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Mathematical reasoninghelps in solving many mathematical puzzle,this can increase our capicity of solving.The operations involved are addition,subtraction,multiplication and division.The puzzles can be solved easily and reasoning can be given based..

Mathematical expressions are the combination of mathematical symbols or numbers or both. Every expression contains important elements called variables. Expression generally does not contain equality sign, if it has an equality sign then it represented as an equation. So, In th..

> = . * M [log | sin (x - a) | - log | sin (x - fc) [ ] + c = log + * sm {a - b) sin (a - b) sm (x - b) 11. ^ = f~-dx [For alternative solution see Q. 3 in solved Ex. 21 J1 + cot x J sin x + cos x x 1 = f si" x (cos x - sin x) ^ _ fsin 2x -2sin 2 x ^ J (cos x + sin x) (cos x - s..

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The standard form of the linear programming problem (LPP) is used to develop the procedure for solving a general programming problem. A general LPP is of the form Maximize (or minimize) Z = c 1 x 1 + c 2 x 2 + + c n x n (1) Subject to a 11 x 1 + a 12 x 2 + . . . . ..

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Finally, lets take a look at comparing two ratios In order to compare two ratios we first need to convert the ratios into fractions with same denominator ( very important ) For example, consider the ratios 2 : 7 and 1 : 3 Here we have to compare the fractions $$ \frac{2}{7}$$ and $$ \frac{1}{3}..

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Solution: Let n(M) = number of students who secured first class marks in Mathematics n(E) = number of students who secured first class marks in English n(M) = 70 n(M $\cup$ E) = 120 n(M $\cap$ E) = 31 Using n(M $\cup$ E) = n(M) + n(E) - n(M $\cap$ E) ..