MHT CET20255 May 2025Evening ShiftMathematicsIndefinite IntegrationActual
If A = [ array lll a & 0 & 0 0 & ~b & 0 0 & 0 & c array ] where a =7^ x , b =7^ 7^x , c =7^ 7^ 7^x then |A| d x , (Where |A| is the determinant of the matrix A ) is equal to
Options
- A7^ 7^x ( 7)^3 +k, where k is constant of integration
- B7^ 7^ 7^x 7 + k , where k is constant of integration
- C7^ 7^ 7^x ( 7)^3 + k , where k is constant of integration
- D7^ 7^ 7^x ( 7)^3+ k , where k is constant of integration
Correct answer
C. 7^ 7^ 7^x ( 7)^3 + k , where k is constant of integration
Step-by-step solution
Determinant Evaluation: The determinant of the diagonal matrix A is the product of its diagonal elements: |A| = a b c = 7^x 7^ 7^x 7^ 7^ 7^x . Integral Setup: The required integral is |A| dx = 7^x 7^ 7^x 7^ 7^ 7^x dx . Substitution: Let u = 7^ 7^ 7^x . Differentiating using the chain rule: du dx = 7^ 7^ 7^x 7 d dx (7^ 7^x ) = 7^ 7^ 7^x 7 (7^ 7^x 7 7^x 7) = 7^ 7^ 7^x 7^ 7^x 7^x ( 7)^3 . This gives dx = du u 7^ 7^x 7^x ( 7)^3 . Integration: Substituting into the integral yields u du u ( 7)^3 = 1 ( 7)^3 du = u ( 7)^3