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→