BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER

ADITYA BIRLA GROUP ADITYA BIRLA CENTRE ‘C’ WING 1ST
BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE PILANI (RAJ) I
BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE PILANI (RAJASTHAN) 1ST

BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE PILANI SECOND SEMESTER
BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE PILANI SEMESTERII 200708
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI HYDERABAD

Birla Institute of Technology and Science, Pilani

Birla Institute of Technology and Science, Pilani

II SEMESTER 2008-09

MATH D021: REMEDIAL MATHEMATICS

Comprehensive Examination-PART A(Closed Book)

Date: 9th May 2009 Time: 2 Hours

Day: Saturday Max. Marks: 65

Note: Answer all questions in sequence.


1(a). Let A={2, 3, 5, 6, 7, 11}, B={1, 3, 5, 9, 11, 15} and C={2, 3, 4, 6, 9, 12, 13}. Verify that: BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER . [4]


1(b). Let BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER be defined by BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER . Show that BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER is invertible and then find BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER . [2]+[1]


2(a). Represent the complex number z = (-1 - iBIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER ) in polar form. [2]


2(b). Write the complex number BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER in x+iy form and then find its multiplicative inverse. [2]+[2]


3(a). Find the sum to infinity of the series:

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER [5]


3(b). Prove that if x is positive then:

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER [4]


4(a). Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour. [3]


4(b). Find the 13th term in the expansion of BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER [3]


5(a). Show that the points (5, 5), (6, 4), (-2, 4) and (7, 1) all lie on a circle. Find its equation, centre and radius. [6]


5(b). Find the coordinates of the vertices, the foci, the eccentricity and the equation of the directrices of the hyperbola 9x2 – 16y2 = 144. [4]


6(a). Find the angle between the lines BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER and the plane 2x + y-3z+4=0.

[5]

6(b). Solve the following differential equation:

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER [4]


7(a). Prove that:

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER [5]


7(b). A coin is tossed and a die is thrown. Find the probability that the outcome will be a head or a number greater than 4 or both. [4]


8(a). Solve the definite integral:

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI II SEMESTER [5]


8(b). Find a vector of magnitude 5 in the direction of the vector from P1 (1, 0, 1) to P2 (3, 2, 0). [4]





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Birla Institute of Technology and Science, Pilani

II SEMESTER 2008-09

MATH D021: REMEDIAL MATHEMATICS

Comprehensive Examination-PART B(OPEN BOOK)

Date: 9/5/2009 Maximum marks: 35

Time: 1 Hour

Note: Answer all questions in sequence.

1. Prove by Mathematical induction that\(x + y) is a factor of the polynomial x2n+1 +

y2n+1 for all integers n". (6)

2. Prove that:

tanA + tanB + tanC = tanA:tanB:tanC

where A + B + C = _. Then by using above result prove that:

cotB:cotC+cotC:cotA+cotA:cotB = 1 (4)+(2)

3. If f(x) = ax+b

x+1 , limx!0f(x) = 2 and limx!1f(x) = 1 then prove that f(􀀀2) = 0.

(6)

4. Show that the function f given by

f(x) = jxj + jx 􀀀 1j; x 2 R

is continuous at x = 0 and x = 1. (6)

5. Find the sum to n terms of the sequence:

3; 33; 333; 3333; : : : (6)

6. If 1; !; !2 are the cube root of unity, then prove that:

(3+3!+5!2)6􀀀(2+6!+2!2)3 = 0 (5)

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