Question 12 Marks
Simplify: $(a + b + c)(a + b - c)$
Answer$ (a+b+c)(a+b-c) $
$ =a(a+b-c)+b(a+b-c)+c(a+b-c) $
$ =a^2+a b-a c+a b+b^2-b c+a c+b c-c^2 $
$ =a^2+a b+a b-a c+a c-b c+b c+b^2-c^2 $
$ =a^2+b^2-c^2+2 a b $
View full question & answer→Question 22 Marks
Simplify: $(1.5x - 4y)(1.5x + 4y + 3) - 4.5x + 12y.$
Answer$ (1.5 x-4 y)(1.5 x+4 y+3)-4.5 x+12 y $
$ =1.5 x(1.5 x+4 y+3)-4 y(1.5 x+4 y+3)-4.5 x+12 y $
$ =2.25 x^2+6 x y+4.5 x-6 x y-16 y^2-12 y-4.5 x+12 y $
$ =2.25 x^2+6 x y-6 x y+4.5 x-4.5 x-16 y^2-12 y+12 y $
$ =2.25 x^2-16 y^2 $
View full question & answer→Question 32 Marks
Simplify: $(x + y) (2x + y) + (x + 2y) (x - y)$
AnswerThe product of $(x + y) (2x + y) +(x + 2y) (x - y)$
$= x(2x + y) + y(2x + y) + x(x - y) + 2y(x - y)$
$= 2x^2+ xy + 2xy + y^2+ x^2- xy + 2xy - 2y^2$
$= (2x^2+ x^2) + (xy + 2xy - xy + 2xy) + (y^2- 2y^2)$ [grouping like terms]
$= 3x^2+ 4xy - y^2$
View full question & answer→Question 42 Marks
Simplify: $(a + b)(c - d) + (a - b)(c + d) + 2(ac + bd)$
Answer$(a + b)(c - d) + (a - b)(c + d) + 2(ac + bd)$
$= a(c - d) + b(c - d) + a(c + d) - b(c + d) + 2ac + 2bd$
$= ac - ad + bc - bd + ac + ad - bc - bd + 2ac + 2bd$
$= ac + ac - ad + ad + bc - bc - bd - bd + 2ac + 2bd$
$= 2ac - 2bd + 2ac + 2bd$
$= 4ac$
View full question & answer→Question 52 Marks
Simplify: $(a^2+ 5) (b^3+ 3) + 5$
AnswerWe have $(a^2+ 5) (b^3+ 3) + 5$
Opening the brackets we get,
$(a^2+ 5) (b^3+ 3) + 5 = a^2(b^3+ 3) + 5 (b^3+ 3) + 5$
$= a^2b^3+ 3a^2+ 5b^3+ 15 + 5$
$= a^2b^3+ 3a^2+ 5b^3+ 20$
View full question & answer→Question 62 Marks
Simplify: $(x^2- 5) (x + 5) + 25$
Answer$(x^2- 5) (x + 5) + 25$
$= x^2(x + 5) - 5(x + 5) + 25$
$= x^2$ $\times$ $x + x^2$ $\times$ $5 - 5$ $\times$ $x - 5$ $\times$ $5 + 25$
$= x^3+ 5x^2- 5x - 25 + 25$
$= x^3+ 5x^2- 5x$
View full question & answer→Question 72 Marks
Multiply the binomials: $\left(\frac{3}{4} a^{2}+3 b^{2}\right)$ and $ 4\left(a^{2}-\frac{2}{3} b^{2}\right)$
Answer$\left(\frac{3}{4} a^{2}+3 b^{2}\right) \times 4\left(a^{2}-\frac{2}{3} b^{2}\right)$
$=\left(\frac{3}{4} a^{2}+3 b^{2}\right) \times\left(4 a^{2}-\frac{8}{3} b^{2}\right)$
$=\frac{3}{4} a^{2} \times\left(4 a^{2}-\frac{8}{3} b^{2}\right)+3 b^{2} \times\left(4 a^{2}-\frac{8}{3} b^{2}\right)$
$=\frac{3}{4} a^{2} \times 4 a^{2}-\frac{3}{4} a^{2} \times \frac{8}{3} b^{2}+3 b^{2} \times 4 a^{2}-3 b^{2} \times \frac{8}{3} b^{2}$
$=3 a^{4}-2 a^{2} b^{2}+12 a^{2} b^{2}-8 b^{4}$
$=3 a^{4}+10 a^{2} b^{2}-8 b^{4}$
View full question & answer→Question 82 Marks
Multiply the binomials $(2pq + 3q^2)$ and $(3pq - 2q^2)$
AnswerProduct of $(2pq + 3q^2)$ and $(3pq - 2q^2)$
$= (2pq + 3q^2)(3pq - 2q^2)$
$= 2pq(3pq - 2q^2) + 3q^2(3pq - 2q^2)$
$= 6p^2q^2- 4pq^3+ 9pq^3- 6q^4$
$= 6p^2q^2+ 5pq^3- 6q^4$
View full question & answer→Question 92 Marks
Multiply the binomials $(2.51 - 0.5m)$ and $(2.51 + 0.5m)$.
AnswerProduct of $(2.51 – 0.5m)$ and $(2.51 + 0.5m)$
$= (2.51 – 0.5m) (2.51 + 0.5m)$
$= (2.51)^2- (0.5m)^2$ [Using identity: $(a + b) (a - b) = a^2- b^2]$
$= 6.3001 - 0.25m^2$
View full question & answer→Question 102 Marks
Multiply the binomials: $(2x + 5)$ and $(4x - 3)$.
Answer$(2x + 5 )$$\times$$(4x - 3)$
$= 2x(4x - 3) + 5(4x - 3)$
$= 2x \times 4x - 2x \times 3 + 5 \times 4x - 5 \times 3$
$= 8x^2- 6x + 20x - 15$
$= 8x^2+ 14x - 15$
View full question & answer→Question 112 Marks
Subtract: $3a(a + b + c) - 2b(a - b + c)$ from $4c(-a + b + c)$.
Answer$ 4 c(-a+b+c)-[3 a(a+b+c)-2 b(a-b+c)] $
$ =-4 a c+4 b c+4 c^2-\left[3 a^2+3 a b+3 a c-2 a b+2 b^2-2 b c\right] $
$ =-4 a c+4 b c+4 c^2-\left[3 a^2+2 b^2+3 a b-2 b c+3 a c-2 a b\right] $
$ =-4 a c+4 b c+4 c^2-\left[3 a^2+2 b^2+a b+3 a c-2 b c\right] $
$ =-4 a c+4 b c+4 c^2-3 a^2-2 b^2-a b-3 a c+2 b c $
$ =-3 a^2-2 b^2+4 c^2-a b+4 b c+2 b c-4 a c-3 a c $
$ =-3 a^2-2 b^2+4 c^2-a b+6 b c-7 a c $
View full question & answer→Question 122 Marks
Add: $2x(z - x - y)$ and $2y(z - y - x)$
Answer$2x(z - x - y) + 2y(z - y - x)$
$=2 x z-2 x^{2}-2 x y+2 y z-2 y^{2}-2 x y$
$=2 x z-2 x y-2 x y+2 y z-2 x^{2}-2 y^{2}$
$=-2 x^{2}-2 y^{2}-4 x y+2 y z+2 z x$
View full question & answer→Question 132 Marks
Simplify $a (a^2+ a + 1) + 5$ and find its value for $a = - 1$
AnswerWe have $a(a^2+ a + 1) + 5$
$a(a^2+ a + 1) + 5$
$= a^3+ a^2+ a + 5$
Substituting $a = -1$ in the expression
$a^3+ a^2+ a + 5$
$= (-1)^3+ (-1)^2- 1 + 5$ [$\because$$(-1)^n= 1$ if $n$ =even, $(-1)^n= -1$ if $n$ = odd]
$= - 1 + 1 - 1 + 5$
$= 4$
View full question & answer→Question 142 Marks
Simplify $a (a^2+ a + 1) + 5$ and find its value for $a = 1$
Answer$a(a^2+ a + 1) + 5$
$= a^3+ a^2+ a + 5$
Substituting $a = 1$ in the expression
$a^3+ a^2+ a + 5$
$= 1^3+ 1^2+ 1 + 5$
$= 8$
View full question & answer→Question 152 Marks
Simplify $3x (4x – 5) + 3$ and find its values for $x = 3$.
AnswerWe have $3x (4x - 5) + 3$
$3x (4x - 5) + 3 = 3x (4x) - 3x(5) + 3$
$= 12x^2- 15x + 3$
Putting $x = 3$ in above equation, we get $12 (3)^2- 15(3) + 3$
$= 12 (9) - 45 + 3$
$= 108 - 42 = 66$
View full question & answer→Question 162 Marks
Complete the product table:
| First expression |
Second expression |
Product |
| $a$ |
$b + c + d$ |
- |
| $x + y - 5$ |
$5\ xy$ |
- |
| $p$ |
$6p^2- 7p + 5$ |
- |
| $4p^2q^2$ |
$p^2- q^2$ |
- |
| $a + b + c$ |
$abc$ |
- |
Answer
| First expression |
Second expression |
Product |
| $a$ |
$b + c + d$ |
$ab + ac + ad$ |
| $x + y - 5$ |
$5\ xy$ |
$5x^2y + 5xy^2- 25xy$ |
| $p$ |
$6p^2- 7p + 5$ |
$6p^3- 7p^2+ 5p$ |
| $4p^2q^2$ |
$p^2- q^2$ |
$4p^4q^2- 4p^2q^4$ |
| $a + b + c$ |
$abc$ |
$a^2bc + ab^2c+ abc^2$ |
View full question & answer→Question 172 Marks
Obtain the product of $xy, yz, zx$
AnswerRequired product
$= (xy)$ $\times$ $(yz)$ $\times$ $(zx)$
$= (x$ $\times$ $x)$ $\times$ $(y$ $\times$ $y)$ $\times$ $(z$ $\times$ $z)$
$= x^2$ $\times$ $y^2$ $\times$ $z^2$
$= x^2y^2z^2$
View full question & answer→Question 182 Marks
Obtain the volume of rectangular box with the length, breadth and height respectively: $xy, 2x^2y, 2xy^2$
AnswerVolume of the rectangular box
= Length $\times$ Breadth $\times$ Height
$= (xy) \times (2x^2y) \times (2xy^2)$
$= (2 \times 2) \times (x \times x^2 \times x) \times (y \times y \times y^2)$
$= 4x^4y^4$
View full question & answer→Question 192 Marks
Obtain the volume of rectangular box with the length, breadth and height respectively: $5a, 3a^2, 7a^4$
AnswerVolume of the rectangular box
= Length $\times$ Breadth $\times$ Height
$= (5a) \times (3a^2) \times (7a^4)$
$= (5 \times 3 \times 7) \times (a \times a^2 \times a^4)$
$= 105a^7$
View full question & answer→Question 202 Marks
Find the areas of rectangle with the monomials as their lengths and breadths respectively: $(3mn, 4np)$
AnswerArea of the rectangle
= Length $\times$ Breadth
$= (3mn) \times (4np)$
$= (3 \times 4) \times (mn) \times (np)$
$= 12 \times m \times (n \times n) \times p$
$= 12\ mn^2p.$
View full question & answer→Question 212 Marks
Find the areas of rectangle with the monomials as their lengths and breadths respectively: $(4x, 3x^2)$
AnswerArea of the rectangle
= Length $\times$ Breadth
$= (4x) \times (3x^2)$
$= (4 \times 3) \times (x \times x^2)$
$= 12 \times x^3= 12x^3$
View full question & answer→Question 222 Marks
Find the areas of rectangle with the monomials as their lengths and breadths respectively: $(20x^2, 5y^2)$
AnswerArea of the rectangle
= Length $\times$ Breadth
$ =\left(20 x^2\right) \times\left(5 y^2\right) $
$ =(20 \times 5) \times\left(x^2 \times y^2\right) $
$ =100 \times\left(x^2 y^2\right) $
$ =100 x^2 y^2 $
View full question & answer→Question 232 Marks
Subtract $4p^2q – 3pq + 5pq^2– 8p + 7q – 10$ from $18 – 3p + 11q + 5pq – 2pq^2+ 5p^2q.$
Answer
$p^2q - 7pq^2+ 8pq - 18q + 5p + 28$ View full question & answer→Question 242 Marks
Subtract $3xy + 5yz – 7zx$ from $5xy – 2yz – 2zx + 10xyz.$
View full question & answer→Question 252 Marks
Subtract $4a – 7ab + 3b +12$ from $12a – 9ab + 5b – 3$.
View full question & answer→Question 262 Marks
Add: $4y(3y^2+ 5y – 7)$ and $2(y^3– 4y^2+ 5)$
AnswerThe first expression $= 4y(3y^2+ 5y – 7) = (4y \times 3y^2) + (4y \times 5y) + (4y \times (–7)) = 12y^3+ 20y^2– 28y$
The second expression $= 2(y^3– 4y^2+ 5) = 2y^3+ 2 \times (– 4y^2) + 2 \times 5 = 2y^3- 8y^2+ 10$
Adding the two expressions,
Thus, the sum is $14y^3+ 12y^2- 28y + 10$ View full question & answer→Question 272 Marks
Simplify the expression and evaluate as directed: $3y(2y – 7) – 3(y – 4) – 63$ for $y = –2$
Answer$3y(2y – 7) – 3(y – 4) – 63$
$= 6y^2- 21y – 3y + 12 – 63$
$= 6y^2– 24y – 51$
For $y = –2$,
$6y^2– 24y – 51 = 6(–2)^2– 24(–2) – 51$
$= 6 × 4 + 24 × 2 \;– 51$
$= 24 + 48 – 51 = 72 – 51 = 21$
View full question & answer→Question 282 Marks
Complete the table for the area of a rectangle with given length and breadth.
| Length |
Breadth |
Area |
| $3x$ |
$5y$ |
$3x$ $\times$ $5y = 15xy$ |
| $9y$ |
$4y^2$ |
_______ |
| $4ab$ |
$5bc$ |
_______ |
| $2l^2m$ |
$3lm^2$ |
_______ |
AnswerAfter doing the product of length and breadth, we get
| Length |
Breadth |
Area |
| $3x$ |
$5y$ |
$3x$ $\times$ $5y = 15xy$ |
| $9y$ |
$4y^2$ |
$9y$ $\times$ $4y^2= 36y^3$ |
| $4ab$ |
$5bc$ |
$4ab$ $\times$ $5bc = 20ab^2c$ |
| $2l^2m$ |
$3lm^2$ |
$2l^2m$ $\times$ $3lm^2= 6l^3m^3$ |
View full question & answer→Question 292 Marks
Subtract the expression $5x^2– 4y^2+ 6y – 3$ from $7x^2– 4xy + 8y^2+ 5x – 3y$
AnswerWriting the two expressions in separate rows, with like terms one below the other, we have
Thus, the difference of the expressions is $2x^2– 4xy + 12y^2+ 5x – 9y + 3$ View full question & answer→Question 302 Marks
Simplify: $(a + b)(2a – 3b + c) – (2a – 3b)c$
Answer$(a + b)(2a – 3b + c) – (2a – 3b)c$
$= a(2a – 3b + c) + b(2a – 3b + c) – (2ac – 3bc)$
$= 2a^2- 3ab + ac + 2ab – 3b^2+ bc – 2ac + 3bc$
$= 2a^2– ab – 3b^2+ (bc + 3bc) + (ac – 2ac)$
$= 2a^2– 3b^2– ab + 4bc – ac$
View full question & answer→Question 312 Marks
Add: $7xy + 5yz – 3zx, 4yz + 9zx – 4y, –3xz + 5x – 2xy$
AnswerWriting the three expressions in separate rows, with like terms one below the other, we have
Thus, the sum of the expressions is $5xy + 9yz + 3zx + 5x – 4y.$ View full question & answer→