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

Prove that in any triangle ABC, cos A = `("b"^2 + "c"^2 - "a"^2)/(2"bc")`, where a, b, c are the magnitudes of the sides opposite to the vertices A, B, C, respectively.

The vector in the direction of the vector `hat"i" - 2hat"j" + 2hat"k"` that has magnitude 9 is ______.

The magnitude of the vector `6hati - 2hatj + 3hatk` is ______.

Read the following passage and answer the questions given below:

1. What is the magnitude of the force of Team A ?
2. Which team will win the game?
3. Find the magnitude of the resultant force exerted by the teams.
   OR
   In what direction is the ring getting pulled?

Evaluate : `int_0^3dx/(9+x^2)`

If a unit vector `veca` makes angles `pi/3` with `hati,pi/4` with `hatj` and acute angles θ with ` hatk,` then find the value of θ.

Write the value of `vec a .(vecb xxveca)`

If `veca=hati+2hatj-hatk, vecb=2hati+hatj+hatk and vecc=5hati-4hatj+3hatk` then find the value of `(veca+vecb).vec c`

If `veca=2hati+hatj+3hatk and vecb=3hati+5hatj-2hatk` ,then find ` |veca xx vecb|`

Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar.

A line passing through the point A with position vector `veca=4hati+2hatj+2hatk` is parallel to the vector `vecb=2hati+3hatj+6hatk` . Find the length of the perpendicular drawn on this line from a point P with vector `vecr_1=hati+2hatj+3hatk`

if `|vecaxxvecb|^2+|veca.vecb|^2=400 ` and `|vec a| = 5` , then write the value of `|vecb|`

Integrate the following w.r.t. x `(x^3-3x+1)/sqrt(1-x^2)`

If `vecr=xhati+yhatj+zhatk` ,find `(vecrxxhati).(vecrxxhatj)+xy`

Find `veca.(vecbxxvecc), " if " veca=2hati+hatj+3hatk, vecb=-hati+2hatj+hatk  " and " vecc=3hati+hatj+2hatk`

Find the equation of the plane through the intersection of the planes 3x – y + 2z – 4 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).

Find the vector equation of the plane passing through the intersection of the planes `vecr.(2hati + 2hatj - 3hatk) = 7, vecr.(2hati + 5hatj + 3hatk) = 9` and through the point (2, 1, 3)

Find the equation of the plane passing through the line of intersection of the planes `vecr.(hati + hatj + hatk) = 1` and `vecr.(2hati + 3hatj -hatk) + 4 = 0` and parallel to x-axis.

Find the equation of the plane which contains the line of intersection of the planes `vecrr.(hati + 2hatj + 3hatk) - 4 = 0, vecr.(2hati + htj - hatk) + 5 = 0`,  and which is perpendicular to the plane `vecr.(5hati + 3hatj - 6hatk) + 8 = 0`.

Using vectors find the area of triangle ABC with vertices A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1).