Inequalities Involving Mathematical Means

Q.

If the cube roots of unity are $1,\omega ,{\omega }^2$, then the roots of the equation ${\left(x-1\right)}^3+8=0$:

1. $-1,1+2\omega ,1+2{\omega }^2$

2. $-1,1-2\omega ,1-2{\omega }^2$

3. $-1,-1,-1$

4. $-1,-1+2\omega ,-1-2{\omega }^2$

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. An integer X on division by 7 leaves remainder 5. Find the remainder when 4x+31 is divided by 14.

Q. If the $A.M.$ between two numbers exceeds their $G.M.$ by $2$ and the $G.M.$ exceeds their $H.M.$ by $8/5$, find the smaller of the two numbers.

Q. Assertion: If p(x) = x³ − ax² + 6x − a is divided by (x − a), then the remainder is 5a.
Reason: If p(x) is divided by (x − a), then the remainder is p(a).

(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true and but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

Q.

The sum of roots of the equation $(x+1)(x+2)+(x+3)(x+4)+(x+5)(x+6)+\dots .(x+9)(x+10)=0$ is

1. 11

2. -11

3. 55

4. -55

Q.

Let a, b, c be the sides of a triangle. The least value of expression$E=\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c}$ is ___

Q. (iii) Q $\equiv$ (6, -5) R $\equiv$ (-10, 2) and m:n =3:4

Q. determine the values of p and q so that the prime factorization of 2520 is expressible as 2^3 * 3^p*q*7

Q. Let a=3-root n, where n is a natural number. If p is the least positive value of a, then find the value of root p +1/ root p

Q. Find the value of A and B 5 first part 5 + 2 root 3 upon 7 + 4 root 3 is equals to a minus b root 3
Q. Find the value of a and b
(i). $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a-b\sqrt{3}$

Q.

Let a, b, c be the sides of a triangle. The least value of expression$E=\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c}$ is ___