AM,GM,HM Inequality

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

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

Q. The minimum value of the sum of real numbers $a^{-5},a^{-4},3a^{-3},1,a^8$ and $a^{10}$ with a > 0 is___

Q. If a, b, c, d are in H.P., then [RPET 1991]

1. a + d > b + c
2. ad > bc
3. None of these

4. Both (a) and (b)

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. 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 x + y + z = 1 and x, y, z are positive numbers such that $(1-x)(1-y)(1-z)\ge kxyz$, then exact value of k is equal to

1. 4
2. 8
3. 16
4. 2

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 $x,y,z\in R^{+}$ then $\frac{xy}{x+y}+\frac{yz}{y+z}+\frac{xz}{x+z}$ is always

1. $\le 2(x+y+z)$

2. $\le \frac{1}{2}(x+y+z)$

3. $\le 4(x+y+z)$

4. None of the above

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.

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

1. 2.

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

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

4. $\sqrt{2}$