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Total No. of Questions : 10
Total No. of Printed Pages : 04
RJ-206
B.Sc. B.Ed. (Secondary) Examination, 2024
(Second Semester)
MATHEMATICS
SEC-I
Analytic Geometry-I
[Time : 3 Hours]
[Maximum Marks : 40
Note : All questions are compulsory. There are Two Sections. Each question in Section A carries 5 marks. There are internal choice within question in Section A. Each question in Section B carries 3 marks.
Note : Attempt all questions. All questions carry equal marks.
Section A
1.
Find the equation of the right circular cylinder which passes through the circle
x² + y² + z² = 9, x - y + z = 3.
2.
Prove that a general equation of second degree in two variables always represent a conic section.
3.
Trace the parabola :
x² - 4xy + 4y² - 12x - 6y - 39 = 0
Also find the focus and vertex.
4.
Derive general polar equation of a straight line.
5.
Derive the distance between the points (r₁, θ₁) and (r₂, θ₂) which is given in polar coordinate.
6.
Derive the polar equation of conic considering the pole as the focus.
7.
Prove that the locus of midpoints of the focal chords of a conic, is a conic.
Prove that the plane
x + 2y - z - 4 = 0
cuts the sphere
x² + y² + z² - x + z - 2 = 0
in a circle of radius unity and find the equation to the sphere which has this circle as one of its great circles.
8.
Derive the equation of polar to a circle. Prove that for a point (α, β) polar is the chord of contact if it lie outside the circle and polar is the tangent at (α, β) if it lie on the circle.
Section B
9.
Find the equation of line in polar coordinate, given that perpendicular distance from the origin to the line is 6 unit and the angle between the perpendicular and the position x-axis is 120°. Then determine if (8, 150°) lies on the line.
10.
Find the equation of the tangent to the conic :
x²/16 + y²/9 = 1
at the point (x, y) = (2√3, 3/2).
Explain Hyperboloid of one sheet and hyperboloid of two sheet.
Prove that equation of cone with vertex at the origin is a Homogeneous equation of 2-degree in three variables.
Find the coordinate of the centre of the conic
14x² - 4xy + 11y² - 44x - 58y + 71 = 0
and hence reduce it to standard form.
Find the distance between (5, 30°) and (7, 20°) given polar coordinate.