Local Maxima

Q.

Find the principal value of ${\sin }^{-1}\left(-\frac{1}{2}\right)$

Q.

What is the value of $\log \left(100\right)$?

Q. The area between the parabolas $y^2=4ax,x^2=4ay$ is

1. $\frac{16a^2}{3}$
2. $\frac{16a^2}{2}$
3. $\frac{10a^2}{3}$
4. $\frac{16a^2}{5}$

Q. The eccentricity of the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$ is

1. $\frac{2\sqrt{5}}{4}$
2. $\frac{2\sqrt{13}}{6}$
3. $\frac{2\sqrt{5}}{6}$
4. $\frac{2\sqrt{13}}{4}$

Q.

If $30,000c{m}^2$ of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Q.

The value of $\tan \left[{\cos }^{-1}\left(-\frac{2}{7}\right)-\frac{\mathrm{\pi }}{2}\right]$ is

1. $\frac{2}{3\sqrt{5}}$

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

3. $\frac{1}{\sqrt{5}}$

4. $\frac{4}{\sqrt{5}}$

Q.

If $\alpha$ and $\beta$ are the roots of the equation $x^2-7x+1=0$, then the value of $1/{(\alpha -7)}^2+1/{(\beta -7)}^2$ is

1. $45$

2. $47$

3. $49$

4. $50$

5. $51$

Q.

If $a,b,c,d\&p$are distinct real numbers such that $(a^2+b^2+c^2)p^2-2(ab+bc+cd)p+(b^2+c^2+d^2)\le 0,$Then $a,b,c\&d$

1. $areinarithmeticprogression$

2. $areingeometricprogression$

3. $areinharmonicprogression$

4. $satisfyab=cd$

Q.

The smallest value of k, for which both the roots of the equation $x^2-8kx+16(k^2-k+1)$ = 0 are real, distinct and have values at least 4, is

1. 1

2. 2

3. 3

4. 0

Q.

If$\alpha$ and$\beta$ are the roots of the equation$2x(2x+1)=1$, $\beta$ then is equal to:

1. $2\alpha (\alpha -1)$

2. $-2\alpha (\alpha +1)$

3. $2{\alpha }^2$

4. $2\alpha (\alpha +1)$

Q.

What does twice differentiable mean?

Q.

If $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right],\hspace{0.17em}A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right],$ and $Q=PA{P}^T$ then find the value of $P^T{Q}^{2005}P$ is equal to

1. $\left[\begin{array}{cc}1& 2005\\ 0& 1\end{array}\right]$

2. $\left[\begin{array}{cc}1& 2005\\ 2005& 1\end{array}\right]$

3. $\left[\begin{array}{cc}1& 0\\ 2005& 1\end{array}\right]$

4. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Q.

If ${}^{n-1}{C}_3{+}^{n-1}{C}_4{>}^n{C}_3$ the $n$ is just greater than

1. $5$

2. $6$

3. $4$

4. $7$

Q. The length of the latus-rectum of the ellipse $3x^2+y^2=12$ is $\underset{–}{}$

Q.

The function $f\left(x\right)=\frac{\log \left(x\right)}{x}$ is increasing in the interval

1. $\left(1,2e\right)$

2. $\left(0,e\right)$

3. $\left(2,2e\right)$

4. $\left(\frac{1}{e},2e\right)$

Q.

For real $\mathrm{x},\mathrm{if}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3+5\mathrm{x}+1,$ then

1. $f$ is one-one but not onto $R$

2. $f$ is onto $R$ but not one-one

3. $f$is one-one and onto $R$

4. $f$is neither one-one nor onto $R$

Q.

$limi{t}_{x\to 2}\left(\frac{3^x+3^{3-x}-12}{3^{-{\displaystyle \frac{x}{2}}}-3^{1-x}}\right)=$

Q.

The value of$i–i^2+i^3–i^4+\dots –i^{100}$ equal is

1. $i$

2. $-i$

3. $0$

4. $1+i$

Q.

If $\mathrm{y}=\sqrt{(\mathrm{x}{\log }_{\mathrm{e}}\mathrm{x})}$ then $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x}=\mathrm{e}$ is

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

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

3. $\sqrt{e}$

4. $e^2$

Q.

If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$are less than $5$, then $\mathrm{k}$ lies in the interval

1. $\left(4,5\right)$

2. $\left(-\infty ,4\right)$

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

4. $\left(5,6\right)$

Q.

Newton-Raphson method is applicable to the solution of which equations?

Q. The maximum value of the function $f\left(x\right)=3x^3-18x^2+27x-40$ on the set $S=\{x\in R:x^2+30\le 11x\}$ is :

1. $122$
2. $-122$
3. $222$
4. $-222$

Q.

If for the real value of x, $\cos \left(\theta \right)=\mathrm{x}+\frac{1}{\mathrm{x}}$, then

1. $\mathrm{\theta }$ is an acute angle

2. $\mathrm{\theta }$ is a right angle

3. $\mathrm{\theta }$ is an obtuse angle

4. No value of $\mathrm{\theta }$ is possible

Q.

The value of $$x$$at which the first derivative of the function${\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)}^2$w.r.t. $$x$$ is $${\displaystyle \frac{3}{4}}$$are

1. $$\pm 2$$

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

3. $$\pm {\displaystyle \frac{2}{\sqrt{3}}}$$

4. $$\pm {\displaystyle \frac{\sqrt{3}}{2}}$$

Q.

If $f\left(x\right)=g\left(x\right)$ and $g\left(x\right)=–f\left(x\right)$for all $x$ and$f\left(2\right)=4=f\left(2\right)$, then $f^2\left(4\right)+g^2\left(4\right)$ is equal to

1. $8$

2. $16$

3. $32$

4. $64$

Q. Let f (x) = sinx + ax + b. Then f(x) = 0 has

1. only one real root which is positive if a > 1, b < 0
2. only one real root which is negative if a > 1, b < 0
3. only one real root which is negative if a < -1, b > 0
4. None of these .

Q. Minimum distance between the curves $y^2=x-1$ and $x^2=y-1$ is $\frac{a\sqrt{2}}{b}$, where $a,b$ are co-prime numbers, then the value of $a+b$ is

Q.

Consider the function $f:(-\infty ,\infty )\to (-\infty ,\infty )$ defined by$f\left(x\right)=\frac{[x^2-ax+1]}{[x^2+ax+1]};0<a<2$. Which of the following is true

1. ${(2+a)}^{2}f\left(1\right)+{(2–a)}^{2}f(-1)=0$

2. ${(2–a)}^{2}f\left(1\right)–{(2+a)}^{2}f(-1)=0$

3. $f\left(1\right)f(-1)={(2–a)}^2$

4. $f\left(1\right)f(-1)={-(2+a)}^2$

Q. The set of all real values of $x$ for which the function $f\left(x\right)=\sqrt{\sin x+\cos x}+\sqrt{7x-x^2-6}$ takes real values is

1. $[1,\frac{3\pi }{4}]\cup [\frac{7\pi }{4},6]$
2. $[1,6]$
3. $[1,\frac{\pi }{4}]\cup [\frac{7\pi }{4},6]$
4. $[1,6]-\{\frac{3\pi }{4},\frac{7\pi }{4}\}$

Q.

The range of$a\in R$ for which the function $f\left(x\right)=(4a–3)(x+{\log }_{e}5)+2(a–7)cot\left(\frac{x}{2}\right){\sin }^2\left(\frac{x}{2}\right),x\ne 2n\pi ,n\in N$has critical points is

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

1. $\left[-\frac{4}{3},2\right]$

2. $[1,\infty )$

3. $(-\infty ,–1]$

4. $(–3,1)$