Question 12 Marks
Factorise the following, using the identity
$a^2- 2ab + b^2= (a - b)^2$.
$x^2- 10x + 25$
Answer$x^2- 10x + 25$
$= x^2- 2 × x × 5 + 5^2$
$= (x - 5)^2$
$= (x - 5)(x - 5)$
View full question & answer→Question 22 Marks
Factorise the following, using the identity $a^2+ 2ab + b^2= (a + b)^2$
$a^2x^2+ 2abxy + b^2y^2$
Answer$ a^2 x^2+2 a b x y+b^2 y^2 $
$ =(a x)^2+2 \times a x \times b y+(b y)^2 $
$ =(a x+b y)^2 $
$= (ax + by)(ax + by)$
View full question & answer→Question 32 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$\frac{1}{36}\text{a}^2\text{b}^2-\frac{16}{49}\text{b}^2\text{c}^2$
Answer$\frac{1}{36}\text{a}^2\text{b}^2-\frac{16}{49}\text{b}^2\text{c}^2$$=\Big(\frac{\text{ab}}{6}\Big)^2-\Big(\frac{\text{4bc}}{7}\Big)^2$
$=\Big(\frac{\text{ab}}{6}+\frac{\text{4bc}}{7}\Big)$
$=\text{b}^2\Big(\frac{\text{a}}{6}+\frac{\text{4c}}{7}\Big)\Big(\frac{\text{a}}{6}-\frac{\text{4c}}{7}\Big)$
View full question & answer→Question 42 Marks
Expand the following, using suitable identities.$\Big(\frac{2\text{a}}{3}+\frac{\text{b}}{3}\Big)\Big(\frac{2\text{a}}{3}-\frac{\text{b}}{3}\Big)$
Answer$\Big(\frac{2\text{a}}{3}+\frac{\text{b}}{3}\Big)\Big(\frac{2\text{a}}{3}-\frac{\text{b}}{3}\Big)$$=\Big(\frac{2\text{a}}{3}\Big)^2-\Big(\frac{\text{b}}{3}\Big)^2$
$=\frac{4}{9}\text{a}^2-\frac{1}{9}\text{b}^2$
View full question & answer→Question 52 Marks
Subtract: $-3p2 + 3pq + 3px$ from $3p(-p - a - r)$
AnswerThe required difference is
$ 3 p(-p-a-r)-\left(-3 p^2+3 p q+3 p x\right) $
$ =-3 p^2-3 a p-3 p r+3 p^2-3 p q-3 p x $
$ =\left(-3 p^2+3 p^2\right)-3 a p-3 p r-3 p q-3 p x $
View full question & answer→Question 62 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b). a^4- (a - b)^4$
Answer$ a^4-(a-b)^4 $
$ =\left(a^2\right)^2-\left[\left(a^2-b^2\right)\right]^2 $
$ =\left[a^2+(a-b)^2\right]\left[a^2-(a-b)^2\right] $
$ =\left[a^2+a^2+b^2-2 a b\right]\left[a^2-\left(a^2-\left(a^2+b^2-2 a b\right)\right]\right. $
$ =\left[2 a^2+b^2-2 a b\right]\left[-b^2+2 a b\right] $
$ =\left(2 a^2+b^2-2 a b\right)(2 a b-b) $
View full question & answer→Question 72 Marks
Factorise the following expressions.
$ 63 p^2 q^2 r^2 s-9 p q^2 r^2 s^2+15 p^2 q r^2 s^2-60 p^2 q^2 r s^2 $
Answer$ 63 p^2 q^2 r^2 s-9 p q^2 r^2 s^2+15 p^2 q r^2 s^2-60 p^2 q^2 r s^2 $
$ =3 \times 21 p^2 q^2 r^2 s-3 \times 3 p q^2 r^2 s^2+3 \times 5 p^2 q r^2 s^2-3 \times 20 p^2 q^2 r s^2 $
View full question & answer→Question 82 Marks
Multiply the following:
$-5a^2bc, 11ab, 13abc^2$
Answer$ -5 a^2 b c, 11 a b, 13 a b c^2 $
$ -5 a^2 b c \times 11 a b \times 13 a b c^2 $
$ =(-5 \times 11 \times 13) a^2 b c \times a b \times a b c^2 $
$ =-715 a^4 b^3 c^3 $
View full question & answer→Question 92 Marks
Multiply the following: $(p + 6), (q - 7)$
Answer$(p + 6), (q - 7) (p + 6) \times (q - 7) $
$=p(q - 7) + 6(q - 7) $
$= pq - 7p + 6q - 42$
View full question & answer→Question 102 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$ 16x^4- 625y^4$
Answer$16x^4- 625y^4$
$= (4x^2)^2- (25y^2)^2$
$= (4x^2+ 25y^2)(4x^2- 25y^2)$
$= (4x^2+ 25y^2)[(2x)^2- (5y)^2]$
$= (4x^2+ 25y^2)(2x + 5y)(2x - 5y)$
View full question & answer→Question 112 Marks
Factorise the following, using the identity $a^2+ 2ab + b^2= (a + b)^2 a^2x^2+ 2ax + 1$
Answer$ a^2 x^2+2 a x+1 $
$ =(a x)^2+2 \times a x \times 1+(1)^2 $
$ =(a x+1)^2 $
$ =(a x+1)(a x+1) $
View full question & answer→Question 122 Marks
Factorise the following, using the identity $a^2- 2ab + b^2= (a - b)^2$.
$9x^2- 12x + 4$
Answer$ 9 x^2-12 x+4 $
$ =(3 x)^2-2 \times 3 x \times 2+2^2 $
$ =(3 x-2)^2 $
$ =(3 x-2)(3 x-2) $
View full question & answer→Question 132 Marks
Using suitable identities, evaluate the following. $(35.4)^2- (14.6)^2$
Answer$(35.4)^2- (14.6)^2$
$= (35.4 + 14.6)(35.4 - 14.6)$
$= 50 \times 20.8$
$= 1040$
View full question & answer→Question 142 Marks
Factorise the following expressions. $l^2m^2n - lm^2n^2- l^2mn^2$
Answer$l^2m^2n - lm^2n^2- l^2mn^2lmn(lm - mn - ln)$
View full question & answer→Question 152 Marks
Using suitable identities, evaluate the following. $(103)^2$
Answer$ (103)^2 $
$ =(100+3)^2 $
$ =(100)^2+3^2+2 \times 100 \times 3 $
$= 10000 + 9 + 600$
$= 10609$
View full question & answer→Question 162 Marks
Using suitable identities, evaluate the following. $104 \times 97$
Answer$104 \times 97$
$= (100 + 4)(100 - 3)$
$= (100)^2+ (4 - 3)100 + 4 × (-3)$
$= 10000 + 100 - 12$
$= 10088$
View full question & answer→Question 172 Marks
Multiply the following: $6mn, 0mn$
Answer$6mn, 0mn$
$6mn \times 0mn$
$= (6 \times 0)mn \times mn$
$=0 \times m^2n^2$
$= 0$
View full question & answer→Question 182 Marks
Add: $2 p^4-3 p^3+p^2-5 p+7,-3 p^4-7 p^3-3 p^2-p-12$
Answer$ \left(2 p^4-3 p^3+p^2-5 p+7\right)+\left(-3 p^4-7 p^3-3 p^2-p-12\right) $
$ =2 p^4-3 p^3+p^2-5 p+7-3 p^4-7 p^3-3 p 2-p-12 $
$ =\left(2 p^4-3 p^4\right)+\left(-3 p^3-7 p^3\right)+\left(p^2-3 p^2\right)+(-5 p-p)+(7-12) $
$ =-p^4-10 p^3-2 p^2-6 p-5 $
View full question & answer→Question 192 Marks
Factorise the following, using the identity $a^2+ 2ab + b^2 = (a + b)^2$
$a^2x^3+ 2abx^2+ b^2x$
Answer$ a^2 x^3+2 a b x^2+b^2 x $
$ =x\left(a^2 x^2+2 a b x+b^2\right) $
$ =x\left[(ax)^2+2 \times a x \times b+b^2\right] $
$ =x(a x+b)^2 $
$=x(ax + b)(ax + b)$
View full question & answer→Question 202 Marks
Factorise the following using the identity $a^2-b^2=(a+b)(a-b)$
$ 4 x^2-49 y^2$
Answer$4x^2- 49y^2$
$(2x)^2- (7y)^2$
$= (2x - 7)(2x + 7y)$
View full question & answer→Question 212 Marks
The height of a triangle is $x^4+ y^4$ and its base is $14xy$. Find the area of the triangle.
AnswerThe height of a triangle $= x^4+ y^4$ and its base = $14xy$
Area of the triangle $=\frac{1}{2}\times\text{base}\times\text{height}$
$=\frac{1}{2}\times14\text{xy}\times(\text{x}^4+\text{y}^4)$
$=7\text{xy}(\text{x}^4+\text{y}^4)$
View full question & answer→Question 222 Marks
Using suitable identities, evaluate the following.
$(1005)^2$
Answer$(1005)^2= (1000 + 5)^2$
$= (1000)^2+ (5)^2+ 2 × 1000 × 5$
$= 1000000 + 25 + 10000$
$= 1010025$
View full question & answer→Question 232 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$\text{x}^2-\frac{\text{y}^2}{100}$
Answer$\text{x}^2-\frac{\text{y}^2}{100}$$=\text{x}^2-\Big(\frac{\text{y}}{10}\Big)^2$
$=\Big(\text{x}+\frac{\text{y}}{10}\Big)\Big(\text{x}-\frac{\text{y}}{10}\Big)$
View full question & answer→Question 242 Marks
Verify the following:
$ (m+n)\left(m^2-m n+n^2\right)=m^3+n^3 $
Answer$ (m+n)\left(m^2-m n+n^2\right)=m^3+n^3 $
$ =m\left(m^2-m n+n^2\right)+n\left(m^2-m n+n^2\right) $
$ =m^3-m^2 n+m n^2+n m^2-m n^2+n^3 $
$ =m^3+n^3 $
= RHS
Hence verified
View full question & answer→Question 252 Marks
Factorise the following expressions.
$6ab + 12bc$
Answer$6ab + 12bc$
$= 6ab + 6 \times 2 \times bc$
$= 6b(a + 2c)$
View full question & answer→Question 262 Marks
Find the value of a, if:
$pqa = (3p + q)^2- (3p - q)^2$
Answer$pqa = (3p + q)^2- (3p - q)^2$
$pqa = [(3p + q) + (3p - q)][(3p + q) - (3p - q)]$
$pqa = [(3p + q + 3p - q)][(3p + q - 3p + q)$
$\text{a}=\frac{6\text{p}\times2\text{q}}{\text{pq}}$
$=\frac{(6\times2)\text{pq}}{\text{pq}}$
$a = 12$
View full question & answer→Question 272 Marks
Perform the following divisions: $\left(3 p q r-6 p^2 q^2 r^2\right) \div 3 p q$
Answer$\left(3 p q r-6 p^2 q^2 r^2\right) \div 3 p q$
$\frac{3\text{pqr}-6\text{p}^2\text{q}^2\text{r}^2}{3\text{pqr}}$
$=\frac{3\text{pqr}}{3\text{pq}}-\frac{6\text{p}^2\text{q}^2\text{r}^2}{3\text{pq}}$
$=\text{r}-\frac{2\times3\times\text{p}\times\text{p}\times\text{q}\times\text{q}\times\text{r}\times\text{r}}{3\times\text{p}\times\text{q}}$
$=\text{r}-2\text{pqr}^2$
View full question & answer→Question 282 Marks
Factorise the following, using the identity $a^2+ 2ab + b^2= (a + b)^2$
$x^2+ 12x + 36$
Answer$ x^2+12 x+36 $
$ =x^2+2 \times 6 \times x+6^2 $
$ =(x+6)^2 $
$ =(x+6)(x+6) $
View full question & answer→Question 292 Marks
Using suitable identities, evaluate the following.
$98 \times 103$
Answer$98 × 103$
$= (100 - 2)(100 + 3)$
$= (100)^2+ (-2 + 3)100 + (-2) × 3$
$= 10000 + 100 - 6$
$= 10094$
View full question & answer→Question 302 Marks
Using suitable identities, evaluate the following.
$(729)^2- (271)^2$
Answer$(729)^2- (271)^2$
$= (729 + 271)(729 - 271)$
$= 1000 \times 458$
$= 458000$
View full question & answer→Question 312 Marks
The radius of a circle is $7ab - 7bc - 14ac$. Find the circumference of the circle. $\Big(\pi=\frac{22}{7}\Big)$
AnswerRadius of the circle $= 7ab - 7bc - 14ac = r$
The circumference of the circle = $2\pi\text{r}$
$=2\times\frac{22}{7}\times(7\text{ab}-\text{7bc}-14\text{ac})$
$=\frac{44}{7}\times7(\text{ab}-\text{bc}-\text{ac})$
$=44[\text{ab}-\text{c}(\text{b}+2\text{a})]$
View full question & answer→Question 322 Marks
Factorise the following.
$y^2- 2y - 15$
Answer$y^2- 2y - 15$
$= y^2+ (3 - 5)y - 15$
$= y^2+ 3y - 5y - 15$
$= y(y + 3) - 5(y + 3)$
$= (y + 3)(y - 5)$
View full question & answer→Question 332 Marks
Factorise the following, using the identity $a^2+ 2ab + b^2= (a + b)^2$
$2x^3+ 24x^2+ 72x$
Answer$ 2 x^3+24 x^2+72 x $
$ =2 x\left(x^2+12 x+36\right) $
$ =2 x\left(x^2+2 \times 6 \times x+6^2\right) $
$ =2 x(x+6)^2 $
$ =2 x(x+6)(x+6) $
View full question & answer→Question 342 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$y^4- 81$
Answer$ y^4-81 $
$=\left(y^2\right)^2-(9)^2 $
$ =\left(y^2+9\right)\left(y^2-9\right) $
$ =\left(y^2+9\right)\left[(y)^2-(3)^2\right] $
$ =\left(y^2+9\right)(y+3)(y-3) $
View full question & answer→Question 352 Marks
Using suitable identities, evaluate the following.
$105 \times 95$
Answer$105 × 95$
$= (100 + 5)(100 - 5)$
$= (100)^2- (5)^2$
$= 10000 - 25$
$= 9975$
View full question & answer→Question 362 Marks
Subtract $b(b^2+ b - 7) + 5$ from $3b^2- 8$ and find the value of expression obtained for $b = -3$.
AnswerRequired difference $= (3b^2- 8)[b(b^2+ b - 7) + 5]$
$ =3 b^2-8-b\left(b^2+b-7\right)-5 $
$ =3 b^2-8-b^3-b^2+7 b-5 $
$ =-b^3+2 b^2+7 b-13 $
Now, if $b = -3$
The value of expression $= -(-3)^2+ 2(-3)^2+ 7(-3) - 13$
$= -(-27) + 2 \times 9 - 21 - 13$
$= 27 + 18 - 21 - 13$
$= 45 - 34$
$= 11$
View full question & answer→Question 372 Marks
Perform the following divisions:
$\left(a x^3-b x^2+c x\right) \div(-d x)$
Answer$\left(a x^3-b x^2+c x\right) \div(-d x)$
$=\frac{\text{ax}^3-\text{bx}^2+\text{cx}}{-\text{dx}}$
$=\frac{\text{ax}^3}{-\text{dx}}+\frac{\text{bx}^2}{\text{dx}}+\frac{\text{cx}}{\text{-dx}}$
$=\frac{\text{a}\times\text{x}\times\text{x}\times\text{x}}{-\text{d}\times\text{x}}+\frac{\text{b}\times\text{x}\times\text{x}}{\text{d}\times\text{x}}+\frac{\text{c}\times\text{x}}{-\text{d}\times\text{x}}$
$=-\frac{\text{a}}{\text{b}}\text{x}^2+\frac{\text{b}}{\text{d}}\text{x}-\frac{\text{c}}{\text{d}}$
View full question & answer→Question 382 Marks
Subtract:
$2ab + 5bc - 7ac from 5ab - 2bc - 2ac + 10abc$
AnswerThe required difference is given by
$(5ab - 2bc - 2ac + 10abc) - (2ab + 5bc - 7ac)$
$= 5ab - 2bc - 2ac + 10abc - 2ab - 5bc + 7ac$
$= (5ab - 2ab) + (-2bc - 5bc) + (-2ac + 7ac) + 10abc$
$= 3ab - 7bc + 5ac + 10abc$
View full question & answer→Question 392 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$\frac{\text{x}^3}{8}-\frac{\text{y}^2}{18}$
Answer$\frac{\text{x}^3}{8}-\frac{\text{y}^2}{18}$$=\frac{1}{2}\Big(\frac{\text{x}^2}{4}+\frac{\text{y}^2}{9}\Big)$
$=\frac{1}{2}\bigg[\Big(\frac{\text{x}}{2}\Big)^2-\Big(\frac{\text{y}}{3}\Big)^2\bigg]$
$=\frac{1}{2}\Big(\frac{\text{x}}{2}+\frac{\text{y}}{3}\Big)\Big(\frac{\text{x}}{2}-\frac{\text{y}}{3}\Big)$
View full question & answer→Question 402 Marks
Multiply the following: $(a^2- b^2), (a^2+ b^2)$
Answer$(a^2- b^2), (a^2+ b^2)$
$ \therefore\left(a^2-b^2\right)\left(a^2+b^2\right) $
$ =a^2\left(a^2+b^2\right)-b^2\left(a^2+b^2\right) $
$ =a^4+a^2 b^2-b^2 a^2-b^4 $
$ =? 4-b^4 $
View full question & answer→Question 412 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$3a^2b^3- 27a^4b$
Answer$ 3 a^2 b^3-27 a^4 b $
$ 3 a^2 b\left(b^2-9 a^2\right) $
$ =3 a^2 b\left[b^2-(3 a)^2\right] $
$ =3 a^2 b(a+3 a)(b-3 a) $
View full question & answer→Question 422 Marks
Using suitable identities, evaluate the following. $52 \times 53$
Answer$52 \times 53$
$= (50 + 2)(50 + 3)$
$= (50)^2+ (2 + 3)50 + 2 × 3$
$= 2500 + 250 + 6$
$= 2756$
View full question & answer→Question 432 Marks
Factorise the following expressions.
$24 x^2 y z^3-6 x y^3 z^2+15 x^2 y^2 z-5 x y z$
Answer$24 x^2 y z^3-6 x y^3 z^2+15 x^2 y^2 z-5 x y z$
$= xyz(24xz^2- 6y^2z + 15xy - 5)$
View full question & answer→Question 442 Marks
Multiply the following:$-\frac{100}{9}\text{rs};\frac{3}{4}\text{r}^3\text{s}^2$
Answer$-\frac{100}{9}\text{rs};\frac{3}{4}\text{r}^3\text{s}^2$$-\frac{100}{9}\text{rs}\times\frac{3}{4}\text{r}^3\text{s}^2$ $=(-\frac{100}{9}\times\frac{3}{4})\text{rs}\times\text{r}^3\text{s}^2$
$=-\frac{25}{3}\times\text{r}^4\text{s}^3$
View full question & answer→Question 452 Marks
Multiply the following:
$ b^3, 3 b^2, 7 a b^5 $
Answer$ b^3, 3 b^2, 7 a b^5 $
$ b^3 \times 3 b^2 \times 7 a b^5 $
$ =(1 \times 3 \times 7) b^3 \times b^2 \times a b^5 $
$ =21 a b^{10} $
View full question & answer→Question 462 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$x^4- y^4$
Answer$ x^4-y^4 $
$ =\left(x^2\right)^2-\left(y^2\right)^2 $
$ =\left(x^2+y^2\right)\left(x^2-y^2\right) $
$ =\left(x^2+y^2\right)(x+y)(x-y) $
View full question & answer→Question 472 Marks
Write the greatest common factor in each of the following terms.
$-18a^2, 108a$
Answer$-18a^2, 108a$
$= -18a = -18 \times a \times a$
$= 108a = 18 \times 10 \times a$
The greatest common factor i.e. $GCF$ is $18a$.
View full question & answer→Question 482 Marks
Factorise the following expressions.
$a^2b + a^2c + ab + ac + b^2c + c^2b$
Answer$ a^2 b+a^2 c+a b+a c+b^2 c+c^2 b $
$ \left(a^2 b+a b+b^2 c\right)+\left(a^2 c+a c+c^2 b\right) $
$ =b\left(a^2+a+b c\right)+c\left(a^2+a+b c\right)$
$ =\left(a^2+a+b c\right)(b+c) $
View full question & answer→Question 492 Marks
Factorise the following.
$x^2+ 15x + 26$
Answer$x^2+ 15x + 26$
$= x^2+ 2x + 12x + 2 × 13$
$= x(x + 2) + 13(x + 2)$
$= (x + 2)(x + 13)$
View full question & answer→Question 502 Marks
Find the value of a, if:
$8a = 35^2- 27^2$
Answer$8a = 35^2- 27^2$
$8a = (35 + 27)(35 - 27)$
$8a= 62 \times 8$
$\text{a}=\frac{62\times8}{8}$
$= 62$
View full question & answer→