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Dibrugarh University Arts Question Papers:MATHEMATICS [(Group – A: Coordinate Geometry and Group – B: Analysis – I (Real Analysis)]' (November)-2014


2014
(November)
MATHEMATICS
(General)
Course: 301
[(Group – A: Coordinate Geometry and
Group – B: Analysis – I (Real Analysis)]
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions
GROUP – A
(Coordinate Geometry)
SECTION – I
(2-Dimension)
1. (a) Write what should be done to remove term from the equation . 1
(b) Write the transformed equation of the straight line when the origin is transformed to the point.
(c) Find the angle through which the axes must be turned to remove the term from the equation.
2. (a) Straight lines represented by pass through the origin. State true or false.
(b) Show that straight lines represented by are parallel.
(c) Show that the equation 4
represents a pair of straight lines.
(d) Show that the area of the triangle formed by the lines and is     5
Or
Show that the lines and form an equilateral triangle.
3. (a) Let and for the conic . Write the name of the conic. 1
(b)  Find the equation of the tangent at to the conic. 2
(c) Write the definition of a diameter of a conic. 1
(d) Show that the equation 3
represents a hyperbola.
(e) Show that the sum of the squares of two conjugate semi-diameters of a conic is constant. 3
Or
Determine the nature of the conic. Also find the centre of the conic.
SECTION – II
(3-Dimension)
4. (a) Find the intercepts made by the plane on the axes. 1
(b) Express the equation of the plane in normal form. 2
(c) Find the equation to the plane through the point and normal to the straight line joining the points and. 3
Or
Find the coordinates of the point, where the line meets the plane.
(d) Find the distance of the point of intersection of the line and the plane from the point.
5. (a) Define skew lines. 1
(b) Find the shortest distance between the planes and. 2
(c) Find the shortest distance between the lines and. 5
GROUP – B
(Analysis – I)
6. (a) If , then write the value of 1
(b) If then find the length of sub tangent of the curve. 1
(c) Find the radius of curvature at any point on the curve 2
(d) If then find the value of 3
(e) Evaluate: 3
Or
If then find the value of
7. (a) Write when a function is derivable in a closed interval . 1
(b) Write the geometrical interpretation of Lagrange mean value theorem. 2
(c) Prove that if a function is continuous on and, then it assumes every value between and. 4
(d) Discuss the applicability of Rolle’s Theorem to in. 3
Or
Expandby Maclaurin’s theorem with Lagrange form of remainder.
8. (a) If , then find 2
(b) If, then show that 3
Or
If then show that
9. (a) Write the condition when 1
(b) Evaluate: 5
(c) Evaluate: 4
Or
Find the length of the curve from 1 to 2.

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