Local Minima

Q. Let $A=\left[\begin{array}{ccc}2& b& 1\\ b& b^2+1& b\\ 1& b& 2\end{array}\right]$ where $b>0$. Then the minimum value of $\frac{det\left(A\right)}{b}$ is :

1. $-2\sqrt{3}$
2. $-\sqrt{3}$
3. $\sqrt{3}$
4. $2\sqrt{3}$

Q.

How do I find the value of $\sin \left(\frac{5\pi }{6}\right)$.

Q.

If $\underset{x\to 0}{lim}\frac{a{e}^x-b\cos x+c{e}^{-x}}{x\sin x}=2$, then $a+b+c$ is equal to:

Q.

Assuming that the petrol burnt in a motor boat varies as the cube of its velocity, the most economical speed, when going against a current of c km/h is

1. (c/2) km/h

2. (5c/2) km/h

3. (3c/4) km/h

4. (3c/2 ) km/h

Q.

The area (in square unit) of the triangle formed by $x+y+1=0$ and the pair of straight lines $x^2-3xy+2y^2=0$ is

1. $\frac{7}{12}$

2. $\frac{5}{12}$

3. $\frac{1}{12}$

4. $\frac{1}{6}$

Q.

Evaluate:$\underset{x\to 0}{\lim }\frac{e^x-e^{\sin x}}{x-\sin x}$

1. $-1$

2. $0$

3. $1$

4. None of these

Q. Let $f\left(x\right)$ be a polynomial function of degree $4,$ vanishes at $x=-1$. If $f\left(x\right)$ has local maxima/minima at $x=1,2,3$ and $\underset{2}{\overset{-2}{\int }}f\left(x\right)dx=\frac{1348}{15}.$ Then

1. the $y$-intercept of the tangent to the curve at $x=-1$ equals $-96$
2. the maximum value of $f\left(x\right)$ is $-63$
3. $\underset{-1}{\overset{1}{\int }}[f\left(x\right)+f(-x)]dx=-\frac{1424}{15}$
4. the minimum value of $f\left(x\right)$ is $-64$

Q. Let $f:R\to [1,\infty )$ be a quadratic surjective function such that $f(2+x)=f(2-x)$ and $f\left(1\right)=2.$ Let $g:(-\infty ,\ln 2]\to [1,5]$ be another function defined as $g\left(\ln x\right)=f\left(x\right),$ then which of the following(s) is/are correct ?

1. minimum value of $g^{\prime }\left(x\right)$ is $-2$.
2. $g^{-1}\left(x\right)=\ln (2+\sqrt{x-1})$
3. $g^{-1}\left(x\right)=\ln (2-\sqrt{x-1})$
4. the sum of the values of $x$ that satisfying the equation $f\left(x\right)=5$ is $4.$

Q. If the complex number z satisifies the condition $\left|z\right|\ge 3$, then the least value of $|z+(1/z\left)\right|$ is equal is

1. 
2. 
3. 
4. None of these

Q.

If $a<\frac{1}{32}$, then the number of solutions of ${\left({\sin }^{-1}x\right)}^3+{\left({\cos }^{-1}x\right)}^3=a{\pi }^3$ is

1. $0$

2. $1$

3. $2$

4. $Infinite$

Q.

The area (in square units) of the quadrilateral formed by the two pairs of lines$l^2{x}^2-m^2{y}^2-n(lx+my)=0$ and$l^2{x}^2-m^2{y}^2+n(lx-my)=0$

1. $\frac{n^2}{2\left|lm\right|}$

2. $\frac{n^2}{\left|lm\right|}$

3. $\frac{n}{2\left|lm\right|}$

4. $\frac{n^2}{4\left|lm\right|}$

Q. Value of $\underset{x\to 0}{lim}\frac{x\log (1+7x)}{1-\cos 3x}=$

1. $\frac{5}{9}$
2. $\frac{14}{9}$
3. $\frac{14}{5}$
4. $\frac{7}{5}$

Q.
Consider the functions $f\left(x\right)=\{\begin{array}{c}x+1,x\le 1\\ 2x+1,1<x\le 2\end{array}$

$g\left(x\right)=\{\begin{array}{c}{x}^2,-1\le x<2\\ x+2,2\le x\le 3\end{array}$

The number of roots of the equation $f\left(g\right(x\left)\right)=2$ is

Q. Let $f\left(x\right)=|x^2-4x+3|$ be a function defined on $x\in [0,4]$ and $\alpha ,\beta ,\gamma$ are the abscissas of the critical points of $f\left(x\right).$ If $m$ and $M$ are the local and absolute maximum values of $f\left(x\right)$ respectively, then the value of ${\alpha }^2+{\beta }^2+{\gamma }^2+m^2+M^2$ is

Q.

$\text{The set of values of}x\text{which satisfy}$
$5x+2<3x+8\text{and}\frac{x+2}{x-1}<4\text{is}$,

1. $(2,3)$

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

3. $(-\infty ,1)$

4. $(1,3)$

Q. The local maxima of the funtion $f\left(x\right)=\sin x+\cos x\forall x\in [0,\frac{\pi }{2}]$ is

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

Q. If $f\left(x\right)={\cos }^{2}x\cdot e^{\tan x},x\in (-\frac{\pi }{2},\frac{\pi }{2})$, then

1. $f^{\prime }\left(x\right)$ has a point of local minimum at $x=\frac{\pi }{4}$
2. $f^{\prime }\left(x\right)$ has a point of local maximum in $(-\frac{\pi }{4},0)$
3. $f^{\prime }\left(x\right)$ has exactly two points of extrema.
4. $f^{\prime \prime }\left(x\right)=0$ has no real roots.

Q.

How do you solve by completing the square: $a{x}^2+bx+c=0$.

Q.

The area (in sq units) of the region bounded by the curves $y^2=4ax$ and $x^2=4ay$, $a>0$ is

1. $16\left(\frac{a^2}{3}\right)$

2. $14\left(\frac{a^2}{3}\right)$

3. $13\left(\frac{a^2}{3}\right)$

4. $16a^2$

5. None of these

Q. $\underset{x\to 0}{lim}\frac{\log (1+x+x^2)+\log (1-x+x^2)}{\sec x-\cos x}=$

Q. Let a function $f:[0,5]\to R$, be continuous, $f\left(1\right)=3$ and $F$ be defined as:
$F\left(x\right)=\underset{1}{\overset{x}{\int }}{t}^{2}g\left(t\right)dt$, where $g\left(t\right)=\underset{1}{\overset{t}{\int }}f\left(u\right)du$. Then for the function $F$, the point $x=1$ is

1. a point of local maxima
2. a point of local minima
3. a point of inflection
4. not a critical point

Q. The coordinates of the point on the curve $x^3=y(x-a{)}^2,a>0$ where the ordinate is minimum

1. $(2a,8a)$
2. $(-2a,\frac{-8a}{9})$
3. $(3a,\frac{27a}{4})$
4. $(-3a,\frac{-27a}{16})$

Q. If $f\left(x\right)=x^2+\frac{1}{x^2}\text{and}g\left(x\right)=x-\frac{1}{x},xϵR-\{-1,0,1\},\text{and}h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ then the local minimum value of $h\left(x\right)$ is

1. $-2\sqrt{2}$
2. 3
3. -3
4. $2\sqrt{2}$

Q. Let $A=\left[\begin{array}{ccc}2& b& 1\\ b& b^2+1& b\\ 1& b& 2\end{array}\right]$ where $b>0$. Then the minimum value of $\frac{det\left(A\right)}{b}$ is :

1. $-2\sqrt{3}$
2. $-\sqrt{3}$
3. $\sqrt{3}$
4. $2\sqrt{3}$

Q. If $2{\sin }^{-1}{x}_0+{\tan }^{-1}{x}_0=\frac{5\pi }{\sqrt{-x_0^2+2x_0+15}},$ then the possible value(s) of $\frac{dy}{dx}$ to the curve $y=\frac{2x}{y+x}$ at $x=x_0$ is (are)

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

Q. If the area enclosed by the parabolas y² = 16ax and x² = 16ay, a > 0 is $\frac{1024}{3}$ square units, find the value of a.

Q. Let $f\left(x\right)=\left|\begin{array}{ccc}\sin x& 0& 2\cos x\\ 0& \sin x& 0\\ 2\cos x& 0& \sin x\end{array}\right|$ where $x\in (0,\pi ).$ Then total number of local maxima and local minima of $f\left(x\right)$ is

Q. If A + B + C = $\pi$, then prove that,
$sin\frac{A}{2}+sin\frac{B}{2}+sin\frac{C}{2}=1+4sin\left[\frac{\pi -A}{4}\right]sin\left[\frac{\pi -B}{4}\right]sin\left[\frac{\pi -C}{4}\right]$

Q. Let $A=\left[\begin{array}{ccc}2& b& 1\\ b& b^2+1& b\\ 1& b& 2\end{array}\right]$ where $b>0$. Then the minimum value of $\frac{det\left(A\right)}{b}$ is :

1. $-2\sqrt{3}$
2. $-\sqrt{3}$
3. $\sqrt{3}$
4. $2\sqrt{3}$

Q. Let $A$ and $B$ be two points with position vector $2\hat{i}+\hat{j}-3\hat{k}$ and $\beta \hat{i}+{\alpha }^2\hat{j}$ respectively such that $|\overrightarrow{AB}|=3$ then $\underset{x\to \beta }{lim}(x-{\alpha }^2{)}^{\frac{1}{x-(\alpha +1)}}$ may be equal to:

1. $1$
2. $e$
3. $e^2$
4. $3$