Exercise 1.3
Question 1
(i) $7 \times(-35)=-245$
(ii) $(-13) \times(-15)=195$
(iii) $(-12) \times(-11) \times(-10)=(-12) \times(110)=-1320$
(iv) $(-13) \times 0 \times(-24)=0 .$
(v) $(-1) \times(-2) \cdot(-3) \times(+4)=-24$
(vi) $(-3) \times(-6) \times(-2) \times(-1)=18 \times 2=36$
Question 2
(i) $37 \times[6+(-3)]$
If a,b,c are integers , then $a \times(b+c)$ = $a \times b+a \times c$
$=37 \times 6+37 \times(-3)$
∴ Both are equal , Verified
(ii) $(-21) \times[(-6)+(-4)]=(-21) \times(-6)+(-21) \times(-4)$
If a ,b , c are integers, then $a \times(b+c)=a \times b+a \times c$
=$(-21) \times(-6)+(-21) \times(-4)$
∴ Verified
Question 3
(i)
$\begin{aligned} 8 \times 53 \times(-125)=& 8 \times(-125) \times 53 \\=&-1000 \times 53 \\=&-53,000 \end{aligned}$
(ii)
$\begin{aligned}(-8) \times(-2) \times 3 \times(-5)=&(-8) \times(-5) \times(-2) \times 3 \\ &=40 \times 6 \\ &=240 . \end{aligned}$
(iii)
$\begin{aligned}(-6) \times 2 \times(-8) \times 5=&(-6) \times(-8) \times 2 \times 5 \\ &=48 \times 10 \\ &=480 . \end{aligned}$
(iv)
$\begin{aligned} 15 \times(-25) \times(-4) \times(-10) &=15 \times(-10) \times(-25) \times(-4) \\ &=-150 \times 100 \\ &=-15,000 \end{aligned}$
(v) $26 \times(-48)+(-48) \times(-36)$
$=(-48)[26+(-36)]$
$=(-48)[26-36]$
$=(-48) \times(-10)$
$=480 .$
(vi) $724 \times(-56)+(-724) \times 44$
$=724 \times(-56)+724 \times(-44)$
$=724 \times[-56+(-44)]$
$=724 \times[-56-44]$
$=724 \times(-100)$
$=-72400$
(vii)
$\begin{aligned}(-47) \times 102=&(-43) \times(100+2) \\=&-47 \times 100+(-47) \times 2 \\=&-4700+(-94) \\=&-4700-94 \\ &=-4794 . \end{aligned}$
(viii) $(-39) \times(-97)$
$(-39) \times(-100+3)$
$(-39) \times(-100)+(-39) \times 3$
$=3900+(-117)$
$=3900-117$
$=3783$
Question 4
(i)
$\begin{aligned}(-4) \times 1 &=44 \\ x &=\frac{44}{-4}=\frac{44 x-1}{-4 x-1} \\ & x=\frac{-44}{4}=-11 \end{aligned}$
(ii)
$\begin{aligned} 7 \times 1 . &=-42 \\ x &=\frac{-42}{7} \\ x &=-6 . \end{aligned}$
(iii)
$\begin{aligned} x \times(-13) &=143 \\ x=& \frac{143}{-13} \times \frac{-1}{-1} \\ x &=\frac{-143}{13} \\ & x=-11 \end{aligned}$
(iv) (-5) $\times$__ =0
Any number Multiplied with '0' We get '0
So 0 is answer
i.e -5$times$ 0= 0
Question 5
Fro every one hour the temperature lowered at a rate of $5^{\circ} \mathrm{C}$
For 8 hours it will be lowered by $5 \times 8=40^{\circ} \mathrm{C}$
Room temperature after free process= 32 - 40
$=-8^{\circ} \mathrm{C}$
Question 6
Total number of question = 10
Marks given for correct answer = 5 and
Marks given for incorrect answer = -2
(i) Rohit gets 4 correct and 5 incorrect answer
Rohit score = 4 $\times$ 5+6 $\times(-2)=20-12=8$
(ii) Seema gets 5 correct and 5 incorrect answer
Seema's score = $5 \times 5+5 \times(-2)=25-10=15$
(iii) As ritu attempted 7 questions and only 2 question are
correct , so number of incorrect question are 7-2=5
Ritu's score = $2 \times 5+5 \times(-2)=10-10=0$
Question 7
(i) Let pair of integers be x, y
Product is - 15 i.e $x y=-15$
$y=\frac{-15}{x}$ ___(1)
Difference = 8
x-y = 8
From (1) $x-\left(\frac{-15}{x}\right)=8$
$\frac{x^{2}+15}{x}=8$
$x^{2}+15=8 x$
$x^{2}-8 x+15=0$
$x^{2}-5 x-3 x+15=0$
$x(x-5)-3(x-5)=0$
$(x-3)(x-5)=0$
$x=3, x=9$
$y=\frac{-15}{3} ; \quad y=-\frac{15}{5}$
$y=-5 ; \quad y=-3$
The pair of integers are -3, -5 , 5, -3
Alternative method for (ii)
Product is - 36
Difference is 15
We know
$12 \times(-3)=-36 ;(-12) \times 3=-36$
$9 \times(-4)=-36 \quad ;(-9) \times 4=-36$
Thus , we have pair of integers $12,-3 ;-12,3 ; 9,-4 ;-9,4$
Such that product of each pair is -36
But Difference of $9,-4=9-(-4)=9+4=13 \mathrm{~cm}$
Difference of $-9,4=-9-4=-13$
Hence the required pair are $12,-3$ or $-12,3$.
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