Method of Intervals

Q.

If x = 9+4 square root 5.Find square root x - square root 1/x

Q. If x = 3 $-$ $\sqrt{18}$ then find the value of $\sqrt{x}-\frac{1}{\sqrt{x}}$

Q. If $(x+1)$ and $(x-1)$ are factor of $P{x}^3+x^2-2x+9$ then value of P are

1. 11
2. 12
   
3. 10
   
4. 8

Q.

What are the conditions under which the logarithm $lo{g}_{2x}(x^2-1)$ is defined ?

1. x > 0

2. x > 1

3. x > 1 and x < -1

4. x < -1

Q. If x=9 + 4 square root of 5, find the value of square root of x - 1/root x

Q.

Solve for $x$:

$3\left(9^x\right)<8\left(3^x\right)+3$

1. (0, 81)

2. $(-\infty ,-2)\cup (-2,-1)\cup (-1,0)$

3. $(-\infty ,1)$

4. None of these.

Q. If p= 5+2 root 6 and x=1 by p, then what will be the value of p square + x square ?

Q. If $x=\left(5+2\sqrt{6}\right)$, show that $\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)=\pm 2\sqrt{3}$.

Q. If $f\left(x\right)=x^4-2x^3+3x^2-ax+b$ is a polynomial such that when it is divided by $(x-1)$ and $(x+1)$, then the remainders are $5$ and $19$ respectively. If $f\left(x\right)$ is divided by $(x-2)$, then the remainder is:

1. $5$
2. $0$
3. $2$
4. $10$

Q. Match the following columns:

Column I		Column II
(a)	$\sqrt[4]{{\left(81\right)}^{-2}}=.......$	(p)	4
(b)	If ${\left(\frac{a}{b}\right)}^{x-2}={\left(\frac{b}{a}\right)}^{x-4}$, then x = ........ .	(q)	$\frac{2}{9}$
(c)	If $x=\left(9+4\sqrt{5}\right)$, then $\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)$= ...... .	(r)	$\frac{1}{9}$
(d)	${\left(\frac{81}{16}\right)}^{-3/4}\times {\left(\frac{64}{27}\right)}^{-1/3}=?$	(s)	3

(a) ......
(b) ......
(c) ......
(d) ......

Q. A natural number is chosen at random from the first 100 natural numbers. Then the probability, for the in-equation $x+\frac{100}{x}>50$ satisfied, is

1. $\frac{1}{4}$
2. $\frac{11}{20}$
3. $\frac{1}{20}$
4. $\frac{1}{3}$

Q. Factorise x⁴ + 2x³ - 3x² - 14x + 24

Q. $\sqrt{{\log }_{1/2}\left({\log }_2\right(x^2+4x+5\left)\right)}$.

Q. Given that ${\log }_{a}A=x$ is similar to $a^x=A$.
If true then write 1 and if false then write 0

Q. If $\left|{\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right|<\frac{\pi }{3}$, then $x$ belongs to the interval

1. $(0,1\sqrt{3})$
2. $\left(\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)$
3. $(-\sqrt{3},\sqrt{3})$
4. none of these

Q. If ${\log }_{4}1/4=x$, then $x$ is equal to____

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

Q. What are the conditions under which the logarithm log ${}_{2x}(x^2$ - 1) is defined ?

1. $x>0$
2. $x>1$
3. $x>1$ and $x<-1$
4. $x<-1$

Q. $x^2-17x+16$

Q. ​if x-1+x = 3, find the value of x⁴+ 1+x⁴