Without expanding prove that , `|{:(a+bx,c+dx,p+qx),(ax+b,cx+d,px+q),(u,v,w):}|=(1-x^2)|{:(a,c,p),(b,d,q),(u,v,w):}|`

Using the property of determinants and without expanding {:|( b+c,q+r,y+z),( c+a,r+p,z+x),( a+b,p+q,x+y) |:}=2 {:|(a,q,x),( b,q,y),(c,r,z)|:}

If u=ax+by+cz , v=ay+bz+cx , w=az+bx+cy , then the value of |{:(a,b,c),(b,c,a),(c,a,b):}|xx|{:(x,y,z),(y,z,x),(z,x,y):}| is

Consider int(x^(3)+3x^(2)+2x+1)/(sqrt(x^(2)+x+1))dx =(ax^(2)+bx+c)sqrt(x^(2)+x+1)+lambda int(dx)/(sqrt(x^(2)+x+1)) Now, match the following lists and then choose the correct code. Codes: {:(,a,b,c,d),((1),q,p,s,r),((2),s,p,q,r),((3),r,q,p,s),((4),q,s,p,r):}

Solve the equations : (b-ax)/(bx-a)=(d-cx)/(dx-c)

Prove that : (v) p^(log_(x)q) = q^(log_(x)p)

Without expanding at any stage, prove that the value of each of the following determinants is zero. (1) |[0,p-q,p-r],[q-p,0,q-r],[r-p,r-q,0]| (2) |[41,1,5],[79,7,9],[29,5,3]| (3) |[1,w,w^2],[w,w^2,1],[w^2,1,w]| , where w is cube root of unity

If P={a,b,c,d,e} and Q={a,e,i,o,u} , prove that P subPcupQ

Match the columns (1) a-s, b-t, c-r, d-p (2) a-t, b-q, c-r, d-p (3) a-t, b-r, c-q, d-p (4) a-t, b-r, c-p, d-q

Long-answer type questions (L.A.) (ax+b)/(a+bx)=(cx+d)/(c+dx)(aneb,cned,xne-(a)/(b),-(c)/(d))

If P={a,b,c,d,e} and Q={a,e,i,o,u} , prove that PcapQ subP.