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→