Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions

By using properties of determinants, show that:

`|(a^2+1, ab, ac),(ab, b^2+1, bc),(ca, cb, c^2+1)| = 1+a^2+b^2+c^2`

Without expanding the determinant, prove that

`|(a, a^2,bc),(b,b^2, ca),(c, c^2,ab)| = |(1, a^2, a^3),(1, b^2, b^3),(1, c^2, c^3)|`

Evaluate `|(x, y, x+y),(y, x+y, x),(x+y, x, y)|`

Evaluate `|(1,x,y),(1,x+y,y),(1,x,x+y)|`

Using properties of determinants, prove that:

`|(alpha, alpha^2,beta+gamma),(beta, beta^2, gamma+alpha),(gamma, gamma^2, alpha+beta)|` =  (β – γ) (γ – α) (α – β) (α + β + γ)

Using properties of determinants, prove that:

`|(x, x^2, 1+px^3),(y, y^2, 1+py^3),(z, z^2, 1+pz^2)|` = (1 + pxyz) (x – y) (y – z) (z – x), where p is any scalar.

Using properties of determinants, prove that:

`|(3a, -a+b, -a+c),(-b+a, 3b, -b+c),(-c+a, -c+b, 3c)|`= 3(a + b + c) (ab + bc + ca)

Using properties of determinants, prove that:

`|(1, 1+p, 1+p+q),(2, 3+2p, 4+3p+2q),(3,6+3p,10+6p+3q)| =  1`                 

Using properties of determinants, prove that

`|(sin alpha, cos alpha, cos(alpha+ delta)),(sin beta, cos beta, cos (beta + delta)),(sin gamma, cos gamma, cos (gamma+ delta))| = 0`

Differentiate the following w.r.t. x:

`e^x/sinx`

Differentiate the following w.r.t. x: 

`e^(sin^(-1) x)`

Differentiate the following w.r.t. x:

`e^(x^3)`

Differentiate the following w.r.t. x: 

sin (tan^–1 e^–x)

Differentiate the following w.r.t. x:

log (cos e^x)

Differentiate the following w.r.t. x:

`e^x + e^(x^2) + "..." + e^(x^5)`

Differentiate the following w.r.t. x:

`sqrt(e^(sqrtx))`, x > 0

Differentiate the following w.r.t. x:

log (log x), x > 1

Differentiate the following w.r.t. x: 

`cos x/log x`, x > 0

Differentiate the following w.r.t. x:

cos (log x + e^x), x > 0

Differentiate the function with respect to x:

(log x)^{log x}, x > 1