Commerce (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

Integrate the rational function:

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

Integrate the rational function:

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

Integrate the rational function:

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

Integrate the rational function:

`1/(x^4 - 1)`

Integrate the rational function:

`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by x^{n − 1} and put xⁿ = t]

Integrate the rational function:

`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]

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

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