Rajasthan Technical University 2009-1st Year B.E , y- - Question Paper
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** B.E. I-Year Ex Mathematics -1
B. E. I - Year Examination, July - 2009
Time: 3 Hours] [Total Marks: 60
[N/lin. Passing Marks:
Attempt five questions in all. Schematic diagrams must be shown wherever necessary. Any data you feel missing may suitably be assumed and stated clearly.
Use of following supporting material is permitted during examination.
(Mentioned in form No. 205)
1 (a) Find the d.c's. l.m.n. of two lines which are connected by the relation l-5m + 3n = 0 and 712 + 5 m2 - 3 n2 = 0.
3
(b) Find the equation of the plane perpendicular to the yz-plans and passing through (1, -2, 4) and (3, -4, 5).
3
(c) Prove that the lines x = ay + b. z = cy + d and x = a'y = b' z = cy+d' are perpendicular if aa' +cc' = -1.
3
(d) Find the equation of a sphere touching three coordinate planes. How many such spheres can be drawn ?
3
OR
1 (a) Find angle between two diagonals of a cube.
- 3
(b) Two systems of rectangular axes have the same origin. If a plane cuts them at distance a, b, c and a', b\ c' from the origin
1111 11
show that t + tt+r = S"fT3 + x-a ir cr a o' c
3
(c) Find the shortest distance between two given lines.
3
(d) Prove that the spheres cutting two given spheres along a great circle pass through two fixed points.
3
Show that a cone of second degree can be found to pass through any five concurrent lines.
3
2 (a) 0)
(c)
(d)
2 (a)
(b)
(c)
(d)
3a)
i
E702]
Find the rank of matrix.
4 2 13
6 3 9 7
2 1 0 1 '
8
By use of matrices, solve the equations :
* + y + * = 9,2* + 5y + 7*=82,2* + y-* = 0.
3
Find eigen values and eigen vectors of matrix :
3 2 4
2 0 2
4 2 9
3
OR
Find equation of right circular cylinder whose axis is
Oil 10 1. 110
Find inverse of the matrix
Show that the only real values of X for which the following equations have non-zero solution is 6. x + iy + Zz = Xx, 3* + > + 2r = Xy, 2x+3y + z = te.
3
Prove if A and B are 2 square matrices of the same order then AB and BA have the eigen values.
Find the asymptotes of the curve x -2xty + xy -x* -xy+z = Q.
(b) Find the multiple points for the curve ys + Sax* + x* = 0 and their nature.
4
(c) Trace the curve y = x* (cubical parabola)
' 4
3 (a) Find the radius of curvature of curve r = a sin 6 at the pole.
(b) Find the point of inflexion of the curve y = +
(c) Trace the curve x* = y* (x +1) .
(a/lf x*yyz* = c show that at x y z ~ (xlogex) .
iOw that sin' 0 cos9 0 attains a maximum when tan'1
(c) Find area included between the curve and its asymptote
*V =o*(>* -**)
4 (a) If 0 = t*e 41; find values of n will make pTl rj- ~$l'
(b) Find maxima and minima of sinx + cosx in the interval
0 y 2 n.
(c) Find length of the arc of parabola xi = 4 ay from its vertex to an extremity of the latus rectum.
[Contd...
E702]
5 (a) Find the volume in the first octant bounded by cylinders x*+y*=o* and x*+zt = ot.
4
(b) Evaluate JJ y dx dy wjjere a is the region bounded by parabolas y%=4ax and x*=4ay.
4
(c) Change the order of integration in the following double integral JJ~f(xy)dxdy.
4
OR
5 (a) Find the surface of solid generated by the revolution the astroid x = a cos* t, y - a sin* t about x-axis.
(b) Evaluate J0** V sin 0 dQ dr Jfi) Evaluate the integral by changing order of integration
l 2500 ]
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