Show that: `|[1 ,1+p,1+p+q],[2, 3+2p,1+3p+2q],[3, 6+3p,1+6p+3q]|=1` .

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 the following: |[1,1+p,1+p+q],[2,3+2p,1+3p+2q],[3,6+3p,1+6p+3q]|=1

Show that |(1, 1+p, 1+p+q), (2, 3+2p, 4+3p+2q), (3, 6+3p, 10+6p+3q)|=1.

Show that |1 1+p1+p+q2 3+2p1+3p+2q3 6+3p 106 p+3q|=1.

If x ,y \ a n d \ z are not all zero and connected by the equations a_1x+b_1y+c_1z=0,a_2x+b_2y+c_2z=0 , and (p_1+lambdaq_1)x+(p_2+lambdaq_2)+(p_3+lambdaq_3)z=0 , show that lambda=-|[a_1,b_1,c_1],[a_2,b_2,c_2],[p_1,p_2,p_3]|-:|[a_1,b_1,c_1],[a_2,b_2,c_2],[q_1,q_2,q_3]|

Show that the points P (-2, 3, 5) , Q (1, 2, 3) and R (7, 0, -1) are collinear.

In the quadratic equation x^2+(p+i q)x+3i=0,p&q are real. If the sum of the squares of the roots is 8 then: p=3,q=-1 b. p=3,q=1 c. p=-3,q=-1 d. p=-3,q=1

If the lines p_1x+q_1y=1,p_2x+q_2y=1a n dp_3x+q_3y=1, be concurrent, show that the point (p_1, q_1),(p_2, q_2)a n d(p_3, q_3) are collinear.

The value of p and q for which the function f(x)""={(sin(p+1)x+sinx)/x , x 0} is continuous for all x in R, are: (1) p=1/2, q=-3/2 (2) p=5/2, q=-1/2 (3) p=-3/2, q=1/2 (4) p=1/2, q=3/2

Find the number of distinct rational numbers x such that oltxlt1 and x=p//q , where p ,q in {1,2,3,4,5,6} .