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

Find the distance of the point (2, 12, 5) from the point of intersection of the line 

`vecr=2hati-4hat+2hatk+lambda(3hati+4hatj+2hatk) `

Using properties of determinants, prove that 

`|[b+c,c+a,a+b],[q+r,r+p,p+q],[y+z,z+x,x+y]|=2|[a,b,c],[p,q,r],[x,y,z]|`

if `A = [(1,2,-3),(5,0,2),(1,-1,1)], B = [(3,-1,2),(4,2,5),(2,0,3)] and C = [(4,1,2),(0,3,2),(1,-2,3)]` then compute (A + B) and (B - C). Also verify that A + (B -C) = (A + B) - C.

If ` A = [(2/3, 1, 5/3), (1/3, 2/3, 4/3),(7/3, 2, 2/3)]` and `B = [(2/5, 3/5,1),(1/5, 2/5, 4/5), (7/5,6/5, 2/5)]` then compute 3A - 5B.

Simplify, `cos theta[(cos theta, sintheta),(-sin theta, cos theta)] + sin theta [(sin theta, -cos theta), (cos theta, sin theta)]`

Show that `[(5, -1),(6,7)][(2,1),(3,4)] != [(2,1),(3,4)][(5,-1),(6,7)]`

Show that `[(1,2,3),(0,1,0),(1,1,0)][(-1,1,0),(0,-1,1),(2,3,4)]!=[(-1,1,0),(0,-1,1),(2,3,4)][(1,2,3),(0,1,0),(1,1,0)]`

Find `A^2 - 5A + 6I if A = [(2,0,1),(2,1,3),(1,-1,0)]`

Using the property of determinants and without expanding, prove that:

`|(a-b,b-c,c-a),(b-c,c-a,a-b),(a-a,a-b,b-c)| = 0`

Using the property of determinants and without expanding, prove that:

`|(2,7,65),(3,8,75),(5,9,86)| = 0`

Using the property of determinants and without expanding, prove that:

`|(1, bc, a(b+c)),(1, ca, b(c+a)),(1, ab, c(a+b))| = 0`

Using the property of determinants and without expanding, prove that:

`|(b+c, q+r, y+z),(c+a, r+p, z +x),(a+b, p+q, x + y )| = 2|(a,p,x),(b,q,y),(c, r,z)|`

By using properties of determinants, show that:

`|(0,a, -b),(-a,0, -c),(b, c,0)| = 0`

By using properties of determinants, show that:

`|(-a^2, ab, ac),(ba, -b^2, bc),(ca,cb, -c^2)| = 4a^2b^2c^2`

By using properties of determinants, show that:

`|(1,a,a^2),(1,b,b^2),(1,c,c^2)| = (a - b)(b-c)(c-a)`

By using properties of determinants, show that:

`|(1,1,1),(a,b,c),(a^3, b^3,c^3)|` = (a-b)(b-c)(c-a)(a+b+c)

By using properties of determinants, show that:

`|(x,x^2,yz),(y,y^2,zx),(z,z^2,xy)| = (x-y)(y-z)(z-x)(xy+yz+zx)`

By using properties of determinants, show that:

`|(x+4,2x,2x),(2x,x+4,2x),(2x , 2x, x+4)| = (5x + 4)(4-x)^2`

By using properties of determinants, show that:

`|(y+k,y, y),(y, y+k, y),(y, y, y+k)| = k^2(3y + k)`

By using properties of determinants, show that:

`|(a-b-c, 2a,2a),(2b, b-c-a,2b),(2c,2c, c-a-b)| = (a + b + c)^2`