Inequalities Involving Mathematical Means

Q.

If $\alpha$ and $\beta$are the roots of $x^2-2x+4=0$, then the value of ${\alpha }^6+{\beta }^6$is

1. $32$

2. $64$

3. $128$

4. $256$

Q.

If$a^2+b^2=1$, then $\frac{(1+b+ia)}{(1+b-ia)}$is equal to?

1. $1$

2. $2$

3. $b+ia$

4. $a+ib$

Q.

The greatest integer among following by which the number 5^5+7^5 is divisible?

a) 6

b) 8

c) 11

d) 12

I am not sure from which chapter this question belongs to.....

Q.

The average of the four digits numbers that can be formed by using each of the digits $3,5,7$, and $9$ exactly once in each number, is?

1. $4444$

2. $5555$

3. $6666$

4. $7777$

Q.

Using Binomial Theorem, evaluate (96)³

Q. The harmonic mean of two numbers is 4 and the arithmetic and geometric means satisfy the relation $2A+G^2=27,$ the numbers are

1. 6, 3
2. 5, 4
3. 5, -2.5
4. -3, 1

Q. The least value of the expression $2{\log }_{10}x-{\log }_x\left(0.01\right)$, for $x>1$, is

1. $2$
2. $10$
3. $-0.01$
4. $4$

Q. A manufacturer can sell 'x' items at the rate of (330-x) each. The cost of producing x items is rupees x​​​​​​2 + 10x + 12. How many items must be sold so that his profit is maximum?

Q.

Simplify the following expression ${\left(\sqrt{5}+\sqrt{2}\right)}^2$

Q.

Using Binomial Theorem, evaluate (102)⁵

Q. If $f\left(x\right)=x^x,x>0$, then

1. $f\left(x\right)$ is increasing on $(1/e,\infty )$
2. $f\left(x\right)$ is decreasing on $(0,1/e)$
3. $x=1/e^2$ is point of local minima
4. local minimum value of $f\left(x\right)=e^{-1/e}$

Q. If roots of the equation $f\left(x\right)=x^6-12x^5+{\mathrm{bx}}^4+{\mathrm{cx}}^3+{\mathrm{dx}}^2+\mathrm{ex}+64=0$ are positive, then the remainder when f(x) is divided by x-1 is

1. 2
2. 1
3. 3
4. 10

Q.

If $a,b,c,d$ are positive numbers, then $ab+cd<2\sqrt{abcd}$

1. False

2. True

Q. If $a,b,c\in R$ and $a^2+b^2+c^2=1$, then $ab+bc+ca$ lies in the nterval

1. $[\frac{1}{2},2]$
2. $[-1,2]$
3. $[-\frac{1}{2},1]$
4. $[-1,\frac{1}{2}]$

Q. If $a_1,a_2,.......,a_n$are positive real numbers whose product is a fixed number c, then the minimum value of $a_1+a_2+.......+a_{n-1}+2a_n$ is

1. $n(2c{)}^{\frac{1}{n}}$
2. $(n+1)c{}^{\frac{1}{n}}$
3. $2n{c}^{\frac{1}{n}}$
4. $(n+1)(2c{)}^{\frac{1}{n}}$

Q.

If roots of the equation
$x^4-8x^3+b{x}^2+cx+16=0$
are positive, then

1. b=8

2. c=-32

3. b=24

4. c=32

Q.

If $a^2+b^2+c^2=1,$ then $bc+ca+ab$ lies in the interval

1. [-1, 2]

2. $[\frac{-1}{2},1]$

3. $[-1,\frac{1}{2}]$

4. $[\frac{-1}{2},3]$

Q.

Prove that the Greatest Integer Function f: R → R given by f(x) = [x], is neither one-once nor onto, where [x] denotes the greatest integer less than or equal to x.

Q. If a, b and c be in G .P. and x, y be the arithmetic means between a, b and b, c respectively then $\frac{a}{x}+\frac{c}{y}$ is-

1. $2$
2. $1$
3. $3$
4. $4$

Q.

Let $\alpha =\frac{(-1+\sqrt{3}i)}{2}$ . If $a=\left(1+\alpha \right)\sum _{k=0}^{100}{\alpha }^{2k}$ and $b=\sum _{k=0}^{100}{\alpha }^{3k}$, then $a$ and $b$ are the roots of the quadratic equation:

1. $x^2+101x+100=0$

2. $x^2+102x+101=0$

3. $x^2-102x+101=0$

4. $x^2-101x+100=0$

Q. What is the domain and range of : $y=\sqrt{2-x}+\sqrt{1+x}$

1. $D:-1\le x\le 2,R:[\sqrt{3},\sqrt{6}]$
2. $D:-2\le x\le 1,R:[\sqrt{3},\sqrt{6}]$
3. none of these
4. $D:-1\le x\le 2,R:[\sqrt{3},\sqrt{12}]$

Q. If $x^2+y^2+z^2=1$, then $yz+zx+xy$ lies in the interval

1. $[\frac{1}{2},2]$
2. $[-1,2]$
3. $[-\frac{1}{2},1]$
4. $[-1,\frac{1}{2}]$

Q.

Show that the statement

p: “If x is a real number such that x³ + 4x = 0, then x is 0” is true by

(i) direct method

(ii) method of contradiction

(iii) method of contrapositive

Q. Find the absolute maximum and minimum values of a function f given by

$f\left(x\right)=12x^{4/3}-6x^{1/3},x\in [-1,1]$

Q. A relation R is defined from set A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10 } as follows : R = { (x, y) : x is relatively prime to y

Q. The product of n positive numbers is unity. Then their sum is

1. a positive integer
2. divisible by n
3. equals to $n+\frac{1}{n}$
4. never less than n

Q.

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x⁴ (5 sin x – 3 cos x)

Q.

List all the elements of the following sets:

(i) A = {x: x is an odd natural number}

(ii) B = {x: x is an integer, }

(iii) C = {x: x is an integer, }

(iv) D = {x: x is a letter in the word “LOYAL”}

(v) E = {x: x is a month of a year not having 31 days}

(vi) F = {x: x is a consonant in the English alphabet which proceeds k}.

Q. The least and the greatest value of $f\left(x\right)=x^2\log x$ in $[1,e]$ are

1. $0,e^2$
2. $\frac{1}{e},e$
3. $\log 2,\log 4$
4. $e^2,e^4$

Q. If $A_1,A_2$ be two arithmetic means and $G_1,G_2$ be two geometric means between two positive numbers $a$ and $b,$ then$\left(\frac{A_1+A_2}{G_1{G}_2}\right)$ is equal to

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