. $\frac{7}{16}\text{sq. units}$

Q. The two curves $x^3-3x{y}^2+2=0$ and $3x^{2}y-y^3=2$

1. touch each other
2. cut at right angle
3. cut at angle $\frac{\pi }{3}$
4. cut at angle $\frac{\pi }{4}$

Q. If the domain of the function $f\left(x\right)=\sqrt{3{\cos }^{-1}\left(4x\right)-\pi }$ is $[a,b],$ then the value of $(4a+64b)$ is

Q.

If $f\left(x\right)=x+\frac{1}{x},x>0$ then its greatest value is

1. $-2$

2. $0$

3. $3$

4. None of these

Q.

The minimum value of $2x+3y$, when $xy=6$ is

1. $9$

2. $12$

3. $8$

4. $6$

Q. Let $f:Z\to Z$ be defined as $f\left(x\right)=2\sin \left(2\pi x\right)-10\tan \left(5\pi x\right)+7\cos \left(4\pi x\right)+3.$ Then which of the following statements is/are TRUE?

1. $f\left(x\right)$ is a periodic function
2. $f\left(x\right)$ is an even function
3. $f\left(x\right)$ is an odd function and its inverse exists
4. $f\left(f\right(f\left(x\right)\left)\right)=f\left(f\right(x\left)\right)$ for all $x\in Z$

Q. A box open from top is made from a rectangular sheet of dimension $a\times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $x$ is equal to

1. $\frac{a+b-\sqrt{a^2+b^2+ab}}{6}$
2. $\frac{a+b-\sqrt{a^2+b^2-ab}}{6}$
3. $\frac{a+b-\sqrt{a^2+b^2-ab}}{12}$
4. $\frac{a+b+\sqrt{a^2+b^2-ab}}{6}$

Q.

A function f is defined by $f\left(x\right)=e^x\sin x$ in $[0,\pi ]$.

Which of the following is not correct?

1. $f$is continuous in $[0,\pi ]$

2. $f$is differentiable in $[0,\pi ]$

3. $f\left(0\right)=f\left(\pi \right)$

4. Rolls theorem is not true in$[0,\pi ]$

Q. A box, constructed from a rectangular metal sheet, is $21$ cm by $16$ cm by cutting equal squares of side $x$ cm from the corners of the sheet and then turning up the projected portions. The value of $x$ (in cm) so that volume of the box is maximum is

1. $1$
2. $2$
3. $3$
4. $4$

Q. If $f\left(x\right)=max\{x^4,x^2,\frac{1}{4}\},\forall x\in [0,\infty ),$ then the sum of square of reciprocal of all the values of $x,$ where $f\left(x\right)$ is not differentiable is:

1. $5$
2. $3$
3. $82$
4. $\frac{5}{4}$

Q.

The vertex connectivity of any tree is

1. One

2. Two

3. Three

4. None of the above

Q. Match the statements given in Column I with the intervals/union of intervals given in Column II

Column I	Column II
(A) The set $\{Re\left(\frac{2/z}{1-z^2}\right);\text{z is a complex number,}f|z|=1,z\ne \pm 1\}\text{is}$	(p) $(-\infty ,-1)\cup (1,\infty )$
(B) The domain of the function $f\left(x\right)={\sin }^{-1}\left(\frac{8(3{)}^{x-2}}{1-3^{2(x-1)}}\right)\text{is}$	(q) $-\infty ,-0\cup (0,\infty )$
(C) If $f\left(\theta \right)=\left|\begin{array}{ccc}1& \tan \theta & 1\\ -\tan \theta & 1& \tan \theta \\ -1& -\tan \theta & 1\end{array}\right|\text{then the set}\left\{f\right(\theta ):0\le \theta <\frac{\pi }{2}\}\text{is}$	(r) [$2,\infty$)
(D) If $f\left(x\right)=x^{\frac{3}{2}}(3x-10),x\ge 0,\text{then}f\left(x\right)$ is increasing in	(s) $(-\infty ,-1]\cup [1,\infty )$
	(t) $(-\infty ,0]\cup [2,\infty )$

1. A(s), B(t), C(r), D(r)

Q. Area of the region in which point p(x, y), {x > 0} lies; such that $y\le \sqrt{16-x^2}$ and $\left|ta{n}^{-1}\left(\frac{y}{x}\right)\right|\le \frac{\pi }{3}$ is

1. None of these
2. $(\frac{16}{3},\pi )$
3. $(\frac{8\pi }{3}+8\sqrt{3})$
   
4. $(4\sqrt{3}-\pi )$

Q. Let $f\left(x\right)=x^4+b{x}^3+1.$ A possible value of $b\in R$ for which the equation $f\left(x\right)=0$ has no real root is

1. $\frac{-11}{5}$
2. $\frac{-3}{2}$
3. $2$
4. $\frac{5}{2}$

Q. If $f\left(x\right)=\frac{ax+b}{(x-1)(x-4)}$ has local maximum point at $(2,-1),$ then $(a+b)$ is equal to

Q.

If at each point of the curve$y=x^3-a{x}^2+x+1$ tangent is inclined at an acute angle with the positive direction of the X-axis, then

1. $a\le \surd 3$

2. $a>0$

3. $–\surd 3<a<\surd 3$

4. None of these

Q. A box is constructed from a rectangular metal sheet of $21\text{cm}$ by $16\text{cm},$ by cutting equal squares of sides $x$ from the corners of the sheet and then turning up the projected portions. The value of $x$ for which volume of the box is maximum, is

1. $1$
2. $2$
3. $3$
4. $4$

Q. The derivative of $f\left(x\right)=3|2+x|$ at $x=-3$ is

1. $3$
2. $-3$
3. $0$
4. does not exist

Q.

In the function$f\left(x\right)=a{x}^3+b{x}^2+11x-6$ satisfies condition of Rolles theorem in $[1,3]$ and $f\prime \left(2+\frac{1}{\sqrt{3}}\right)=0$, then value of $a$ and $b$ are respectively

1. $1,-6$

2. $-2,1$

3. $-1,2$

4. $-1,6$

Q. Let $f$ be a function which is continuous in $[0,1]$ and differentiable in $(0,1)$ such that $f\left(1\right)=0$, then there exists some $c\in (0,1)$ such that:

1. $c{f}^{\prime }\left(c\right)-f\left(c\right)=0$
2. $f^{\prime }\left(c\right)-cf\left(c\right)=0$
3. $f^{\prime }\left(c\right)+cf\left(c\right)=0$
4. $c{f}^{\prime }\left(c\right)+f\left(c\right)=0$

Q. Which of the following statements is (are) CORRECT?

1. Rolle's theorem is applicable to the function $F\left(x\right)=1-\sqrt[5]{5x^6}$ on the interval $[-1,1].$
2. The domain of definition of the function $F\left(x\right)=\frac{{\log }_4(5-[x-1]-[x{]}^2)}{x^2+x-2},$ where $\left[x\right]$ denotes the greatest integer function, is $(-3,-2)\cup (-2,1)\cup (1,2).$
3. The value of $a$ for which the function $F\left(\theta \right)=a\sin \theta +\frac{1}{3}\sin 3\theta$ has an extremum at $\theta =\frac{\pi }{3}$ is $-2.$
4. The value of $\sum _{k=1}^{2010}\frac{\{x+k\}}{2010},$ where $\left\{x\right\}$ denotes the fractional part of $x,$ is $\left\{x\right\}.$

Q.

The period of $f\left(x\right)={10}^{\left\{{\cos }^2\left(\mathrm{\pi x}\right)+x-\left\{x\right\}\right\}}+{\sin }^2\left(\mathrm{\pi x}\right)$ is

1. $1$

2. $2$

3. $3$

4. $\mathrm{\pi }$

Q.

Critical points are the points where f'(x) is zero or it doesn't exist

1. True

2. False

Q. The least value of '$a$' for which equation $\frac{4}{\sin x}+\frac{1}{1-\sin x}=a$ has at least one solution in the interval $[0,\frac{\pi }{2}]$ is

1. $3$
2. $11$
3. $7$
4. $9$

Q. $\text{Let}f\left(x\right)=\frac{\sin \pi x}{x^2},x>0.$
$\text{Let}{x}_1<x_2<x_3<...<x_n<...$ be all the points of local maximum of $f$ and $y_1<y_2<y_3<...<y_n<...$ be all the points of local minimum of $f.$
Then which of the following options is/are correct?

1. $x_1<y_1$
2. $x_{n+1}-x_n>2$ for every $n$
3. $x_n\in (2n,2n+\frac{1}{2})$ for every $n$
4. $|x_n-y_n|>1$ for every $n$