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]] `

Delta=[[a+bx,c+dx,p+qxax+b,cx+d,px+qu,v,w]]=(1-x^(2))det[[a,c,pb,d,qu,c,w]]

Prove that [a b c][u v w]=|[a*u, b*u, c*u], [a*v, b*v, c*v], [a*w, b*w, c*w]|

Consider the following relations from A to B where A={u, v, w, x, y, z} and B={p, q, r, s}. 1. {(u,p),(v,p),(w,p),(x,q),(y,q),(z,q)} 2. {(u,p),(v,q),(w,r),(z,s)} 3. {(u,s),(v,r),(w,q),(u,p),(v,p),(z,q)} 4. {(u,q),(v,p),(w,s),(x,r),(y,q),(z,s)} Which of the above relations are not functions ?

Prove that d/(dx)|(u_1,v_1,w_1),(u_2,v_2,w_2),(u_3,v_3,w_3)|=|(u_1,v_1,w_1),(u_2,v_2,w_2),(u_4,v_4,w_4)| where u,v,w are functions of x and (du)/(dx)=u_1,(d^2u)/(dx^2)=u_2, etc.

(C) less than q?(12p-? 1.) The 288th term of the series a, b, b, c, c, c, d, d, d, d, e, e, e, e, e, f, f, f, f, fs (A) u (B) v (C) w

If hat u , hat v , hat w be three non-coplanar unit vectors with angles between hat u& hat v is alpha between hat v& hat w is beta and between hat w& hat u is gamma . If vec a , vec b , vec c are the unit vectors along angle bisectors of alpha,beta,gamma respectively, then prove that [ vec ax vec b vec bx vec c vec cx vec a]=1/(16)[ hat u hat v hat w]^2sec^2(alpha/2)sec^2(beta/2)sec^2(gamma/2)

Illustration Based upon ∫√ ax2+bx+c dx, ∫(px+q)√ax2+bx+c dx and ∫px2+qx+r/ax2+bx+c

Prove the following,using properties of determinants |[a+bx^(^^)2,c+dx^(^^)2,p+qx^(^^)2][ax^(^^)2+b,cx^(^^)2+d,px^(^^)2+q