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

The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .

Find : 

`∫ sin(x-a)/sin(x+a)dx`

Using properties of determinants, prove that

`|[b+c , a ,a  ] ,[ b , a+c, b ] ,[c , c, a+b ]|` = 4abc 

Using properties of determinant prove that 

`|(b+c , a , a), (b , c+a, b), (c, c, a+b)|` = 4abc

Using properties of determinants, prove the following:

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

If x^y - y^x = a^b, find `(dy)/(dx)`.

If `|vec"a"| = 4, |vec"b"| = 3` and `vec"a".vec"b" = 6 sqrt(3)`, then find the value of `|vec"a" xx vec"b"|`.

If `"x" = "e"^(cos2"t")  "and"  "y" = "e"^(sin2"t")`, prove that `(d"y")/(d"x") = - ("y"log"x")/("x"log"y")`.

Solve for x : `|("a"+"x","a"-"x","a"-"x"),("a"-"x","a"+"x","a"-"x"),("a"-"x","a"-"x","a"+"x")| = 0`, using properties of determinants. 

Find the coordinates of the foot of perpendicular and perpendicular distance from the point P(4,3,2) to the plane x + 2y + 3z = 2. Also find the image of P in the plane.

The product of any matrix by the scalar ______ is the null matrix.

The value of `|(1, 1, 1),(""^"n""C"_1, ""^("n" + 2)"C"_1, ""^("n" + 4)"C"_1),(""^"n""C"_2, ""^("n" + 2)"C"_2, ""^("n" + 4)"C"_2)|` is 8.

Evaluate: `|(x^2 - x + 1, x - 1),(x + 1, x + 1)|`

Evaluate: `|("a" + x, y, z),(x, "a" + y, z),(x, y, "a" + z)|`

Evaluate: `|(0, xy^2, xz^2),(x^2y, 0, yz^2),(x^2z, zy^2, 0)|`

Evaluate: `|(3x, -x + y, -x + z),(x - y, 3y, z - y),(x - z, y - z, 3z)|`

Evaluate: `|(x + 4, x, x),(x, x + 4, x),(x, x, x + 4)|`

Evaluate: `|("a" - "b" - "c", 2"a", 2"a"),(2"b", "b" - "c" - "a", 2"b"),(2"c", 2"c", "c" - "a" - "b")|`

Prove that: `|(y^2z^2, yz, y + z),(z^2x^2, zx, z + x),(x^2y^2, xy, x + y)|` = 0

Prove that: `|(y + z, z, y),(z, z + x, x),(y, x, x + y)|` = 4xyz