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

Integrate the function in (x² + 1) log x.

Integrate the function in e^x (sinx + cosx).

Integrate the function in `(xe^x)/(1+x)^2`.

Integrate the function in `e^x (1 + sin x)/(1+cos x)`.

Integrate the function in `e^x (1/x - 1/x^2)`.

Integrate the function in `((x- 3)e^x)/(x - 1)^3`.

Integrate the function in e^2x sin x.

Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.

`intx^2 e^(x^3) dx` equals: 

`int e^x sec x (1 +   tan x) dx` equals:

Find the projection of \[\vec{b} + \vec{c}  \text { on }\vec{a}\]  where \[\vec{a} = 2 \hat{i} - 2 \hat{j} + \hat{k} , \vec{b} = \hat{i} + 2 \hat{j} - 2 \hat{k} \text{ and } \vec{c} = 2 \hat{i} - \hat{j} + 4 \hat{k} .\]

If \[\vec{a} = 5 \hat{i} - \hat{j} - 3 \hat{k} \text{ and } \vec{b} = \hat{i} + 3 \hat{j} - 5 \hat{k} ,\] then show that the vectors \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} \] are orthogonal.

A unit vector \[\vec{a}\] makes angles \[\frac{\pi}{4}\text{ and }\frac{\pi}{3}\] with \[\hat{i}\] and \[\hat{j}\]  respectively and an acute angle θ with \[\hat{k}\] .  Find the angle θ and components of \[\vec{a}\] .

If two vectors \[\vec{a} \text{ and } \vec{b}\] are such that \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 1 \text{ and } \vec{a} \cdot \vec{b} = 1,\]  then find the value of \[\left( 3 \vec{a} - 5 \vec{b} \right) \cdot \left( 2 \vec{a} + 7 \vec{b} \right) .\] 

If \[\vec{a}\] is a unit vector, then find \[\left| \vec{x} \right|\]  in each of the following. 

\[\left( \vec{x} - \vec{a} \right) \cdot \left( \vec{x} + \vec{a} \right) = 8\] 

If \[\vec{a}\] is a unit vector, then find \[\left| \vec{x} \right|\]  in each of the following. 

\[\left( \vec{x} - \vec{a} \right) \cdot \left( \vec{x} + \vec{a} \right) = 12\] 

Find \[\left| \vec{a} \right| \text{ and } \left| \vec{b} \right|\] if 

\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 12 \text{ and } \left| \vec{a} \right| = 2\left| \vec{b} \right|\]

Find  \[\left| \vec{a} \right| \text{ and } \left| \vec{b} \right|\] if 

\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 8 \text{ and } \left| \vec{a} \right| = 8\left| \vec{b} \right|\]

Find \[\left| \vec{a} \right| and \left| \vec{b} \right|\] if 

\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 3\text{  and } \left| \vec{a} \right| = 2\left| \vec{b} \right|\]

Find \[\left| \vec{a} - \vec{b} \right|\] if 

\[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} \cdot \vec{b} = 8\]