Rolle's Theorem

Q.

If $\alpha ,\beta$ are roots of the equation $2x^2-35x+2=0$, then the value of ${(2\alpha -35)}^3\times {(2\beta -35)}^3$ is equal to?

1. $10$

2. $100$

3. $64$

4. $1$

Q.

Let $f$ be the function given by $f\left(x\right)=x^3-6x^2-15x$. What is the absolute maximum value of $f$ on the interval $\left[0,6\right]$?

1. $0$

2. $4$

3. $5$

4. $6$

5. $8$

Q.

If the line, $2x-y+3=0$ is at a distance $\frac{1}{\sqrt{5}}$ and $\frac{2}{\sqrt{5}}$ from the lines $4x-2y+\alpha =0$ and $6x-3y+\beta =0$, respectively, then the sum of all possible values of $\alpha$ and $\beta$ is

Q. Given the function $f\left(x\right)=x^3-2x^2-x+1$. Then the value(s) of $c$ satisfying the conditions of the mean value theorem for the function on the interval $[-2,2]$, is

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

Q. The maximum value of function $x^3-12x^2+36x+17$ in the interval [1, 10] is

1. 17
2. None of these
3. 77
4. 177

Q.

The length of perpendicular from a point $(1,2)$ to the straight line $3x+4y+4=0$

1. $5$

2. $3$

3. $\frac{7}{5}$

4. None of these

Q. Given $P\left(x\right)=x^4+a{x}^3+b{x}^2+cx+d$ such that $x=0$ is the only real root of $P^{\prime }\left(x\right)=0.$ If $P(-1)<P\left(1\right),$ then in the interval $[-1,1],$

1. $P(-1)$ is the minimum and $P\left(1\right)$ is the maximum value of $P\left(x\right)$
2. $P(-1)$ is not minimum but $P\left(1\right)$ is the maximum value of $P\left(x\right)$
3. $P(-1)$ is the minimum and $P\left(1\right)$ is not the maximum value of $P\left(x\right)$
4. neither $P(-1)$ is the minimum nor $P\left(1\right)$ is the maximum value of $P\left(x\right)$

Q. If |z+1/z|=a then find the maximum and minimum values of |z|

Q. Let $f\left(x\right)=\frac{1}{3}x\sin x-(1-\cos x)$. The smallest positive integer $k$ such that $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x^k}\ne 0$ is

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

Q.

The least value of 'a' for which $\frac{4}{sinx}+\frac{1}{1-sinx}=a$ has at least one solution in the interval $(0,\pi /2)$ is

1. 1

2. 9

3. 8

4. 4

Q. The equation $x\log x=3–x$

1. has no root in $(1,3)$
2. has exactly one root in $(1,3)$
3. $x\log x–(3–x)>0$ in $[1,3]$
4. $x\log x–(3–x)<0$ in $[1,3]$

Q.

What are the two fundamental theorems of calculus?

Q.

Let $f:\left[0,2\right]\to R$be a twice differentiable functions such that $f"\left(x\right)>0$ for all $x\in \left(0,2\right)$. If $\varphi \left(x\right)=f\left(x\right)+f\left(2-x\right).$ then $\varphi$ is:

1. Increasing in $\left(0,1\right)$ and decreasing in $\left(1,2\right)$

2. Decreasing in $\left(0,1\right)$ and increasing in $\left(1,2\right)$

3. Increasing in $\left(0,2\right)$

4. Decreasing in $\left(0,2\right)$

Q. The function $f\left(x\right)=(3x-7)x^{2/3},x\in R$ is increasing for all $x$ lying in

1. $(-\infty ,-\frac{14}{15})\cup (0,\infty )$
2. $(-\infty ,\frac{14}{15})$
3. $(-\infty ,0)\cup (\frac{14}{15},\infty )$
4. $(-\infty ,0)\cup (\frac{3}{7},\infty )$

Q. Let $f$ be real valued function on $R$ defined as $f\left(x\right)=x^4(1-x{)}^2.$ Then which of the following statements is (are) CORRECT?

1. $f^{\prime }\left(x\right)=0$ for some $x\in (0,1)$
2. $f^{\prime \prime }\left(x\right)$ vanishes exactly twice in $R$
3. $f\left(x\right)$ is an even function
4. $f\left(x\right)$ is monotonically increasing in $(0,\frac{2}{3})\cup (1,\infty )$

Q.

Between any two real roots of the equation $e^x$ sin x = 1, the equation $e^x$ cos x = - 1 has

1. Atleast one root

2. Exactly one root

3. Atmost one root

4. No root

Q.

It is a continuous function $f$ defined on the real line $R$, assume positive and negative values in $R$ then the equation $f\left(x\right)=0$ has root in $R$.

For example, if it is known that a continuous function $f$ on $R$ is positive at some point and its minimum value is negative then the equation $f\left(x\right)=0$ has a root in $R$.

Consider $f\left(x\right)=k{e}^x–x$ for all real $x$where $k$ is a real constant. The positive value of k for which $k{e}^x–x=0$ has only one root is

1. $\frac{1}{e}$

2. $1$

3. $e$

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

Q. [x]+[2x]+[4x]+[8x]+[16x]+[32x]=12345. What is the real value of x for which the above equation holds true.

Q. Let $M$ and $N$ be the number of points on the curve $y^5–9xy+2x=0$, where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of $M+N$ equals ________.

Q.

The least value of n for which $(n-2{)}^2+8x+n+4>si{n}^{-1}(sin12)+co{s}^{-1}(cos12)\forall xϵR$ and $nϵN$ is

1. 4

2. 5

3. 6

4. 7

Q. \(\int \dfrac{dx}{\sqrt{9x - 4x^2}}\) equals:
\( (A) \dfrac{1}{9} \sin^{-1} \left(\dfrac{9x -8}{8} \right ) + C\)
\( (B) \dfrac{1}{2} \sin^{-1} \left(\dfrac{8x - 9}{9} \right ) + C\)
\((C) \dfrac{1}{3} \sin^{-1} \left(\dfrac{9x - 8}{8} \right ) + C\)
\( (D) \dfrac{1}{2} \sin^{-1} \left(\dfrac{9x -8}{8} \right ) + C\)

Q.

Show that the function defined by f(x) = cos $x^2$ is a continuous function.

Q.

The lines $x=a{t}^2$ meets the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, in the real points, if

1. $\left|t\right|<2$

2. $\left|t\right|\le 1$

3. $\left|t\right|\ge 1$

4. none of these

Q. Let $f,g:[-1,2]\to R$ be continuous functions which are twice differentiable on the interval $(-1,2).$ Let the values of $f$ and $g$ at the points $-1,0$ and $2$ be as given in the following table:

	$x=-1$	$x=0$	$x=2$
$f\left(x\right)$	$3$	$6$	$0$
$g\left(x\right)$	$0$	$1$	$-1$

In each of the interval $(-1,0)$ and $(0,2)$ the function $(f-3g{)}^{\prime \prime }$ never vanishes. Then the correct statement(s) is(are)

1. $f^{\prime }\left(x\right)-3g^{\prime }\left(x\right)=0$ has exactly three solutions in$(-1,0)\cup (0,2)$
2. $f^{\prime }\left(x\right)-3g^{\prime }\left(x\right)=0$ has exactly one solution in $(-1,0)$
3. $f^{\prime }\left(x\right)-3g^{\prime }\left(x\right)=0$ has exactly one solution in $(0,2)$
4. $f^{\prime }\left(x\right)-3g^{\prime }\left(x\right)=0$ has exactly two solutions in $(-1,0)$ and exactly two solutions in $(0,2)$

Q. If $f:R\to R$ is a twice differentiable function such that $f^{\prime \prime }\left(x\right)>0$ for all $x\in R$, and $f\left(\frac{1}{2}\right)=\frac{1}{2},f\left(1\right)=1,\text{then}$

1. $f^{\prime }\left(1\right)\le 0$
2. $0<f^{\prime }\left(1\right)\le \frac{1}{2}$
3. $\frac{1}{2}<f^{\prime }\left(1\right)\le 1$
4. $f^{\prime }\left(1\right)>1$

Q. Find the intervals in which the function $f$ given by $f\left(x\right)=x^2-4x+6$ is
(a) increasing (b) decreasing

Q.

If the function f (x) = x² - 8x + 12 satisfies the condition of Rolle's Theorem on (2, 6), find the value of c such that f '(c) = 0

1. 4

2. 6

3. 2

4. 8

Q.

Consider the point $(x,y)$ lying on the graph of the line line .$2x+4y=5.$ Let $L$ be the distance from the point $(x,y)$ to the origin $(0,0)$ Write $L$ as a function of $x$.

Q.

The point where the function $f\left(x\right)=x^2-5x-6$ satisfies the condition of Rolle’s theorem is

1. $x=5$

2. $x=\frac{5}{2}$

3. $x=6$

4. None of these

Q. Consider the function $f\left(x\right)=x^3-p{x}^2+qx$ defined on $[1,3].$ If $f$ satisfies the hypothesis of Rolle's theorem such that $c=\frac{3}{2},$ then the value of $4p+q+16$ is