5 Marks Questions - MATHS STD 8 Questions - Vidyadip

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5 Marks Questions

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16 questions · self-marked practice — reveal the answer and mark yourself.

Question 15 Marks

(i) Add $p(p-q), q(q-r)$ and $r(r-p).$
(ii) Add $2 x(z-x-y)$ and $2 y(z-y-x).$
(iii) Subtract $3 I(I-4 m+5 n)$ from $4 I(10 n-3 m+2I).$
(iv) Subtract $3 a(a+b+c)-2 b(a-b+c)$ from $4 c(-a+b+c).$

Answer

$\text{(i) First expression }=p(p-q)$
$=p\times p-p \times q=p^{2}-pq$
$\text {Second expression} =q(q-r) $
$=q \times q-q \times r=q^2-q r $
$\text {Third expression} =r(r-p) $
$=r \times r-r \times p=r^2-r p$
On adding above expressions, we get

$\text{(ii) }2 x z-2 x^2-4 x y+2 y z-2 y^2$
$\begin{aligned} \text {(iii) First expression } & =3 l(l-4 m+5 n) \\ & =(3 l) \times(l)-(3 l) \times(4 m)+(3 l) \times(5 n) \\ & =3 l^2-12 l m+15 l n\end{aligned}$
$\text {Second expression}=4 l(10 n-3 m+2 l) $
$ =(4 l) \times(10 n)-(4 l) \times(3 m)+(4 l) \times(2 l) $
$=40 l n-12 l m+8 l^2$
On subtracting first expression from second expression, we get

(iv) $-3a^{2}-2b^{2}-ab-7ac+6bc+4c^{2}$

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Question 25 Marks

Complete the table.

	First expression	Second expression	Product
(i)	$a$	$b+c+d$
(ii)	$x+y-5$	$5xy$
(iii)	$p$	$6 p^2-7 p+5$
(iv)	$4 p^2 q^{2}$	$p^2-q^2$
(v)	$a+b+c$	$abc$

Answer

(i) $a(b+c+d)=a \times b+a \times c+a \times d=a b+a c+a d$
(ii) $(x+y-5) \times 5 x y=5 x y \times(x+y-5)\qquad$ [by commutative law]
$=(5 x y) \times(x)+(5 x y) \times(y)+(5 x y) \times(-5)$
$=5 x^2 y+5 x y^2-25 x y$
(iii) $p\left(6 p^2-7 p+5\right) =(p) \times\left(6 p^2\right)+(p) \times(-7 p)+(p) \times 5 $
$=6 p^3-7 p^2+5 p$
(iv) $\left(4 p^2 q^2\right) \times\left(p^2-q^2\right)=\left(4 p^2 q^2\right) \times(p)^2-\left(4 p^2 q^2\right) \times q^2 $
$=4 p^4 q^2-4 p^2 q^4$
(v) $(a+b+c) \times a b c=a b c \times(a+b+c)\qquad$ [by commutative law]
$=(a b c) \times(a)+(a b c) \times(b)+(a b c) \times(c) $
$ =a^2 b c+a b^2 c+a b c^2$

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Question 35 Marks

The given figure shows the dimensions of a wall having a window and a door of a room. Write an algebraic expression for the area of the wall to be painted.

Answer

According to the given figure,
$\text {Area of wall} =\text { Length } \times \text { Breadth } $
$ =(5 x+2) \times 5 x $
$ =5 x \times 5 x+5 x \times 2 $
$ =25 x^2+10 x \text { sq unit. }$
Area of window $=2 x \times x=2 x^2$ sq unit
and area of door $=3 x \times x=3 x^2$ sq unit
Now, total area of window and door
$=2 x^2+3 x^2=5 x^2$ sq unit
Thus, remaining area of wall to be painted
$=\left(25 x^2+10 x\right)-\left(5 x^2\right)$
$=25 x^2+10 x-5 x^2=20 x^2+10 x$
$=10 x(2 x+1)$ sq unit

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Question 45 Marks

Find the product of $(\text x-2\text y)(\text x+2\text y)\left(\text x^2+4\text y^2\right)$ at $\text x=1$ and $\text y=1.$

Answer

$(x -2 y)(x+2 y)\left(x^2+4 y^2\right) $
$ =(x-2 y)(x+2 y)\left(x^2+4 y^2\right) $
$ =x(x+2 y)-2 y(x+2 y)\left(x^2+4 y^2\right) $
$ =\left[x^2+2 x y-2 x y-4 y^2\right]\left(x^2+4 y^2\right) $
$ =\left(x^2-4 y^2\right)\left(x^2+4 y^2\right) $
$ =x^2\left(x^2+4 y^2\right)-4 y^2\left(x^2+4 y^2\right) $
$ =x^4+4 x^2 y^2-4 x^2 y^2-16 y^4 $
$ =x^4-16 y^4$
Now, at $x=1$ any $y=1$
$x^4-16 y^4$
$=(1)^4-16(1)^4$
$=-15$
Alternate method
$(x-2 y)(x+2 y)\left(x^2+4 y^2\right)$
$=[(x-2 y)(x+2 y)]\left(x^2+4 y^2\right)$
$=\left[x^2-(2 y)^2\right]\left(x^2+4 y^2\right)\left[\because(a+b)(a-b)=\left(a^2-b^2\right)\right]$
$=\left(x^2-4 y^2\right)\left(x^2+4 y^2\right)$
$=\left(x^2\right)^2-\left(4 y^2\right)^2$
$=x^4-16 y^4$

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Question 55 Marks

Multiply $\left(3\text y^5-7\text y^3+2\text y^2-\text y+4\right)$ by $\left(\text y^3-2\text y^2+3\text y-1\right).$

Answer

$\left(3 y^5-7 y^3+2 y^2-y+4\right) \times\left(y^{3}-2 y^2+3 y-1\right)$
$=3 y^5\left(y^3-2 y^2+3 y-1\right)-7 y^3\left(y^3-2 y^2+3 y-1\right)+2 y^2\left(y^3-2 y^2+3 y-1\right)-y\left(y^{3}-2 y^2+3 y-1\right)+4\left(y^3-2 y^2+3 y-1\right)$
$=3 y^8-6 y^7+9 y^6-3 y^5-7 y^6+14 y^5-21 y^4+7 y^3+2 y^5-4 y^4+6 y^3-2 y^2-y^4+2 y^3-3 y^2+y+4 y^3-8 y^2+12 y-4$
$=3 y^8-6 y^7+(9-7) y^6+(-3+14+2) y^5+(-4-21-1) y^4+(7+6+2+4) y^3+(-2-3-8) y^2+13 y-4$
$=3 y^8-6 y^7+2 y^6+13 y^5-26 y^4+19 y^3-13 y^2+13 y-4$

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Question 65 Marks

Subtract the following polynomials :
$5 x^2 y+3 x y+4 y x^2-2 y^2 x$ from $x^2 y-x y+5 y x^2$

Answer

$5 x^2 y+3 x y+4 y x^2-2 y^2 x$ from $x^2 y-x y+5 y x^2$
$=x^2 y-x y+5 y x^2-5 x^2 y-3 x y-4 y x^2 +2 y^2 x$
$=-3 x^2 y-4 x y+2 y^2 x$

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Question 75 Marks

Subtract the following polynomials :
$2 x^2 y^2-3 x y+4$ from $4 x^2 y^2+10 x y$

Answer

$\begin{aligned} 2 x^2 y^2 & -3 x y+4 \text { from } 4 x^2 y^2+10 x y \\ & =4 x^2 y^2+10 x y-\left(2 x^2 y^2-3 x y+4\right) \\ & =4 x^2 y^2+10 x y-2 x^2 y^2+3 x y-4 \\ & =2 x^2 y^2+13 x y-4\end{aligned}$

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Question 85 Marks

Subtract the following polynomials :
$3 x y+5 y z-7 x z+1$ from $-4 x y+2 y z-2 x z+5 x y z+1$

Answer

$3 x y+5 y z-7 x z+1 \text { from }-4 x y+2 y z-2 x z+5 x y z+1 $
$ =-4 x y+2 y z-2 x z+5 x y z+1-(3 x y+5 y z-7 x z+1) $
$ =-4 x y+2 y z-2 x z+5 x y z+1-3 x y -5 y z+7 x z-1$
$ =5 x z+5 x y z-7 x y-3 y z$

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Question 95 Marks

Subtract the following polynomials :
$(7 x+2)$ from $(-6 x+8)$

Answer

$\begin{aligned}(7 x+2) & \text { from }(-6 x+8) \\ & =(-6 x+8)-(7 x+2) \\ & =-6 x+8-7 x-2 \\ & =-13 x+6\end{aligned}$

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Question 105 Marks

Complete the following table.

Answer

(i) First expression : $9 x y$
Second expression : $-4 x(x+y)$
Product : $9 x y \times[-4 x(x+y)]$
$=-36 x^2 y(x+y)$
$=-36 x^2 y \times x+\left(-36 x^2 y \times y\right)$
$=-36 x^3 y-36 x^2 y^2$
(ii) First expression: pq
Second expression : $-9 p^2-7 q^2$
Product: $p q\left(-9 p^2-7 q^2\right)$
$=-p q \times 9 p^2-7 q^2 \times p q$
$=-9 p^3 q-7 p q^3$
(iii) First expression : $a^2+b^2+c^2$
Second expression : $-5 a b c$
Product: $-5 a b c \times\left(a^2+b^2+c^2\right)$
$=-5 a b c \times a^2-5 a b c \times b^2-5 a b c \times c^2$
$=-5 a^3 b c-5 a b^3 c-5 a b c^3$
(iv) First expression : $9 x+5 y-2$
Second expression : $x^2 y^2$
Product : $x^2 y^2 \times(9 x+5 y-2)$
$=x^2 y^2 \times 9 x+x^2 y^2 \times 5 y-x^2 y^2 \times 2$
$=9 x^3 y^2+5 x^2 y^3-2 x^2 y^2$
(v) First expression : $-\frac{5}{3} p^2 q^2$
Second expression : $6 p q-9 q^2$
Product $:-\frac{5}{3} p^2 q^2 \times\left(6 p q-9 q^2\right)$
$=-\frac{5}{3} p^2 q^2 \times 6 p q+\frac{5}{3} p^2 q^2 \times 9 q^2$
$=-10 p^3 q^3+15 p^2 q^4$

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Question 115 Marks

Multiply the following :
$\left(3 x^2+4 x-8\right),\left(2 x^2-4 x+3\right)$

Answer

We have, $\left(3 x^2+4 x-8\right)$ and $\left(2 x^2-4 x+3\right)$
$\therefore\left(3 x^2+4 x-8\right)\left(2 x^2-4 x+3\right)$
$=3 x^2\left(2 x^2-4 x+3\right)+4 x\left(2 x^2-4 x+3\right) -8\left(2 x^2-4 x+3\right)$
$=6 x^4-12 x^3+9 x^2+8 x^3-16 x^2+12 x-16 x^2+32 x-24$
$=6 x^4-12 x^3+8 x^3+9 x^2-16 x^2 -16 x^2 +12 x+32 x-24\quad$ [grouping like terms]
$=6 x^4-4 x^3-23 x^2+44 x-24$

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Question 125 Marks

Multiply the following :
$\left(x^2-5 x+6\right),(2 x+7)$

Answer

We have, $\left(x^2-5 x+6\right)$ and $(2 x+7)$
$\therefore\left(x^2-5 x+6\right)(2 x+7)$
$=x^2(2 x+7)-5 x(2 x+7)+6(2 x+7)$
$=2 x^3+7 x^2-10 x^2-35 x+12 x+42$
$=2 x^3-3 x^2-23 x+42$

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Question 135 Marks

Multiply the following :
$\left(\frac{3}{2} p^2+\frac{2}{3} q^2\right),\left(2 p^2-3 q^2\right)$

Answer

We have, $\left(\frac{3}{2} p^2+\frac{2}{3} q^2\right)$ and $\left(2 p^2-3 q^2\right)$
$\therefore\left(\frac{3}{2} p^2+\frac{2}{3} q^2\right)\left(2 p^2-3 q^2\right)= \frac{3}{2} p^2\left(2 p^2-3 q^2\right) +\frac{2}{3} q^2\left(2 p^2-3 q^2\right)$
$=\frac{3}{2} p^2 \times 2 p^2-\frac{9}{2} p^2 q^2+\frac{4}{3} q^2 p^2-2 q^4$
$=3 p^4+\left(\frac{4}{3}-\frac{9}{2}\right) p^2 q^2-2 q^4$
$=3 p^4+\left(\frac{8-27}{6}\right) p^2 q^2-2 q^4$
$=3 p^4-\frac{19}{6} p^2 q^2-2 q^4$

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Question 145 Marks

Multiply the following :
$x^2 y^2 z^2,(x y-y z-z x)$

Answer

We have, $x^2 y^2 z^2$ and $(x y-y z-z x)$
$\therefore x^2 y^2 z^2 \times(x y-y z-z x)=x^2 y^2 z^2 \times x y$ $-x^2 y^2 z^2 \times y z-x^2 y^2 z^2 \times z x$
$=\left(x^2 \times x\right) \times\left(y^2 \times y\right) \times z^2-x^2 \times\left(y^2 \times y\right)$ $\times\left(z^2 \times z\right)-\left(x^2 \times x\right) \times y^2 \times\left(z^2 \times z\right)$
$=x^3 y^3 z^3-x^2 y^3 z^3-x^3 y^2 z^3$

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Question 155 Marks

Multiply the following :
$15 x y^2, 17 y z^2$

Answer

We have, $15 x y^2$ and $17 y z^2$
$15 x y^2 \times 17 y z^2=15 \times 17 \times x \times\left(y^2 \times y\right) \times z^2$
$=255 x y^3 z^2$

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Question 165 Marks

Multiply the following :
$-7 p q^2 r^3,-13 p^3 q^2 r$

Answer

We have, $-7 p q^2 r^3$ and $-13 p^3 q^2 r$
$\therefore\left(-7 p q^2 r^3\right) \times\left(-13 p^3 q^2 r\right)$
$=(-7) \times(-13) \times\left(p \times p^3\right) \times\left(q^2 \times q^2\right) \times\left(r^3 \times r\right)$
$=91 p^4 q^4 r^4$

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5 Marks Questions - MATHS STD 8 Questions - Vidyadip