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

Evaluate the following:

`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`

`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.

`int_0^(pi/2) sqrt(1 - sin2x)  "d"x` is equal to ______.

`int_0^(pi/2)  cos x "e"^(sinx)  "d"x` is equal to ______.

If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.

The value of λ for which the two vectors `2hati - hatj + 2hatk` and `3hati + λhatj + hatk` are perpendicular is ______.

The number of vectors of unit length perpendicular to the vectors `vec"a" = 2hat"i" + hat"j" + 2hat"k"` and `vec"b" = hat"j" + hat"k"` is ______.

Find the distance of the point whose position vector is `(2hat"i" + hat"j" - hat"k")` from the plane `vec"r" * (hat"i" - 2hat"j" + 4hat"k")` = 9

Find the distance of the point (– 2, 4, – 5) from the line `(x + 3)/3 = (y - 4)/5 = (z + 8)/6`

Find the coordinates of the point where the line through (3, – 4, – 5) and (2, –3, 1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0)

A plane meets the co-ordinates axis in A, B, C such that the centroid of the ∆ABC is the point (α, β, γ). Show that the equation of the plane is `x/alpha + y/beta + z/ϒ` = 3

The distance of a point P(a, b, c) from x-axis is ______.

Find the distance of a point (2, 4, –1) from the line `(x + 5)/1 = (y + 3)/4 = (z - 6)/(-9)`

Distance of the point (α, β, γ) from y-axis is ____________.

The distance of the plane `vec"r" *(2/7hat"i" + 3/4hat"j" - 6/7hat"k")` = 1 from the origin is ______.

Find the foot of the perpendicular from the point (1, 2, 0) upon the plane x – 3y + 2z = 9. Hence, find the distance of the point (1, 2, 0) from the given plane.

Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`

The equations of motion of a rocket are:
x = 2t,y = –4t, z = 4t, where the time t is given in seconds, and the coordinates of a ‘moving point in km. What is the path of the rocket? At what distances will the rocket be from the starting point O(0, 0, 0) and from the following line in 10 seconds? `vecr = 20hati - 10hatj + 40hatk + μ(10hati - 20hatj + 10hatk)`

Let `veca = hati + hatj, vecb = hati - hatj` and `vecc = hati + hatj + hatk`. If `hatn` is a unit vector such that `veca.hatn` = 0 and `vecb.hatn` = 0, then find `|vecc.hatn|`.

If `veca` and `vecb` are unit vectors inclined at an angle 30° to each other, then find the area of the parallelogram with `(veca + 3vecb)` and `(3veca + vecb)` as adjacent sides.