Conditional Identities

Q.

Simplify each of the following expressions:

(i) (ii)

(iii) (iv)

Q. If a + 2b + c = 0; then evaluate $a^3+8b^3+c^3$

1. 3abc
2. 6abc
3. abc
4. 0

Q. Find the values of a and b in each of the following.
$\frac{3+\sqrt{2}}{3-\sqrt{2}}=a+b\sqrt{2}$

Q. Write the following cubes in expanded from:
(i) $(2x+1{)}^3$ (ii) $(2a-3b{)}^3$ (iii) ${\left[\frac{3}{2}x+1\right]}^3$ (iv) ${[x-\frac{2}{3}y]}^3$

Q.

Question 46
In the following question, state whether the given statement is true (T) or false (F).
LXXIV=74

Q.

Simplify :

6 minus 4 route 3 upon 6 plus 4 route 3 .

Q.

Fill in the boxes: (a) (b) (c)

Q. What is the value of $3^3+5^3-8^3$?

1. -160
2. 260
3. -360
4. 160

Q. The value of
$(2x-3y{)}^3+(x+3y{)}^3+3(2x-3y{)}^2(x+3y)+3(2x-3y\left)\right(x+3y{)}^2=$

1. 
2. 
3. 
4. 

Q. The value of 64^-1/3 (64^1/3-64^2/3), is

(a) 1

(b) $\frac{1}{3}$

(c) −3

(d) −2

Q. If $A+B+C=\pi$, prove that $\tan \frac{A}{2}\tan \frac{B}{2}+\tan \frac{B}{2}\tan \frac{C}{2}+\tan \frac{C}{2}\tan \frac{A}{2}=1$.

Q.

Express the given formula in term of y.

$\frac{T}{T_i}=1+\frac{y-1}{2}$

1. 2(T_i /T - 1 ) + 1

2. 2((T/T_i)​ - 1) - 1

3. 2(T/T_i ) - 1

4. 2((T/T_i )​ + 1 ) + 1

Q.

$\frac{a^m}{a^n}$=$a^{m-n}$, m>n

1. True

2. False

Q. The value of $\sqrt{5+2\sqrt{6}}$ is

(a) $\sqrt{3}-\sqrt{2}$

(b) $\sqrt{3}+\sqrt{2}$

(c) $\sqrt{5}+\sqrt{6}$

(d) none of these

Q. The value of $\sqrt{3-2\sqrt{2}}$ is

(a) $\sqrt{2}-1$

(b) $\sqrt{2}+1$

(c) $\sqrt{3}-\sqrt{2}$

(d) $\sqrt{3}+\sqrt{2}$

Q. Simplify $\sqrt{3-2\sqrt{2}}$.

Q. Simplify:
(i) $\left(4+\sqrt{2}\right)\left(4-\sqrt{2}\right)$
(ii) $\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)$
(iii) $\left(6-\sqrt{6}\right)\left(6+\sqrt{6}\right)$
(iv) $\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{2}-\sqrt{3}\right)$
(v) ${\left(\sqrt{5}-\sqrt{3}\right)}^2$
(vi) ${\left(3-\sqrt{3}\right)}^2$

Q. Evaluate:
(i) (64)^−1/2
(ii) (8)^−1/3
(iii) (81)^−1/4

Q.

Multiply:
(i) $\sqrt{3}(\sqrt{7}-\sqrt{3})$

(ii) $(\sqrt{5}-\sqrt{7})\sqrt{2}$

(iii) $(3\sqrt{2}-\sqrt{3})(4\sqrt{3}-\sqrt{2})$

Q.

3+2+90

Q. $$\begin{gathered} Solveu\sin gBODMAS \\ 100\times \left(-10\right)+\left[400÷\left\{100-\left(50-\overline{5\times 10}\times 10\right)\right\}\right] \\ 5+\left[14÷\left(-7\right)-\left\{6\times \left(\overline{5\times 1}\times 4\right)\right\}\right] \end{gathered}$$

Q. Find the values of a and b in each of the following.
$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a-b\sqrt{3}$

Q.

What are the factors of $({101}^2-{99}^2)$?

1. 101

2. 99

3. 2

4. 34

Q. If $x+\frac{1}{x}=4$, then find $x^3+\frac{1}{x^3}$.

Q. If $a=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$ and b = $a=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$ find the value a² + b² − 5ab.

Q. Evaluate: {(28)³ + (−15)³ + (−13)³}.

Q. If $a+\frac{1}{a}=p$ and $a\ne 0$; then $a^3+\frac{1}{a^3}$ is equal to

1. $P\left(P^2-3\right)$
2. $P^2\left(P-3\right)$
3. p - 3
4. p( p - 3 )

Q. Simplify:
(i)
(ii)
(iii)
(iv)
(v)
(vi)

Q. If x^{​​​​​​2} +1/x^{​​​​​​2}= 7, then find x^{​​​​​​3}+1/x^{​​​​​​3}.
How to do this sum?

Q. In $△ABC$, show that $\frac{acosA+bcosB+ccosC}{a+b+c}=\frac{r}{R}$