This chemistry question involves key chemical concepts and calculations. The detailed solution below walks through each step, from identifying the reaction type to computing the final answer.
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(1) (ক) Step 1: ^4 x = (^2 x)^2 = ( (1 - 2x)/(2) )^2 ^4 x = ( (1 - 2x)/(2) )^2 = (1 - 2 2x + ^2 2x)/(4) Step 2: ^2 2x = (1 + 4x)/(2) ^4 x = (1 - 2 2x + 1 + 4x)/(2)4 = (1 - 2 2x + 1)/(2) + ( 4x)/(2)4 = (3)/(2) - 2 2x + ( 4x)/(2)4 = (3)/(8) - (1)/(2) 2x + (1)/(8) 4x Step 3: Integrate term by term from 0 to ()/(2) _0^/2 ^4 x \, dx = _0^/2 ( (3)/(8) - (1)/(2) 2x + (1)/(8) 4x ) dx = [ (3)/(8) x - (1)/(4) 2x + (1)/(32) 4x ]_0^/2 = (3)/(8) · ()/(2) - 0 = (3)/(16) ** (3)/(16) (1) (খ) Step 1: ^5 x = ^4 x x = (1 - ^2 x)^2 x Let u = x, du = x \, dx ^5 x \, dx = (1 - u^2)^2 \, du Step 2: Expand (1 - u^2)^2 = 1 - 2u^2 + u^4 (1 - 2u^2 + u^4) \, du = u - (2)/(3) u^3 + (1)/(5) u^5 + C Step 3: Substitute back x - (2)/(3) ^3 x + (1)/(5) ^5 x + C ** x - (2)/(3) ^3 x + (1)/(5) ^5 x + C (1) (গ) Step 1: Complete the square in denominator x^2 + 4x + 3 = (x+2)^2 - 4 + 3 = (x+2)^2 - 1 (dx)/(sqrt((x+2)^2 - 1)) Step 2: Let u = x+2, du = dx (du)/(sqrt(u^2 - 1)) = ^-1 u + C or |u + sqrt(u^2 - 1)| + C ** | x + 2 + sqrt(x^2) + 4x + 3 | + C (2) (ক) (i) Step 1: y = ((x + a))/((x - a)) Use quotient rule: (dy)/(dx) = ( u' v - u v' )/(v^2) where u = (x+a), v = (x-a) u' = (x+a), v' = (x-a) Step 2: Numerator: (x+a) (x-a) - (x+a) (-(x-a)) = (x+a)(x-a) + (x+a)(x-a) = ( (x+a) - (x-a) ) = 2a Step 3: Denominator: ^2 (x-a) (dy)/(dx) = ( 2a)/(^2 (x - a)) ** ( 2a)/(^2)(x - a) (2) (ক) (ii) Step 1: y = x^ x Take : y = x x Step 2: Differentiate: (1)/(y) y' = x x + x · (1)/(x) y' = y ( x x + ( x)/(x) ) = x^ x ( x x + ( x)/(x) ) ** x^ x ( x x + ( x)/(x) ) (2) (ক) (iii) Step 1: y = ( x + x) (dy)/(dx) = (1)/( x + x) · (d)/(dx) ( x + x) Step 2: Derivative of x + x = x x + ^2 x = x ( x + x) Step 3: So (dy)/(dx) = ( x ( x + x) )/( x + x ) = x ** x (3) (ক) Step 1: A = 2 & 3 \\ 1 & 4 |A| = 2 · 4 - 3 · 1 = 8 - 3 = 5 5 (3) (খ) Assuming inverse of same A. Step 1: A^-1 = (1)/(|A|) 4 & -3 \\ -1 & 2 = (1)/(5) 4 & -3 \\ -1 & 2 (1)/(5) 4 & -3 \\ -1 & 2 (3) (গ) Assuming product of two matrices, say P = 1 & 2 \\ 3 & 4 , Q = 5 & 6 \\ 7 & 8 Step 1: PQ_11 = 1·5 + 2·7 = 5+14=19 PQ_12 = 1·6 + 2·8 = 6+16=22 PQ_21 = 3·5 + 4·7 = 15+28=43 PQ_22 = 3·6 + 4·8 = 18+32=50 19 & 22 \\ 43 & 50