AM,GM,HM Inequality

Q.

Find the cube root by prime factorization $17576$

Q.

Find the cube root of $91125$ by prime factorization method.

Q.

Find the cube root of $8000$.

Q.

Find the cube root by prime factorization of $1331$

Q.

Find the cube root of $125$ number by prime factorization method.

Q. The sum of the integers from $1$ to $100$ which are not divisible by $3$ or $5$ is

1. $2489$
2. $2317$
3. $2632$
4. $4735$

Q.

Identify the pairs of like terms among the following.

$a^{2}b$ and $a{b}^2$

Q. IF $3p^2=5p+2$ and $3q^2=5q+2$ where $p\ne q$, then the equation whose roots are $3p-2q$ and $3q-2p$ is

1. $3x^2-5x-100=0$
2. $5X^2+3x+100=0$
3. $5x^2-3x-100=0$
4. $3x^2-5x+100=0$

Q. Find the sum and product of the roots of equation $3x^2+5=0$

Q. Match each of the following GPs with the correct next terms.

1. 32
2. 8
3. 81
4. $x^{4}y$

Q.

Which of the following is not a geometric progression?

1. 3, 6, 12, 24

2. 3, 9, 27, 81

3. 3, 3, 3, 3

4. 3, 6, 9, 12

Q. If $p,q$ are the distinct roots of the equation $x^2+px+q=0$, then

1. $p=1,q=-2$
2. $p=-2,q=0$
3. $p=0,q=1$
4. $p=-2,q=1$

Q. Let $x_1$ and $x_2$ be the roots of the quadratic equation $x^2+px+q=0$.
If $x_1=\frac{x_2+4}{2x_2-1}$, then the value of $(2q+p)$ is equal to

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

Q. If $P\left(x\right)=a{x}^2+bx+c$ and $Q\left(x\right)=-a{x}^2+dx+c$ , $ac\ne 0$, then the equation $P\left(x\right).Q\left(x\right)=0$ has

1. two imaginary roots
2. more than two imaginary roots
3. atleast two real roots
4. no real roots

Q. If $\alpha$ and $\beta$ be two real roots of the equation $x^3+p{x}^2+qx+r=0$ satisfying the relation $\alpha \beta +1=0$, then prove that $r^2+pr+q+1=0$.

Q.

If $\left(2+i\sqrt{3}\right)$ is a root of the equation $x^2+px+q=0$ where $p$ and $q$ are real, then find $\left(p,q\right)$.

Q. If ${\cos }^{-1}\frac{p}{a}+{\cos }^{-1}\frac{q}{b}=\alpha$, then prove that $\frac{p^2}{a^2}-\frac{2pq}{ab}\cos \alpha +\frac{q^2}{b^2}={\sin }^2\alpha$

Q. If $p,q,r$ are positive and are in A.P., the roots of quadratic equation $p{x}^2+qx+r=0$ are all real for

1. $|\frac{p}{r}-7|\ge 4\sqrt{3}$
2. No $p$ and $r$
3. All $p$ and $r$
4. $|\frac{r}{p}-7|\ge 4\sqrt{3}$

Q. Find the values of $p$ and $q$ for which $x=\frac{2}{3}$ and $x=-3$ are the roots of the equation $p{x}^2+7x+q=0$

1. $p=-3,q=-6$
2. $p=-3,q=6$
3. $p=3,q=-6$
4. $p=3,q=6$

Q.

The sum of $x^2-y^2$ and $y^2-x^2$ is

1. $0$

2. $2x^2-2y^2$

3. $2y^2-2x^2$

4. $x^4-y^4$

Q. Given that $sinA=\frac{1}{2}$ and $cosB=\frac{1}{\sqrt{2}}$, then the value of $(A+B)$ is ___________.
(Here, 0 < A + B ≤ 90°)

1. 45°

2. 30°

3. 15°

4. 75°

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

Q. If one root of $x^2+px-10=0$ is $2$ while the equation $x^2-px+q=0$ has equal roots, then the value of $q$ is

1. $\frac{9}{16}$
2. $9$
3. $\frac{4}{9}$
4. $\frac{9}{4}$

Q. If every pair from among the equations $x^2+px+qr=0,x^2+qx+rp=0,x^2+rx+pq=0$ has a common root then the product of three common roots is

1. $pqr$
2. $2pqr$
3. $p^2{q}^2{r}^2$
4. None of these

Q. If the roots of the equation $x^2-2ax+a^2+a-3=0$ are less than $3$ then

1. $a<2$
2. $2\le a\le 3$
3. $3<a\le 4$
4. $a>4$

Q. A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the question?