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$$-\frac{100}{9}\text{rs}\times\frac{3}{4}\text{r}^3\text{s}^2$ $=(-\frac{100}{9}\times\frac{3}{4})\text{rs}\times\text{r}^3\text{s}^2$
$=-\frac{25}{3}\times\text{r}^4\text{s}^3$
View full question & answer→Question 452 Marks
Multiply the following:
$ b^3, 3 b^2, 7 a b^5 $
Answer$ b^3, 3 b^2, 7 a b^5 $
$ b^3 \times 3 b^2 \times 7 a b^5 $
$ =(1 \times 3 \times 7) b^3 \times b^2 \times a b^5 $
$ =21 a b^{10} $
View full question & answer→Question 462 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$x^4- y^4$
Answer$ x^4-y^4 $
$ =\left(x^2\right)^2-\left(y^2\right)^2 $
$ =\left(x^2+y^2\right)\left(x^2-y^2\right) $
$ =\left(x^2+y^2\right)(x+y)(x-y) $
View full question & answer→Question 472 Marks
Write the greatest common factor in each of the following terms.
$-18a^2, 108a$
Answer$-18a^2, 108a$
$= -18a = -18 \times a \times a$
$= 108a = 18 \times 10 \times a$
The greatest common factor i.e. $GCF$ is $18a$.
View full question & answer→Question 482 Marks
Factorise the following expressions.
$a^2b + a^2c + ab + ac + b^2c + c^2b$
Answer$ a^2 b+a^2 c+a b+a c+b^2 c+c^2 b $
$ \left(a^2 b+a b+b^2 c\right)+\left(a^2 c+a c+c^2 b\right) $
$ =b\left(a^2+a+b c\right)+c\left(a^2+a+b c\right)$
$ =\left(a^2+a+b c\right)(b+c) $
View full question & answer→Question 492 Marks
Factorise the following.
$x^2+ 15x + 26$
Answer$x^2+ 15x + 26$
$= x^2+ 2x + 12x + 2 × 13$
$= x(x + 2) + 13(x + 2)$
$= (x + 2)(x + 13)$
View full question & answer→Question 502 Marks
Find the value of a, if:
$8a = 35^2- 27^2$
Answer$8a = 35^2- 27^2$
$8a = (35 + 27)(35 - 27)$
$8a= 62 \times 8$
$\text{a}=\frac{62\times8}{8}$
$= 62$
View full question & answer→Question 512 Marks
Expand the following, using suitable identities.$\Big(\frac{4}{5}\text{a}+\frac{5}{4}\text{b}\Big)^2$
Answer$\Big(\frac{4}{5}\text{a}+\frac{5}{4}\text{b}\Big)^2$$=\Big(\frac{4}{5}\text{a}\Big)^2+\Big(\frac{5}{4}\text{b}\Big)^2+2\times\frac{4}{5}\text{a}\times\frac{5}{4}\text{b}$
$=\frac{16}{25}\text{a}^2+\frac{25}{16}\text{b}^2+2\text{ab}$
View full question & answer→Question 522 Marks
Expand the following, using suitable identities.
$(7x + 5)^2$
Answer$(7x + 5)^2$
$= (7x)^2+ 5^2+ 2 × 7x × 5$
$= 49x^2+ 25 + 70x$
View full question & answer→Question 532 Marks
Using suitable identities, evaluate the following.
$(132)^2- (68)^2$
Answer$(132)^2- (68)^2$
$= (132 + 68)(132 - 68)$
$= 200 \times 64$
$= 12800$
View full question & answer→Question 542 Marks
Add:
$x y^2 z^2+3 x^2 y^2 z-4 x^2 y z^2,-9 x^2 y^2 z+3 x y^2 z^2+x^2 y z^2$
Answer$ \left(x y^2 z^2+3 x^2 y^2 z-4 x^2 y z^2\right)+\left(-9 x^2 y^2 z+3 x y^2 z^2+x^2 y z^2\right) $
$ =x y^2 z^2+3 x^2 y^2 z-4 x^2 y z^2-9 x^2 y^2 z+3 x y^2 z^2+x^2 y z^2 $
$ =\left(x y^2 z^2+3 x y^2 z^2\right)+\left(3 x^2 y^2 z-9 x^2 y^2 z\right)+\left(-4 x^2 y z^2+x^2 y z^2\right) $
$ =4 x y^2 z^2-6 x^2 y^2 z^2-3 x^2 y z^2 $
View full question & answer→Question 552 Marks
If $x - y = 13$ and $xy = 28$, then find $x^2+ y^2$.
AnswerGiven, $x - y = 13$ and $xy = 28$
Since,
$ (13)^2=x^2+y^2-2 x y $
$ x^2+y^2=(13)^2+56 $
$ x^2+y^2=169+56 $
$ x^2+y^2=225 $
View full question & answer→Question 562 Marks
Factorise the following.
$a^2- 16p - 80$
Answer$a^2- 16p - 80$
$= a^2- (20 - 4)p - 80$
$= a^2- 20p + 4p - 80$
$= a(a - 20) + 4(p - 20)$
$= (a - 20)(a + 4)$
View full question & answer→Question 572 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$ x^4-y^4+x^2-y^2 $
Answer$ x^4-y^4+x^2-y^2 $
$ =\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)+\left(x^2-y^2\right) $
$ =\left(x^2-y^2\right)\left(x^2+y^2+1\right) $
$ =(x+y)(x-y)\left(x^2+y^2+1\right) $
View full question & answer→Question 582 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$ (a - b)^2 - (b - c)^2$
Answer$(a - b)^2 - (b - c)^2$
$= (a - b + b - c)(a - b - b + c)(a - c)(a - 2b + c)$
View full question & answer→Question 592 Marks
Add: $3a(a - b + c), 2b(a - b + c)$
Answer$ 3 a(a-b+c)+2 b(a-b+c) $
$ =\left(3 a^2-3 a b+3 a c\right)+\left(2 a b-2 b^2+2 b c\right) $
$ =3 a^2-3 a b+2 a b+3 a c+2 b c-2 b^2 $
$ =3 a^2-a b+3 a c+2 b c-2 b^2 $
View full question & answer→Question 602 Marks
Factorise the following.
$x^2- 17x + 60$
Answer$x^2- 17x + 60$
$= x2 - (12 + 5)x + 60$
$= x2 - 12x - 5x + 60$
$= x(x - 12) - 5(x - 12)$
$= (x - 12)(x - 5)$
View full question & answer→Question 612 Marks
Add: $9ax + 3by - cz, -5by + ax + 3cz$
AnswerWe have, $(9ax + 3by - cz) + (-5by + ax + 3cz) $
$= 9ax + 3by - cz - 5by + ax + 3cz$
$ = (9ax + ax) + (3by - 5by) + (-cz + 3cz) $
$= 10ax - 2by + 2cz$
View full question & answer→Question 622 Marks
Perform the following divisions: $(-qrxy + pryz - rxyz) ÷ (-xyz)$
Answer$(-qrxy + pryz - rxyz) ÷ (-xyz)$$=\frac{-\text{qrxy}+\text{pryz}-\text{rxyz}}{-\text{xyz}}$
$=\frac{-\text{qrxy}}{-\text{xyz}}+\frac{\text{pryz}}{-\text{xyz}}-\frac{\text{rxyz}}{-\text{xyz}}$
$=\frac{\text{qr}}{\text{z}}-\frac{\text{pr}}{\text{x}}+\text{r}$
View full question & answer→Question 632 Marks
Factorise the following, using the identity $a^2+ 2ab + b^2= (a + b)^2$
$9x^2+ 24x + 16$
Answer$ 9 x^2+24 x+16 $
$ =(3 x)^2+2 \times 3 x \times 4+4^2 $
$ =(3 x+4)^2 $
$ =(3 x+4)(3 x+4) $
View full question & answer→Question 642 Marks
Write the greatest common factor in each of the following terms.
$21pqr$, $-7 p^2 q^2 r^2, 49 p^2 q r$
Answer21pqr, $-7 p^2 q^2 r^2, 49 p^2 q r$
$21 p q r=3 \times 7 \times p \times q \times r $
$ -7 p^2 q^2 r^2=-7 \times p \times p \times q \times q \times r \times r$
$ 49 p^2 q r=7 \times 7 \times p \times p \times q \times r$
The greatest common factor i.e. $GCF$ is $7$.
View full question & answer→Question 652 Marks
Factorise the following expressions.
$4 x y^2-10 x^2 y+16 x^2 y^2+2 x y$
Answer$4 x y^2-10 x^2 y+16 x^2 y^2+2 x y$
$=2 \times 2 x y^2-2 \times 5 \times x^2 y+2 \times 8 \times x^2 y^2+2 x y$
View full question & answer→Question 662 Marks
Factorise the following, using the identity $a^2+2 a b+b^2=(a+b)^2$
$x^2+14 x+49$
Answer$x^2+14 x+49$
$=x^2+2 \times 7 \times x+7^2$
$=(x+7)^2$
$= (x + 7)(x + 7)$
View full question & answer→Question 672 Marks
Factorise the following, using the identity $\mathrm{a}^2-2 \mathrm{ab}+\mathrm{b}^2=(\mathrm{a}-\mathrm{b})^2$.
$y^2-14 y+49$
Answer$y^2-14 y+49$
$ =y^2-2 \times y \times 7+7^2 $
$ =(y-7)^2 $
$=(y-7)(y-7)$
View full question & answer→Question 682 Marks
Factorise the expressions and divide them as directed:
$\left(x^2-22 x+117\right) \div(x-13)$
Answer$\left(x^2-22 x+117\right) \div(x-13)$
$=\frac{\text{x}^2-22\text{x}+117}{\text{x}-13}$
$=\frac{\text{x}^2-13\text{x}-9\text{x}+117}{\text{x}-13}$
$=\frac{\text{x}(\text{x}-13)-9(\text{x}-13)}{\text{x}-13}$
$=\frac{(\text{x}-13)(\text{x}-9)}{\text{x}-13}$
$=\text{x}-9$
View full question & answer→Question 692 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$ 9 x^2-(3 y+z)^2 $
Answer$ 9 x^2-(3 y+z)^2 $
$ =9 x^2-(3 y+z)^2 $
$ =(3 x)^2-(3 y+z)^2 $
$ =(3 x+3 y+z)(3 x-3 y-z) $
View full question & answer→Question 702 Marks
Factorise the following, using the identity $a^2-2 a b+b^2=(a-b)^2$.
$p^2 y^2-2 p y+1$
Answer$p^2 y^2-2 p y+1$
$ =(p y)^2-2 \times p y \times 1+1^2 $
$ =(p y-1)^2 $
$ =(p y-1)(p y-1)$
View full question & answer→Question 712 Marks
Factorise the following expressions.
$ 2 a x^2+4 a x y+3 b x^2+2 a y^2+6 b x y+3 b y^2 $
Answer$ 2 a x^2+4 a x y+3 b x^2+2 a y^2+6 b x y+3 b y^2 $
$ =\left(2 a x^2+2 a y^2+4 a x y\right)+\left(3 b x^2+3 b y^2+6 b x y\right) $
$ =2 a\left(x^2+y^2+2 x y\right)+3 b\left(x^2+y^2+2 x y\right) $
$ =(2 a+3 b)(x+y)^2 $
View full question & answer→Question 722 Marks
Factorise the following expressions.
$ a^3+a^2+a+1 $
Answer$ a^3+a^2+a+1 $
$=a^2(a+1)+1(a+1) $
$ =(a+1)\left(a^2+1\right) $
View full question & answer→Question 732 Marks
If $m - n = 16$ and $m^2+ n^2= 400$, then find $mn$.
AnswerGiven, $m - n = 16$ and $m^2+ n^2= 400$
Since,
$ (m-n)^2=m^2+n^2-2 m n $
$ (16)^2=400-2 m n $
$ 2 m n=400-(16)^2 $
$2mn = 400 - 256$
$\text{mn}=\frac{144}{2}$
$= 72$
View full question & answer→Question 742 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$49x^2- 36y^2$
Answer$49x^2- 36y^2$
$= (7x)^2- (6y)^2$
$= (7x - 6y)(7x + 6y)$
View full question & answer→Question 752 Marks
Expand the following, using suitable identities.
$ \left(a^2+b^2\right)^2 $
Answer$ \left(a^2+b^2\right)^2 $
$ =\left(a^2\right)^2+\left(b^2\right)^2+2 a^2 \times b^2 $
$ =a^4+b^4+2 a^2 b^2 $
View full question & answer→Question 762 Marks
Simplify:
$ \left(x^2-4\right)+\left(x^2+4\right)+16 $
Answer$ \left(x^2-4\right)+\left(x^2+4\right)+16 $
$ =x^2-4+x^2+4+16=2 x^2+16 $
View full question & answer→Question 772 Marks
Find the value of a, if:
$p q^2 a=(4 p q+3 q)^2-(4 p q-3 q)^2$
Answer$p q^2 a=(4 p q+3 q)^2-(4 p q-3 q)^2$
$= [(4pq + 3q) + (4pq - 3q)][(4pq + 3q) - (4pq - 3q)]$
$= (4pq + 3q + 4pq - 3q)(4pq + 3q - 4pq + 3q)$
$= 8pq × 6q p^2a$
$= 48pq62$
$\text{a}=\frac{48\text{pq}^2}{\text{pq}^2}=48$
View full question & answer→Question 782 Marks
Factorise the following, using the identity $a^2+2 a b+b^2=(a+b)^2$
$4 x^2+4 x+1$
Answer$4 x^2+4 x+1$
$ =(2 x)^2+2 \times 2 x \times 1+1^2 $
$ =(2 x+1)^2 $
$ =(2 x+1)(2 x+1)$
View full question & answer→Question 792 Marks
The area of a square is $9 x^2+24 x y+16 y^2$. Find the side of the square.
AnswerArea of square $=9 x^2+24 x y+16 y^2$
$=(3 x)^2+2 \times 3 x \times 4 y+(4 y)^2$
$=(3 x+4 y)^2$
The side of the square is $3x + 4y$
View full question & answer→Question 802 Marks
Factorise the following expressions.
$ 3 p q r-6 p^2 q^2 r^2-15 r^2 $
Answer$ 3 p q r-6 p^2 q^2 r^2-15 r^2 $
$ 3 p q r-3 \times 2 p^2 q^2 r^2-3 \times 5 r^2 $
$ =3 r\left(p q-2 p^2 q^2 r-5 r\right) $
View full question & answer→Question 812 Marks
Multiply the following:
$ x^2 y^2 z^2,(x y-y z+z x) $
Answer$ x^2 y^2 z^2,(x y-y z+z x) $
$ x^2 y^2 z^2 \times(x y-y z+z x) $
$ =x^2 y^2 z^2 \times x y-x^2 y^2 z^2 \times x y-x^2 y^2 z^2 \times y z+x^2 y^2 z^2 \times z x $
$ =x^3 y^3 z^3-x^2 y^3 z^3+x^2 y^2 z^3 $
View full question & answer→Question 822 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b)$.
$ x^2- 9$
Answer$ x^2- 9$
$x^2- 3^2= (x - 3)(x + 3)$
View full question & answer→Question 832 Marks
Multiply the following:
$ \left(x^2-5 x+6\right),(2 x+7) $
Answer$ \left(x^2-5 x+6\right),(2 x+7) $
$ \left(x^2-5 x+6\right)(2 x+7) $
$ =x^2(2 x+7)-5 x(2 x+7)+6(2 x+7) $
$ =2 x^3+7 x^2-10 x^2-35 x+12 x+42 $
$ =2 x^3-3 x^2-23 x+42 $
View full question & answer→Question 842 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$ 16 x^4-81 $
Answer$ 16 x^4-81 $
$ =\left(4 x^2\right)^2-9^2=\left(4 x^2+9\right)\left(4 x^2-9\right)$
$=\left(4 x^2+9\right)\left[(2 x)^2-3^2\right] $
$ =\left(4 x^2+9\right)(2 x+3)(2 x-3) $
View full question & answer→Question 852 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$\frac{4\text{x}^2}{9}-\frac{9\text{y}^2}{16}$
Answer$\frac{4\text{x}^2}{9}-\frac{9\text{y}^2}{16}$$=\Big(\frac{2\text{x}}{3}\Big)^2-\Big(\frac{3\text{y}}{4}\Big)^2$
$=\Big(\frac{2\text{x}}{3}+\frac{3\text{y}}{4}\Big)\Big(\frac{2\text{x}}{3}-\frac{3\text{y}}{4}\Big)$
View full question & answer→Question 862 Marks
Find the value of a, if:
$9a = 76^2- 67^2$
Answer$9a = 76^2- 67^2$
$9a = (76 + 67)(76 - 67)$
$9a = 143 × 9$
$\text{a}=\frac{143\times9}{9}$
$= 143$
View full question & answer→Question 872 Marks
Factorise the following.
$ y^2+4 y-21 $
Answer$ y^2+4 y-21 $
$ =y^2+(7-3) y-21 $
$ =y^2+7 y-3 y-21 $
$ =y(y+7)-3(y+7) $
$ =(y+7)(y-3) $
View full question & answer→Question 882 Marks
Factorise the following.
$p^2+ 14p + 13$
Answer$p^2+ 14p + 13$
$= p^2+ 13p + p + 13 × 1$
$= p(p + 13) + 1(p + 13)$
$= (p + 13)(p + 1)$
View full question & answer→Question 892 Marks
Expand the following, using suitable identities.
$x^2y^2= (xy)^2$
View full question & answer→Question 902 Marks
The following expressions are the areas of rectangles. Find the possible lengths and breadths of these rectangles.
$x^2+ 19x - 20$
Answer$x^2+ 19x - 20$ We factorise the given expression,
$ =x^2+(20-1) x-20 $
$ =x^2+20 x-x-20 $
$= x(x + 20) - 1(x + 20)$
$= (x + 20)(x - 1)$
View full question & answer→Question 912 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b)$.
$9x^2- 1$
Answer$9x^2- 1$
$= (3x)^2- 1^2$
$= (3x - 1)(3x + 1)$
View full question & answer→Question 922 Marks
Factorise the following expressions. $lx + my + mx + ly$
Answer$lx + my + mx + ly = x(l + m) + y(m + l) = (l + m)(x + y)$
View full question & answer→Question 932 Marks
Expand the following, using suitable identities.$\Big(\frac{4}{5}\text{p}+\frac{5}{3}\text{q}\Big)^2$
Answer$\Big(\frac{4}{5}\text{p}+\frac{5}{3}\text{q}\Big)^2$$=\Big(\frac{4}{5}\text{p}\Big)^2+\Big(\frac{5}{3}\text{q}\Big)^2+2\times\frac{4}{5}\text{p}\times\frac{5}{3}\text{q}$
$=\frac{16}{25}\text{p}^2+\frac{25}{9}\text{q}^2+\frac{8}{3}\text{pq}$
View full question & answer→Question 942 Marks
Multiply the following:
$ 3 x^2 y^2 z^2, 17 x y z $
Answer$ 3 x^2 y^2 z^2, 17 x y z $
$ 3 x^2 y^2 z^2 \times 17 x y z $
$ =(3 \times 17) x^2 y^2 z^2 \times x y z $
$ =51 x^3 y^3 z^3 $
View full question & answer→Question 952 Marks
Verify the following:
$\left(a^2-b^2\right)\left(a^2+b^2\right)+\left(b^2-c^2\right)\left(b^2+c^2\right)+\left(c^2-a^2\right)+\left(c^2+a^2\right)=0$
Answer$\left(a^2-b^2\right)\left(a^2+b^2\right)+\left(b^2-c^2\right)\left(b^2+c^2\right)+\left(c^2-a^2\right)+\left(c^2+a^2\right)=0$
Taking $LHS$
$ =\left(a^2-b^2\right)\left(a^2+b^2\right)+\left(b^2-c^2\right)\left(b^2+c^2\right)+\left(c^2-a^2\right)+\left(c^2+a^2\right) $
$=\left(a^4-b^4+b^4-c^4+c^4-a^4\right)=0$
$= RHS$
Hence verified
View full question & answer→Question 962 Marks
The area of a circle is given by the expression $\pi\text{x}^2+6\pi\text{x}+9\pi$. Find the radius of the circle.
AnswerArea of a circle $\pi\text{x}^2+6\pi\text{x}+9\pi$
$=\pi(\text{x}^2+6\text{x}+9)$
$\pi\text{r}^2=\pi(\text{x}^2+3\text{x}+3\text{x}+9)$
$\pi\text{r}^2=\pi[\text{x}(\text{x}+3)+3(\text{x}+3)]$
$=\pi(\text{x}+3)(\text{x}+3)$
$=\pi(\text{x}+3)^2$ $\pi\text{r}^2=\pi(\text{x}+3)^2$
On comparing both sides, $r^2= (x + 3)^2$
$r = x + 3$
Hence, the radius of circle is $x + 3$
View full question & answer→Question 972 Marks
The area of a rectangle is $x^2+ 7x + 12$. If its breadth is $(x + 3)$, then find its length.
AnswerArea of a rectangle = $x^2+ 7x + 12$ and breadth $= x + 3$
Let the length of rectangle be $l$
Area of rectangle $= l x b$
$x^2+ 7x + 12 = l × (x + 3)$
$=\text{l}=\frac{\text{x}^2+7\text{x}+12}{\text{x}+3}$
$=\frac{\text{x}^2+4\text{x}+3\text{x}+12}{\text{x}+2}$
$=\frac{\text{x}(\text{x}+4)+3(\text{x}+4)}{\text{x}+3}$
$=\frac{(\text{x}+4)(\text{x}+3)}{\text{x}+3}$
$=\text{x}+4$
Hence, the length of rectangle $= x + 4$
View full question & answer→Question 982 Marks
Factorise the following, using the identity $\mathrm{a}^2-2 \mathrm{ab}+\mathrm{b}^2=(\mathrm{a}-\mathrm{b})^2$.
$4 a^2-4 a b+b^2$
Answer$4 a^2-4 a b+b^2$
$ =(2 a)^2-2 \times 2 a \times b+b^2 $
$ =(2 a-b)^2 $
$ =(2 a-b)(2 a-b)$
View full question & answer→Question 992 Marks
Simplify:
$ (p q-q r)^2+4 p q^2 r $
Answer$ (p q-q r)^2+4 p q^2 r $
$ =p^2 q^2+q^2 r^2-2 p q^2 r+4 p q^2 r $
$ =p^2 q^2+q^2 r^2+2 p q^2 r $
View full question & answer→Question 1002 Marks
Factorise the following expressions.
$ x^3 y^2+x^2 y^3-x y^4+x y$
Answer$ x^3 y^2+x^2 y^3-x y^4+x y$
$=x y\left(x^2 y+x y^2-y^3+1\right)$
View full question & answer→Question 1012 Marks
Write the greatest common factor in each of the following terms.
$11 \mathrm{x}^2, 12 \mathrm{y}^2$
Answer$11 \mathrm{x}^2, 12 \mathrm{y}^2$
The $GCF$ of $11,12 \& 1$
Also, there is no common factor between $x^2$ and $y^2$
Hence, $GCF$ of $11 x^2$ and $12 y^2$ is $1$.
View full question & answer→Question 1022 Marks
Factorise the following, using the identity $\mathrm{a}^2+2 \mathrm{a} b+\mathrm{b}^2=(\mathrm{a}+\mathrm{b})^2$
$9 x^2+30 x+25$
Answer$9 x^2+30 x+25$
$ =(3 x)^2+2 \times 3 x \times 5+5^2$
$=(3 x+5)^2 $
$ =(3 x+5)(3 x+5)$
View full question & answer→Question 1032 Marks
Simplify:
$\left(s^2 \mathrm{t}+\mathrm{tq}^2\right)^2-(2 \mathrm{stq})^2$
Answer$\left(s^2 \mathrm{t}+\mathrm{tq}^2\right)^2-(2 \mathrm{stq})^2$
$ =\left(s^2 t\right)^2+\left(t q^2\right)^2+2 \times s^2 t \times t q^2-4 s^2 t^2 q^2 $
$=s^4 t^2+t^2 q^4+2 s^2 t^2 q^2-4 s^2 t^2 q^2 $
$ =s^4 t^2+t^2 q^4-2 s^2 t^2 q^2 $
View full question & answer→Question 1042 Marks
Factorise the following expressions.
$y^2+ 8zx - 2xy - 4yz$
Answer$y^2+ 8zx - 2xy - 4yz$
$= y^2- 2xy + 8zx - 4yz$
$= y(y - 2x) - 4z(y - 2x)$
$= (y - 2x)(y - 4z)$
View full question & answer→Question 1052 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$4x^2- 25y^2$
Answer$4x^2- 25y^2$
$(2x)^2- (5y)^2$
$= (2x - 5y)(2x + 5)$
View full question & answer→Question 1062 Marks
Expand the following, using suitable identities.$\Big(\frac{2}{3}\text{x}-\frac{3}{2}\text{y}\Big)^2$
Answer$\Big(\frac{2}{3}\text{x}-\frac{3}{2}\text{y}\Big)^2$$=\Big(\frac{2}{3}\text{x}\Big)^2+\Big(\frac{3}{2}\text{y}\Big)^2-2\times\frac{2}{3}\text{x}\times\frac{3}{2}\text{y}$
$=\frac{4}{9}\text{x}^2+\frac{9}{4}\text{y}^2-2\text{xy}$
View full question & answer→Question 1072 Marks
Multiply the following:
$-7st, -1, -13st^2$
Answer$-7st, -1, -13st^2$
$-7st × (-1) × (-13st^2)$
$= [-7 × (-1) × (-13)]st × (st^2)$
$= -91s^2t^3$
View full question & answer→Question 1082 Marks
If $p + q = 12$ and $pq = 22$, then find $p^2+ q^2$.
AnswerGiven, $p + q = 12$ and $pq = 22$
Since,
$(p+q)^2=p^2+q^2+2 p q $
$ (12)^2=p^2+q^2+2(22) $
$ p^2+q^2=(12)^2-44 $
$ p^2+q^2=144-44 $
$= 100$
View full question & answer→Question 1092 Marks
Using suitable identities, evaluate the following.
$(52)^2$
Answer$(52)^2$
$= (50 + 2)^2$
$= (50)^2+ (2)^2+ 2 × 50 × 2$
$= 2500 + 4 + 200$
$= 2704$
View full question & answer→Question 1102 Marks
Expand the following, using suitable identities.
$ (0.9 p-0.5 q)^2 $
Answer$ (0.9 p-0.5 q)^2 $
$ =(0.9 p)^2+(0.5 q)^2-2 \times 0.9 p \times 0.5 q $
$ =0.81 p^2+0.25 q^2-0.9 p q $
View full question & answer→Question 1112 Marks
Factorise the following. $x^2+ 9x + 20$
Answer$x^2+ 9x + 20$
$= x^2+ 5x + 4x + 5 × 4$
$= x(x + 5) + 4(x + 5)$
$= (x + 5)(x + 4)$
View full question & answer→Question 1122 Marks
Write the greatest common factor in each of the following terms.
$l^2m^2n,l^2mn^2, l^2mn^2$
Answer$l^2m^2n,l^2mn^2, l^2mn^2$
$l^2m^2n= l × l × m × m × n$
$l^2mn^2= l × m × m × n$
$l^2mn^2= l × l × m × n × n$
View full question & answer→Question 1132 Marks
Using suitable identities, evaluate the following. $101 \times 103$
Answer$101 × 103$
$= (100 + 1)(100 + 3)$
$= (100)^2+ (1 + 3)100 + 3 × 1$
$= 10000 + 400 + 3$
$= 10403$
View full question & answer→Question 1142 Marks
Multiply the following: $abc, (bc + ca)$
Answer$abc, (bc + ca)$
$abc \times (bc + ca)$
$= abc \times bc + abc \times ca$
$=a b^2 c^2+a^2 b c^2$
View full question & answer→Question 1152 Marks
Expand the following, using suitable identities.$\Big(\frac{2\text{x}}{3}-\frac{2}{3}\Big)\Big(\frac{2\text{x}}{3}+\frac{2\text{a}}{3}\Big)$
Answer$\Big(\frac{2\text{x}}{3}-\frac{2}{3}\Big)\Big(\frac{2\text{x}}{3}+\frac{2\text{a}}{3}\Big)$$=\Big(\frac{2\text{x}}{3}\Big)^2+\Big(-\frac{2}{3}+\frac{2\text{a}}{3}\Big)+\Big(-\frac{2}{3}+\frac{2\text{a}}{3}\Big)\frac{2\text{x}}{3}+\Big(-\frac{2}{3}\times\frac{2\text{a}}{3}\Big)$
$=\frac{4\text{x}^2}{9}+\frac{2\text{a}-2}{3}\times\frac{2}{3}\text{x}-\frac{4}{9}\text{a}$
$=\frac{4\text{x}^2}{9}+\frac{4}{9}(\text{a}-1)\text{x}-\frac{4}{9}\text{a}$
View full question & answer→Question 1162 Marks
Factorise $p^4+ q^4+ p^2q^2$.
AnswerWe have,
$ p^4+q^4+p^2 q^2 $
$ =p^4+q^4+2 p^2 q^2-2 p^2 q^2+p^2 q^2 $
$ =p^4+q^4+2 p^2 q^2-p^2 q^2 $
$ =\left[\left(p^2\right)^2+\left(q^2\right)^2+2 p^2 q^2\right]-p^2 q^2 $
$ =\left(p^2+q^2\right)^2-(p q)^2 $
$ =\left(p^2+q^2+p q\right)\left(p^2+q^2-p q\right) $
View full question & answer→Question 1172 Marks
Simplify:
$ (2.5 m+1.5 q)^2+(2.5 m-1.5 q)^2 $
Answer$ (2.5 m+1.5 q)^2+(2.5 m-1.5 q)^2 $
$ =(2.5 m)^2+(1.5 q)^2+2 \times 2.5 m \times 1.5 q+(2.5 m)^2+(1.5 q)^2-2 \times(2.5 m) \times(1.5 q) $
$ =6.25 m^2+2.25 q^2+6.25 m^2+2.25 q^2 $
$ =(6.25+6.25) m^2+(2.25+2.25) q^2 $
$ =12.5 m^2+4.5 q^2 $
View full question & answer→Question 1182 Marks
Factorise the following, using the identity $a^2+2 a b+b^2=(a+b)^2$
$16 x^2+40 x+25$
Answer$16 x^2+40 x+25$
$=(4 x)^2+2 \times 4 x \times 5+5^2$
$=(4 x+5)^2$
$=(4 x+5)(4 x+5)$
View full question & answer→Question 1192 Marks
Multiply the following:
$7pqr, (p - q + r)$
Answer$7pqr, (p - q + r)$
$7pqr \times (p - q + r)$
$= 7qr \times p - 7pqr \times q + 7qr \times r$
$=7 p^2 q r-7 p q^2 r+7 p q r^2$
View full question & answer→Question 1202 Marks
If $a + b = 25$ and $a^2+ b^2= 225$, then find $ab$.
AnswerGiven,
$a + b = 25$ and $a^2+ b^2= 225$
$(a + b)^2= a^2+ b^2+ 2ab (25)^2$
$= 225 + 2ab 2ab$
$= (25)^2- 225 2ab$
$= 625 - 225 2ab$
$= 400$
$2\text{ab}=\frac{400}{2}$
$= 200$
View full question & answer→Question 1212 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$ 28 a y^2-175 a x^2 $
Answer$ 28 a y^2-175 a x^2 $
$ =7 a\left(4 y^2-25 x^2\right) $
$ =7 a\left[(2 y)^2-(5 x)^2\right] $
$ =7 a(2 y-5 x)(2 y+5 x) $
View full question & answer→Question 1222 Marks
Using suitable identities, evaluate the following.
$(69.3)^2-(30.7)^2$
Answer$(69.3)^2-(30.7)^2$
$= (69.3 + 30.7)(69.3 - 30.7)$
$= 100 \times 38.6$
$= 3860$
View full question & answer→Question 1232 Marks
Add:
$ \left(5 x^2-3 x y+4 y^2-9\right)+\left(7 y^2+5 x y-2 x^2+13\right) $
Answer$ \left(5 x^2-3 x y+4 y^2-9\right)+\left(7 y^2+5 x y-2 x^2+13\right) $
$ =5 x^2-3 x y+4 y^2-9+7 y^2+5 x y-2 x^2+13 $
$ =\left(5 x^2-2 x^2\right)+(-3 x y+5 x y)+\left(4 y^2+7 y^2\right)+(-9+13)$
$ =3 x^2+2 x y+11 y^2+4 $
View full question & answer→Question 1242 Marks
Simplify:
$(1.5 p+1.2 q)^2-(1.5 p-1.2 q)^2$
Answer$(1.5 p+1.2 q)^2-(1.5 p-1.2 q)^2$
$= [(1.5p + 1.2q) + (1.5p - 1.2q)][(1.5p + 1.2q) - (1.5p − 1.2q)]$
$= [(1.5p + 1.5p) + (1.2q - 1.2q)][(1.5p - 1.5p) + (1.2q + 1.2q)]$
$= 3p × 2.4q$
$= 7.2pq$
View full question & answer→Question 1252 Marks
Factorise the following, using the identity $a^2-2 a b+b^2=(a-b)^2.$
$9\text{y}^2-4\text{xy}+\frac{\text{y}^2}{9}$
Answer$9\text{y}^2-4\text{xy}+\frac{\text{y}^2}{9}$$=\big(3\text{y}\big)^2-2\times3\text{y}\times\frac{2}{3}\text{x}+\Big(\frac{2}{3}\text{x}\Big)^2$
$=\Big(3\text{y}-\frac{2}{3}\text{x}\Big)^2$
$=\Big(3\text{y}-\frac{2}{3}\text{x}\Big)\Big(3\text{y}-\frac{2}{3}\text{x}\Big)$
View full question & answer→Question 1262 Marks
Factorise the following using the identity $a^2-b^2=(a+b)(a-b)$.
$\frac{\text{x}^3\text{y}}{9}-\frac{\text{xy}^3}{16}$
Answer$\frac{\text{x}^3\text{y}}{9}-\frac{\text{xy}^3}{16}$$=\text{xy}\Big(\frac{\text{x}^2}{9}-\frac{\text{y}^2}{16}\Big)$
$=\text{xy}\bigg[\Big(\frac{\text{x}}{3}\Big)^2-\Big(\frac{\text{y}}{4}\Big)^2\bigg]$
$=\text{xy}\Big(\frac{\text{x}}{3}+\frac{\text{y}}{4}\Big)\Big(\frac{\text{x}}{3}-\frac{\text{y}}{4}\Big)$
View full question & answer→Question 1272 Marks
Multiply the following:
$ -3 x^2 y,(5 y-x y) $
Answer$ -3 x^2 y,(5 y-x y) $
$ -3 x^2 y \times(5 y-x y) $
$ =-3 x^2 y \times 5 y+3 x^2 y \times x y $
$=-15 x^2 y^2+3 x^3 y^2 $
View full question & answer→Question 1282 Marks
The base of a parallelogram is ($2x + 3$ units) and the corresponding height is ($2x - 3$ units). Find the area of the parallelogram in terms of $x$. What will be the area of parallelogram of $x = 30$ units?
AnswerThe base & the corresponding height of a parallelogram are $(2? + 3)$ units & $(2x - 3)$units, respectively.
Area of a parallelogram = Base $\times $ Height
$= (2x + 3) × (2x - 3)$
$= (2x)^2- (3)^2$
$= (4x - 9)^2$ units
Now, if $x = 10.$
Then, the area of parallelogram $= 4 × (10)^2- 9$
$= 400 - 9$
$= 391$ sq. units
View full question & answer→Question 1292 Marks
Factorise the expressions and divide them as directed: $\left(x^3+x^2-132 x\right) \div x(x-11)$
Answer$\left(x^3+x^2-132 x\right) \div x(x-11)$
$=\frac{\text{x}^3+\text{x}^2-132\text{x}}{\text{x}(\text{x}-11)}$
$=\frac{\text{x}(\text{x}^2+\text{x}-132)}{\text{x}(\text{x}-11)}$
$=\frac{\text{x}^2+12\text{x}-11\text{x}-132}{\text{x}-11}$
$=\frac{\text{x}(\text{x}+12)-11(\text{x}+12)}{\text{x}-11}$
$=\frac{(\text{x}+12)(\text{x}-11))}{\text{x}-11}$
$=\text{x}+12$
View full question & answer→Question 1302 Marks
Expand the following, using suitable identities.$\Big(\frac{4\text{x}}{5}+\frac{\text{y}}{4}\Big)\Big(\frac{4\text{x}}{5}+\frac{3\text{y}}{4}\Big)$
Answer$\Big(\frac{4\text{x}}{5}+\frac{\text{y}}{4}\Big)\Big(\frac{4\text{x}}{5}+\frac{3\text{y}}{4}\Big)$$=\Big(\frac{4\text{x}}{5}\Big)^2+\Big(\frac{\text{y}}{4}+\frac{3\text{y}}{4}\Big)\frac{4\text{x}}{5}+\frac{\text{y}}{4}\times\frac{3\text{y}}{4}$
$=\frac{16}{25}\text{x}^2+\frac{4\text{xy}}{5}+\frac{3\text{y}^2}{16}$
View full question & answer→Question 1312 Marks
Multiply the following:
$ a, a^5, a^6 $
Answer$ a, a^5, a^6 $
$ a \times a^5 \times a^6=a^{1+5+6} $
$ =a^{12} $
View full question & answer→Question 1322 Marks
Expand the following, using suitable identities. $(2x - 5y)(2x - 5y)$
Answer$(2x - 5y)(2x - 5y)$
$ =(4 x)^2+(5 y)^2-2 \times 2 x \times 5 y $
$ =16 x^2+25 y^2-20 x y $
View full question & answer→Question 1332 Marks
Simplify:$\Big(\frac{3}{4}\text{x}-\frac{4}{3}\text{y}\Big)^2+2\text{xy}$
Answer$\Big(\frac{3}{4}\text{x}-\frac{4}{3}\text{y}\Big)^2+2\text{xy}$$=\Big(\frac{3}{4}\text{x}\Big)^2+\Big(\frac{4}{3}\text{y}\Big)^2-2\times\frac{3}{4}\text{x}\times\frac{4}{3}\text{y}+2\text{xy}$
$=\frac{9}{16}\text{x}^2+\frac{16}{9}\text{y}^2-2\text{xy}+2\text{xy}$
$=\frac{9}{16}\text{x}^2+\frac{16}{9}\text{y}^2$
View full question & answer→Question 1342 Marks
Simplify:$\big(\frac{7}{9}\text{a}+\frac{9}{7}\text{b}\Big)^2-\text{ab}$
Answer$\big(\frac{7}{9}\text{a}+\frac{9}{7}\text{b}\Big)^2-\text{ab}$$=\Big(\frac{7}{9}\text{a}\Big)^2+\Big(\frac{9}{7}\text{b}\Big)^2+2\times\frac{7}{9}\text{a}\times\frac{9}{7}\text{b}-\text{ab}$
$=\frac{49}{81}\text{a}^2+\frac{81}{49}\text{b}^2+2\text{ab}-\text{ab}$
$=\frac{49}{81}\text{a}^2+\text{ab}+\frac{81}{49}\text{b}^2$
View full question & answer→Question 1352 Marks
Expand the following, using suitable identities.
$ \left(x^2 y-x y^2\right)^2 $
Answer$ \left(x^2 y-x y^2\right)^2 $
$ =\left(x^2 y\right)^2+\left(x y^2\right)^2-2\left(x^2 y\right)^2\left(x y^2\right) $
$ =x^4 y^2+x^2 y^4-2 x^3 y^3 $
View full question & answer→Question 1362 Marks
Multiply the following: $(pq - 2r), (pq - 2r)$
Answer$(pq - 2r), (pq - 2r)$
$(pq - 2r)(pq - 2r)$
$= pq(pq - 2r) - 2r(pq - 2r)$
$ =p^2 q^2-2 p q r-2 r p a+4 r^2 $
$ =p^2 q^2-4 p q r+4 r^2 $
View full question & answer→Question 1372 Marks
Simplify:
$ (a b-c)^2+2 a b c $
Answer$ (a b-c)^2+2 a b c $
$ =(a b)^2+c^2-2 a b c+2 a b c $
$ =a^2 b^2+c^2 $
View full question & answer→Question 1382 Marks
If $\text{x}-\frac{1}{\text{x}}=7$ then find the value of $\text{x}^2+\frac{1}{\text{x}^2}$.
AnswerGiven,$\text{x}-\frac{1}{\text{x}}=7$
Since,$\Big(\text{x}-\frac{1}{\text{x}}\Big)^2=\text{x}^2+\frac{1}{\text{x}^2}-2.\text{x}.\frac{1}{\text{x}}$
$7^2=\text{x}^2+\frac{1}{\text{x}^2}-2$
$\text{x}^2+\frac{1}{\text{x}^2}=49+2$
$\text{x}^2+\frac{1}{\text{x}^2}=51$
View full question & answer→Question 1392 Marks
Write the greatest common factor in each of the following terms.
$ 3 x^3 y^2 z,-6 x y^3 z^2, 12 x^2 y z^3 $
Answer$ 3 x^3 y^2 z,-6 x y^3 z^2, 12 x^2 y z^3 $
$ 3 ?^3 ?^2 ?=3 \times x \times x \times x \times y \times y \times z $
$ -6 x y^3 z^2=-3 \times 2 \times x \times y \times y \times y \times z \times z $
$ 12 x^2 y z^3=3 \times 4 \times x \times x \times y \times z \times z \times z G C F=3 x y z $
View full question & answer→Question 1402 Marks
Using suitable identities, evaluate the following.
$(98)^2$
Answer$ (98)^2$
$ =(100-2)^2 $
$ =(100)^2+(2)^2-2 \times 100 \times 2 $
$ =10000+4-400 $
$ =9604$
View full question & answer→Question 1412 Marks
Expand the following, using suitable identities. $(2x + 9)(2x - 7)$
Answer$(2x + 9)(2x - 7)$
$= (2x + 9)[2x + (-7)]$
$= (2x)^2+ [9 + (-7)] 2x + 9 × (-7)$
$= 4x^2+ 4x - 6$
View full question & answer→Question 1422 Marks
Factorise the following expressions.
$ a^3 x-x^4+a^2 x^2-a x^3 $
Answer$ a^3 x-x^4+a^2 x^2-a x^3 $
$ =x\left(a^3-x^2+a^2 x-a x^2\right) $
$ =x\left(a^3+a^2 x-x^3-ax^2\right) $
$ =x\left[a^2(a+x)-x^2(x+?)\right] $
$ =x\left[(x+a)\left(a^2-x^2\right)\right] $
$ =x\left(a^2-x^2\right)(a+x) $
View full question & answer→Question 1432 Marks
Factorise the following expressions.
$ a x^3-b x^2+c x $
Answer$ a x^3-b x^2+c x $
$ =x\left(a x^2-b x+c\right) $
View full question & answer→Question 1442 Marks
Multiply the following:
$ 15 x y^2, 17 y z^2 $
Answer$ 15 x y^2, 17 y z^2 $
$ 15 x y^2 \times 17 y z^2 $
$ =(15 \times 17) xy^2 \times y z^2 $
$=255 x y^3 z^3 $
View full question & answer→Question 1452 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$ y^4-625 $
Answer$ y^4-625 $
$ =\left(y^2\right)^2-(25)^2 $
$ =\left(y^2+25\right)\left(y^2-25\right) $
$ =\left(y^2+25\right)\left(y^2-5^2\right) $
$ =\left(y^2+25\right)(y+5)(y-5) $
View full question & answer→Question 1462 Marks
Factorise the following expressions. $-xy - ay$
Answer$-xy - ay = -y(x + a)$
View full question & answer→Question 1472 Marks
Subtract: $7p(3q + 7p)$ from $8p(2p - 7q)$
AnswerThe required difference is given by
$8p(2p - 7q) - 7p(3q + 7p)$
$ =16 p^2-56 p q-21 p q-49 p^2 $
$ =\left(16 p 2-49 p^2\right)+(-56 p q-21 p q) $
$ =-33 p^2-77 p q $
View full question & answer→Question 1482 Marks
Simplify:
$ (4.5 a+1.5 b)^2+(4.5 b+1.5 a)^2$
Answer$ (4.5 a+1.5 b)^2+(4.5 b+1.5 a)^2$
$ =(4.5 a)^2+(1.5 b)^2+2 \times 4.5 a \times 1.5 b+(4.5 b)^2+(1.5 a)^2+2 \times 4.5 b \times 1.5 a $
$ =20.25 a^2+2.25 b^2+13.5 a b+20.25 b^2+2.25 a^2+13.5 a b $
$ =40.5 a^2+4.5 b^2+27 a b $
View full question & answer→Question 1492 Marks
The sum of first n natural numbers is given by the expression $\frac{\text{n}^2}{2}+\frac{\text{n}}{2}$. Factorise this expression.
Answer The sum of first n natural numbers $=\frac{\text{n}^2}{2}+\frac{\text{n}}{2}$
Factorisation of given expression $=\frac{1}{2}(\text{n}^2+\text{n})=\frac{1}{2}\text{n}(\text{n}+1)$ View full question & answer→Question 1502 Marks
Factorise the following, using the identity $a^2+ 2ab + b^2= (a + b)^2$
$x^2+ 6x + 9$
Answer$x^2+ 6x + 9$
$= x^2+ 2 × 3 × x + 3^2$
$= (x + 3)^2$
$= (x + 3)(x + 3)$
View full question & answer→Question 1512 Marks
Using suitable identities, evaluate the following. $9.8 \times 10.2$
Answer$9.8 \times 10.2$
$= (10 - 0.2)(10 + 0.2)$
$= 102 - (0.2)^2$
$= 100 - 0.04$
$= 99.96$
View full question & answer→Question 1522 Marks
Simplify:
$ (3 x+2 y)^2+(3 x-2 y)^2 $
AnswerWe have,
$ (3 x+2 y)^2+(3 x-2 y)^2 $
$ =(3 x)^2+(2 y)^2+2 x 3 x \times 2 y+(3 x)^2+(2 y)^2-2 \times 3 x \times 2 y $
$ =9 x^2+4 y^2+12 x y+9 x^2+4 y^2-12 x y $
$ =\left(9 x^2+9 x^2\right)+\left(4 y^2+4 y^2\right)+12 x y-12 x y $
$ =18 x^2+8 y^2 $
View full question & answer→Question 1532 Marks
Using suitable identities, evaluate the following. $47 \times 53$
Answer$47 \times 53$
$= (50 - 3)(50 + 3)$
$= (50)^2- (3)^2$
$= 2500 - 9$
$= 2491$
View full question & answer→Question 1542 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$25 a x^2-25 a$
Answer$25 a x^2-25 a$
$= 25a(x - 1)(x + 1)$
View full question & answer→Question 1552 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$x^4- 1$
Answer$ x^4-1 $
$ =\left(x^2\right)^2-1 $
$ =\left(x^2+1\right)\left(x^2-1\right) $
$ =\left(x^2+1\right)(x+1)(x-1)$
View full question & answer→Question 1562 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b)$.
$1331x^3y - 11y^3x$
Answer$ 1331 x^3 y-11 y^3 x $
$ =(11)^3 x^3 y-11 y^3 x $
$ =11 x y\left(11^2 x^2-y^2\right) $
$ =11 x y\left[(11 x)^2-y^2\right] $
$ =11 x y(11 x+y)(11 x-y) $
View full question & answer→Question 1572 Marks
Expand the following, using suitable identities.
$ \left(x^2+y^2\right)\left(x^2-y^2\right) $
Answer$ \left(x^2+y^2\right)\left(x^2-y^2\right) $
$ =\left(x^2\right)^2-\left(y^2\right)^2 $
$ =x^4-y^4 $
View full question & answer→Question 1582 Marks
The following expressions are the areas of rectangles. Find the possible lengths and breadths of these rectangles.
$x^2+ 9x + 20$
Answer$x^2+ 9x + 20$
We factorise the given expression,
$= x^2+ (5 + 4)x + 20$
$= x^2+ 5x + 4x + 20$
$= x(x + 5) + 4(x + 5)$
$= (x + 5)(x + 4)$
View full question & answer→Question 1592 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$8a^3- 2a$
Answer$8a^3- 2a$
$= 2a(4a^2- 1)$
$= 2a[(2a)^2- (1)^2]$
$= 2a(2a + 1)(2a - 1)$
View full question & answer→Question 1602 Marks
The following expressions are the areas of rectangles. Find the possible lengths and breadths of these rectangles.
$x^2- 7x + 10$
Answer$x^2- 7x + 10$
We factorise the given expression,
$= x^2- (5 + 2)x + 10$
$= x^2- 5x - 2x + 10$
$=x(x - 5) - 2(x - 5)$
$= (x - 5)(x - 2)$
View full question & answer→Question 1612 Marks
If $a^2+ b^2= 74$ and $ab = 35$, then find $a + b$.
AnswerGiven, $a^2+ b^2= 74$ and $ab = 35$
Since,
$(a + b)^2= a^2+ b^2+ 2ab (a + b)^2$
$= 74 + 2 \times 35$
$(a + b)^2= 144$
$\text{a}+\text{b}=\sqrt{144}$
$= 12$
View full question & answer→Question 1622 Marks
Expand the following, using suitable identities.
$(xy + yz)^2$
Answer$ (x y+y z)^2 $
$ =(x y)^2+(y z)^2+2 \times x y \times y z $
$ =x^2 y^2+y^2 z^2+2 x y^2 z $
View full question & answer→Question 1632 Marks
Factorise $\text{x}^2+\frac{1}{\text{x}^2}+2-3\text{x}-\frac{3}{\text{x}}$.
Answer$\text{x}^2+\frac{1}{\text{x}^2}+2-3\text{x}-\frac{3}{\text{x}}$ (given)$=\text{x}^2+\frac{1}{\text{x}^2}+2.\text{x}.\frac{1}{\text{x}}-3\Big(\text{x}+\frac{1}{\text{x}}\Big)$
$=\Big(\text{x}+\frac{1}{\text{x}}\Big)^2-3\Big(\text{x}+\frac{1}{\text{x}}\Big)$
$=\Big(\text{x}+\frac{1}{\text{x}}\Big)\Big(\text{x}+\frac{1}{\text{x}}-3\Big) (\text{taking}\Big(\text{x}+\frac{1}{\text{x}}\Big)\text{is common})$
View full question & answer→Question 1642 Marks
Find the value of:$\frac{6.25\times6.25-1.75\times1.75}{4.5}$
Answer$\frac{6.25\times6.25-1.75\times1.75}{4.5}$$=\frac{(6.25)^2-(1.75)^2}{4.5}$
$=\frac{(6.25+1.75)(6.25+1.75)}{4.5}$
$=\frac{8\times4.5}{4.5}=8$
View full question & answer→Question 1652 Marks
Write the greatest common factor in each of the following terms.
$2xy, -y^2, 2x^2y$
Answer$ 2 x y,-y^2, 2 x^2 y $
$ 2 x y=2 \times x \times y $
$ -y^2=-y \times y $
$ 2 x^2 y=2 \times x \times x \times y $
The greatest common factor i.e. $GCF$ is $y$
View full question & answer→Question 1662 Marks
Factorise the following.
$x^2+ 4x - 77$
Answer$x^2+ 4x - 77$
$= x^2+ (11 - 7)x - 77$
$= x^2+ 11x - 7x - 77$
$= x(x + 11) - 7(x + 11)$
$= (x + 11)(x - 7)$
View full question & answer→Question 1672 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b). $
$p^5- 16p$
Answer$ p^5-16 p $
$ =p\left(p^4-16\right) $
$ =p\left[\left(p^2\right)^2-4^2\right] $
$ =p\left(p^2+4\right)\left(p^2-4\right)$
$=p\left(p^2+4\right)\left(p^2-2^2\right) $
$ =p\left(p^2+4\right)(p+2)(p-2) $
View full question & answer→Question 1682 Marks
Simplify:
$ (a-b)\left(a^2+b^2+a b\right)-(a+b)\left(a^2+b^2-a b\right) $
Answer$(a-b)\left(a^2+b^2+a b\right)-(a+b)\left(a^2+b^2-a b\right) $
$ =a\left(a^2+b^2+a b\right)-b\left(a^2+b^2+a b\right)-a\left(a^2+b^2-a b\right)-b\left(a^2+b^2-a b\right) $
$ =a^3+a b^2+a^2 b-b a^2-b^3-a b^2-a^3-a b^2+a^2 b-b a^2-b^3+a b^2 $
$ =\left(a^3-a^3\right)+\left(-b^3-b^3\right)+\left(a b^2-a b^2\right)+\left(a^2 b-a^2 b+a^2 b-a^2 b\right) $
$ =0-2 b^3+0+0+0 $
$ =-2 b^3$
View full question & answer→Question 1692 Marks
Verify the following:
$ (p-q)\left(p^2+p q+q^2\right)=p^3-q^3 $
Answer$ (p-q)\left(p^2+p q+q^2\right)=p^3-q^3 $
$ =p\left(p^2+p q+q^2\right)-q\left(p^2+p q+q^2\right) $
$ =p^3+p^2 q+p q^2-q p^2-p q^2-q^3 $
$ =p 3-q 3 $
= RHS
Hence verified
View full question & answer→Question 1702 Marks
Write the greatest common factor in each of the following terms.
$ 3 x^2 y, 18 x y^2,-6 x y $
Answer$ 3 x^2 y, 18 x y^2,-6 x y $
$ 3 x^2 y=3 \times x \times x \times y $
$ 18 x y^2=3 \times 6 \times x \times y \times y $
$-6xy = (-3) × 2 × x × y$
The greatest common factor i.e. $GCF$ is $3$.
View full question & answer→Question 1712 Marks
Factorise the following, using the identity $a^2-2 a b+b^2=(a-b)^2$.
$\frac{\text{x}^2}{4}-2\text{x}+4$
Answer$\frac{\text{x}^2}{4}-2\text{x}+4$$=\Big(\frac{\text{x}}{4}\Big)^2-2\times\frac{\text{x}}{2}\times2+2^2$
$=\Big(\frac{\text{x}}{2}-2^2\Big)$
$=\Big(\frac{\text{x}}{2}-2\Big)\Big(\frac{\text{x}}{2}-2\Big)$
View full question & answer→Question 1722 Marks
Simplify:$ (b^2- 49)(b + 7) + 343$
Answer$ \left(b^2-49\right)(b+7)+343 $
$ =b^2(b+7)-49(b+7)+343 $
$ =b^3+7 b^2-49 b-343+343 $
$ =b^3-49 b+7 b^2 $
View full question & answer→Question 1732 Marks
Using suitable identities, evaluate the following.
$(9.9)^2$
Answer$(9.9)^2$
$= (10 - 0.1)^2$
$= 10^2+ (0.1)^2- 2 × 10 × 0.1$
$= 100 + 0.01 - 2$
$= 98.01$
View full question & answer→Question 1742 Marks
Expand the following, using suitable identities. $(x + 3)(x + 7)$
Answer$(x + 3)(x + 7)$
$= x^2+ (3 + 7)x + 3 × 7$
$= x^2+ 10x + 21$
View full question & answer→Question 1752 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$ (x+y)^4-(x-y)^4 $
Answer$ (x+y)^4-(x-y)^4 $
$ =\left[(x+y)^2\right]^2-\left[(x-y)^2\right]^2 $
$ =\left[(x+y)^2+(x-y)^2\right]\left[(x+y)^2-(x-y)^2\right] $
$ =\left(x^2+y^2+2 x y+x^2+y^2-2 x y\right)(x+y+x-y)(x+y-x+y) $
$ =\left(2 x^2+2 y^2\right)(2 x)(2 y) $
$ =2\left(x^2+y^2\right)(2 x)(2 y) $
$ =8 x y\left(x^2+y^2\right) $
View full question & answer→Question 1762 Marks
The area of a square is given by $ 4 x^2+12 x y+9 y^2. $ Find the side of the square.
AnswerArea of a square = $ 4 x^2+12 x y+9 y^2 $
We factorise the given expression,
$ 4 x^2+12 x y+9 y^2 $
$ =(2 x)^2+2 \times 2 x \times 3 y+(3 y)^2 $
$ =(2 x+3 y)^2 $
Area of a square having side length a is $a^2.$
Hence, side of given square is $2x + 3y$.
View full question & answer→Question 1772 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$\frac{\text{x}^2}{9}-\frac{\text{y}^2}{25}$
Answer$\frac{\text{x}^2}{9}-\frac{\text{y}^2}{25}$$=\Big(\frac{\text{x}}{3}\Big)^2-\Big(\frac{\text{y}}{5}\Big)^2$
$=\Big(\frac{\text{x}}{3}-\frac{\text{y}}{5}\Big)\Big(\frac{\text{x}}{3}+\frac{\text{y}}{5}\Big)$
View full question & answer→Question 1782 Marks
Factorise the following expressions.
$ax^2y - bxyz - ax^2z + bxy^2$
Answer$ax^2y - bxyz - ax^2z + bxy^2$
$= x(axy - byz - axz + by^2)$
$= x(axy - axz - byz + by^2)$
$= x[ax(y - z) + by(-z + y)]$
$= x[(ax + by)(y - z)]$
View full question & answer→Question 1792 Marks
The following expressions are the areas of rectangles. Find the possible lengths and breadths of these rectangles.
$x^2- 6x + 8$
Answer$x^2- 6x + 8$
We factorise the given expression,
$x^2- 6x + 8$
$= x^2- (4 + 2)x + 8$
$= x^2- 4x - 2x + 8$
$= x(x - 4) - 2(x - 4)$
$= (x - 4)(x - 2)$
View full question & answer→Question 1802 Marks
The following expressions are the areas of rectangles. Find the possible lengths and breadths of these rectangles.
$x^2- 3x + 2$
Answer$x^2- 3x + 2$
We factorise the given expression,
$ =x^2-(2+1) x+2 $
$ =x^2-2 x-x+2 $
$ =x(x-2)-1(x-2) $
$ =(x-2)(x-1) $
View full question & answer→Question 1812 Marks
Add:
$3a(2b + 5c), 3c(2a + 2b)$
Answer$3a(2b + 5c) + 3c(2a + 2b)$
$= (6ab + 15ac) + (6ac + 6bc)$
$= 6ab + 15ac + 6ac + 6bc$
$= 6ab + 21ac + 6bc$
View full question & answer→Question 1822 Marks
Multiply the following:
$ -7 p q^2 r^3,-13 p^3 q^2 r $
Answer$-7 p q^2 r^3,-13 p^3 q^2 r $
$ \left(-7 p q^2 r^3\right) \times\left(-13 p^3 q^2 r\right) $
$ =(-7) \times(-13) p^4 q^4 r^4 $
$ =91 p^4 q^4 r^4 $
View full question & answer→Question 1832 Marks
Write the greatest common factor in each of the following terms.
$13 x^2 y, 169 x y$
Answer$13 x^2 y, 169 x y$
$13 x^2 y = 13 × x × x × y$
$169xy = 13 × 13 × x × y$
$GCF = 13xy$
View full question & answer→Question 1842 Marks
Factorise the following, using the identity $a^2+2 a b+b^2=(a+b)^2$
$\frac{\text{x}^2}{4}+2\text{x}+4$
Answer$\frac{\text{x}^2}{4}+2\text{x}+4$$=\Big(\frac{\text{x}}{2}\Big)^2+2\times\frac{\text{x}}{2}\times2+2^2$
$=\Big(\frac{\text{x}}{2}+2\Big)^2$
$=\Big(\frac{\text{x}}{2}+2\Big)\Big(\frac{\text{x}}{2}+2\Big)$
View full question & answer→Question 1852 Marks
Factorise the following, using the identity $a^2-2 a b+b^2=(a-b)^2$.
$p^2-2 p+1$
Answer$p^2-2 p+1$
$=p^2-2 \times p \times 1+1^2 $
$ =(p-1)^2 $
$ =(p-1)(p-1)$
View full question & answer→Question 1862 Marks
Subtract: $6x^2- 4xy + 5y^2$ from $8y^2+ 6xy - 3x^2$
AnswerThe required difference is given by
$ \left(8 y^2+6 x y-3 x^2\right)-\left(6 x^2-4 x y+5 y^2\right) $
$ =8 y^2+6 x y-3 x^2-6 x^2+4 x y-5 y^2 $
$ =\left(8 y^2-5 y^2\right)+(6 x y+4 x y)-\left(3 x^2+6 x^2\right) $
$ =3 y^2+10 x y-9 x^2 $
View full question & answer→Question 1872 Marks
Factorise the following expressions. $2x^2- 2y + 4xy - x$
Answer$2x^2- 2y + 4xy - x$
$ = 2x^2- x - 2y + 4xy$
$ = x(2x - 1) - 2y(1 - 2x)$
$ = x(2x - 1) + 2y(2x - 1)$
$ = (2x - 1)(x + 2y)$
View full question & answer→Question 1882 Marks
Factorise the following, using the identity $a^2+2 a b+b^2=(a+b)^2$
$4 x^2+12 x+9$
Answer$4x^2+ 12x + 9$
$= (2x)^2+ 2 × 2x × 3 + 3^2$
$= (2x + 3)^2$
$= (2x + 3)(2x + 3)$
View full question & answer→Question 1892 Marks
Factorise the following.
$ p^2-13 p-30 $
Answer$ p^2-13 p-30 $
$ p^2-(15-2) p-30 $
$= p 2-15 p+2 p-30 $
$= p(p-15)+2(p-15) $
$= (p-15)(p+2) $
View full question & answer→Question 1902 Marks
Using suitable identities, evaluate the following.
$10.1 × 10.2$
Answer$10.1 × 10.2$
$= (10 + 0.1)(10 + 0.2)$
$= (10)^2+ (0.1 + 0.2)10 + (0.1)(0.2)$
$= 100 + 0.3 × 10 + 0.02$
$= 103.02$
View full question & answer→Question 1912 Marks
Simplify:
$ (3 x+2 y)^2-(3 x-2 y)^2 $
Answer$ (3 x+2 y)^2-(3 x-2 y)^2 $
$ =[(3 x+2 y)+(3 x-2 y)][(3 x+2 y)-(3 x-2 y)] $
$ =(3 x+2 y+3 x-2 y)(3 x+2 y-3 x+2 y) $
$ =6 x \times 4 y $
$ =(6 \times 4) \times x y $
$ =24 x y $
View full question & answer→Question 1922 Marks
Factorise the following. $18 + 11x + x$
Answer$18 + 11x + x$
$= x^2+ 11x + 18$
$= x^2+ (9 + 2)x + 18$
$= x^2+ 9x + 2x + 18$
$= x(x + 9) + 2(x + 9)$
$= (x + 9)(x + 2)$
View full question & answer→Question 1932 Marks
Factorise the following, using the identity $a^2+ 2ab + b^2= (a + b)^2$
$x^2+ 2x + 1$
Answer$x^2+ 2x + 1$
$= x^2+ 2 × 1 × x + 1^2$
$= (x + 1)^2$
$= (x + 1)(x + 1)$
View full question & answer→Question 1942 Marks
Using suitable identities, evaluate the following. $(49)^2$
Answer$(49)^2$
$= (50 - 1)^2$
$= (50)^2+ 12 - 2 × 50 × 1$
$= 2500 + 1 - 100 = 2401$
View full question & answer→Question 1952 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$\text{y}^3-\frac{\text{y}}{9}$
Answer$\text{y}^3-\frac{\text{y}}{9}$$= \text{y}\Big(\text{y}^2-\frac{1}{9}\Big)$
$=\text{y}\bigg[\text{y}^2-\Big(\frac{1}{3}\Big)^2\bigg]$
$=\text{y}\Big(\text{y}+\frac{1}{3}\Big)\Big(\text{y}-\frac{1}{3}\Big)$
View full question & answer→Question 1962 Marks
Factorise the following, using the identity $a^2- 2ab + b^2= (a - b)^2$.
$a^2y^3- 2aby^2+ b^2y$
Answer$ a^2 y^3-2 a b y^2+b^2 y $
$ =y\left(a^2 y^2-2 a b y-b^2\right) $
$ =y\left[(a y)^2-2 \times a y \times b+b^2\right] $
$ =y(a y-b)^2 $
$ =y(a y-b)(a y-b) $
View full question & answer→Question 1972 Marks
Factorise the following. $x^2- 10x + 21$
Answer$x^2- 10x + 21$
$ =x^2-(7+3) x+21 $
$ =x^2-7 x-3 x+21 $
$ =x(x-7)-3(x-7) $
$ =(x-7)(x-3) $
View full question & answer→Question 1982 Marks
Using suitable identities, evaluate the following. $(995)^2$
Answer$(995)^2$
$= (1000 - 5)^2$
$= (1000)^2+ (5)^2- 2 × 1000 × 5$
$= 1000000 + 25 - 10000$
$= 990025$
View full question & answer→Question 1992 Marks
Factorise the following expressions. $2 a^3-3 a^2 b+5 a b^2-a b$
Answer$2 a^3-3 a^2 b+5 a b^2-a b$
$ a\left(2 a^2-3 a b+5 b^2-b\right)$
View full question & answer→Question 2002 Marks
Factorise the following using the identity $a^2-b^2=(a+b)(a-b)$.
$\frac{\text{x}^2}{25}-625$
Answer$\frac{\text{x}^2}{25}-625$$=\Big(\frac{\text{x}}{5}\Big)^2-(25)^2$
$=\Big(\frac{\text{x}}{5}-25\Big)\Big(\frac{\text{x}}{5}-25\Big)$
View full question & answer→Question 2012 Marks
Write the greatest common factor in each of the following terms. $qrxy, pryz, rxyz$
Answer$qrxy, pryz, rxyz qrxy$
$= q × r × x × y pryz$
$= p × r × y × z rxyz$
$= r × x × y × z GCF = ry$
View full question & answer→Question 2022 Marks
Factorise the following, using the identity $a^2- 2ab + b^2= (a - b)^2$.
$x^2- 8x + 16$
Answer$ x^2-8 x+16 $
$ =x^2-2 \times x \times 4+4^2 $
$ =(x-4)^2 $
$ =(x-4)(x-4) $
View full question & answer→Question 2032 Marks
Factorise the following, using the identity $a^2+2 a b+b^2=(a+b)^2 $
$a^2 x^2+2 a b x+b^2$
Answer$a^2 x^2+2 a b x+b^2$
$=(a x)^2+2 \times a x \times b+b^2$
$=(a x+b)^2$
$=(a x+b)(a x+b)$
View full question & answer→Question 2042 Marks
Factorise the following, using the identity $a^2-2 a b+b^2=(a-b)^2 .$
$a^2 y^2-2 a b y+b^2$
Answer$a^2 y^2-2 a b y+b^2$
$(a y)^2-2 \times a y \times b+b^2$
$(a y-b)^2$
$=(a y-b)(a y-b)$
View full question & answer→Question 2052 Marks
Factorise the following. $y^2+ 18x + 65$
Answer$y^2+18 x+65$
$=y^2+13 y+15 y+5 \times 13$
$=y(y+13)+5(y+13)$
$=(y+13)(y+5)$
View full question & answer→Question 2062 Marks
Add: $7 a^2 b c,-3 a b c^2, 3 a^2 b c, 2 a b c^2$
AnswerWe have,
$7 a^2 b c,-3 a b c^2, 3 a^2 b c, 2 a b c^2$
$=7 a^2 b c-3 a b c^2+3 a^2 b c+2 a b c^2$
$=10 a^2 b c+(-a b c)^2$
$=10 a^2 b c-a b c^2$
View full question & answer→Question 2072 Marks
Using suitable identities, evaluate the following. $(339)^2- (161)^2$
Answer$(339)^2-(161)^2$
$=(339+161)(339-161)$
$=500 \times 178$
$=89000$
View full question & answer→Question 2082 Marks
Write the greatest common factor in each of the following terms.
$63 p^2 a^2 r^2 s,-9 p q^2 r^2 s^2, 15 p^2 q r^2 s^2,-60 p^2 a^2 r s^2$
Answer$63 p^2 a^2 r^2 s,-9 p q^2 r^2 s^2, 15 p^2 q r^2 s^2,-60 p^2 a^2 r s^2$
$63 p^2 a^2 r^2 s = 3 × 3 × 7 × p × p × a × a × r × r × r$
$-9 p q^2 r^2 s^2= -3 × 3 × p × q × q × r × r × s × s$
$15 p^2 q r^2 s^2= 3 × 5 × p × p × q × r × r × s × s × s$
$-60 p^2 a^2 r s^2= 2 × 2 × 3 × 5 × p × p × a × a × r × s × s$
$GCF = 3prs$
View full question & answer→Question 2092 Marks
Factorise the following, using the identity $a^2+ 2ab + b^2= (a + b)^2$
$ 4 x^4+12 x^3+9 x^2 $
Answer$ 4 x^4+12 x^3+9 x^2 $
$=x^2\left(4 x^2+12 x+9\right) $
$ =x^2\left[(2 x)+2 \times 2 x \times 3+3^2\right] $
$ =x^2(2 x+3)^2 $
$ =x^2(2 x+3)(2 x+3) $
View full question & answer→Question 2102 Marks
Factorise the following, using the identity $a^2+ 2ab + b^2= (a + b)^2$
$9\text{x}^2+2\text{xy}+\frac{\text{y}^2}{9}$
Answer$9\text{x}^2+2\text{xy}+\frac{\text{y}^2}{9}$$=(3\text{x})^2+2\times3\text{x}\times\frac{\text{y}}{3}+\Big(\frac{\text{y}}{3}\Big)^2$
$=\Big(3\text{x}+\frac{\text{y}}{3}\Big)^2$
$=\Big(3\text{x}+\frac{\text{y}}{3}\Big)\Big(3\text{x}+\frac{\text{y}}{3}\Big)$
View full question & answer→Question 2112 Marks
Multiply the following: $(ab + c), (ab + c)$
Answer$(ab + c), (ab + c)$
$(ab + c)(ab + c)$
$= ab(ab + c) + c(ab + c)$
$= a^2b^2+ abc + cab + c^2$
$= a^2b^2+ 2abc + c^2$
View full question & answer→Question 2122 Marks
Using suitable identities, evaluate the following.
$(9.7)^2- (0.3)^2$
Answer$(9.7)^2- (0.3)^2$
$= (9.7 + 0.3)(9.7 - 0.3)$
$= 10 \times 9.4$
$= 94$
View full question & answer→Question 2132 Marks
Factorise the following, using the identity $a^2- 2ab + b^2= (a - b)^2$.
$4y^2- 12y + 9$
Answer$4 y^2-12 y+9$
$=(2 y)^2-2 \times 2 y \times 3+3^2$
$=(2 y-3)^2$
$=(2 y-3)(2 y-3)$
View full question & answer→Question 2142 Marks
Factorise the following using the identity $a^2- b^2= (a + b)(a - b).$
$\frac{2\text{p}^2}{25}-32\text{q}^2$
Answer$\frac{2\text{p}^2}{25}-32\text{q}^2$$= 2\Big(\frac{\text{p}^2}{25}-16\text{q}^2\Big)$
$=2\Big[\Big(\frac{\text{p}^2}{5}\Big)-\big(4\text{q}^2\big)\Big]$
$=2\Big(\frac{\text{p}}{5}+4\text{q}\Big)\Big(\frac{\text{p}}{5}-4\text{q}\Big)$
View full question & answer→