Explanation:वजाबाकी/बेरीज छेद समान नसतांना
\[1)\: \frac{x+y}{x-y}-\frac{x-y}{x+y}=?\]
Explanation: \[\frac{x+y}{x-y}-\frac{x-y}{x+y}\](तिरपा गुणाकार)\[\frac{\left ( x+y \right )^{2}-\left ( x-y \right )^{2}}{\left (x-y \right )\left ( x+y \right )}\\ =\frac{4xy}{x^{2}-y^{2}}\left [ \frac{\left ( a+b \right )^{2}-\left ( a-b \right )^{2}=4ab}{\left ( a+b \right )\left ( a-b \right )=a^{2}-b^{2}} \right ]\]
\[2)\: 1-x+\frac{x^{2}}{1+x}\]
Explanation: \[\frac{1-x}{1}+\frac{x^{2}}{1+x}\] (तिरपा गुणाकार) \[\frac{x^{2}+\left ( 1^{2}-x^{2} \right )}{1+x}=\frac{x^{2}+1^{2}-x^{2}}{1+x}=\frac{1}{1+x}\]
\[3) \frac{m+3}{m^{2}+4m+3}+\frac{m+2}{m^{2}+m-2}\]
Explanation:\[\frac{m+3}{\left ( m+1 \right )\left (m+3 \right )}+\frac{m+2}{\left ( m+2 \right )\left ( m-1 \right )}\\ \frac{1}{m+1}+\frac{1}{m-1}\] (तिरपा गुणाकार)\[\frac{m-1+m+1}{\left ( m+1 \right )\left ( m-1 \right )}=\frac{2m}{m^{2}-1^{2}}\\ =\frac{2m}{m^{2}-1}\\ = (m+1)(m-1) = m^{2}-1^{2}\\ = (a+b)(a-b) = a^{2}-b^{2}\]
\[4) \: \frac{x-3}{x^{2}-4x+3}-\frac{x^{2}-x+1}{x^{3}+1}\]
Explanation: \[\: \frac{x-3}{x^{2}-4x+3}-\frac{x^{2}-x+1}{x^{3}+1}\\ \frac{x-3}{\left ( x-3 \right )\left ( x-1 \right )}-\frac{x^{2}-x+1}{\left ( x+1 \right )\left ( x^{2}-x+1 \right )}\\\rightarrow (a^{3}+b^{3} =(a+b)(a2-ab+b^{2})\\\frac{1}{x-1}+\frac{1}{x+1}\](तिरपा गुणाकार)\[\frac{x+1-\left ( x-1 \right )}{\left ( x-1 \right )\left ( x+1 \right )}=\frac{x+1-x+1}{x^{2}-1^{2}}=\frac{2}{x^{2}-1}\]
\[5)\: \frac{x}{x+y}+\frac{y}{x-y}=?\]
Explanation: \[\frac{x\left ( x-y \right )+y\left ( x+y \right )}{\left ( x+y \right )\left ( x-y \right )}\\ =\frac{x^{2}-xy+xy+y^{2}}{x^{2}+y^{2}}\\ =\frac{x^{2}+y^{2}}{x^{2}-y^{2}}\]
\[6)\: \frac{x^{2}-9}{x^{2}+x-12}-\frac{x^{2}+8x+15}{x^{2}+9x+20}\]
Explanation: \[\frac{x^{2}-9}{x^{2}+x-12}-\frac{x^{2}+8x+15}{x^{2}+9x+20}\\ \frac{\left ( x-3 \right )\left ( x+3 \right )}{\left ( x+4 \right )\left ( x-3 \right )}-\frac{\left ( x+5 \right )\left ( x+3 \right )}{\left ( x+5 \right )\left ( x+4 \right )}\\ \frac{x+3}{x+4}-\frac{x+3}{x+4}\: \: \: \:\rightarrow (a-b = 0)\\ = 0\]
\[7)\: \: 2.5x = 1.5y \: \: , \frac{x+y}{x-y} = ?\]
Explanation: \[\frac{x}{y} = \frac{1.5}{2.5}\\ \frac{1.5+2.5}{1.5-2.5}=\frac{4}{1}\\ = -4\]
\[8)\: \: \frac{x^2-5x+3}{(x+2)(x-1)}\]
Explanation: ही राशी कोणत्या किंमतीसाठी होईल.
उत्तर:- अर्थहीन बैजीक राशी = बैजीक राशीची किंमत शुन्य होणे होय.
उदा. x = -2 किंवा x = 1 ठेवल्यास
x = -2 ठेवून \[\frac{-2^{2}-2\times 5+3}{\left ( 2-2 \right )\left ( 2-1 \right )}=\frac{4-10+3}{0\times 1}=\frac{-3}{0}=0\]म्हणजे x = -2 असतांना अर्थहीन होते.
9) \[\frac{(2x^2-3x+4)}{x-5} -\frac{x^2+3x-1}{x-5}\]
Explanation:\[\frac{2x^2-3x+4-(x^2+3x-1)}{x-5}=\frac{2x^2-3x+4-x^2-3x+1}{x-5}\\ \frac{x^2-6x+5}{x-5}=\frac{(x-5)(x-1)}{(x-5)}= x-1\]
10)\[\frac{a^2-6a+8}{a^2-3a+2} = ?\]
Explanation: \[\frac{a^2-6a+8}{a^2-3a+2} = \frac{(a-4)(a-2)}{(a-2)(a-1)}=\frac{a-4}{a-1}\]