Find all the 8^th roots of . Also show any one of them in an Argand diagram.

(a) Show that the perpendiculars drawn from the origin to tangent planes to the cone  lie on the cone 

Find the values of a ∈R for which ia is a solution of . Also find all the roots of this equation.

(d) Find the equation of the cylinder with base 

Consider the linear system

Give the two reasons for Cramer’s Rule being applicable for solving this system. Also use the rule to solve the linear system.

(c) Find the angle between the lines of intersection of the cone  and the plane 

(b) Find the equation of the sphere touching the plane  at (3, −1, −1) and cutting the sphere    orthogonally.

Give the following:i) a 2× 4 matrix;
 ii) the transpose of the matrix in (i) above;
 iii) a system of linear equations represented by AX = B, where A is the matrix in (ii) above. 

In the context of your IGNOU studies, give the following:

i) an example of an implication;
 ii) the converse of your statement in (i) above;
iii) the contrapositive of your statement in (i) above;
iv) a statement using ∀ ;
v) a statement using ∃ . 

Express the following situation in a Venn diagram:
In a survey of 60 women, it is found that 25 have studied upto Class 12 only, 10 have studied till Class 10 only, 26 got scholarships, 9 of those studying till Class 12 got scholarships, 8 of those studying till Class 10 got scholarships, and 11 had completed their BA degree. 

(a) Show that the plan  is a tangent plane to the sphere 

(c) Find the distance of the origin from the plane which passes through 2, 1, 8 , 1, 0, 2 and (−3, 4, 6) . 

(b) Find the equation of the plane which passes through the line of intersection of the  planes and  and which is perpendicular to the plane 

i) Find A× B, and the number of elements in it, where
  and
 
 ii) Given any two sets C and D , under what conditions on them will C× D and D×C have the same number of elements? Give reasons for your answer. 

(a) Find the equations of the line through (1,3, 4 ) and parallel to the line joining the  points (−4, 5, 3) and (8, 9, 7).

(d) Prove that the length of the chord of a parabola which passes through the focus and which is inclined at 30° to the axis of the parabola is four times the length of the latus rectum.

(c) Find the eccentricity, foci, centre and directrices of the ellipse  Also give a rough sketch of it. 

(b) (i) Show that  represents the equation of a line passing  through (2, 3) and (−4, 7).

(ii) Prove that the equation of a line through  and  can be 

expressed in the form  

If A and B are the set of even integers and set of odd integers, respectively, find A ∪ B and 

(a) Let See Answer →