Operations on Binomial Surds

Q.

$\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{8}}+\frac{1}{\sqrt{8}+\sqrt{9}}=$__________.

1. $1$

2. $2$

3. $-1$

4. $0$

Q.

$\text{If}x=\frac{\sqrt{5}-2}{\sqrt{5}+2}$, then $x+\frac{1}{x}$ = ______.

1. $8\sqrt{5}$

2. 12

3. 18

4. 9

Q.

Simplify $\left(\frac{2+\sqrt{5}}{2-\sqrt{5}}\right)+\left(\frac{2-\sqrt{5}}{2+\sqrt{5}}\right)$.

1. -18

2. + 18

3. $18+8\sqrt{5}$

4. $18-8\sqrt{5}$

Q.

If $Q$ denotes the set of all rational numbers and $f\left(\frac{p}{q}\right)=\sqrt{p^2-q^2}$ for any $\frac{p}{q}\in Q$, then observe the following statements.

I. $f\left(\frac{p}{q}\right)$is real for each $\frac{p}{q}\in Q$.

II. $f\left(\frac{p}{q}\right)$ is a complex number for each $\frac{p}{q}\in Q$.

1. Both I and II are correct

2. I is correct, II is incorrect

3. I is incorrect, II is correct

4. Both I and II are incorrect

Q.

If 'p' and 'q' are rational numbers, then find 'p' and 'q' given that $\frac{3+\sqrt{7}}{3-\sqrt{7}}=p+q\sqrt{7}$

1. $p=8,q=3$

2. $p=-8,q=3$

3. $p=-8,q=-3$

4. $p=8,q=-3$

Q.

If 'a' and 'b' are rational numbers, find 'a' and 'b' given that $\frac{3}{\sqrt{3}-\sqrt{2}}=a\sqrt{3}-b\sqrt{2}$.

1. $a=3,b=3$

2. $a=3,b=-3$

3. $a=1,b=-3$

4. $a=3,b=1$

Q. Simplify the following expression :

ii) $(5+\sqrt{7})(5-\sqrt{7})$

Q.

$\text{If}x=9+4\sqrt{5},\text{then}\sqrt{x}-\frac{1}{x}=$ _______.

1. $4$

2. $16$

3. $2\sqrt{5}$

4. $1$

Q. Simplify : $(11+\sqrt{7})(11-\sqrt{7})$

1. 114
2. 124
3. 164
4. 109

Q.

If $\frac{\sqrt{3}-1}{\sqrt{3}+1}=a+b\sqrt{3}$, then the values of ${}^{\prime }{a}^{\prime }$ and ${}^{\prime }{b}^{\prime }$ are _________.

1. $a=2,b=1$

2. $a=-2,b=1$

3. $a=4,b=-1$

4. $a=2,b=-1$

Q.

Question 2 (ii)
Simplify $(3+\sqrt{3})(3-\sqrt{3})$ .

Q.

Without actually calculating the cubes, find the value of the following: ${(-12)}^3$+ ${\left(7\right)}^3$ + ${\left(5\right)}^3$

Q.

If $x=\frac{\sqrt{5}-2}{\sqrt{5}+2}$ then $x^2$ = ___

1. $161-72\sqrt{5}$

2. 161

3. $161+72\sqrt{5}$

4. $9-4\sqrt{2}$

Q.

If $x^2=11+2\sqrt{30}$, then $x$ = ______

1. $\sqrt{5}+\sqrt{6}$

2. $\sqrt{5}-\sqrt{6}$

3. $4\sqrt{30}$

4. $22$

Q. Give a rational approximation to $\sqrt{3}$ correct upto two decimal places.

Q.

If $x=3+\sqrt{8}$

then $x^4+\frac{1}{x^4}=$ ______

1. 34

2. 1154

3. 1156

4. 36

Q. What is the least rationalising factor of $\sqrt{44}$?

Q.

$\text{If}\frac{5-\sqrt{3}}{2+\sqrt{3}}=x+y\sqrt{3}$, then___________.

1. $x=-13,y=7$

2. $x=13,y=-7$

3. $x=13,y=7$

4. $x=-13,y=-7$

Q. If $x=5-2\sqrt{6},$ find $x^2+\frac{1}{x^2}$.

Q.

$\sqrt{3+2\sqrt{2}}$ = __________________

1. $1-\sqrt{2}$

2. $1+\sqrt{2}$

3. $\sqrt{1+\sqrt{2}}$

4. $\sqrt{2}$

Q. If $\sqrt{3}=1.732,\sqrt{5}=2.236$, then the value of $\frac{6}{\sqrt{5}-\sqrt{3}}$ is

1. 10.905
2. 11.904
3. 11.905
4. None

Q.

If $x=7+4\sqrt{3}$, the value of $\sqrt{x}$ is ________.

1. $4$

2. $2-\sqrt{3}$

3. $4+\sqrt{3}$

4. $2+\sqrt{3}$

Q. $\sqrt[2010]{20102\sqrt{7}-3\sqrt{3}}\times \sqrt[4020]{402055+12\sqrt{21}}=?$

1. $1$
2. $-1$
3. $0$
4. $2$

Q. If $x=\frac{\sqrt{5}-2}{\sqrt{5}+2}$ and $y=\frac{\sqrt{5}+2}{\sqrt{5}-2}$. Find the value of $x+y$?

1. 18

Q. $\frac{\sqrt{3}+\frac{1}{\sqrt{3}}}{1-\sqrt{3}}-\frac{\sqrt{3}-\frac{1}{\sqrt{3}}}{1+\sqrt{3}}=1$

1. True
2. False

Q.

If $x=2+\sqrt{3}$, then $x^3+\frac{1}{x^3}$ = __.

1. 64

2. 52

3. 48

4. 4

Q.

Find the value of $(3+\sqrt{5})(3-\sqrt{5})$ .

1. 1

2. 2
3. 3
4. 4

Q.

The number ${(1+\sqrt{3})}^2$ is a/an ____________.

1. irrational number

2. rational number

3. natural number

4. integer

Q. If $x=\sqrt{7}+\sqrt{6}$, then ${(x+\frac{1}{x})}^2$=

1. $2(\sqrt{7}+\sqrt{6})$
2. $2\sqrt{6}$
3. $28$
4. $2\sqrt{7}$

Q. Simplify: $(2\sqrt{2}+5\sqrt{5})$$(4\sqrt{2}+3\sqrt{5})$

1. $91+26\sqrt{10}$
2. $81+20\sqrt{10}$
3. $81+10\sqrt{10}$
4. $91+10\sqrt{10}$