LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIRST SEMESTER – APRIL 2012
MT 1500 – ALGEBRA, ANALY. GEO., CALCULUS & TRIGONOMETRY
Date : 28-04-2012 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
PART – A
Answer ALL the questions: (10 X 2 = 20 Marks)
- Find the nth derivative of .
- Find the slope of the straight line .
- Write the formula for the radius of curvature in Cartesian form.
- Define Cartesian equation of the circle of the curvature.
- If ,are the roots of the equation x3+px2+qx+r=0. Find the value of .
- Diminish the roots x4+x3-3x2+2x-4 =0 by 2.
- Evaluate
- Prove that
- Define Pole and Polar of a ellipse.
- In the hyperbola 16x2-9y2 = 144, find the equation of the diameter conjugate to the diameter x =2y.
PART – B
Answer any FIVE questions: (5 X 8 = 40 Marks)
- Find the nth derivative of .
- Find the angle between the radius vector and tangent for the curve at
.
- Solve the equation x3-4x2-3x+18=0 given that two of its roots are equal.
- Solve the equation x4-5x3+4x2+8x-8=0 given that 1-is a root.
- Expand in terms .
- Separate real and imaginary parts .
- P and Q are extremities of two conjugate diameters of the ellipse and S is a focus. Prove that
- The asymptotes of a hyperbola are parallel to 2x+3y=0 and 3x-2y =0 . Its centre is at (1,2) and it passes through the point (5,3). Find its equation and its conjugate.
PART – C
Answer any TWO questions: (2 x 20=40 Marks)
- (a) If , show that
(b) Prove that the sub-tangent at any point on is constant ant the subnormal is
(10 +10)
- (a) Find the radius of curvature at any point on the curve
(b) Show that the evolute of the cycloid is another
cycloid . (10+10)
- (a) Solve 6x5+11x4-33x3-33x2+11x+6=0.
(b) Find by Horner’s method, the roots of the equation which lies between 1 and 2
correct to two decimal places. (10+10)
- (a) Prove that
(b) Prove that the Product of the perpendicular drawn from any point on a hyperbola to its
asymptotes is constant. (10+10)
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