Special Products

Q.

The imaginary part of ${\left(3+2\sqrt{-54}\right)}^{\frac{1}{2}}-{\left(3-2\sqrt{-54}\right)}^{\frac{1}{2}}$ can be

1. $\sqrt{6}$

2. $-2\sqrt{6}$

3. $6$

4. $-\sqrt{6}$

Q. Evaluate the following:

(ii) $(98{)}^3$

Q. $(x+2y)(x^2-2xy+4y^2)$ can also be written as :

1. ($x^3+2y^3$)
2. ($x^3+8y^3$)
3. ($x^2+2y^2$)
4. ($x^3+y^3$)

Q.

The value of $\frac{{23}^3+{27}^3}{{23}^2-23\times 27+{27}^2}$ is :

1. $46$
2. $52$
3. $50$
4. $48$

Q.

Use suitable identities to find the following products:
(i) $(x+4)(x+10)$
(ii) $(x+8)(x-1)$
(iii) ($y^2$+$\frac{3}{2}$)($y^2$−$\frac{3}{2}$)

Q. The expansion of ($x+2$)($x+3$)($x+4$) is ______.

1. $x^3$ + $9$$x^2$ + $26$$x$ + $24$
2. $x^3$ + $18$$x^2$ + $26$$x$ + $24$
3. $x^3$ + $9$$x^2$ + $26$$x$ + $26$
4. $x^3$ + $9$$x^2$ + $24$$x$ + $24$

Q. Question 10
Factorise each of the following:
$\left(i\right)27y^3+125z^3$
$\left(ii\right)64m^3-343n^3$

Q. If $x=4+\sqrt{15},$ then find $x+\frac{1}{x}$ is equal to

1. $8$
2. $4$
3. $10$
4. $6$

Q.

If ${(x}^4+x^{2}y+y^2)$ is one of the factors of an expression which is the difference of two cubes, then the other factor is ___.

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

Q. Simplify : $(\sqrt[4]{46}-\sqrt[4]{45})(\sqrt[4]{46}+\sqrt[4]{45})(\sqrt{6}+\sqrt{5})$

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

Q. The expansion of $(x-2)(x+5)(x+7)$ is :

1. $x^3$+ $10$$x^2$ + $11$$x$ - $70$
2. $x^3$- $10$$x^2$ + $11$$x$ - $70$
3. $x^3$+ $10$$x^2$ + $11$$x$ + $70$
4. $x^3$+ $10$$x^2$ - $11$$x$ - $70$

Q. The expansion of ($x-1$)($x-3$)($x+4$) is _______.

1. $x^3$- $13$$x$ + $12$
2. $x^3$- $13$$x$ + $24$
3. $x^3$ + $9$$x^2$ - $13$$x$ + $12$
4. $x^3$ + $9$$x^2$ + $26$$x$ + $24$

Q. The expansion of ($x-6$)($x+5$)($x-4$) is _____.

1. $x^3$ + $9$$x^2$ + $26$$x$ + $24$
2. $x^3$ - $5$$x^2$ - $26$$x$ + $120$
3. $x^3$ + $18$$x^2$ + $26$$x$ + $24$
4. $x^3$ + $9$$x^2$ + $26$$x$ + $120$

Q. If $a-b=-8$ and $ab=-12$, then $a^3-b^3=$

Q. Expand using identity : $(3x+5y+2z)(3x-5y+2z)$

1. $22x^2+12xz+4z^2+25y^2$
2. $16x^2-12xz+4z^2-25y^2$
3. $9x^2+12xz+4z^2-25y^2$
4. $13x^2-12xz+4z^2-25y^2$

Q. Identify the commutative property of addition from the given options:

1. $(\sqrt{2}+\sqrt{3})+\sqrt{5}=\sqrt{2}+(\sqrt{3}+\sqrt{5})$
2. $\sqrt{2}(\sqrt{3}+\sqrt{5})=(\sqrt{2}\times \sqrt{3})+(\sqrt{2}\times \sqrt{5})$
3. $\sqrt{2}+\sqrt{3}=\sqrt{3}+\sqrt{2}$
4. $\sqrt{2}\times \frac{1}{\sqrt{2}}=1$

Q. Evaluate: $(t-8)(t+8)$

1. $t^2-4$
2. $t^2-64$
3. $t+64$
4. $t^2-8$

Q. Evaluate the following:

(iv) $(10.4{)}^3$

Q.

What is $(x-y{)}^3$?

1. $x^3-y^3+3xy(x-y)$

2. $x^3+y^3-3xy(x+y)$

3. $x^3-y^3-3xy(x-y)$

4. $x^3+y^3+3xy(x+y)$

Q. $\text{If}(a^3+b^3)=243\text{and}(a+b)=9,\text{then the value of}(a^2-ab+b^2)\text{is}$

1. 9
2. 27
3. 18
4. 36

Q. The expansion of ($x+2$)($x+3$)($x+4$) is ______.

1. $x^3$ + $9$$x^2$ + $26$$x$ + $24$
2. $x^3$ + $18$$x^2$ + $26$$x$ + $24$
3. $x^3$ + $9$$x^2$ + $26$$x$ + $26$
4. $x^3$ + $9$$x^2$ + $24$$x$ + $24$

Q. The value of (-3)${}^3$ + (7)${}^3$ + (-4)${}^3$ is :

1. 252
2. 264
3. 121
4. 343

Q. The value of the expression $\frac{10\times 10\times 10+5\times 5\times 5}{10\times 10-10\times 5+5\times 5}$ is :

1. 10
2. 5
3. 20
4. 15

Q.

If ${(x}^4+x^{2}y+y^2)$ is one of the factors of an expression which is the difference of two cubes, then the other factor is ___.

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

Q. Simplify: $(x+3)(2x+5)$

1. $x^2+11x+15$
2. $2x^2+11x+5$
3. $2x^2-1x+15$
4. $2x^2+11x+15$

Q. $(3\sqrt{2}-2\sqrt{3})(2\sqrt{3}+3\sqrt{2})=$

Q. If $(a^2+ab+b^2)=39$ and $(a-b)=3$, then the value of $(a^3-b^3)$ is :

1. 111
2. 117
3. 171
4. 187

Q. ${11}^3+{12}^3+{13}^3$ is divisible by

1. 11
2. 16
3. 13
4. 36

Q. Value of the expression $\frac{8.73\times 8.73\times 8.73+4.27\times 4.27\times 4.27}{8.73\times 8.73-8.73\times 4.27+4.27\times 4.27}$ is

1. $8.73$
2. $4.27$
3. $4.46$
4. $13$

Q. Find the values of $a$ and $b,$ if $\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{2}-1}{\sqrt{2}+1}=a+\sqrt{2}b$