LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

**B.Sc.** DEGREE EXAMINATION – **MATHEMATICS**

FIRST SEMESTER – NOVEMBER 2012

# MT 1502 – ALGEBRA AND CALCULUS – I

Date : 08/11/2012 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

__PART – A__

**ANSWER ALL QUESTIONS: (10×2=20)**

** **

- Find the n
^{th}derivative of - Show that, in the curve, the polar sub tangent varies as the square of the

radius vector and the polar subnormal is a constant.

- Write the conditions for maxima and minima of two variables.
- What is the radius of curvature of the curve at the point (1, 1)?
- Find the co-ordinates of the centre of curvature of the curve at the point

(2, 1).

- Form a rational cubic equation which shall have the roots 1,
- If are the roots of the biquadratic equation

find

- State Newton’s theorem on the sum of the powers of the roots.
- State Descartes’ rule of signs for negative roots.
- Determine if Cardon’s method can be applied to solve the equation

__PART – B__

**ANSWER ANY FIVE QUESTIONS: (5×8=40)**

- a) Find the angle at which the radius vector cuts the curve
- b) Find the slope of the tangent with the initial line for the cardioid

at (4 + 4)

- Discuss the maxima and minima of the function

- Prove that the (p-r) equation of the cardioid is and hence

prove that its radius of curvature is

- Show that the evolute of the cycloid ; is another

cycloid.

- Solve the equation
- Show that the sum of the eleventh powers of the roots of is zero.
- a) If are the roots of the equation find the value of

- b) Determine completely the nature of the roots of the equation

(5 + 3)

- If be a real root of the cubic equation of which the coefficients

are real, show that the other two roots of the equation are real, if

__PART – C__

** **

**ANSWER ANY TWO QUESTIONS: (2 x 20 = 40)**

** **

- a) Find the n
^{th }differential coefficient of . - b) If, prove that (10 +10)

- A tent having the form of a cylinder surmounted by a cone is to contain a given

volume. If the canvass required is minimum, show that the altitude of the cone is

twice that of the cylinder.

- a) Find the asymptotes of

- b) Show that the roots of the equation are in Arithmetical

progression if Show that the above condition is satisfied by the

equation and hence solve it. (10 + 10)

- Determine the root of the equation which lies between 1 and 2

correct to three places of decimals by Horner’s method.