AM,GM,HM Inequality

Q.

A man has 7 friends. In how many ways he can invite one or more of them for a tea party.

1. 128

2. 127

3. 256

4. 130

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 polynomials with real coefficients defined by $S=\left\{{(x^2–1)}^2(a_0+a_{1}x+a_2{x}^2+a_3{x}^3):a_0,a_1,a_2,a_3\in R\right\}$

For a polynomial $f$, let $f_1$ and $f_2$ denote its first and second order derivatives, respectively.

Then the minimum possible value of $(m_{f1}+m_{f2})$, where $f\in S$, is _____

Q.

If the arithmetic, geometric and harmonic means between two positive real numbers be $A,G$ and $H$, then

1. $A^2=GH$

2. $H^2=AG$

3. $G=AH$

4. $G^2=AH$

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.

$li{m}_{x\to 0}\frac{(e^{1/x}-1)}{\left(e^{1/x}\right)+1}$

1. 1

2. 0

3. -1

4. Does not exist

Q.

If x be real, then the minimum value of $x^2-8x+17$ is

1. 2

2. -1

3. 0

4. 1

Q.

Differentiate $x{e}^x$ using first principle.

Q.

Prove that the following statement is true : If x, $y\in Z$ such that x and y are odd, then xy is odd.

Q.

If $a,b,c$are in GP and $x,y$ are arithmetic mean of $a,b$and $b,c$respectively, then $\frac{1}{x}+\frac{1}{y}$ is equal to

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

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

3. $\frac{b}{3}$

4. $\frac{b}{2}$

Q.

Three numbers, of which the third is equal to $36$, form a geometric progression. If $36$ is replaced with $20$, then the three numbers form an arithmetic progression. Find these three numbers.

Q.

If $\frac{1}{b-c}$, $\frac{1}{c-a}$, $\frac{1}{a-b}$ be consecutive terms of an A.P., then $(b-c{)}^2,(c-a{)}^2,(a-b{)}^2$ will be in ___.

1. H.P

2. G.P

3. None of these

4. A.P

Q.

In how many ways 4 Indian and 4 Pakistani army Generals can be seated, half on each side of a long table, so that no two Indian Generals may be together?

(i) Find the number of sitting arrangements.

(ii) Do you feel that periodic meeting between senior officers and politicians are in the interest of both the countries for keeping peace and goodwill in the region? Express your views briefly.

Q.

If the sum of the roots of the equation a$x^2$ + bx + c = 0 is equal to the sum of the squares of their reciprocals, then $\frac{a}{c}$, $\frac{b}{a}$, $\frac{c}{b}$ are in :

1. G.P.

2. H.P.

3. None

4. A.P.

Q. Using Binomial theorem, evaluate $(99{)}^5$

Q.

If $A(a{t}^2,2at),B(\frac{a}{t^2},\frac{-2a}{t})$ and C(a, 0), then 2a is equal to

1. A.M. of CA and CB

2. G.M. of CA and CB

3. H.M. of CA and CB

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. 2
2. 4
3. 8
4. 16

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.

In any $\Delta ABC,$ prove that

$\frac{sin(B-C)}{sin(B+C)}=\frac{(b^2-c^2)}{a^2}$

Q. A man starts repaying a loan as first installment of Rs. $100$. If he increases the installment by Rs $5$ every month. what amount he will pay in the ${30}^{th}$ installment?

Q.

Given HM of 2 numbers = 4 and 2A + $G^2$ = 27. Use the relation $G^2$ = AH with 2A + $G^2$ = 27, to find A & G.​

1. , 3

2. 3, 9

3. , 3

4. , 9

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. $f\left(x\right)=a{x}^3+b{x}^2+cx-3$ be a polynomial with real coefficient. Let $x_1,x_2,x_3,(x_i>0)$ be the roots of $f\left(x\right)=0$, such that $\sum _{i,j,k=1i\ne j\ne k}^3\frac{x_i}{x_j+x_k}=\frac{3}{2}$. Then which of following(s) is/are possible ?

1. $a=\frac{1}{9}$
2. $a=24$
3. $f\left(x\right)=0$ has atleast one repeated root.
4. $b>0$

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 $x,y,z\in R^{+}$ 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 $a,b,c,d\in R^{+}$ 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. 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.

If the aritmetic, geometric and harmonic menas between two positive real numbers be A, G and H, then

1. 

2. 

3. 

4. 

Q.

If $A_1,A_2,A_3$ be the area of the incircle and exercise, then $\frac{1}{{\sqrt{A}}_1}+\frac{1}{{\sqrt{A}}_2}+\frac{1}{{\sqrt{A}}_3}$ is equal to

1. 

2. 

3. 

4. None of these

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 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