AM,GM,HM Inequality

Q.

If $f\left(x\right)=\frac{{\sin }^{2}x}{1+\cot x}+\frac{{\cos }^{2}x}{1+\tan x}$, then $f\left(\frac{\mathrm{\pi }}{4}\right)$ is equal to

1. $\sqrt{3}$

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

3. $0$

4. $-\sqrt{3}$

Q. Let $a,b,c,d$ be in arithmetic progression with common difference $\lambda .$ If $\left|\begin{array}{ccc}x+a-c& x+b& x+a\\ x-1& x+c& x+b\\ x-b+d& x+d& x+c\end{array}\right|=2,$ then value of ${\lambda }^2$ is equal to

Q.

The maximum value of the term independent of ‘$t$’ in the expansion of ${\left[t{x}^{\frac{1}{5}}+\left\{\frac{{\left(1-x\right)}^{{\displaystyle \frac{1}{10}}}}{t}\right\}\right]}^{10}$ where $x\in \left(0,1\right)$ is :

1. $\frac{10!}{\sqrt{3}{\left(5!\right)}^2}$

2. $\frac{2·10!}{3{\left(5!\right)}^2}$

3. $\frac{10!}{3{\left(5!\right)}^2}$

4. $\frac{2·10!}{3\sqrt{3}{\left(5!\right)}^2}$

Q.

If $a,b,c,d$ are four positive real numbers such that $abcd=1$, then the minimum value of $(1+a)(1+b)(1+c)(1+d)$ is equal to_________?

Q.

The number of solutions of the equation ${\sin }^{-1}\left(\left[x^2+\frac{1}{3}\right]\right)+{\cos }^{-1}\left(\left[x^2-\frac{2}{3}\right]\right)=x^2$ for $x\in \left[-1,1\right]$, and $\left[x\right]$ denotes the greatest integer less than or equal to $x$, is:

1. $0$

2. $2$

3. $4$

4. Infinite

Q.

Prove that $ta{n}^{2}A+co{t}^{2}A+2=cose{c}^{2}Ase{c}^{2}A$

Q. Show that :
$\left(i\right){\sin }^{-1}\left(2x\sqrt{1-x^2}\right)=2{\sin }^{-1}x,-\frac{1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}}$

$\left(ii\right){\sin }^{-1}\left(2x\sqrt{1-x^2}\right)=2{\cos }^{-1}x,\frac{1}{\sqrt{2}}\le x\le 1$

Q.

In a triangle $PQR$, let $\overrightarrow{a}=\overrightarrow{QR}$, $\overrightarrow{b}=\overrightarrow{RP}$ and $\overrightarrow{c}=\overrightarrow{PQ}$. If $\left|\overrightarrow{a}\right|=3$, and $\left|\overrightarrow{b}\right|=4$ and $\frac{\overrightarrow{a}·\left(\overrightarrow{c}-\overrightarrow{b}\right)}{\overrightarrow{c}·\left(\overrightarrow{a}-\overrightarrow{b}\right)}=\left|\frac{\overrightarrow{a}}{\overrightarrow{a}+\overrightarrow{b}}\right|$. Then the value of ${\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2$ is?

Q. The number of ordered triplets of natural numbers $(a,b,c)$ for which $abc\le 11$ is

1. $52$
2. $53$
3. $55$
4. $56$

Q.

If the first and ${(2n-1)}^{th}$ term of a A.P., G.P. and H.P. are equal and their $n^{th}$ terms are $a,b,c$ respectively, then

1. $a=b=c$

2. $a\ge b\ge c$

3. $a+c=b$

4. $ac-b^2=0$

Q. If $△ABC$ is an acute angle triangle, then the minimum value of $\tan A+\tan B+\tan C$ is

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

Q.

Find the Cube Root of $1000000$

Q. If $a,b,c$ are non zero real numbers, then minimum value of the expression$\left(\frac{(a^4+a^2+1)(b^4+7b^2+1)(c^4+11c^2+1)}{\left(a^2{b}^2{c}^2\right)}\right)$ is

1. $415$
2. $451$
3. $315$
4. $351$

Q.

If $G_1$ and $G_2$ are two geometric means and $A$ the arithmetic mean inserted between two numbers, then the value of $\frac{{G_1}^2}{G_2}+\frac{{G_2}^2}{G_1}$ is

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

2. $A$

3. $2A$

4. None of these

Q.

Triangle $ABC$ has $AB=90,BC=50$ and $CA=70.$ A circle is drawn with centre $P$ on $AB$ such that $CA$ and $CB$ are tangents to the circle. Find $\frac{2}{3}AP.$
(correct answer + 3, wrong answer 0)

Q.

The feasible solution for a LPP is shown in following figure. Let Z=3x-4y be the objective funciton. Minimum of Z occurs at

(a)(0, 0)
(b)(0, 8)
(c)(5, 0)
(d)(4, 10)

Q. If $f\left(x\right)=\ln x;$ where $x\in R^{+},$ then

1. $f(x+y)=f\left(x\right)+f\left(y\right)$
2. $f(x+y)=f\left(x\right)\cdot f\left(y\right)$
3. $f(x\cdot y)=f\left(x\right)+f\left(y\right)$
4. $f(x\cdot y)=f\left(x\right)\cdot f\left(y\right)$

Q. A straight line through the vertex P of a traingle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then

1. $\frac{1}{PS}+\frac{1}{ST}<\frac{1}{\sqrt{QS\times SR}}$
2. $\frac{1}{PS}+\frac{1}{ST}>\frac{2}{\sqrt{QS\times SR}}$
3. $\frac{1}{PS}+\frac{1}{ST}<\frac{4}{QR}$
4. $\frac{1}{PS}+\frac{1}{ST}>\frac{4}{QR}$

Q. If $a,b,c\in R^{+}$ and $a^{2}bc+4abc+4bc=81$ such that ($2a+b+c+4$) assumes its least value, then

1. $a$ is a prime number
2. $a+b+c$ is a prime number
3. $b+c-a$ is a prime number
4. $a^3+b^3+c^3$ is a product of two prime numbers

Q. If a, b, c, d are positive real numbers such that a+b+c+d = 2, then M = (a+b)(c+d) satisfies the relation

1. 
2. 
3. 
4. 

Q. For a polynomial $g\left(x\right)$ with real coefficients, let $m_g$ denote the number of distinct real roots of $g\left(x\right)$. Suppose $S$ is the set of polynomial with real coefficients defined by
$S=\left[\right(x^2-1{)}^2(a_0+a_{1}x+a_2{x}^2+a_3{x}^3):a_0,a_1,a_2,a_3\in R]$
For a polynomial $f$, let $f^{\prime }$ and $f^{\prime \prime }$ denote first and second order derivatives, resepectively. Then the minimum possible value of $(m_{f^{\prime }}+m_{f^{\prime \prime }}),$ where $f\in S$, is

Q. Let $f\left(x\right)=2x^3-3(2+p)x^2+12px+\ln (16-p^2).$ If $f\left(x\right)$ has exactly one local maximum and one local minimum, then the number of possible integral values of $p$ is

Q.

How do you add two polar forms?

Q. $\text{If log (a + c) + log (a - 2b +c) = 2log(a -c), then a, b, c are in}$

1. $H.P.$
2. None of these
3. $G.P.$
4. $A.P.$

Q. If $S$ denotes the set of all real values of $x$ such that $(x^{2010}+1)(1+x^2+x^4+\cdots +x^{2008})=2010x^{2009}$, then the number of elements in set $S$ is

Q.

If $H_1$ and $H_2$ are two harmonic means between two positive numbers $a$ and $b\left(a\ne b\right)$. $A$ and $G$ are the arithmetic and geometric means between $a$ and $b$, then $\frac{H_2+H_1}{H_2{H}_1}$ is

1. $\frac{A}{G}$

2. $\frac{2A}{G}$

3. $\frac{A}{G^2}$

4. $\frac{2A}{G^2}$

Q. Let $p,q,r\in R^{+}$ such that $27pqr\ge (p+q+r{)}^3$ and $3p+4q+5r=12.$ Then the value of $p+q+r$ is

1. $3$
2. $6$
3. $2$
4. $12$

Q.

A unit vector in the plane of $i+2j+k$ and $i+j+2k$ and perpendicular to $2i+j+k$ is?

1. $j-k$

2. $\frac{\left(i+j\right)}{\sqrt{2}}$

3. $\frac{\left(j-k\right)}{\sqrt{2}}$

4. $5\left(j-k\right)$

Q.

Write down the coefficient of $x$ in the following :

$-3x^{2}y$

Q. If $a_1,a_2;g_1,g_2$ and $h_1,h_2$ are two arithmetic, geometric and harmonic means respectively between two quantities $a$ and $b$, then $ab$ is equal to

1. $g_1{g}_2$
2. $h_1{h}_2$
3. $a_1{h}_2$
4. $a_2{h}_1$