WBBSE Class-8 (VIII) Koshe dekhi 15 Somadhan || Koshe dekhi 15 WBBSE Class 8 || বীজগাণিতিক সংখ্যামালার সরলীকরণ কষে দেখি 15 || WBBSE Class-8 (VIII) Koshe dekhi 15 Somadhan || Gonitprava Class 8 Chapter 15 Solution || গণিতপ্রভা অষ্টম শ্রেণি (ক্লাস-৮) সমাধান || West Bengal Board Class 8 Math Book Solution || অধ্যায় ১৫ পশ্চিমবঙ্গ মধ্যশিক্ষা পর্ষদ
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Koshe dekhi 15 WBBSE Class 8 || বীজগাণিতিক সংখ্যামালার সরলীকরণ কষে দেখি 15 || WBBSE Class-8 (VIII) Koshe dekhi 15 Somadhan || West Bengal Board Class 8 Book Math Solution || অধ্যায় ১৫ পশ্চিমবঙ্গ মধ্যশিক্ষা পর্ষদ
কষে দেখি - 15
1. নীচের সম্পর্কগুলি দেখি ও কোনটি সত্য ও কোনটি মিথ্যা লিখি।
(i) \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\)
ডানপক্ষ \( =\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}= \) বামপক্ষ
\(\therefore\) সম্পর্কটি সত্য।
\(\therefore\) সম্পর্কটি সত্য।
Koshe dekhi 15 WBBSE Class 8 || বীজগাণিতিক সংখ্যামালার সরলীকরণ কষে দেখি 15 || WBBSE Class-8 (VIII) Koshe dekhi 15 Somadhan || West Bengal Board Class 8 Book Math Solution || অধ্যায় ১৫ পশ্চিমবঙ্গ মধ্যশিক্ষা পর্ষদ
(ii) \(\frac{a}{x+y}=\frac{a}{x}+\frac{a}{y}\)
ডানপক্ষ \( =\frac{a}{x}+\frac{a}{y}=a\left(\frac{1}{x}+\frac{1}{y}\right) \neq
\frac{a}{x+y}= \) বামপক্ষ
\(\therefore\) বামপক্ষ \( \neq \) ডানপক্ষ
\(\therefore\) সম্পর্কটি মিথ্যা।
\(\therefore\) বামপক্ষ \( \neq \) ডানপক্ষ
\(\therefore\) সম্পর্কটি মিথ্যা।
(iii) \(\frac{x-y}{a-b}=\frac{y-x}{b-a}\)
বামপক্ষ \( =\frac{x-y}{a-b}=\frac{-(y-x)}{-(b-a)}=\frac{y-x}{b-a}= \) ডানপক্ষ
\(\therefore\) সম্পর্কটি সত্য।
\(\therefore\) সম্পর্কটি সত্য।
(iv)\(\frac{1}{x}+\frac{1}{y}=\frac{1}{x+y}\)
বামপক্ষ \( =\frac{1}{x}+\frac{1}{y}=\frac{y+x}{x y} \neq \frac{1}{x+y}= \)
ডানপক্ষ
\(\therefore\) বামপক্ষ \( \neq \) ডানপক্ষ
\(\therefore\) সম্পর্কটি মিথ্যা।
\(\therefore\) বামপক্ষ \( \neq \) ডানপক্ষ
\(\therefore\) সম্পর্কটি মিথ্যা।
2. নীচের বীজগাণিতিক ভগ্নাংশগুলি লঘিষ্ঠ আকারে প্রকাশ করি।
(i) \(\frac{63 a^{3} b^{4}}{77 b^{5}}\)
\( \frac{63 a^{3} b^{4}}{77 b^{5}}=\frac{9 a^{3}}{11 b}\quad \left[\because b^{4}
\div b^{5}=b^{4-5}=b^{-1}=\frac{1}{b}\right] \)
(ii) \(\frac{18 a^{4} b^{5} c^{2}}{21 a^{7} b^{2}}\)
\( \frac{18 a^{4} b^{5} c^{2}}{21 a^{7} b^{2}}=\frac{6 b^{3} c^{2}}{7 a^{3}}\quad
\left[\because a^{4} \div a^{7}=a^{-3}=\frac{1}{a^{3}} ; b^{5} \div b^{2}=b^{3}\right] \)
(iii) \(\frac{x^{2}-3 x+2}{x^{2}-1}\)
\( \frac{x^{2}-3 x+2}{x^{2}-1}=\frac{x^{2}-2
x-x+2}{(x+1)(x-1)}=\frac{x(x-2)-1(x-2)}{(x+1)(x-1)} \)
\( =\frac{(x-1)(x-2)}{(x-1)(x+1)}=\frac{x-2}{x+1} \)
\( =\frac{(x-1)(x-2)}{(x-1)(x+1)}=\frac{x-2}{x+1} \)
(iv) \(\frac{a+1}{a-2} \times \frac{a^{2}-a-2}{a^{2}+a}\)
\( \frac{a+1}{a-2} \times \frac{a^{2}-a-2}{a^{2}+a}=\frac{a+1}{a-2} \times
\frac{a^{2}-2 a+a-2}{a(a+1)} \)
\( =\frac{a+1}{a-2} \times \frac{a(a-2)+1(a-2)}{a(a+1)}\)
\(=\frac{a+1}{(a-2)} \times \frac{(a-2)(a+1)}{a(a+1)}=\frac{a+1}{a} \)
\( =\frac{a+1}{a-2} \times \frac{a(a-2)+1(a-2)}{a(a+1)}\)
\(=\frac{a+1}{(a-2)} \times \frac{(a-2)(a+1)}{a(a+1)}=\frac{a+1}{a} \)
(v) \(\frac{p^{3}+q^{3}}{p^{2}-q^{2}} \div \frac{p+q}{p-q}\)
\( \frac{p^{3}+q^{3}}{p^{2}-q^{2}} \div
\frac{p+q}{p-q}=\frac{p^{3}+q^{3}}{p^{2}-q^{2}} \times \frac{p-q}{p+q} \)
\( =\frac{(p+q)\left(p^{2}-p q+q^{2}\right)}{(p+q)(p-q)} \times \frac{(p-q)}{(p+q)}\)
\(=\frac{p^{2}-p q+q^{2}}{p+q} \)
\( =\frac{(p+q)\left(p^{2}-p q+q^{2}\right)}{(p+q)(p-q)} \times \frac{(p-q)}{(p+q)}\)
\(=\frac{p^{2}-p q+q^{2}}{p+q} \)
(vi) \(\frac{x^{2}-x-6}{x^{2}+4 x-5} \times \frac{x^{2}+6 x+5}{x^{2}-4 x+3}\)
\( \frac{x^{2}-x-6}{x^{2}+4 x-5} \times \frac{x^{2}+6 x+5}{x^{2}-4 x+3} \)
\( =\frac{x^{2}-3 x+2 x-6}{x^{2}+5 x-x-5} \times \frac{x^{2}+5 x+x+5}{x^{2}-3 x-x+3}\)
\(=\frac{x(x-3)+2(x-3)}{x(x+5)-1(x+5)} \times \frac{x(x+5)+1(x+5)}{x(x-3)-1(x-3)}\)
\(=\frac{(x-3)(x+2)}{(x+5)(x-1)} \times \frac{(x+5)(x+1)}{(x-3)(x-1)}\)
\(=\frac{(x+2)(x+1)}{(x-1)^{2}} \)
\( =\frac{x^{2}-3 x+2 x-6}{x^{2}+5 x-x-5} \times \frac{x^{2}+5 x+x+5}{x^{2}-3 x-x+3}\)
\(=\frac{x(x-3)+2(x-3)}{x(x+5)-1(x+5)} \times \frac{x(x+5)+1(x+5)}{x(x-3)-1(x-3)}\)
\(=\frac{(x-3)(x+2)}{(x+5)(x-1)} \times \frac{(x+5)(x+1)}{(x-3)(x-1)}\)
\(=\frac{(x+2)(x+1)}{(x-1)^{2}} \)
(vii) \(\frac{a^{2}-a b+{b}^{2}}{a^{2}+a b} \div \frac{a^{3}+b^{3}}{a^{2}-b^{2}}\)
\( \frac{a^{2}-a b+b^{2}}{a^{2}+a b} \div \frac{a^{3}+b^{3}}{a^{2}-b^{2}}\)
\(=\frac{a^{2}-a b+b^{2}}{a(a+b)} \div \frac{(a+b)\left(a^{2}-a b+b^{2}\right)}{(a+b)(a-b)}\)
\(=\frac{a^{2}-a b+b^{2}}{a(a+b)} \times \frac{(a+b)(a-b)}{(a+b)\left(a^{2}-a b+b^{2}\right)}\)
\(=\frac{a-b}{a(a+b)} \)
\(=\frac{a^{2}-a b+b^{2}}{a(a+b)} \div \frac{(a+b)\left(a^{2}-a b+b^{2}\right)}{(a+b)(a-b)}\)
\(=\frac{a^{2}-a b+b^{2}}{a(a+b)} \times \frac{(a+b)(a-b)}{(a+b)\left(a^{2}-a b+b^{2}\right)}\)
\(=\frac{a-b}{a(a+b)} \)
3. নীচের বীজগাণিতিক ভগ্নাংশগুলি সরলতম আকারে প্রকাশ করি।
(i) \(\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}\)
\( \frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}\)
\(=\frac{c}{a b c}+\frac{a}{a b c}+\frac{b}{a b c} \)
\( =\frac{c+a+b}{a b c}\)
\(=\frac{a+b+c}{a b c} \)
\(=\frac{c}{a b c}+\frac{a}{a b c}+\frac{b}{a b c} \)
\( =\frac{c+a+b}{a b c}\)
\(=\frac{a+b+c}{a b c} \)
(ii) \(\frac{a-b-c}{a}+\frac{a+b+c}{a}\)
\( \frac{a-b-c}{a}+\frac{a+b+c}{a}\)
\(=\frac{a-b-c+a+b+c}{a}\)
\(=\frac{2 a}{a}\)
\(=2 \)
\(=\frac{a-b-c+a+b+c}{a}\)
\(=\frac{2 a}{a}\)
\(=2 \)
Koshe dekhi 15 WBBSE Class 8 || বীজগাণিতিক সংখ্যামালার সরলীকরণ কষে দেখি 15 || WBBSE Class-8 (VIII) Koshe dekhi 15 Somadhan || West Bengal Board Class 8 Book Math Solution || অধ্যায় ১৫ পশ্চিমবঙ্গ মধ্যশিক্ষা পর্ষদ
(iii) \(\frac{x^{2}+a^{2}}{a b}+\frac{x-a}{a x}-\frac{x^{3}}{b}\)
\( \frac{x^{2}+a^{2}}{a b}+\frac{x-a}{a x}-\frac{x^{3}}{b}\)
\(=\frac{x\left(x^{2}+a^{2}\right)+b(x-a)-a x x^{3}}{a b x} \)
\( =\frac{x^{3}+a^{2} x+b x-a b-a x^{4}}{a b x} \)
\(=\frac{x\left(x^{2}+a^{2}\right)+b(x-a)-a x x^{3}}{a b x} \)
\( =\frac{x^{3}+a^{2} x+b x-a b-a x^{4}}{a b x} \)
(iv) \(\frac{2 a^{2} b}{3 b^{2} c} \times \frac{c^{4}}{3 a^{3}} \div \frac{4 b
c^{3}}{9 a^{2}}\)
\( \frac{2 a^{2} b}{3 b^{2} c} \times \frac{c^{4}}{3 a^{3}} \div \frac{4 b c^{3}}{9
a^{2}}=\frac{2 a^{2} b}{3 b^{2} c} \times \frac{c^{4}}{3 a^{3}} \times \frac{9 a^{2}}{4 b c^{3}}
\)
\( =\frac{a^{4} b c^{4}}{2 a^{3} b^{3} c^{4}}\)
\(=\frac{a}{2 b^{2}}\quad \left[\because a^{4-3}=a, b^{1-3}=b^{-2}\right] \)
\( =\frac{a^{4} b c^{4}}{2 a^{3} b^{3} c^{4}}\)
\(=\frac{a}{2 b^{2}}\quad \left[\because a^{4-3}=a, b^{1-3}=b^{-2}\right] \)
(v) \(\frac{1}{x^{2}-3 x+2}+\frac{1}{x^{2}-5 x+6}+\frac{1}{x^{2}-4 x+3}\)
\( \frac{1}{x^{2}-3 x+2}+\frac{1}{x^{2}-5 x+6}+\frac{1}{x^{2}-4 x+3}\)
\(=\frac{1}{x^{2}-2 x-x+2}+\frac{1}{x^{2}-3 x-2 x+6}+\frac{1}{x^{2}-3 x-x+3}\)
\(=\frac{1}{x(x-2)-1(x-2)}+\frac{1}{x(x-3)-2(x-3)}+\frac{1}{x(x-3)-1(x-3)}\)
\(=\frac{1}{(x-2)(x-1)}+\frac{1}{(x-3)(x-2)}+\frac{1}{(x-3)(x-1)}\)
\( =\frac{x-3+x-1+x-2}{(x-1)(x-2)(x-3)}=\frac{3 x-6}{(x-1)(x-2)(x-3)} \)
\(=\frac{3(x-2)}{(x-1)(x-2)(x-3)}=\frac{3 }{(x-1)(x-3)} \)
\(=\frac{1}{x^{2}-2 x-x+2}+\frac{1}{x^{2}-3 x-2 x+6}+\frac{1}{x^{2}-3 x-x+3}\)
\(=\frac{1}{x(x-2)-1(x-2)}+\frac{1}{x(x-3)-2(x-3)}+\frac{1}{x(x-3)-1(x-3)}\)
\(=\frac{1}{(x-2)(x-1)}+\frac{1}{(x-3)(x-2)}+\frac{1}{(x-3)(x-1)}\)
\( =\frac{x-3+x-1+x-2}{(x-1)(x-2)(x-3)}=\frac{3 x-6}{(x-1)(x-2)(x-3)} \)
\(=\frac{3(x-2)}{(x-1)(x-2)(x-3)}=\frac{3 }{(x-1)(x-3)} \)
(vi)\(\frac{1}{x-1}+\frac{1}{x+1}+\frac{2 x}{x^{2}+1}+\frac{4 x^{3}}{x^{4}+1}\)
\( \frac{1}{x-1}+\frac{1}{x+1}+\frac{2 x}{x^{2}+1}+\frac{4 x^{3}}{x^{4}+1} \)
\( =\frac{x+1+x-1}{(x-1)(x+1)}+\frac{2 x}{x^{2}+1}+\frac{4 x^{3}}{x^{4}+1} \)
\( =\frac{2 x}{x^{2}-1}+\frac{2 x}{x^{2}+1}+\frac{4 x^{3}}{x^{4}+1} \)
\( =2 x \cdot \frac{x^{2}+1+x^{2}-1}{\left(x^{2}-1\right)\left(x^{2}+1\right)}+\frac{4 x^{3}}{x^{4}+1} \)
\( =2 x \cdot \frac{2 x^{2}}{\left(x^{2}\right)^{2}-(1)^{2}}+\frac{4 x^{3}}{x^{4}+1}=\frac{4 x^{3}}{x^{4}-1}+\frac{4 x^{3}}{x^{4}+1}\)
\(=4 x^{3} \cdot \frac{x^{4}+x+x^{4}-1}{\left(x^{4}-1\right)\left(x^{4}+1\right)}=4 x^{3} \frac{2 x^{4}}{\left(x^{4}\right)^{2}-(1)^{2}}=\frac{8 x^{7}}{x^{8}-1} \)
\( =\frac{x+1+x-1}{(x-1)(x+1)}+\frac{2 x}{x^{2}+1}+\frac{4 x^{3}}{x^{4}+1} \)
\( =\frac{2 x}{x^{2}-1}+\frac{2 x}{x^{2}+1}+\frac{4 x^{3}}{x^{4}+1} \)
\( =2 x \cdot \frac{x^{2}+1+x^{2}-1}{\left(x^{2}-1\right)\left(x^{2}+1\right)}+\frac{4 x^{3}}{x^{4}+1} \)
\( =2 x \cdot \frac{2 x^{2}}{\left(x^{2}\right)^{2}-(1)^{2}}+\frac{4 x^{3}}{x^{4}+1}=\frac{4 x^{3}}{x^{4}-1}+\frac{4 x^{3}}{x^{4}+1}\)
\(=4 x^{3} \cdot \frac{x^{4}+x+x^{4}-1}{\left(x^{4}-1\right)\left(x^{4}+1\right)}=4 x^{3} \frac{2 x^{4}}{\left(x^{4}\right)^{2}-(1)^{2}}=\frac{8 x^{7}}{x^{8}-1} \)
(vii)\(\frac{b^{2}-5 b}{3 b-4 a} \times \frac{9 b^{2}-16 a^{2}}{b^{2}-25} \div
\frac{3 b^{2}+4 a b}{a b+5 a}\)
\( \frac{b^{2}-5 b}{3 b-4 a} \times \frac{9 b^{2}-16 a^{2}}{b^{2}-25} \div \frac{3
b^{2}+4 a b}{a b+5 a}\)
\(=\frac{b(b-5)}{3 b-4 a} \times \frac{(3 b)^{2}-(4 a)^{2}}{(b)^{2}-(5)^{2}} \times \frac{a b+5 a}{3 b^{2}+4 a b}\)
\(=\frac{b(b-5)}{(3 b-4 a)} \times \frac{(3 b+4 a)(3 b-4 a)}{(b+5)(b-5)} \times \frac{a(b+5)}{b(3 b+4 a)}\)
\(=a \)
\(=\frac{b(b-5)}{3 b-4 a} \times \frac{(3 b)^{2}-(4 a)^{2}}{(b)^{2}-(5)^{2}} \times \frac{a b+5 a}{3 b^{2}+4 a b}\)
\(=\frac{b(b-5)}{(3 b-4 a)} \times \frac{(3 b+4 a)(3 b-4 a)}{(b+5)(b-5)} \times \frac{a(b+5)}{b(3 b+4 a)}\)
\(=a \)
(viii) \(\frac{b+c}{(a-b)(a-c)}+\frac{c+a}{(b-a)(b-c)}+\frac{a+b}{(c-a)(c-b)}\)
\( \frac{b+c}{(a-b)(a-c)}+\frac{c+a}{(b-a)(b-c)}+\frac{a+b}{(c-a)(c-b)}\)
\(=\frac{-(b+c)}{(a-b)(c-a)}+\frac{-(c+a)}{(a-b)(b-c)}+\frac{-(a+b)}{(c-a)(b-c)} \)
\( =\frac{-(b+c)}{(a-b)(c-a)}-\frac{(c+a)}{(a-b)(b-c)}-\frac{(a+b)}{(c-a)(b-c)}\)
\(=-\left[\frac{(b+c)(b-c)+(c+a)(c-a)+(a+b)(a-b)}{(a-b)(b-c)(c-a)}\right]\)
\( =-\left[\frac{b^{2}-c^{2}+c^{2}-a^{2}+a^{2}-b^{2}}{(a-b)(b-c)(c-a)}\right] \)
\(=\frac{0}{(a-b)(b-c)(c-a)}=0 \)
\(=\frac{-(b+c)}{(a-b)(c-a)}+\frac{-(c+a)}{(a-b)(b-c)}+\frac{-(a+b)}{(c-a)(b-c)} \)
\( =\frac{-(b+c)}{(a-b)(c-a)}-\frac{(c+a)}{(a-b)(b-c)}-\frac{(a+b)}{(c-a)(b-c)}\)
\(=-\left[\frac{(b+c)(b-c)+(c+a)(c-a)+(a+b)(a-b)}{(a-b)(b-c)(c-a)}\right]\)
\( =-\left[\frac{b^{2}-c^{2}+c^{2}-a^{2}+a^{2}-b^{2}}{(a-b)(b-c)(c-a)}\right] \)
\(=\frac{0}{(a-b)(b-c)(c-a)}=0 \)
(ix)
\(\frac{b+c-a}{(a-b)(a-c)}+\frac{c+a-b}{(b-c)(b-a)}+\frac{a+b-c}{(c-a)(c-b)}\)
\( \frac{b+c-a}{(a-b)(a-c)}+\frac{c+a-b}{(b-c)(b-a)}+\frac{a+b-c}{(c-a)(c-b)}
\)
\( =\frac{-b-c+a}{(a-b)(c-a)}+\frac{-c-a+b}{(b-c)(a-b)}+\frac{-a-b+c}{(c-a)(b-c)}\)
\(=\frac{(-b-c+a)(b-c)+(-c-a+b)(c-a)+(-a-b+c)(a-b)}{(a-b)(b-c)(c-a)}\)
\( =\frac{a(b-c)-(b+c)(b-c)+b(c-a)-(c+a)(c-a)+c(a-b)-(a+b)(a-b)}{(a-b)(b-c)(c-a)} \)
\( =\frac{a b-a c-b^{2}+c^{2}+b c-a b-c^{2}+a^{2}+a c-b c-a^{2}+b^{2}}{(a-b)(b-c)(c-a)}\)
\(=\frac{0}{(a-b)(b-c)(c-a)}=0 \)
\( =\frac{-b-c+a}{(a-b)(c-a)}+\frac{-c-a+b}{(b-c)(a-b)}+\frac{-a-b+c}{(c-a)(b-c)}\)
\(=\frac{(-b-c+a)(b-c)+(-c-a+b)(c-a)+(-a-b+c)(a-b)}{(a-b)(b-c)(c-a)}\)
\( =\frac{a(b-c)-(b+c)(b-c)+b(c-a)-(c+a)(c-a)+c(a-b)-(a+b)(a-b)}{(a-b)(b-c)(c-a)} \)
\( =\frac{a b-a c-b^{2}+c^{2}+b c-a b-c^{2}+a^{2}+a c-b c-a^{2}+b^{2}}{(a-b)(b-c)(c-a)}\)
\(=\frac{0}{(a-b)(b-c)(c-a)}=0 \)
Koshe dekhi 15 WBBSE Class 8 || বীজগাণিতিক সংখ্যামালার সরলীকরণ কষে দেখি 15 || WBBSE Class-8 (VIII) Koshe dekhi 15 Somadhan || West Bengal Board Class 8 Book Math Solution || অধ্যায় ১৫ পশ্চিমবঙ্গ মধ্যশিক্ষা পর্ষদ
(x)
\(\frac{\frac{a^{2}}{x-a}+\frac{b^{2}}{x-b}+\frac{c^{2}}{x-c}+a+b+c}{\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}}\)
\(
\frac{\frac{a^{2}}{x-a}+\frac{b^{2}}{x-b}+\frac{c^{2}}{x-c}+a+b+c}{\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}}
\)
\( =\frac{\frac{a^{2}}{x-a}+a+\frac{b^{2}}{x-b}+b+\frac{c^{2}}{x-c}+c}{\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}} \)
\( =\frac{\frac{a^{2}+a x-a^{2}}{x-a}+\frac{b^{2}+b x-b^{2}}{x-b}+\frac{c^{2}+c x-c^{2}}{x-c}}{\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}} \)
\( =\frac{\frac{a x}{x-a}+\frac{b x}{x-b}+\frac{c x}{x-c}}{\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}}=\frac{x\left(\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}\right)}{\left(\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}\right)}=x \)
\( =\frac{\frac{a^{2}}{x-a}+a+\frac{b^{2}}{x-b}+b+\frac{c^{2}}{x-c}+c}{\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}} \)
\( =\frac{\frac{a^{2}+a x-a^{2}}{x-a}+\frac{b^{2}+b x-b^{2}}{x-b}+\frac{c^{2}+c x-c^{2}}{x-c}}{\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}} \)
\( =\frac{\frac{a x}{x-a}+\frac{b x}{x-b}+\frac{c x}{x-c}}{\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}}=\frac{x\left(\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}\right)}{\left(\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}\right)}=x \)
(xi)
\(\left(\frac{a^{2}+b^{2}}{a^{2}-b^{2}}-\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\right)
\div\left(\frac{a+b}{a-b}-\frac{a-b}{a+b}\right) \times\left(\frac{a}{b}+\frac{b}{a}\right)\)
\( \left(\frac{a^{2}+b^{2}}{a^{2}-b^{2}}-\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\right)
\div\left(\frac{a+b}{a-b}-\frac{a-b}{a+b}\right) \times\left(\frac{a}{b}+\frac{b}{a}\right) \)
\( =\frac{\left(a^{2}+b^{2}\right)^{2}-\left(a^{2}-b^{2}\right)^{2}}{\left(a^{2}-b^{2}\right)\left(a^{2}+b^{2}\right)} \div\left\{\frac{(a+b)^{2}-(a-b)^{2}}{(a+b)(a-b)}\right\} \times\left(\frac{a^{2}+b^{2}}{a b}\right)\)
\(=\frac{4 a^{2} b^{2}}{\left(a^{2}-b^{2}\right)\left(a^{2}+b^{2}\right)} \div \frac{4 a b}{a^{2}-b^{2}} \times \frac{a^{2}+b^{2}}{a b}\)
\(=\frac{4 a^{2} b^{2}}{\left(a^{2}+b^{2}\right)\left(a^{2}-b^{2}\right)} \times \frac{\left(a^{2}-b^{2}\right)}{4 a b} \times \frac{\left(a^{2}+b^{2}\right)}{a b}=1\)
\( =\frac{\left(a^{2}+b^{2}\right)^{2}-\left(a^{2}-b^{2}\right)^{2}}{\left(a^{2}-b^{2}\right)\left(a^{2}+b^{2}\right)} \div\left\{\frac{(a+b)^{2}-(a-b)^{2}}{(a+b)(a-b)}\right\} \times\left(\frac{a^{2}+b^{2}}{a b}\right)\)
\(=\frac{4 a^{2} b^{2}}{\left(a^{2}-b^{2}\right)\left(a^{2}+b^{2}\right)} \div \frac{4 a b}{a^{2}-b^{2}} \times \frac{a^{2}+b^{2}}{a b}\)
\(=\frac{4 a^{2} b^{2}}{\left(a^{2}+b^{2}\right)\left(a^{2}-b^{2}\right)} \times \frac{\left(a^{2}-b^{2}\right)}{4 a b} \times \frac{\left(a^{2}+b^{2}\right)}{a b}=1\)
(xii) \(\frac{b+c}{b c}(b+c-a)+\frac{c+a}{c a}(c+a-b)+\frac{a+b}{a b}(a+b-c)\)
\( \frac{b+c}{b c}(b+c-a)+\frac{c+a}{c a}(c+a-b)+\frac{a+b}{a b}(a+b-c) \)
\( =\frac{(b+c)(b+c-a)}{b c}+\frac{(c+a)(c+a-b)}{c a}+\frac{(a+b)(a+b-c)}{a b}\)
\(=\frac{a(b+c)(b+c-a)+b(c+a)(c+a-b)+c(a+b)(a+b-c)}{a b c}\)
\(=\frac{(a b+a c)(b+c-a)+(b c+b a)(c+a-b)+(a c+b c)(a+b-c)}{a b c}\)
\(=\frac{a b^{2}+2 a b c-a^{2} b+a c^{2}-a^{2} c+b c^{2}+2 a b c-b^{2} c+a^{2} b-a b^{2}+a^{2} c+2 a b c-a c^{2}+b^{2} c-b c^{2}}{a b c}=\frac{6 a b c}{a b c}=6 \)
বিকল্প পদ্ধতি :
\( \frac{b+c}{b c}(b+c-a)+\frac{c+a}{c a}(c+a-b)+\frac{a+b}{a b}(a+b-c) \)
\( =\left(\frac{b}{b c}+\frac{c}{b c}\right)(b+c-a)+\left(\frac{c}{c a}+\frac{a}{c a}\right)(c+a-b)+\left(\frac{a}{a b}+\frac{b}{a b}\right)(a+b-c) \)
\( =\left(\frac{1}{c}+\frac{1}{b}\right)(b+c-a)+\left(\frac{1}{a}+\frac{1}{c}\right)(c+a-b)+\left(\frac{1}{b}+\frac{1}{a}\right)(a+b-c)\)
\(=\frac{1}{c}(b+c-a)+\frac{1}{b}(b+c-a)+\frac{1}{a}(c+a-b)+\frac{1}{c}(c+a-b)+\frac{1}{b}(a+b-c)+\frac{1}{a}(a+b-c)\)
\(=\frac{1}{c}(b+c-a+c+a-b)+\frac{1}{b}(b+c-a+a+b-c)+\frac{1}{a}(c+a-b+a+b-c)\)
\(=\frac{1}{c} \times 2 c+\frac{1}{b} \times 2 b+\frac{1}{a} \times 2 a=2+2+2=6 \)
\( =\frac{(b+c)(b+c-a)}{b c}+\frac{(c+a)(c+a-b)}{c a}+\frac{(a+b)(a+b-c)}{a b}\)
\(=\frac{a(b+c)(b+c-a)+b(c+a)(c+a-b)+c(a+b)(a+b-c)}{a b c}\)
\(=\frac{(a b+a c)(b+c-a)+(b c+b a)(c+a-b)+(a c+b c)(a+b-c)}{a b c}\)
\(=\frac{a b^{2}+2 a b c-a^{2} b+a c^{2}-a^{2} c+b c^{2}+2 a b c-b^{2} c+a^{2} b-a b^{2}+a^{2} c+2 a b c-a c^{2}+b^{2} c-b c^{2}}{a b c}=\frac{6 a b c}{a b c}=6 \)
বিকল্প পদ্ধতি :
\( \frac{b+c}{b c}(b+c-a)+\frac{c+a}{c a}(c+a-b)+\frac{a+b}{a b}(a+b-c) \)
\( =\left(\frac{b}{b c}+\frac{c}{b c}\right)(b+c-a)+\left(\frac{c}{c a}+\frac{a}{c a}\right)(c+a-b)+\left(\frac{a}{a b}+\frac{b}{a b}\right)(a+b-c) \)
\( =\left(\frac{1}{c}+\frac{1}{b}\right)(b+c-a)+\left(\frac{1}{a}+\frac{1}{c}\right)(c+a-b)+\left(\frac{1}{b}+\frac{1}{a}\right)(a+b-c)\)
\(=\frac{1}{c}(b+c-a)+\frac{1}{b}(b+c-a)+\frac{1}{a}(c+a-b)+\frac{1}{c}(c+a-b)+\frac{1}{b}(a+b-c)+\frac{1}{a}(a+b-c)\)
\(=\frac{1}{c}(b+c-a+c+a-b)+\frac{1}{b}(b+c-a+a+b-c)+\frac{1}{a}(c+a-b+a+b-c)\)
\(=\frac{1}{c} \times 2 c+\frac{1}{b} \times 2 b+\frac{1}{a} \times 2 a=2+2+2=6 \)
(xiii) \(\frac{y^{2}+y z+z^{2}}{(x-y)(x-z)}+\frac{z^{2}+z x+x^{2}}{(y-z)(y-
x)}+\frac{x^{2}+x y+y^{2}}{(z-x)(z-y)}\)
\( \frac{y^{2}+y z+z^{2}}{(x-y)(x-z)}+\frac{z^{2}+z
x+x^{2}}{(y-z)(y-x)}+\frac{x^{2}+x y+y^{2}}{(z-x)(z-y)} \)
\( =\frac{-\left(y^{2}+y z+z^{2}\right)}{(x-y)(z-x)}+\frac{-\left(z^{2}+z x+x^{2}\right)}{(y-z)(x-y)}+\frac{-\left(x^{2}+x y+y^{2}\right)}{(z-x)(y-z)}\)
\(=-\left[\frac{(y-z)\left(y^{2}+y z+z^{2}\right)+(z-x)\left(z^{2}+z x+x^{2}\right)+(x-y)\left(x^{2}+x y+y^{2}\right)}{(x-y)(y-z)(z-x)}\right]\)
\(=-\left[\frac{y^{3}-z^{3}+z^{3}-x^{3}+x^{3}-y^{3}}{(x-y)(y-z)(z-x)}\right]=-\frac{0}{(x-y)(y-z)(z-x)}=0 \)
\( =\frac{-\left(y^{2}+y z+z^{2}\right)}{(x-y)(z-x)}+\frac{-\left(z^{2}+z x+x^{2}\right)}{(y-z)(x-y)}+\frac{-\left(x^{2}+x y+y^{2}\right)}{(z-x)(y-z)}\)
\(=-\left[\frac{(y-z)\left(y^{2}+y z+z^{2}\right)+(z-x)\left(z^{2}+z x+x^{2}\right)+(x-y)\left(x^{2}+x y+y^{2}\right)}{(x-y)(y-z)(z-x)}\right]\)
\(=-\left[\frac{y^{3}-z^{3}+z^{3}-x^{3}+x^{3}-y^{3}}{(x-y)(y-z)(z-x)}\right]=-\frac{0}{(x-y)(y-z)(z-x)}=0 \)
Koshe dekhi 15 WBBSE Class 8 || বীজগাণিতিক সংখ্যামালার সরলীকরণ কষে দেখি 15 || WBBSE Class-8 (VIII) Koshe dekhi 15 Somadhan || West Bengal Board Class 8 Book Math Solution || অধ্যায় ১৫ পশ্চিমবঙ্গ মধ্যশিক্ষা পর্ষদ
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