Question

Assertion :If $\Delta \left(x\right)=\left|\begin{array}{cc}f\left(x\right)& g\left(x\right)\\ m_1& m_2\end{array}\right|$ then
$\int \Delta \left(x\right)=\left|\begin{array}{cc}\int f\left(x\right)dx& \int g\left(x\right)dx\\ m_1& m_2\end{array}\right|$ Reason: $\int \lambda f\left(x\right)dx=\lambda \int f\left(x\right)dx$

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A),
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A)is true but (R} is false,
• D. (A)is false but (R} is true.

Answer 1

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A),

$\Delta \left(x\right)=\left|\begin{array}{cc}f\left(x\right)& g\left(x\right)\\ m_1& m_2\end{array}\right|$
$=m_{2}f\left(x\right)-m_{1}g\left(x\right)$
$\therefore \int \Delta \left(x\right)dx=\int \left\{m_{2}f\right(x)-m_{1}g(x\left)\right\}dx$
$=m_2\int f\left(x\right)dx-m_1\int g\left(x\right)dx$ $=\left|\begin{array}{cc}\int f\left(x\right)dx& \int g\left(x\right)dx\\ m_1& m_2\end{array}\right|$
Also, $\int \lambda f\left(x\right)dx=\lambda \int f\left(x\right)dx$
$\therefore$Assertion A) and Reason (R) are individually true and Reason (R) is correct explanation of Assertion (A).