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 n
^{th}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 x
^{3}+px^{2}+qx+r=0. Find the value of . - Diminish the roots x
^{4}+x^{3}-3x^{2}+2x-4 =0 by 2. - Evaluate
- Prove that
- Define Pole and Polar of a ellipse.
- In the hyperbola 16x
^{2}-9y^{2}= 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 n
^{th}derivative of . - Find the angle between the radius vector and tangent for the curve at

.

- Solve the equation x
^{3}-4x^{2}-3x+18=0 given that two of its roots are equal. - Solve the equation x
^{4}-5x^{3}+4x^{2}+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.

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__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 6x
^{5}+11x^{4}-33x^{3}-33x^{2}+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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