Question

solve all the questions

Answer 1

please reply with good feedback

Answer 2

☞(15.)
$15. \\ if \: possible \: let \: \sqrt{2} \: \: be \: rational \\ let \: its \: simplest \: form \: be \: \frac{a}{b} where \: a \: and \: b \: are \: positive \: integers \: having \: no \: common \:factor \: other \: than \: one \\ then \: \: \: \: \: \: \: \: \: \: \: \sqrt{2} = \frac{a}{b} \\ a = \sqrt{2} \: b \: \: \: \: \: \: \: ......(1)\\ squring \: on \: both \: sides \\ a ^{2} = 2b ^{2} \\ 2b ^{2} is \: divisible \: by \: 2 \\ a ^{2} is \: divisible \: by \: 2 \\ a \: is \: also \: divisible \: by \: 2 \\ now \\ let \: a = 2c \\ \sqrt{2} \: b = 2c \: \: \: \: ( \: \: from \: (1) \: = > \: a = \sqrt{2} \: \: b )\\ \: on \: squaring \: both \: sides \\ (\sqrt{2} \: b) ^{2} = (2c)^{2} \\ 2b ^{2} = 4c ^{2} \\ b ^{2} = 2c^{2} \\ 2c ^{2} is \: divisible \: by \: 2 \\ b ^{2} is \: divisible \: by \: 2 \\ b \: is \: also \: divisible \: by \: 2 \\ a \: and \: b \: are \: divisible \: by \: 2 \\ this \: contradicts \: our \: supposition \: that \frac{a}{b} \: is \: written \: in \: the \: simplest \: form \\ hence \: our \: supposition \: is \: wrong \\ ∴ \sqrt{2} \: \: is \: irrational \: number$

(10.)
We have (3×5×7)+7=105+7=112
=2×2×2×2×7, which has more than two factors
Thus, the given number is a composite number.
☞(11.)
Smallest Composite Number = 4
=2×2
Smallest Prime Number = 2
HCF of (4,2) = 2

☞(12.)
120= 2×2×2×3×5
144 =2×2×2×2×3×3
HCF of (120,144) = 2×2×2×3=24
LCM of (120,144) =2×2×2×2×3×3×5 = 720

☞(13.)
Length = 8m 25cm = 825 cm
Breadth = 6m 75cm = 675 cm
Height = 4m 50 cm = 450 cm
Now we have to find the HCF of these numbers
825=3×5×5×11
675 = 3×3×3×5×5
450 = 2×3×3×5×5
Longest rod = HCF of (825,675 and 450) = 3×5×5 =75
☞(14.)==>
HCF×LCM = Product of two numbers
11×253×R=253×440
R = (253×440)/(11×253)
R=40

☞(16) same as 15
check=>
$(2 \sqrt{3} + \sqrt{5} )(2 \sqrt{3} - \sqrt{5} ) \\ (2 \sqrt{3} ) ^{2} - ( \sqrt{5} ) ^{2} \\ 12 - 5 = 7 \\ it \: is \: rational \: number$


☞18.
$\frac{43}{2 ^{4} \times 5 ^{3} } =\frac{43 \times 5}{2 ^{4} \times 5 ^{3} \times 5 } = \frac{215}{(2 \times 5) ^{4} } = \frac{215}{10 ^{4} } = \frac{215}{10000} = 0.0215$