Question 13 Marks
Simplify:$a^5 ÷ a^3+ 3a \times 2a$
Answer$a^5 ÷ a^3+ 3a \times 2a$
$= a^{5−3} + 3a \times 2a$
$= a^2 + 6a^2$
$= 7a^2$
View full question & answer→Question 23 Marks
Simplify: $3 \mathrm{a} \times[8 \mathrm{~b} \div 4-6\{\mathrm{a}-(5 \mathrm{a}-\overline{3 \mathrm{~b}-2 \mathrm{a}})\}]$
Answer$ 3 a \times[8 b \div 4-6\{a-(5 a-\overline{3 b-2 a})\}]$
$ =3 a \times\left[\frac{8 b}{4}-6\{a-(5 a-3 b+2 a)\}\right]$
$ =3 a \times[2 b-6\{a-7 a+3 b\}]$
$ =3 a \times[2 b-6\{-6 a+3 b\}]$
$ =3 a \times[2 b+36 a-18 b]$
$ =3 a \times[36 a-16 b]$
$ =108 a^2-48 a b$
View full question & answer→Question 33 Marks
Simplify: $(y^3 − 5y^2) ÷ y \times (y − 1)$
Answer${l}\left(y^3-5 y^2\right) \div y \times(y-1)$
$=\frac{y^3-5 y^2}{y} \times y-1$
$=\left(y^2-5 y\right) \times(y-1)$
$=y^2(y-1)-5 y(y-1)$
$=y^3-y^2-5 y^2+5 y$
$=y^3-6 y^2+5 y$
View full question & answer→Question 43 Marks
Simplify: $(x^5 ÷ x^2) \times y^2\times y^3$
Answer$(x^5 ÷ x^2) \times y^2\times y^3$
$= x^{5−2}\times y^{2+3}$
$= x^3y^5$
View full question & answer→Question 53 Marks
Simplify: $x^5 ÷ (x^2 \times y^2) \times y^3$
Answer$x^5 \div\left(x^2 \times y^2\right) \times y^3 $
$ =\frac{x^5}{x^2 y^2} \times y^3 $
$=x^{5-2}-y^{3-2}$
$ =x^3 y$
View full question & answer→Question 63 Marks
Simplify : $\mathrm{a}-[\mathrm{a}-\overline{\mathrm{b}+\mathrm{a}}-\{\mathrm{a}-(\mathrm{a}-\overline{\mathrm{b}-\mathrm{a}})\}]$
Answer$ a-[a-\overline{b+a}-\{a-(a-\overline{b-a})\}]$
$ =a-[a-b-a-\{a-(a-b+a)\}]$
$ =a-[-b-\{a-a+b-a\}]$
$ =a-[-b-b+a]$
$ =a+b+b-a$
$ =2 b$
View full question & answer→Question 73 Marks
Simplify : $\mathrm{p}^2 \mathrm{x}-2\left\{\mathrm{px}-3 \mathrm{x}\left(\mathrm{x}^2-\overline{3 \mathrm{a}-\mathrm{x}^2}\right)\right\}$
Answer$ p^2 x-2\left\{p x-3 x\left(x^2-\overline{3 a-x^2}\right)\right\}$
$ =p^2 x-2\left\{p x-3 x\left(x^2-3 a+x^2\right)\right\}$
$ =p^2 x-2\left\{p x-3 x\left(2 x^2-3 a\right)\right\}$
$ =p^2 x-2\left\{p x-6 x^3+9 a x\right\}$
$ =p^2 x-2 p x+12 x^3-18 a x$
View full question & answer→Question 83 Marks
Simplify : $3 x-[3 x-\{3 x-(3 x-\overline{3 x-y})\}]$
Answer$ 3 x-[3 x-\{3 x-(3 x-\overline{3 x-y})\}]$
$ =3 x-[3 x-\{3 x-(3 x-3 x+y)\}]$
$ =3 x-[3 x-\{3 x-y\}]$
$ =3 x-[3 x-3 x+y]$
$ =3 x-y$
View full question & answer→Question 93 Marks
Simplify : $2\{m-3(n+\overline{m-2 n})\}$
Answer$ 2\{m-3(n+\overline{m-2 n})\}$
$ =2\{m-3(n+m-2 n)\}$
$ =2\{m-3(m-3(m-n)\}$
$ =2\{m-3 m+3 n\}$
$ =2\{3 n-2 m\}$
$ =6 n-4 m$
View full question & answer→Question 103 Marks
Simplify : $−3 (1 − x^2) − 2{x^2 − (3 − 2x^2)}$
Answer$−3 (1 − x^2) − 2 {x^2− (3 − 2x^2)}$
$= −3 + 3x^2 − 2 {x^2 − 3 + 2x^2}$
$= −3 + 3x^2 − 2{3x^2 − 3}$
$= −3 + 3x^2 − 6x^2 + 6$
$= 3 − 3x^2$
View full question & answer→Question 113 Marks
Simplify : $x-y-\{x-y-(x+y)-\overline{x-y}\}$
Answer$ x-y-\{x-y-(x+y)-\overline{x-y}\}$
$ =x-y-\{x-y-(x+y)-x+y\}$
$ =x-y-\{x-y-x-y-x+y\}$
$ =x-y-x+y+x+y+x-y$
$ =2 x$
View full question & answer→Question 123 Marks
Divide: $ x^2 + 3x − 54$ by $x − 6$
View full question & answer→Question 133 Marks
Divide: $6x^3− 13x^2 − 13x + 30 by 2x^2 − x − 6$
View full question & answer→Question 143 Marks
Divide: $12x^2 + 7xy − 12y^2 by 3x + 4y$
View full question & answer→Question 153 Marks
Divide: $a^2 + 7a + 12$ by $a + 4$
View full question & answer→Question 163 Marks
Divide: $−14x^6y^3− 21x^4y^5 + 7x^5y^4$ by $7x^2y^2$
Answer$ \frac{-14 x^6 y^3-21 x^4 y^5+7 x^5 y^4}{7 x^2 y^2}$
$ =\frac{-14 x^6 y^3}{7 x^2 y^2}-\frac{21 x^4}{7 x^2 y^2}+\frac{7 x^5 y^4}{7 x^2 y^2}$
$ =-2 x^{6-2} y^{3-2}-3 x^{4-2} y^{5-2}+x^{5-2} y^{4-2}$
$ =-2 x^4 y-3 x^2 y^3+x^3 y^2$
View full question & answer→Question 173 Marks
Divide: $15a^3b^4 − 10a^4b^3 − 25a^3b^6$ by $−5a^3b^2$
Answer$\frac{15 a^3 b^4-10 a^4 b^3-25 a^3 b^6}{-5 a^3 b^2}$
$=\frac{15 a^3 b^4}{-5 a^3 b^2}-\frac{10 a^4 b^3}{-5 a^3 b 2}-\frac{25 a^3 B^6}{-5 a^3 b^2}$
$=-3 b^{4-2}+2 a^{4-3} b^{3-2}+5 b^{6-2}$
$=-3 b^2+2 a b+5 b^4$
View full question & answer→Question 183 Marks
Divide: $15a^4b$ by $−5a^3b$
Answer$\frac{15 a^4 b}{-5 a^3 b}=\left(\frac{15}{-5}\right)\left(\frac{a^4}{a^3}\right)\left(\frac{b}{b}\right)$
$ =-3 a^{4-3} b^{1-1}$
$ =-3 a b^0$
$ =-3 a \times 1 \ldots \ldots .\left(\because b^0=1\right)$
$ =-3 a$
View full question & answer→Question 193 Marks
Evaluate $(x^5) \times (3x^2) \times (-2x)$ for $x = 1.$
AnswerFor $x = 1$
$(x^5) \times (3x^2) \times (−2x)$
$(1^5) \times (3 \times 1^2) \times (−2 \times 1)$
$1 \times 3 \times (−2)$
$= −6$
View full question & answer→Question 203 Marks
Evaluate $(3x^4y^2) (2x^2y^3)$ for $x = 1$ and $y = 2.$
Answer$(3x^4y^2) (2x^2y^3)$
$(3 \times 1^4 \times 2^2) \times (2 \times 1^2 \times 2^3)$
$(3 \times 1 \times 4) \times (2 \times 1 \times 8)$
$= 12 \times 16$
$= 192$
View full question & answer→Question 213 Marks
Find the value of $(3x^3) \times (-5xy^2) \times (2x^2yz^3)$ for $x = 1, y = 2$ and $z = 3.$
AnswerFor $x = 1, y = 2$ and $z = 3$
$(3x^3) \times (−5xy^2) \times (2x^2yz^3)$
$(3 \times 1^3) \times (−5 \times 1 \times 2^2) \times (2 \times 1^2 \times 2 \times 3^3)$
$3 \times (−5 \times 4) \times (2 \times 1 \times 2 \times 27)$
$3 \times (−20) \times 108$
$= −6480$
View full question & answer→Question 223 Marks
The base and the altitude of a triangle are $(3x – 4y)$ and $(6x + 5y)$ respectively. Find its area.
AnswerReqd. Area $=\frac{1}{2} ($base$) \times ($altitude$)$
$ =\frac{1}{2}(3 x-4 y)(6 x+5 y)$
$ =\frac{1}{2}\left(18 x^2+15 x y-24 x y-20 y^2\right)$
$ =\frac{1}{2}\left(18 x^2-9 x y-20 y^2\right)$ sq. unit.
View full question & answer→Question 233 Marks
The adjacent sides of a rectangle are $x^2 – 4xy + 7y^2$ and $x^3 – 5xy^2$. Find its area.
AnswerReqd. area $= (x^2 – 4xy + 7y^2) (x^3 – 5xy^2)$
$= x^2 (x^3– 5xy^2) – 4xy (x^3 – 5xy^2) + 7y^2 (x^3 – 5xy^2)$
$= x^5 – 5x^3y^2 – 4x^4y + 20x^2y^3 + 7x^3y^2– 35xy^4$
$= x^5+ 2x^3y^2 – 4x^4y + 20x^2y^3– 35xy^4$
$= (x^5 – 4x^4y + 2x^3y^2 + 20x^2y^3– 35xy^4)$ sq. unit.
View full question & answer→Question 243 Marks
Simplify : $(3y + 4z) (3y – 4z) + (2y + 7z) (y + z)$
Answer$(3y + 4z) (3y – 4z) + (2y + 7z) (y + z)$
$= 3y (3y – 4z) + 4z (3y – 4z) + 2y (y + z) (y + z)$
$= 9y^2 – 12yz + 12yz – 16z^2 + 2y^2 + 2yz + 7yz + 7z^2$
$= (9+2)y^2+ (–12 + 12 + 2 + 7)yz + (–16 + 7)z^2$
$= 11y^2 + 9yz – 9z^2$
View full question & answer→Question 253 Marks
Simplify : $(5a + 5b – c) (2b – 3c)$
Answer$(5a + 5b – c) (2b – 3c)$
$= 5a(2b – 3c) + 5b(2b – 3c) – c(2b – 3c)$
$= 10ab – 15ac + 10b^2 – 15ac – 2bc + 3c^2$
$= 10ab + 10b^2 – 17bc – 15ac + 3c^2$
View full question & answer→Question 263 Marks
Multiply: $5p^2+ 25pq + 4q^2 by 2p^2 − 2pq +3q^2$
Answer$5p^2 + 25pq + 4q^2$
$\times 2p^2 − 2pq + 3q^2$
$10p^4 + 50p^3q + 8p^2q^2$
$−10p^3q − 50p^2q^2 − 8pq^3$
$+15p^2q^2 + 75pq^3 + 12q^4$
$10p^4 + 40p^3q − 27p^2q^2 + 67pq^3 + 12q^4$
View full question & answer→Question 273 Marks
Multiply:$2y − 4y^3 + 6y^5 by y^2 + y − 3$
Answer$2y − 4y^3 + 6y^5$
$\times y^2 + y − 3$
$2y^3 − 4y^5 + 6y^7$
$+ 2y^2 − 4y^4 + 6y^6$
$− 6y + 12y^3− 18y^5$
$6y^7 + 6y^6−(4+18)y^5 − 4y^4 + (2+12)y^3 + 2y^2 − 6y$
$= 6y^7 + 6y^6 − 22y^5 − 4y^4 + 14y^3 + 2y^2 − 6y$
View full question & answer→Question 283 Marks
Multiply: $6x^3 − 5x + 10$ by $4 − 3x^2$
Answer$6x^3 − 5x + 10$
$\times 4 − 3x^2$
$24x^3 − 20x + 40$
$− 18x^5 + 15x^3 − 30x^2$
$− 18x^5 + 39x^3− 30x^2− 20x + 40$
View full question & answer→Question 293 Marks
Multiply: $2x^2 – 4x + 5 by x^2 + 3x – 7$
Answer$2x^2 – 4x + 5 by x^2 + 3x – 7$
$(2x^2 – 4x + 5) \times (x^2 + 3x – 7)$
$2x^2(x^2 + 3x − 7) − 4x(x^2 + 3x − 7) + 5(x^2 + 3x − 7)$
$2x^4 + 6x^3− 14x^2 − 4x^3 − 12x^2 + 28x + 5x^2+ 15x − 35$
$2x^4 + 6x^3 − 4x^3 − 14x^2 − 12x^2 + 5x^2+ 28x + 15x − 35$
$2x^4 + 2x^3− 21x^2 + 43x − 35$
View full question & answer→Question 303 Marks
Multiply: $2 x+\frac{1}{2} y$ and $2 x-\frac{1}{2} y$
Answer$ \left(2 x+\frac{1}{2} y\right)\left(2 x-\frac{1}{2} y\right)$
$ =2 x\left(2 x-\frac{1}{2} y\right)+\frac{1}{2} y\left(2 x-\frac{1}{2} y\right)$
$ =4 x^2-x y+x y-\frac{1}{4} y^2$
$ =4 x^2-\frac{1}{4} y^2$
View full question & answer→Question 313 Marks
Multiply: $2 \mathrm{a}^3-3 \mathrm{a}^2 \mathrm{~b}$ and $-\frac{1}{2} a \mathrm{~b}^2$
Answer$ \left(2 a^3-3 a^2 b\right)\left(-\frac{1}{2} a b^2\right)$
$ =-\frac{1}{2} a b^2\left(2 a^3-3 a^2 b\right)$
$ =2 a^3 \times-\frac{1}{2} a b^2-3 a^2 b \times-\frac{1}{2} a b^2$
$ =-a^4 b^2+\frac{3}{2} a^3 b^3$
View full question & answer→Question 323 Marks
Multiply: $-\frac{2}{3} a^7 b^2$ and $-\frac{9}{4} a b^5$
Answer$ \left(-\frac{2}{3} a^7 b^2\right)\left(-\frac{9}{4} a b^5\right)$
$ =\left(-\frac{2}{3} \times \frac{-9}{4}\right)\left(a^7 \times a\right)\left(b^2 \times b^5\right)$
$ =\frac{3}{2} a^8 b^7$
View full question & answer→Question 333 Marks
Evaluate: $(2x – 5y)(2x + 3y)$ for $x = 2$ and $y = 3.$
Answer$(2x – 5y) (2x + 3y)$
$= 2x \times 2x − 5y \times 2x + 2x \times 3y − 5y \times 3y$
$= 4x^2− 10xy + 6xy − 15y^2$^
Hence, $x = 2$ and $y = 3$
$4x^2 − 10xy + 6xy − 15y^2$
$= 4(2)^2 − 10(2)(3) + 6(2)(3) − 15(3)^2$
$= 16 − 60 + 36 − 135$
$= 52 − 195$
$= −143$
View full question & answer→Question 343 Marks
Evaluate:$(3x – 2)(x + 5)$ for $x = 2$.
AnswerFor $x = 2$
$(3x – 2) (x + 5)$
$(3 \times 2 − 2) (2 + 5)$
$(6 − 2) \times 7$
$4 \times 7$
$= 28$
View full question & answer→Question 353 Marks
If $x = 2$ and $y = 1$; find the value of $(−4x^2y^3) \times (−5x^2y^5).$
AnswerFor $x = 2$ and $y = 1$
$(−4x^2y^3) \times (−5x^2y^5)$
$(−4 \times 2^2 \times 1^3) \times (−5 \times 2^2 \times 1^5)$
$(−4 \times 4 \times 1) \times (−5 −4 \times 1)$
$−16 \times −20$
$= 320$
View full question & answer→Question 363 Marks
Subtract the sum of $5y^2 + y – 3$ and $y^2 – 3y + 7$ from $6y^2 + y – 2.$
Answer$5y^2 + y – 3$
$y^2 – 3y + 7$
$6y^2 – 2y + 4 6y^2 + y – 2$
$6y^2 – 2y + 4$
$– + –$
$3y – 6$
View full question & answer→Question 373 Marks
Take $m^2 + m + 4$ from $−m^2 + 3m + 6$ and the result from $m^2 + m + 1.$
Answer$− m^2 + 3m + 6$
$+ m^2 + m + 4$
$− − −$
$− 2m^2+ 2m + 2$
A.T.Q. $m^2 + m + 1$
$− 2m^2 + 2m + 2$
$+ − −$
$3m^2 − m − 1$
View full question & answer→Question 383 Marks
The perimeter of a triangle is $8y^2 – 9y + 4$ and its two sides are $3y^2 – 5y$ and $4y^2 + 12$. Find its third side.
AnswerPerimeter of the triangle $=$ Sum of three sides
$= 8y^2 – 9y + 4$
Sum of two sides $= 3y^2 – 5y + 4y^2 + 12$
$= 7y^2 − 5y + 12$
$\therefore (8y^2 – 9y + 4) – (7y^2 – 5y + 12)$
$= 8y^2 – 9y + 4 – 7y^2 + 5y – 12$
$= y^2 – 4y – 8$
Hence third side $= y^2 – 4y – 8$
View full question & answer→Question 393 Marks
The sides of a triangle are $x^2 – 3xy + 8, 4x^2 + 5xy – 3$ and $6 – 3x^2 + 4xy.$ Find its perimeter.
AnswerRequired perimeter $=$ Sum of three sides
$= x^2 − 3xy + 8 + 4x^2 + 5xy − 3 + 6 − 3x^2 + 4xy$
$= x^2 + 4x^2 − 3x^2 − 3xy + 5xy + 4xy + 8 − 3 + 6$
$= 2x^2 + 6xy + 11$
View full question & answer→