B.TECH/AEIE/BT/CE/CHE/CSE/ECE/EE/IT/ME/2ND SEM/MATH 1201/2017 B.
TECH/AEIE/BT/CE/CHE/CSE/ECE/EE/IT/ME/2ND SEM/MATH 1201/2017
B.TECH/AEIE/BT/CE/CHE/CSE/ECE/EE/IT/ME/2ND SEM/MATH 1201/2017
Group – D
MATHEMATICS - II
dx
6. (a) Evaluate
x 2
2x 2
, if it exists. (MATH 1201)
Time Allotted : 3 hrs Full Marks : 70
(b) Assuming the convergence of the integral prove that,
Figures out of the right margin indicate full marks.
3
x
xe dx . Candidates are required to answer Group A and
3
0
any 5 (five) from Group B to E, taking at least one from each group.
6 + 6 = 12
Candidates are required to give answer in their own words as far as
7. (a) Evaluate: L 1 log s a practicable.
s b
Group – A
(b) Solve by Laplace Transform: (Multiple Choice Type Questions)
y ( t ) y ( t ) sin 2 t , where y ( 0 ) y ( 0 ) 1 .
6 + 6 = 12 1. Choose the correct alternative for the following: 10 × 1 = 10
(i) 1
Group – E sin 2 x
D2 1
8. (a) Show that the straight lines whose direction cosines are given by (a) (b) (c) − (d)
al bm cn 0 and ul 2 vm 2 wn 2 0 are parallel if
d2y
a2 b2 c2 (ii) The general solution of 0 is
0. dx 2
u v w
(a) y a (b) y ax b
(b) A variable plane at a constant distance p from the origin 0 meets the
(c) y ax 2 bx c (d) y a cos x b sin x.
axes at A, B, C. Show that the locus of the centroid of the tetrahedron
1 1 1 16 (where a , b, c are arbitrary constants).
OABC is 2 2 2 2 .
x y z p dy
(iii) Integrating factor of x y xe x is
6 + 6 = 12 dx
9. (a) Find the equations of the lines of greatest slope and least slope on (a) x (b) e x (c) 1 (d) .
the plane 3 x 4 y 5 z 5 0 drawn through the point (1, 2, 2), (iv) The minimum number of edges in a connected graph having 21
given that the plane 4 x 5 y 6 z 6 0 is horizontal. vertices is
(a) 18 (b) 20 (c) 10 (d) 11.
(b) Show that the equation to the plane containing the line
y z x z (v) The number of pendant vertices in a binary tree with n vertices is
1, x 0 and parallel to the line 1, y 0 is (a) n-1 (b) n (c) n+1 (d) (n+1)/2.
b c a c
x y z
1 0 and if 2d is the shortest distance, prove that
(vi)
L e t sin 2t
a b c (a) 1 (b) 2
1 1 1 1 s 2 2s 5 s 2 2s 5
2 2 2. s s 1
d 2
a b c (c) (d) .
s 2 2s 5 s2 2s 5
6 + 6 = 12
MATH 1201 4 MATH 1201 1
B.TECH/AEIE/BT/CE/CHE/CSE/ECE/EE/IT/ME/2ND SEM/MATH 1201/2017 B.TECH/AEIE/BT/CE/CHE/CSE/ECE/EE/IT/ME/2ND SEM/MATH 1201/2017
(vii) 1 Group – C
L1
( p 1)( p 3) 4. (a) What will be the least possible number of simple regular graphs
(a) 1 e t e 3t (b) e e
t 3 t
(c) 1 e t e 3t (d) 1 3t t
e e . having 20 edges? Justify your answer.
2 2 2
(b) Draw the digraph with the following adjacency matrix:
(viii) 7 2 is equal to 0
0 1 1 0
1 1 0 0 0
(a) 5 (b) 3 (c) 15 (d) . 0 0 0 0 1
8 4 8 2
0 0 0 0 0
(ix) o
A directed line makes angles 60 and 45 with the axes of x and y o 0 1 1 0 0
respectively. What angle does it make with the axis of z?
(c) Let G be a graph with 15 vertices and 4 components. Prove that, G
(a) 60 o (b) 45 o (c) 30 o (d) 90 o.
has at least one component having at least 4 vertices.
(x) The length of the perpendicular from the origin upon the plane 5 + 4 + 3 = 12
2 x 6 y 3 z 5 0 is
(a) 5/7 (b) 6/7 (c) 3/7 (d) 5. 5. (a) Use Kruskal’s algorithm to find a minimal spanning tree for the
following graph:
Group – B V2
2
2. (a) The equation x2 y2
1 (where a and b are fixed constants 8 V1 8 V6
a2 b2 3 7
and is an arbitrary parameter which can assume all real values) 2 12
represents a family of confocal conics. Obtain the differential V3 V4
equation of this family. 5 V5 4
(b) Determine the exactness of the following differential equation: (b) Find and hence draw all possible spanning trees in the following
1 graph:
y 1 cos y d x x log x x sin y d y 0
x d c
Solve: e p p 0
y 3
(c)
4 + 2 + 6 = 12
d2y dy a b
3. (a) Solve: x 2 2
x 4 y x sin(log x )
dx dx (c) Using BFS and DFS algorithms, find spanning trees of the following
(b) Solve the following differential equation using the method of graph:
variation of parameters: a b
2
d y 2
y e
dx 2
1 ex
6 + 6 = 12
c d
4 + 3 + 5 = 12
MATH 1201 2 MATH 1201 3
