Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions

Integrate the rational function:

`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`

Integrate the rational function:

`(2x)/((x^2 + 1)(x^2 + 3))`

Integrate the rational function:

`1/(x(x^4 - 1))`

Integrate the rational function:

`1/(e^x -1)`[Hint: Put e^x = t]

`int (xdx)/((x - 1)(x - 2))` equals:

`int (dx)/(x(x^2 + 1))` equals:

Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`

Consider f: `R_+ -> [-5, oo]` given by `f(x) = 9x^2 + 6x - 5`. Show that f is invertible with `f^(-1) (y) ((sqrt(y + 6)-1)/3)`

Hence Find

1) `f^(-1)(10)`

2) y if `f^(-1) (y) = 4/3`

where R₊ is the set of all non-negative real numbers.

Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`

Evaluate the following determinant:

\[\begin{vmatrix}x & - 7 \\ x & 5x + 1\end{vmatrix}\]

Evaluate the following determinant:

\[\begin{vmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{vmatrix}\]

Evaluate the following determinant:

\[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]

Evaluate the following determinant:

\[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix}\]

Evaluate

\[\begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}^2 .\]

Show that

\[\begin{vmatrix}\sin 10^\circ & - \cos 10^\circ \\ \sin 80^\circ & \cos 80^\circ\end{vmatrix} = 1\]

Evaluate

\[\begin{vmatrix}2 & 3 & - 5 \\ 7 & 1 & - 2 \\ - 3 & 4 & 1\end{vmatrix}\] by two methods.

Evaluate
\[∆ = \begin{vmatrix}0 & \sin \alpha & - \cos \alpha \\ - \sin \alpha & 0 & \sin \beta \\ \cos \alpha & - \sin \beta & 0\end{vmatrix}\]

\[∆ = \begin{vmatrix}\cos \alpha \cos \beta & \cos \alpha \sin \beta & - \sin \alpha \\ - \sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{vmatrix}\]

If \[A = \begin{bmatrix}2 & 5 \\ 2 & 1\end{bmatrix} \text{ and } B = \begin{bmatrix}4 & - 3 \\ 2 & 5\end{bmatrix}\] , verify that |AB| = |A| |B|.

If A \[\begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4\end{bmatrix}\] , then show that |3 A| = 27 |A|.