Question 12 Marks
Expand : $(5x + 3y)^3$
Answer$(5x + 3y)^3= (5x)^3 + (3y)^3 + 3 (5x) (3y) (5x + 3y)$
$= 125x^3 + 27y^3 + 45xy (5x + 3y)$
$= 125x^3+ 27y^3+ 225x^2y + 135xy^2$
View full question & answer→Question 22 Marks
Expand : $(2a – 5b – 4c)^2$
Answer$(2a – 5b – 4c)^{2}$
$= (2a)^2 + (–5b)^2 + (–4c)^2 + 2 (2a) (–5b) + 2 (–5b) (–4c) + 2 (–4c) (2a)$
$= 4a^2 + 25b^2 + 16c^2 – 20ab + 40bc – 16ca$
View full question & answer→Question 32 Marks
Expand :$ (3x – 4y + 5z)^2$
Answer$(3x – 4y + 5z)^{2}$
$= (3x)^2+ (– 4y)^2 + (5z)^2+ 2(3x) (– 4y) + 2(– 4y) (5z) + 2(5z) (3x)$
$= 9x^2 + 16y^2 + 25z^2 – 24xy – 40yz + 30zx$
View full question & answer→Question 42 Marks
If $3 x-\frac{1}{3 x}=5$, find : $81 x^4+\frac{1}{81 x^4}$
Answer$81 x^4+\frac{1}{81 x^4}=\left(9 x^2+\frac{1}{9 x^2}\right)^2-2$
$=(27)^2-2$
$=729-2$
$=727$
View full question & answer→Question 52 Marks
If $3 \mathrm{x}-\frac{1}{3 \mathrm{x}}=5$, find: $9 \mathrm{x}^2+\frac{1}{9 \mathrm{x}^2}$
Answer$ 9 x^2+\frac{1}{9 x^2}=\left(3 x-\frac{1}{3 x}\right)^2+2$
$ =(5)^2+2$
$ =25+2$
$ =27$
View full question & answer→Question 62 Marks
If $2 a+\frac{1}{2 a}=8$, find : $16 a^4+\frac{1}{16 a^4} C$
Answer$ 16 a^4+\frac{1}{16 a^4}=\left(4 a^2+\frac{1}{4 a^2}\right)^2-2.4 a^2 \cdot \frac{1}{4 a^2}$
$ =(62)^2-2$
$ =3844-2$
$ =3842$
View full question & answer→Question 72 Marks
If $a^2 + b^2 = 41$ and $ab = 4$, find : $a – b$
Answer$(a-b)^2=a^2+b^2-2 a b $
$ =41-2(4) $
$ =41-8 $
$ =33$
$\therefore a-b=\sqrt{33}$
View full question & answer→Question 82 Marks
If $\mathrm{m}-\frac{1}{\mathrm{~m}}=5$, find $: \mathrm{m}^4+\frac{1}{\mathrm{~m}^4}$
Answer$ m^4+\frac{1}{m^4}=\left(m^2+\frac{1}{m^2}\right)^2-2$
$ =(27)^2-2$
$ =729-2$
$ =727$
View full question & answer→Question 92 Marks
If $\mathrm{m}-\frac{1}{\mathrm{~m}}=5$, find $: \mathrm{m}^2+\frac{1}{\mathrm{~m}^2}$
Answer$ \mathrm{m}^2+\frac{1}{\mathrm{~m}^2}=\left(\mathrm{m}-\frac{1}{\mathrm{~m}}\right)^2+2$
$ =(5)^2+2$
$ =25+2$
$ =27$
View full question & answer→Question 102 Marks
If $a+\frac{1}{a}=2$, find : $a^4+\frac{1}{a^4}$
Answer$ a^4+\frac{1}{a^4}=\left(a^2+\frac{1}{a^2}\right)^2-2$
$ =(2)^2-2$
$ =4-2$
$ =2$
View full question & answer→Question 112 Marks
If $a+\frac{1}{a}=2$, find $: a^2+\frac{1}{a^2}$
Answer$ a^2+\frac{1}{a^2}=\left(a+\frac{1}{a}\right)^2-2$
$ =(2)^2-2$
$ =4-2$
$ =2$
View full question & answer→Question 122 Marks
Find the square of $9.7$
Answer$(9.7)^2 = (10 − 0.3)^2$
$= (10)^2 + (0.3)^2 − 2 (10) (0.3)$
$= 100 + 0.9 − 6$
$= 100.09 − 6.00$
$= 94.09$
View full question & answer→Question 132 Marks
Find the square of $391$
Answer$(319)^2 = (400 − 9)^2$
$= (400)^2 + 9^2 + 2 (400) (9)$
$= 160000 + 81 − 7200$
$= 152881$
View full question & answer→Question 142 Marks
Find the square of $8 x+\frac{3}{2} y$
Answer$ \left(8 x+\frac{3}{2} y\right)^2$
$=(8 x)^2+\left(\frac{3}{2} y\right)^2+2 \times 8 x \times \frac{3}{2} y$
$ =64 x^2+\frac{9}{4} y^2+24 x y$
$ =64 x^2+24 x y+\frac{9}{4} y^2$
View full question & answer→Question 152 Marks
Find the square of $5 x+\frac{1}{5 x}$
Answer$ \left(5 \mathrm{x}+\frac{1}{5 \mathrm{x}}\right)^2$
$=(5 \mathrm{x})^2+\frac{1}{(5 \mathrm{x})^2}+2 \times 5 \mathrm{x} \times \frac{1}{5 \mathrm{x}}$
$ =25 \mathrm{x}^2+\frac{1}{25 \mathrm{x}^2}+2$
$ =25 \mathrm{x}^2+2+\frac{1}{25 \mathrm{x}^2}$
View full question & answer→Question 162 Marks
Find the square of $2 m^2-\frac{2}{3} n^2$
Answer$ 2 m^2-\frac{2}{3} n^2$
$ \left(2 m^2-\frac{2}{3} n^2\right)^2$
$=\left(2 m^2\right)^2+\left(\frac{2}{3} n^2\right)^2-2 \times 2 m^2 \times \frac{2}{3} n^2$
$ =4 m^4+\frac{4}{9} n^4-\frac{8}{3} m^2 n^2$
$ =4 m^4-\frac{8}{3} m^2 n^2+\frac{4}{9} n^2$
View full question & answer→Question 172 Marks
Find the square of $3 x+\frac{2}{y}$
Answer$ 3 x+\frac{2}{y}$
$ \left(3 x+\frac{2}{y}\right)^2=(3 x)^2+\left(\frac{2}{y}\right)^2+2(3 x)\left(\frac{2}{y}\right)$
$ =9 x^2+\frac{4}{y^2}+\frac{12 x}{y}$
View full question & answer→Question 182 Marks
Evaluate: $(1.6x + 0.7y) (1.6x − 0.7y)$
Answer$(1.6x + 0.7y) (1.6x − 0.7y)$
$= (1.6x)^2 − (0.7y)^2......[ \because (a − b) (a +b) = a^2 − b^2]$
$= 2.56x^2 − 0.49y^2$
View full question & answer→Question 192 Marks
Evaluate: $(4x^2 − 5y^2) (4x^2 + 5y^2)$
Answer$(4x^2 − 5y^2) (4x^2 + 5y^2)$
$= (4x)^2 − (5y^2)^2$
$= 16x^4 − 25y^4 ......[ \because (a − b) (a + b) = a^2 − b^2]$
View full question & answer→Question 202 Marks
Evaluate: $\left(2 \mathrm{a}+\frac{1}{2 \mathrm{a}}\right)\left(2 \mathrm{a}-\frac{1}{2 \mathrm{a}}\right)$
Answer$ \left(2 a+\frac{1}{2 a}\right)\left(2 a-\frac{1}{2 a}\right)$
$ =(2 a)^2-\left(\frac{1}{2 a}\right)^2 \ldots \ldots .\left[\because(a-b)(a+b)=a^2 b^2\right]$
$ =4 a^2-\frac{1}{4 a^2}$
View full question & answer→Question 212 Marks
Evaluate: $(6 − 5xy) (6 + 5xy)$
Answer$(6 − 5xy) (6 + 5xy)$
$= (6)^2− (5xy)^2$
$= 36 − 25x^2y^2 .........[ \because (a − b) (a + b) = a^2b^2]$
View full question & answer→Question 222 Marks
Evaluate: $\left(\frac{4}{7} \mathrm{a}+\frac{3}{4} \mathrm{~b}\right)\left(\frac{4}{7} \mathrm{a}-\frac{3}{4} \mathrm{~b}\right)$
Answer$ \left(\frac{4}{7} a+\frac{3}{4} b\right)\left(\frac{4}{7} a-\frac{3}{4} b\right)$
$ =\left(\frac{4}{7} a\right)^2-\left(\frac{3}{4} b\right)^2 \ldots \ldots\left[\because(a-b)(a+b)=a^2 b^2\right]$
$ =\frac{16}{49} a^2-\frac{9}{16} b^2 S$
View full question & answer→Question 232 Marks
Evaluate: $\left(2 \mathrm{x}-\frac{3}{5}\right)\left(2 \mathrm{x}+\frac{3}{5}\right)$
Answer$ \left(2 \mathrm{x}-\frac{3}{5}\right)\left(2 \mathrm{x}+\frac{3}{5}\right)$
$ =(2 \mathrm{x})^2-\left(\frac{3}{5}\right)^2 \ldots \ldots .\left[\because(\mathrm{a}-\mathrm{b})(\mathrm{a}+\mathrm{b})=\mathrm{a}^2 \mathrm{~b}^2\right]$
$ =4 \mathrm{x}^2-\frac{9}{25}$
View full question & answer→Question 242 Marks
Evaluate: $(5xy − 7) (7xy + 9)$
Answer$(5xy − 7) (7xy + 9)$
$= 5xy (7xy + 9) − 7 (7xy + 9)$
$= 35x^2y^2+ 45xy − 49xy − 63$
$= 35x^2y^2 − 4xy − 63$
View full question & answer→Question 252 Marks
The difference between the two numbers is $5$ and their products are $14$. Find the difference between their cubes.
AnswerLet $x$ and $y$ be two numbers, then $x – y = 5$ and $xy = 14\therefore x^3 − y^3 = (x − y)^3 + 3xy(x − y)$
$= (5)^3 + 3 \times 14 \times 5$
$= 125 + 210$
$= 335$
View full question & answer→Question 262 Marks
Evaluate: $(4 − ab) (8 + ab)$
Answer$(4 − ab) (8 + ab)$
$= 4 (8 + ab) − ab (8 + ab)$
$= 32 + 4ab − 8ab − a^2b^2$
$= 32 − 4ab − a^2b^2$
View full question & answer→Question 272 Marks
Evaluate: $(a^2 + 5) (a^2 − 3)$
Answer$(a^2 + 5) (a^2 − 3)$
$= a^2 (a^2 − 3) + 5 (a^2 − 3)$
$= a^4− 3a^2 + 5a^2 − 15$
$= a^4 + 2a^2 − 15$
View full question & answer→Question 282 Marks
Evaluate: $(2 − z) (15 − z)$
Answer$(2 − z) (15 − z)$
$= 2(15 − z) −z(15 − z)$
$= 30 − 2z − 15z + z^2$
$= 30 − 17z + z^2$
View full question & answer→Question 292 Marks
If $a + b = 8$ and $ab = 15$, find :$ a^3 + b^3.$
Answer$a^3 + b^3$
$= (a + b)^3 − 3ab (a + b)$
$= (8)^3 − 3(15) (8)$
$= 512 − 360$
$= 152$
View full question & answer→Question 302 Marks
Evaluate: $(9 − y) (7 + y)$
Answer$(9 − y) (7 + y)$
$= 9(7 + y) − y (7 + y)$
$= 63 + 9y − 7y − y^2$
$= 63 + 2y − y^2$
View full question & answer→Question 312 Marks
Evaluate: $(2a + 0.5) (7a − 0.3)$
Answer$(2a + 0.5) (7a − 0.3)$
$= 2a (7a − 0.3) + 0.5 (7a − 0.3)$
$= 14a^2− 0.6a + 3.5a − 0.15$
$= 14a^2 + 2.9a − 0.15$
View full question & answer→Question 322 Marks
Find the cube of: $3b − 2a$
Answer$(3b − 2a)^3$
$= (3b)^3 − (2a)^3 − 3\times 3b\times 2a(3b − 2a)$
$= 27b^3 − 8a^3 − 18ab (3b − 2a)$
$= 27b^3− 8a^3 − 54ab^2 + 36a^2b$
$= 27b^3− 54b^2a + 36ba^2 − 8a^3$
View full question & answer→Question 332 Marks
Find the cube of: $2a + 3b$
Answer$(2a + 3b)^3$
$= (2a)^3 + (3b)^3+ 3\times 2a\times 3b(2a + 3b)$
$= 8a^3 + 27b^3 + 18ab (2a + 3b)$
$= 8a^3+ 27b^3 + 36a^2b + 54ab^2$
$= 8a^3+ 36a^2b + 54ab^2 + 27b^3$
View full question & answer→Question 342 Marks
Find the cube of: $2a − 1$
Answer$(2a − 1)^3$
$= (2a)^3 − (1)^3 − 3\times 2a\times 1(2a − 1)$
$= 8a^3− 1 − 6a (2a − 1)$
$= 8a^3 − 1 − 12a^2 + 6a$
$= 8a^3 − 12a^2 + 6a − 1$
View full question & answer→Question 352 Marks
Find the cube of: $a + 2$
Answer$(a + 2)^3$
$= (a)^3+ (2)^3+ 3\times a\times 2(a + 2)$
$= a^3 + 8 + 6a(a+ 2)$
$= a^3+ 8 + 6a^2+ 12a$
$= a^3+ 6a^2 + 12a+ 8$
View full question & answer→Question 362 Marks
Expand : $\left(2 a-\frac{1}{2 a}\right)^3$
Answer$ \left(2 a-\frac{1}{2 a}\right)^3$
$=(2 a)^3-\left(\frac{1}{2 a}\right)^3-3 \times 2 a \times \frac{1}{2 a}\left(2 a-\frac{1}{2 a}\right)$
$ =8 a^3-\frac{1}{8 a^3}-3\left(2 a-\frac{1}{2 a}\right)$
$ =8 a^3-\frac{1}{8 a^3}-6 a+\frac{3}{2 a}$
View full question & answer→Question 372 Marks
Expand : $(x + 5y)^3$
Answer$(x+ 5y)^{3}$
$= (x)^3+ (5y)^3 + 3\times x \times 5y (x + 5y)$
$= x^3 + 125y^3 + 15xy (x + 5y)$
$= x^3 + 125y^3 + 15x^2y + 75xy^2$
View full question & answer→Question 382 Marks
Expand :$(3x − 2y)^3$
Answer$(3x − 2y)^2$
$= (3x)^3− (2y)^3− 3 \times 3x \times 2y (3x − 2y)$
$= 27x^3− 8y^3 − 18xy (3x − 2y)$
$= 27x^3− 8y^3− 54x^2y + 36xy^2$
View full question & answer→Question 392 Marks
Expand : $(a − 2b)^3$
Answer$(a − 2b)^3$
$= (a)^3− (2b)^3− 3\times a\times 2b (a − 2b) .........[(a − b)^3= a^3− b^3− 3ab(a − b)]$
$= a^3− 8b^3− 6ab (a − 2b)$
$= a^3 − 8b^3 − 6a^2b + 12ab^2$
View full question & answer→Question 402 Marks
Expand : $(2a + b)^3$
Answer$(2a + b)^3$
$= (2a)^3 + (b)^3 + 3\times 2a\times b(2a + b) ......[(a + b)^3$
$= a^3+ b^3 + 3ab(a + b)]$
$= 8a^3 + b^3 + 6ab (2a + b)$
$= 8a^3+ b^3+ 12a^2b + 6ab^2$
View full question & answer→Question 412 Marks
Evaluate using the expansion of $(a + b)^2$ or $(a – b)^2: (20.7)^2$
Answer$(20.7)^2$
$= (20 + 0.7)^2$
$= (20)^2 + (0.7)^2 + 2 (20) (0.7)$
$= 400 + 0.49 + 28$
$= 428 + 0.49$
$= 428.49$
View full question & answer→Question 422 Marks
Evaluate using the expansion of $(a + b)^2$ or $(a – b)^2:(9.4)^2$
Answer$(9.4)^2= (10 − 0.6)^2$
$= (10)^2 + (0.6)^2 − 2 (10) (0.6)$
$= 100 + 0.36 − 12$
$= 88 + 0.36$
$= 88.36$
View full question & answer→Question 432 Marks
Evaluate using the expansion of $(a + b)^2$ or $(a – b)^2: (188)^2$
Answer$(188)^2 = (200 − 12)^2$
$= (200)^2+ (12)^2 − 2(200) (12)$
$= 40000 + 144 − 4800$
$= 40144 − 4800$
$= 35344$
View full question & answer→Question 442 Marks
Evaluate using the expansion of $(a + b)^2 $ or $ (a – b)^2 :(415)^2$
Answer$(415)^2 = (400 + 15)^2$
$= (400)^2+ (15)^2+ 2(400)(15)$
$= 160000 + 225 + 12000$
$= 172225$
View full question & answer→Question 452 Marks
Evaluate using the expansion of $(a + b)^2$ or $(a – b)^2 :(92)^2$
Answer$(92)^2 = (100 − 8)^{2}$
$= (100)^2+ (8)^2 − 2(100) (8)$
$= 10000 + 64 − 1600$
$= 10064 − 1600$
$= 8464$
View full question & answer→Question 462 Marks
Evaluate using the expansion of $(a + b)^2$ or $(a – b)^2 :(208)^2$
Answer$(208)^2 = (200 + 8)^2$
$= (200)^2+ (8)^2 + 2(200) (8)$
$= 40000 + 64 + 3200$
$= 43264$
View full question & answer→Question 472 Marks
Find the square of $x+\frac{1}{x}-1$
Answer$ \left(\mathrm{x}+\frac{1}{\mathrm{x}}-1\right)^2$
$=(\mathrm{x})^2+\left(\frac{1}{\mathrm{x}}\right)^2+(-1)^2+2 \times \mathrm{x} \times \frac{1}{\mathrm{x}}+2 \times \frac{1}{\mathrm{x}} \times(-1)+2(-1) \times \mathrm{x}$
$ =\mathrm{x}^2+\frac{1}{\mathrm{x}^2}+1+2-\frac{2}{\mathrm{x}}-2 \mathrm{x}$
$ =\mathrm{x}^2+\frac{1}{\mathrm{x}^2}+3-\frac{2}{\mathrm{x}}-2 \mathrm{x}$
View full question & answer→Question 482 Marks
Find the square of $2x − 3y + z$
Answer$(2x − 3y + z)^2 $
$= (2x)^2 + (−3y)^2+ (z)^2 + 2 \times 2x \times −3y + 2(−3y) \times z + 2 \times z \times 2x$
$= 4x^2 + 9y^2+ z^2− 12xy − 6yz + 4zx$
View full question & answer→Question 492 Marks
Find the square of $5-x+\frac{2}{x}$
Answer$ \left(5-x+\frac{2}{x}\right)^2$
$=(5)^2+(-x)^2+\left(\frac{2}{x}\right)^2+2 \times 5 \times(-x)+2(-x) \times \frac{2}{x}+2 \times \frac{2}{x} \times 5$
$ =25+x^2+\frac{4}{x^2}-10 x-4+\frac{20}{x}$
$ =21+x^2+\frac{4}{x^2}-10 x+\frac{20}{x}$
View full question & answer→Question 502 Marks
Find the square of $2 x+\frac{1}{x}+1$
Answer$ \left(2 \mathrm{x}+\frac{1}{\mathrm{x}}+1\right)^2$
$=(2 \mathrm{x})^2+\left(\frac{1}{\mathrm{x}}\right)^2+(1)^2+2 \times 2 \mathrm{x} \times \frac{1}{\mathrm{x}}+2 \times \frac{1}{\mathrm{x}} \times 1+2 \times 1 \times 2 \mathrm{x}$
$ =4 \mathrm{x}^2+\frac{1}{\mathrm{x}^2}+1+4+\frac{2}{\mathrm{x}}+4 \mathrm{x}$
$ =4 \mathrm{x}+\frac{1}{\mathrm{x}^2}+5+\frac{2}{\mathrm{x}}+4 \mathrm{x}$
View full question & answer→