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

Find a vector in the direction of vector \[2 \hat{i} - 3 \hat{j} + 6 \hat{k}\] which has magnitude 21 units.

If in a ∆ABC, A = (0, 0), B = (3, 3 \[\sqrt{3}\]), C = (−3\[\sqrt{3}\], 3), then the vector of magnitude 2 \[\sqrt{2}\] units directed along AO, where O is the circumcentre of ∆ABC is 

\[\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx\]

Evaluate `int_1^4 ( 1+ x +e^(2x)) dx` as limit of sums.

Evaluate `int_(-1)^2 (7x - 5)"d"x` as a limit of sums

If f and g are continuous functions in [0, 1] satisfying f(x) = f(a – x) and g(x) + g(a – x) = a, then `int_0^"a" "f"(x) * "g"(x)"d"x` is equal to ______.

Evaluate the following as limit of sum:

`int _0^2 (x^2 + 3) "d"x`

Evaluate the following as limit of sum:

`int_0^2 "e"^x "d"x`

Evaluate the following:

`int_0^2 ("d"x)/("e"^x + "e"^-x)`

Evaluate the following:

`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`

Evaluate the following:

`int_0^1 (x"d"x)/sqrt(1 + x^2)`

Evaluate the following:

`int_0^pi x sin x cos^2x "d"x`

Evaluate the following:

`int_0^(1/2) ("d"x)/((1 + x^2)sqrt(1 - x^2))`  (Hint: Let x = sin θ)

Evaluate the following:

`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2)  "d"x`

The value of `int_(-pi)^pi sin^3x cos^2x  "d"x` is ______.

Find all vectors of magnitude `10sqrt(3)` that are perpendicular to the plane of `hat"i" + 2hat"j" + hat"k"` and `-hat"i" + 3hat"j" + 4hat"k"`

Prove that in a ∆ABC,  `sin"A"/"a" = sin"B"/"b" = sin"C"/"c"`, where a, b, c represent the magnitudes of the sides opposite to vertices A, B, C, respectively.

The magnitude of the vector `6hat"i" + 2hat"j" + 3hat"k"` is ______.

A vector `vec"r"` is inclined at equal angles to the three axes. If the magnitude of `vec"r"` is `2sqrt(3)` units, find `vec"r"`.

Find a vector of magnitude 6, which is perpendicular to both the vectors `2hat"i" - hat"j" + 2hat"k"` and `4hat"i" - hat"j" + 3hat"k"`.