Extrema

Q.

If the sum and the product of the polynomial ${\mathrm{ax}}^2-5\mathrm{x}+\mathrm{c}$ is equal to $10$ each. Find the values of $a$ and $c$.

Q.

Let $f:R\to R$be defined as $f\left(x\right)=2x-1$and $g:R-\left\{1\right\}\to R$ be defined as $g\left(x\right)=\frac{\left(x-\frac{1}{2}\right)}{(x-1)}$. Then the composition function $f\left(g\right(x)$is:

1. Both one-one and onto

2. onto but not one-one

3. Neither one-one nor onto

4. one-one but not onto

Q.

If $\mathrm{f}\left(\mathrm{x}+\left(\frac{1}{\mathrm{x}}\right)\right)={\mathrm{x}}^2+\left(\frac{1}{{\mathrm{x}}^2}\right)$, then $f\left(x\right)$ will be

Q.

Let the function $f:[0,1]\to R$be defined by $f\left(x\right)=\frac{4^x}{(4x+2)}$. Then the value of $f\left(\frac{1}{40}\right)+f\left(\frac{2}{40}\right)+f\left(\frac{3}{40}\right)+\dots +f\left(\frac{39}{40}\right)–f\left(\frac{2}{2}\right)$is _____.

Q.

Let $f:\left[0,1\right]\to R$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f\left(0\right)=f\left(1\right)=0$ and satisfies$f"\left(x\right)-2f\left(x\right)+f\left(x\right)\ge e^x$, $x\in \left[0,1\right]$

1. $0<f\left(x\right)<\infty$

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

3. $-\frac{1}{4}<f\left(x\right)<1$

4. $-\infty <f\left(x\right)<0$

Q. Let $A=\left[\begin{array}{cc}2& 3\\ a& 0\end{array}\right],$ $a\in R$ be written as $P+Q$ where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix. If $det\left(Q\right)=9,$ then the modulus of the sum of all possible values of determinant of $P$ is equal to

1. $24$
2. $18$
3. $45$
4. $36$

Q. Let a function $g:[0,4]\to R$ be defined as
$g\left(x\right)=\{\begin{array}{ccc}\underset{0\le t\le x}{max}\{t^3-6t^2+9t-3\},& 0\le x\le 3& \\ 4-x,& 3<x\le 4& \end{array},$
then the number of points in the interval $(0,4)$ where $g\left(x\right)$ is NOT differentiable, is

Q.

Simplest form of $\frac{2}{\sqrt{2+\left(\sqrt{2+\left(\sqrt{\left(2+2\cos 4x\right)}\right)}\right)}}$ is

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

2. $\sec \left(x\right)$

3. $\csc \left(x\right)$

4. $1$

Q. If $x=-1$ and $x=2$ are extreme points of $f\left(x\right)=\alpha \log |x|+\beta x^2+x$ then :

1. $\alpha =-6,\beta =\frac{1}{2}$
2. $\alpha =-6,\beta =-\frac{1}{2}$
3. $\alpha =2,\beta =-\frac{1}{2}$
4. $\alpha =2,\beta =\frac{1}{2}$

Q.

What is the domain of definition of $f\left(x\right)=\frac{\left[{\log }_2\left(x+3\right)\right]}{\left(x^2+3x+2\right)}$?

1. $R-\left\{-1,-2\right\}$

2. $\left(-2,\infty \right)$

3. $R-\left\{-1,-2,-3\right\}$

4. $\left(-3,\infty \right)-\left\{-1,-2\right\}$

Q. Let $f$ be a function defined on $R$ such that
$f\text{'}\left(x\right)=2010(x-2009)(x-2010{)}^2$$(x-2011{)}^3(x-2012{)}^4,$ for all $x\in R.$ If $g$ is a function defined on $R$ with values in the interval $(0,\infty )$ such that $f\left(x\right)=\ln \left(g\right(x\left)\right)$ for all $x\in R$, then the number of points in $R$ at which $g$ has a local maximum is

Q. If $f\left(x\right)$ is an odd differentiable function defined on $(-\infty ,\infty )$ such that $f^{\prime }\left(3\right)=2$ , then $f^{\prime }(-3)$ equal to

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

Q. Let p(x) be a polynomial of degree 4 having extremum at x=1, 2 and $\underset{x\to 0}{lim}(1+\frac{p\left(x\right)}{x^2})=2$. Then the value of p(2) is ___

Q.

Let $f\left(x\right)$ be a polynomial function of the second degree. If $f\left(1\right)=f(-1)$ and $a_1,a_2,a_3$ are in AP, then $f\left(a_1\right),f\left(a_2\right),f\left(a_3\right)$ are in

1. AP

2. GP

3. HP

4. None of these

Q. In a third order matrix $A,a_{ij}$ denotes the element in the $i-\text{th}$ row and $j-\text{th}$ column.
If $a_{ij}=0$ for $i=j$
$=1$ for $i>j$
$=–1$ for $i<j$
Then the matrix is

1. skew symmetric
2. symmetric
3. not invertible
4. non-singular

Q.

The solution set of ${\left(\frac{3}{5}\right)}^x=x-x^2-9$ is

1. $\left\{0\right\}$

2. $\left\{1\right\}$

3. $\mathrm{\varphi }$

4. None of these

Q.

If $x+\frac{1}{x}=2$ then $x^2+\frac{1}{x^2}=?$

Q. Let $f\left(x\right)=\sin x+|\cos x|-|sinx-|cosx||$, $xϵ(0,2\pi )$ then the number of points of non-differentiability of $f\left(x\right)$ is

Q. A circle of radius $2\sqrt{10}$ cm is divided by a chord of length $12$ cm into two segments. Then maximum area of a rectangle that can be inscribed into the smaller circular segment is

1. $3\sqrt{15}$
2. $2\sqrt{15}$
3. $4\sqrt{15}$
4. $6\sqrt{15}$

Q. If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f\left(x\right)=\underset{0}{\overset{x^2}{\int }}\frac{t^2-5t+4}{2+e^t}dt$, then the ordered pair $(m,n)$ is equal to

1. $(2,2)$
2. $(3,2)$
3. $(3,4)$
4. $(2,3)$

Q.

The function f(x) = e^x

1. Has a local maximum but no local minimum value

2. Has both , a local minimum and a local maximum value

3. Has a local minimum but no local maximum value

4. Has neither a local minimum nor a local maximum value

Q.

If $f\left(x\right)=x^3-6x^2-36x+2$ is decreasing function, then $x\in$

1. $\left(6,\infty \right)$

2. $\left(-\infty ,-2\right)$

3. $\left(-2,6\right)$

4. None of these

Q. 5. Find the maximum and minimum values of sin inverse x + tan inverse x

Q. Dear sir/madam,
how to find the integral of sin³ x/(cos⁴x+3cos²x+1)tan^-1 (secx + cosx)dx
It was there in the objective test 1, but I didn't understand the given solution. So can you please give me a detailed solution to this problem.

Q. $\mathrm{If}x=a\left(\cos t+t\sin t\right)\mathrm{and}y=a\left(\sin t-t\cos t\right),\mathrm{then}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\frac{d^{2}y}{d{x}^2}\mathrm{at}t=\frac{\mathrm{\pi }}{4}.$

Q. (i) ${\sin }^{-1}\left(\sin \frac{\mathrm{\pi }}{6}\right)$
(ii) ${\sin }^{-1}\left(\sin \frac{7\mathrm{\pi }}{6}\right)$
(iii) ${\sin }^{-1}\left(\sin \frac{5\mathrm{\pi }}{6}\right)$
(iv) ${\sin }^{-1}\left(\sin \frac{13\mathrm{\pi }}{7}\right)$
(v) ${\sin }^{-1}\left(\sin \frac{17\mathrm{\pi }}{8}\right)$
(vi) ${\sin }^{-1}\left\{\left(\sin -\frac{17\mathrm{\pi }}{8}\right)\right\}$
(vii) ${\sin }^{-1}\left(\sin 3\right)$
(viii) ${\sin }^{-1}\left(\sin 4\right)$
(ix) ${\sin }^{-1}\left(\sin 12\right)$
(x) ${\sin }^{-1}\left(\sin 2\right)$

Q. Let $f\left(x\right)=\left|\right(x-1\left)(x^2-2x-3)\right|+x-3,x\in R$. If $m\text{and}M$ are respectively the number of points of local minimum and local maximum of $f$ in the interval $(0,4)$, then $m+M$ is equal to

Q. ${\int }_0^{2\mathrm{\pi }}\sqrt{1+\sin \frac{x}{2}}dx$

Q. State whether $-2$ is a point of local maxima or a point of local minima on the curve $y=2x^3+3x^2-12x+12.$

Q.

If $f\left(x\right)=alo{g}_e|x|+b{x}^2+x$ has extremum at x = 1 and x = 3, then

1. a = -3/4, b= -1/8

2. a=3/4, b= -1/8

3. None of these

4. a =-3/4, b=1/8