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

Prove that, for any three vectors \[\vec{a} , \vec{b} , \vec{c}\] \[\left[ \vec{a} + \vec{b} , \vec{b} + \vec{c} , \vec{c} + \vec{a} \right] = 2 \left[ \vec{a} , \vec{b} , \vec{c} \right]\].

 If the line \[\frac{x - 3}{2} = \frac{y + 2}{- 1} = \frac{z + 4}{3}\]  lies in the plane  \[lx + my - z =\]   then find the value of  \[l^2 + m^2\] .

Evaluate : \[\int\frac{dx}{\sin^2 x \cos^2 x}\] .

Show that the vectors \[\vec{a,} \vec{b,} \vec{c}\] are coplanar if and only if \[\vec{a} + \vec{b}\], \[\vec{b} + \vec{c}\] and \[\vec{c} + \vec{a}\] are coplanar.

`int_0^(2a)f(x)dx`

\[\int\limits_0^4 x\sqrt{4 - x} dx\]

\[\int\limits_1^2 x\sqrt{3x - 2} dx\]

\[\int\limits_1^5 \frac{x}{\sqrt{2x - 1}} dx\]

\[\int\limits_0^1 \cos^{- 1} x dx\]

\[\int\limits_0^1 \tan^{- 1} x dx\]

\[\int\limits_0^1 \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) dx\]

\[\int\limits_0^1 \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) dx\]

\[\int\limits_0^{1/\sqrt{3}} \tan^{- 1} \left( \frac{3x - x^3}{1 - 3 x^2} \right) dx\]

\[\int\limits_0^1 \frac{1 - x}{1 + x} dx\]

\[\int\limits_0^{\pi/3} \frac{\cos x}{3 + 4 \sin x} dx\]

\[\int\limits_0^{\pi/2} \frac{\sin^2 x}{\left( 1 + \cos x \right)^2} dx\]

\[\int\limits_0^{\pi/2} \frac{\sin x}{\sqrt{1 + \cos x}} dx\]

\[\int\limits_0^{\pi/2} \frac{\cos x}{1 + \sin^2 x} dx\]

\[\int\limits_0^\pi \sin^3 x\left( 1 + 2 \cos x \right) \left( 1 + \cos x \right)^2 dx\]

\[\int\limits_0^\infty \frac{x}{\left( 1 + x \right)\left( 1 + x^2 \right)} dx\]