AM,GM,HM Inequality

Q.

If $\omega =\frac{1+\sqrt{3}i}{2}$, then ${\left(3+\omega +3{\omega }^2\right)}^4$ is

1. $16$

2. $-16$

3. $16\omega$

4. $16{\omega }^2$

Q.

Simplify ${\left(x+2\right)}^3=2x\left(x^2-1\right)$ and check whether it is a quadratic equation

Q. If $x,y,z$ are positive numbers, then the minimum value of $(x+y)(y+z)(z+x)(\frac{1}{x}+\frac{1}{y})(\frac{1}{y}+\frac{1}{z})(\frac{1}{z}+\frac{1}{x})$ is

1. $32$
2. $64$
3. $128$
4. $356$

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. Let $f\left(x\right)=0$ be a cubic equation with positive and distinct roots $\alpha ,\beta ,\gamma$ such that $\beta$ is harmonic mean between the roots of $f^{\prime }\left(x\right)=0.$ If $r=\left[\frac{2\beta }{\alpha +\gamma }\right]+\left[\frac{2\alpha \gamma }{\alpha \beta +\beta \gamma }\right],$ then the value of $\sum _{i=1}^3{i}^r$ is
( Here, $[.]$ denotes the greatest integer function.)

Q. The minimum value of $f\left(x\right)=a^{a^x}+a^{1-a^x}$, where $a,x\in R$ and $a>0$, is equal to:

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

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. Let $A=\left[a_{ij}\right]$ be a real matrix of order $3\times 3$, such that $a_{i1}+a_{i2}+a_{i3}=1$, for $i=1,2,3.$ Then, the sum of all the entries of the matrix $A^3$ is equal to:

Q. If $a,b,c$ are positive numbers such that $a+b+c=18,$ then the maximum value of $a^2{b}^3{c}^4$ is

1. $2^{19}\cdot 3^3$
2. $2^{20}\cdot 3^4$
3. $2^{18}\cdot 3^4$
4. $2^{19}\cdot 3^4$

Q. If $P=\{x:x<3,x\in N\},Q=\{x:x\le 2,x\in W\},$ then $(P\cup Q)\times (P\cap Q)$ is

1. $\left\{\right(0,1),(0,2),(1,1),(1,2),(2,1),(2,2\left)\right\}$
2. $\left\{\right(0,1),(0,2),(0,3),(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)\}$
3. $\left\{\right(0,1),(0,2),(0,3\left)\right\}$
4. $\varphi$

Q. The product of $n$ positive numbers is unity, then their sum will be

1. a positive integer
2. equal to ${}^{\prime }{n^2}^{\prime }$
3. never less than ${}^{\prime }{n}^{\prime }$
4. divisible by ${}^{\prime }{n}^{\prime }$

Q. If $x,y,z$ are positive numbers, then the minimum value of $(x+y)(y+z)(z+x)(\frac{1}{x}+\frac{1}{y})(\frac{1}{y}+\frac{1}{z})(\frac{1}{z}+\frac{1}{x})$ is

1. $32$
2. $64$
3. $128$
4. $356$

Q.

If $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c},$ then a, b, c, are in

1. A.P

2. G.P

3. none of these

4. H.P

Q. If $f\left(x\right)=\frac{1}{x},g\left(x\right)=\frac{1}{x^2}$ and $h\left(x\right)=x^2$, then which of the following is always true?

1. $h\left(g\right(x\left)\right)=\frac{1}{x^2}\forall x\ne 0,fog\left(x\right)=x^2$
2. $fog\left(x\right)=x^2\forall x\ne 0,h\left(g\right(x\left)\right)=\left(g\right(x){)}^2\forall x\ne 0$
3. None of these
4. $fog\left(x\right)=x^2\forall x\ne 0,h\left(g\right(x\left)\right)=\frac{1}{x^2}$

Q. Minimum value of the expresssion $3^{2x}+3^{-2x}$ is

Q. Let $b_i>1fori=1,2,...,101.$ Suppose $lo{g}_e{b}_1.lo{g}_e{b}_2,......,lo{g}_e{b}_{101}$ are in Arithmetic Progression (A.P) with the common diffrence $lo{g}_{e}2.$ Suppose $a_1,a_2,.....,a_{101}$ are in A.P. such that $a_1=b_1$ and $a_{51}=b_{51}.$ If $t=b_1+b_2+...+b_{51}$ and $s=a_1+a_2+......+a{5}_3.$ then

1. $s>tand{a}_{101}>b_{101}$
2. $s>tand{a}_{101}<b_{101}$
3. $s<tand{a}_{101}>b_{101}$
4. $s<tand{a}_{101}<b_{101}$

Q. If a, b, c, d be in H.P., then

1. $a^2+c^2>b^2+d^2$
2. $a^2+d^2>b^2+c^2$
3. $ac+bd>b^2+c^2$
4. $ac+bd>b^2+d^2$

Q. If $x+y+z=1$, $x,y,z>0$. Then greatest value of $x^2{y}^3{z}^4$ is

1. $\frac{2^9}{3^5}$
2. $\frac{2^{10}}{3^{15}}$
3. $\frac{2^{15}}{3^{10}}$
4. $\frac{2^{10}}{3^{10}}$

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

Q. Let $a,b,c,d\in R^{+}$ and $256abcd\ge (a+b+c+d{)}^4$ and $3a+b+2c+5d=11$ then $a^3+b+c^2+5d$ is

1. $15$
2. $8$
3. $11$
4. $20$

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 p and q are positive real numbers such that $p^2+q^2=1,$ then the maximum value of (p+q) is

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

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

3. $\sqrt{2}$

4. 2.

Q. Let $b_i>1$ for i = 1, 2, .... , 101. Suppose $lo{g}_e{b}_1,lo{g}_e{b}_2,\dots \dots lo{g}_e{b}_{101}$ are in AP with the common difference $lo{g}_{e}2$. Suppose $a_1,a_2,\dots \dots a_{101}$ are in AP, such that $a_1=b_1$ and $a_{51}=b_{51}$. If $t=b_1+b_2+\dots \dots +b_{51}$ and $s=a_1+a_2+\dots \dots +a_{51}$, then

1. $s>t$ and $a_{101}>b_{101}$
2. $s>t$ and $a_{101}<b_{101}$
3. $s<t$ and $a_{101}>b_{101}$
4. $s<t$ and $a_{101}<b_{101}$

Q. The product of $n$ positive numbers is unity, then their sum will be

1. a positive integer
2. divisible by ${}^{\prime }{n}^{\prime }$
3. equal to ${}^{\prime }{n^2}^{\prime }$
4. never less than ${}^{\prime }{n}^{\prime }$

Q.

If $A$ is the area and $2s$ the sum of 3 sides of triangle then

1. $A\le \frac{s^2}{3\left(\sqrt{3}\right)}$

2. $A\le \frac{s^2}{2}$

3. $A>\frac{s^2}{\left(\sqrt{3}\right)}$

4. None of these

Q. If a, b, c be in H.P., then

1. $a^2+c^2>b^2$
2. $a^2+b^2>2c^2$
3. $a^2+c^2>2b^2$
4. $a^2+b^2>c^2$

Q. For two positive real number's $a$ and $b$, If the A.M. exceeds their G.M. by $2$ and the G.M. exceeds their H.M. by $\frac{8}{5}$, then the value of $a+b$ is

1. $20$
2. $30$
3. $40$
4. $50$

Q. If ${\sin }^4\alpha +4{\cos }^4\beta +2=4\sqrt{2}\sin \alpha \cos \beta$;
$\alpha ,\beta \in [0,\pi ]$, then $\cos (\alpha +\beta )-\cos (\alpha -\beta )$ is equal to :

1. $\sqrt{2}$
2. $-\sqrt{2}$
3. $0$
4. $-1$