Prove that `|{:(a+bx,c+dx,p+qx),(ax+b,cx+d,px+q),(mu,vartheta,omega):}|=(1-x^2)|{:(a,c,p),(b,d,q),(mu,vartheta,omega):}|`

Prove that: |{:(a, b, c), (x, y, z), (p, q, r):}|=|{:(y, b, q), (x, a, p), (z, c, r):}|

Using the property of determinants andd without expanding in following exercises 1 to 7 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):}|

Let Delta=|{:(a,p,x),(b,q,y),(c,r,z):}|=16" then "Delta_1=|{:(p+x,a+x,a+p),(q+y,b+y,b+q),(r+z,c+z,c+r):}|=32

If f(x) =a+bx+cx^(2) and alpha,beta "and" gamma are the roots of the equation x^(3)=1 , then |{:(a,b,c),(b,c,a),(c,a,b):}| is equal to

If y =(ax +b)/( cx + d) , then prove that 2y_1y_3=3(y_2)^2

If g(x)=(f(x))/((x-a)(x-b)(x-c)),w h e r ef(x) is a polynomial of degree < 3, then prove that (dg(x))/(dx)=|{:(1,a, f(a)(x-a)^(-2)),(1,b,f(b)(x-b)^(-2)), (1,c,f(c)(x-c)^(-2)):}|/|{:(a^2,a,1),(b^2,b,1),(c^2,c,1):}|

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax+b)^(n)(cx+d)^(m)

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (px^(2)+qx+r)/(ax+b)

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax^(2)+sinx)(p+qcosx)