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: . [4]
1(b). Let be defined by . Show that is invertible and then find . [2]+[1]
2(a). Represent the complex number z = (-1 - i ) in polar form. [2]
2(b). Write the complex number in x+iy form and then find its multiplicative inverse. [2]+[2]
3(a). Find the sum to infinity of the series:
[5]
3(b). Prove that if x is positive then:
[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 [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 and the plane 2x + y-3z+4=0.
[5]
6(b). Solve the following differential equation:
[4]
7(a). Prove that:
[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:
[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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