AM,GM,HM Inequality

Q. Suppose p denotes the perimeter of a triangle with sides a, b and c and s is the semi perimeter. If A denotes the area of the triangle, then which of the given options is correct?

1. $\frac{\sqrt{p(p+a)(p+b)(p+c)}}{A}$ = 4
2. $\frac{\sqrt{p(p-2a)(p-2b)(p-2c)}}{A}$ = 4
3. $\frac{\sqrt{p(p-2a)(p-2b)(p-2c)}}{A}$ = 9
   
4. $\frac{\sqrt{p(p+a)(p+b)(p+c)}}{A}$ = 9
   

Q. If one root is square of the other root of the equation $x^2+px+q=0$, then find the relation between $p$ and $q$.

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

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

Q.

Why addition and multiplication follow the property of closure, but not subtraction and division? Explain with examples. [2 MARKS]

Q. If variable takes values 0, 1, 2, ......, n with frequencies $q^n$. $\frac{n}{1}$ $q^{n-1}$ p , $\frac{n(n-1)}{1.2}$ $q^n-2$ $p^2$.........., $p^n$ where p+q = 1, then the mean is

1. $pq$
2. $np$
3. $nq$
4. $n{p}^2$

Q. If $p^{1-2x}\times \sqrt{p}=q^{1-2x}\times \sqrt{q};p\ne q$., then the value of $x$ is $\frac{3}{m}$. The value of $m$ is:

Q. The NAV of a unit in mutual fund scheme is $Rs.10.65$ then find the amount required to buy $500$ such units.

1. $5325$
2. $532500$
3. $5235$
4. $53250$

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.

Question 81

In the following question, state whether the given statements are true (T) or false (F).

$a^p\times b^q=(ab{)}^{pq}$

Q.

Add the following :

$$\begin{gathered} 3x^2+3y^2+4xy \\ 2x^2+5y^2-xy \\ -4x^2-7y^2-xy \\ \overline{} \end{gathered}$$

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

1. a + d > b + c
2. ad > bc

3. Both (a) and (b)

4. None of these

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. 2
3. 8
4. 16