Addition of Rational Expressions

Q.

Solve the following linear equation:

$\frac{n}{2}-\frac{3n}{4}+\frac{5n}{6}=21$

Q. The variance of first n natural numbers is $\frac{n^2-1}{12}$.

1. 
2. 

Q.

If $y=\log \left(\frac{(1+\sqrt{x})}{(1-\sqrt{x})}\right)$, then $\frac{dy}{dx}$

1. $\frac{\sqrt{x}}{(1-x)}$

2. $\frac{1}{\sqrt{x}(1-x)}$

3. $\frac{\sqrt{x}}{(1+x)}$

4. $\frac{1}{\sqrt{x}(1+x)}$

Q. If $\sqrt{2}=1.414,\sqrt{3}=1.732,\sqrt{5}=2.236$ and $\sqrt{6}=2.449,$ find the value of $\frac{2+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2+\sqrt{3}}+\frac{\sqrt{3}-1}{\sqrt{3}+1}$ .

1. 14.268
2. 18.428
3. 16.629
4. 14.662

Q.

If $x=\frac{1}{\sqrt{3}+1}$ then the value of $x+\frac{1}{x}$ is

1. 

2. 

3. 

4. 

Q.

If the product of two decimal numbers is $38.745$ and one of the numbers is $2.7$ find the other.

Q.

Question 117

Match the following :
$\begin{array}{cc}& \\ \text{Column I}& \text{Column II}\\ \text{(a) a}\times 1& \text{(i) Additive inverse of a}\\ \text{(b) 1}& \text{(ii) Additive identity}\\ \text{(c) (-a)}÷\text{(-b)}& \text{(iii) Multiplicative identity}\\ \text{(d) a}\times (-1)& \text{(iv) a}÷\text{(-b)}\\ \text{(e) a}\times 0& \text{(v) a}÷\text{b}\\ \text{(f) (-a)}÷\text{b}& \text{(vi) a}\\ \text{(g) 0}& \text{(vii) -a)}\\ \text{(h) a}÷\text{(-a)}& \text{(viii) 0}\\ \text{(i) -a}& \text{(ix) -1}\end{array}$

Q.

Question 176

By what number should $(-8{)}^{-3}$ be multiplied so that the product may be equal to $(-6{)}^{-3}$ ?

Q.

Choose the correct choice in the following and justify:
(ii) ${11}^{th}termoftheA.P-3,\frac{-1}{2},2is$

A) 28
B) 22
C) -38
D) 44

Q. $x^2=y+z;y^2=x+z;z^2=x+y$
$\frac{1}{(x+1)}+\frac{1}{(y+1)}+\frac{1}{(z+1)}$

Q.

If $(10{)}^9+2(11{)}^1(10{)}^8+3(11{)}^2(10{)}^7+...+10(11{)}^9=k(10{)}^9,$ then k is equals to:

1. 100

2. 110

3. $\frac{121}{10}$

4. $\frac{441}{100}$

Q.

Question 31

Fill in the blanks to make the statement true.

(-a) + b = b + additive inverse of ___ .

Q. Simplify $\frac{x+2}{x+3}+\frac{x-1}{x-2}.$

1. $\frac{2x^2+2x-7}{x^2+x-6}$
2. $\frac{2x^2+2x-7}{x^2+x+6}$
3. $\frac{2x^2+2x+7}{x^2+x+6}$
4. $\frac{2x^2+2x-7}{x^2-x+6}$

Q. Simplify
$(a+b)(2a-3b+c)-(2a-3b)c$.

Q.

How do you solve $log\left(x\right)=-0.123$?

Q.

Write in factorised form: $\frac{2}{x^2}-\frac{5}{x}+2=0$

Q.

$\frac{\left(x+1\right)}{\left(x-1\right)\left(x-2\right)\left(x-3\right)}=$

1. $\frac{1}{\left(x-1\right)}-\frac{3}{\left(x-2\right)}+\frac{2}{\left(x-3\right)}$

2. $\frac{1}{\left(x-1\right)}+\frac{3}{\left(x-2\right)}+\frac{1}{\left(x-3\right)}$

3. $\frac{-3}{\left(x-1\right)}+\frac{1}{\left(x-2\right)}+\frac{2}{\left(x-3\right)}$

4. None of these

Q.

( b + c )( c + a )( a + b ) is greater than

1. $abc$

2. $2abc$

3. $4abc$

4. $8abc$

Q. If x = $3+2\sqrt{2}$, then find the value of
$(\sqrt{x}-\frac{1}{\sqrt{x}})$

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

Q.

What is the difference of the rational expression below?

$\frac{14x}{17}-\frac{3x}{17}$

1. $\frac{17x}{17}$

2. $\frac{11x}{17}$

3. $\frac{42x}{17}$

4. $\frac{42x}{289}$

Q.

If $x=2+\sqrt{3},xy=1,$ then find $\frac{x}{2-x}+\frac{y}{2-y}$

1. $2\sqrt{3}$

2. $-2$

3. $\frac{4}{\sqrt{3}}$

4. $-3$

Q.

If $x=\frac{1}{\sqrt{3}+1}$ then the value of $x+\frac{1}{x}$ is

1. 

2. 

3. 

4. 

Q. Find the sum of $\frac{x-2}{x+3}$ and $\frac{x+2}{x+3}.$ [2 Marks]

Q.

Question 95

Solve the following:

Express $\frac{16}{81}and\frac{-16}{81}$ as powers of rational number.

Q. Solve : $\frac{(5x+2{)}^2-(5x-2{)}^2}{(5x+2{)}^2+(5x-2{)}^2}=\frac{12}{13}$

Q.

If ${\sum }_{k=1}^n$ ${\sum }_{m=1}^k$ $m^2$ = a$n^4$ + b$n^3$ + c$n^2$ + dn + e. then b =

1. 1 / 12

2. 1 / 3

3. 5 / 12

4. 1 / 6

5. 5 / 6

Q. Find the sum of $\frac{x+1}{x+2}$ and $\frac{x^2-1}{x^2+1}.$

1. $\frac{2x^3+3x^2+1}{x^3+2x^2+x+2}$
2. $\frac{2x^3-3x^2-1}{x^3-2x^2-x+2}$
3. $\frac{3x^3+3x^2-1}{2x^3+2x^2+x+2}$
4. $\frac{2x^3+3x^2-1}{x^3+2x^2+x+2}$

Q. $x=\frac{1}{2+\sqrt{3}}$ Find the values of $3x^4+2x^3+10x^2$

Q. $\frac{y}{y-2}+\frac{1}{y+2}+\frac{3y}{4-y^2}$ = .

1. $\frac{y^2-2}{y^2-4}$
2. $\frac{y^2-2}{y^2+2}$
3. $\frac{y+2}{y-2}$
4. $\frac{1}{y+2}$

Q.

What is the least common denominator of the rational expressions below?

$\frac{1}{2x^2+8x}+\frac{10}{x^2}$

1. $2(x+4)$

2. $2x^2(x+4)$

3. $2x(x+4)$

4. $x$