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Roll No.
Total No. of Questions : 10
Total No. of Printed Pages : 04
RJ-207
B.Sc.-B.Ed. (Secondary) Examination, 2024
(Second Semester)
MATHEMATICS
SEC-II
Analytic Geometry-II
Time : 3 Hours
[Maximum Marks : 40
Note : All questions are compulsory. There are two sections. Each question in Section A carries 5 marks. There is internal choice with question in Section A. Each question in Section B carries 3 marks.
1.
Prove that the lines drawn from the origin parallel to the normals to ax2 + by2 + cz2 = 1 at the point of intersection with the plane lx + my + nz = p generate the cone.
p2 (x2⁄a + y2⁄b + z2⁄c) = (lx⁄a + my⁄b + nz⁄c)2
(5)
Or
Find the equation to the quadratic cone generated by six normals to the conicoid ax2 + by2 + cz2 = 1 through the fixed point (f, g, h). (5)
Find the equation to the quadratic cone generated by six normals to the conicoid ax2 + by2 + cz2 = 1 through the fixed point (f, g, h). (5)
2.
Prove that the section of the conicoid ax2 + by2 + cz2 = 1 by a tangent plane to the cone
x2⁄b+c + y2⁄c+a + z2⁄a+b = 0
is a rectangular hyperbola.
(5)
Or
Find the locus of the centers of sections of the ellipsoid :
Find the locus of the centers of sections of the ellipsoid :
x2⁄a2 + y2⁄b2 + z2⁄c2 = 1
whose area is constant (= πk2).
(5)
3.
Prove that the axes of the section of the cone ax2 + by2 + cz2 = 0 by the plane lx + my + nz = p are given by :
l2⁄aP2+p2 + m2⁄bP2+p2 + n2⁄cP2+p2 = 0
where P2 = x2⁄a + m2⁄b + n2⁄c
Or
Find the length of semi-axis of the cone : 2x2 + y2 - z2 = 1, 3x2 + 4y + 5z = 0. (3)
Find the length of semi-axis of the cone : 2x2 + y2 - z2 = 1, 3x2 + 4y + 5z = 0. (3)
4.
Write the equation of discriminating cubic. Write the different equation if all the roots of discriminating cubic are different and none of them are zero.
(5)
Or
Write the procedure to reduce the equation, if two roots of discriminating cubic area same (equal) and third root is not equal to zero. (5)
Write the procedure to reduce the equation, if two roots of discriminating cubic area same (equal) and third root is not equal to zero. (5)
5.
Reduce the following equation : x2 + 3y2 + 3z2 - 2yx - 2y + 6z + 3 = 0
(5)
Or
Show that : 5x2 - 4y2 + 5z2 + 4yz - 14zx - 4xy + 16x - 16y - 32z + 8 = 0 represent a pair of planes. (3)
Show that : 5x2 - 4y2 + 5z2 + 4yz - 14zx - 4xy + 16x - 16y - 32z + 8 = 0 represent a pair of planes. (3)
6.
Find the equation of tangent planes to 2x2 - 6y2 + 3z2 = 5 which passes through x + 9y - 3z = 0 = 3x - 3y + 6z - 5.
(3)
7.
Find the point where the plane x + 2y - 2z = 4 touch the paraboloid 3x2 + 4y2 = 24z.
(3)
8.
Find the equation of Tangent plane to the conicoid :
3x2 - 5y2 + 2z2 = 0
at the point (1, 1, 0).
(3)
9.
Define Director Sphere.
(3)
10.
Derive the equation of polar plane at (α, β, γ) for the conicoid ax2 + by2 + cz2 = 1.
(3)