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

Prove that the line through A(0, – 1, – 1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(– 4, 4, 4).

Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0.

The plane ax + by = 0 is rotated about its line of intersection with the plane z = 0 through an angle α. Prove that the equation of the plane in its new position is `"a"x + "b"y +- (sqrt("a"^2 + "b"^2) tan alpha)`z = 0.

Find the equation of the plane through the intersection of the planes `vec"r" * (hat"i" + 3hat"j") - 6` = 0 and `vec"r" * (3hat"i" - hat"j" - 4hat"k")` = 0, whose perpendicular distance from origin is unity.

Solve the following matrix equation for x: `[x 1] [[1,0],[−2,0]]=0`

If A =  `([cos alpha, sin alpha],[-sinalpha, cos alpha])` , find α satisfying 0 < α < `pi/r`when `A+A^T=sqrt2I_2` where A^T is transpose of A.

Find the angle between the vectors `hati-hatj and hatj-hatk`

If `A=[[1,2,2],[2,1,2],[2,2,1]]` ,then show that `A^2-4A-5I=0` and hence find A^-1.

If `A=([2,0,1],[2,1,3],[1,-1,0])` find A² - 5A + 4I and hence find a matrix X such that  A² - 5A + 4I + X = 0

If `veca and vecb` are perpendicular vectors, `|veca+vecb| = 13 and |veca| = 5` ,find the value of `|vecb|.`

Let `A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]`

Find  A + B

Compute the following:

`[(a,b),(-b, a)] + [(a,b),(b,a)]`

Compute the following:

`[(a^2+b^2, b^2+c^2),(a^2+c^2, a^2+b^2)] + [(2ab , 2bc),(-2ac, -2ab)]`

Compute the following: 

`[(-1,4, -6),(8,5,16),(2,8,5)] + [(12,7,6),(8,0,5),(3,2,4)]`

Compute the following:

`[(cos^2x, sin^2 x),(sin^2 x ,cos^2 x)]+[(sin^2 x, cos^2 x), (cos^2 x, sin^2 x)]`

If F(x) = `[(cosx, -sinx,0), (sinx, cosx, 0),(0,0,1)]`  show that F(x)F(y) = F(x + y)

Show that the vectors `2hati - 3hatj + 4hatk` and `-4hati + 6hatj -  8hatk` are collinear.

Evaluate the product `(3veca - 5vecb).(2veca + 7vecb)`.

Find `|vecx|`, if for a unit vector veca , `(vecx -  veca).(vecx + veca) = 12`.

If  `veca.veca = 0` and `veca . vecb = 0,` then what can be concluded about the vector `vecb`?