DU SOL Question Paper 2018 BA Mathematics - Algebra and Calculus

[This question paper contains 4 printed pages.]

4478 Ytttrr Roll ....
.'.\o.
B.A. (Prog.)/I G-II

Paper Code:A-155

MATHEMATICS - Paper I

(Algebra and Calculus)

Time : 3 Hours Muximunt Murks : 75

(trlrrite your Roll No. on the top in'tntetliatel.v

on receipt of this qtrcstion paper.)

Note;- The mqximum mtrrks printed on the tluestir.tn
paper ure upplicable./br the stutlents cf Categrtry 'A'. These
murks will, however, be scalecl up prctytortionntely in respccl
o.f'the .stttdents of SOL ot the time o.f'ltttsting of utt'urcls for
compi I tttion of result.

All questions ore compulsory uncl hove equul mnrks.

Attempt ctny two parts from eoch tluestion.

1. (a) Show that vectors (1,2, 7), (2,1.0), (i, -1, 2) form a
basis of R3.

(b) ls the following system of equations

2x+3y+42:ll
P.T.O

4478 2

,r+ 5.y+ 7z:15
3r + llv + 132: 25
consistent? If yes, then solve it.
(c) Veiify Cayley Hamilton Theorem for the matrix

[oo']
o=lt r ol
12141
Hence compute A I
the inverse of A.

2. (a) If

sin d+ sin Q + sin r1r = 0,

cos d * cos 4i * cos r1t : 0,
show that

cos 30 * cos 3 0 + cos 3tlz :3 cos(O Q + Vr ) and

sin 3d+ sin 3 Q + sin 3rY : 3 sin (0 Q + V/ ).
(b) Solve the equation:
3.tr + 11,rr + 12x + 4:0

the roots being in H.P.(Harmonic Progression).
(c) If d, B, 7 are the roots of the equation:
x3 + qr + r :0, such that no two of them are equal 1n
magnitude but opposite in sign. Find the values of:

4478

(r) \-l
- p*y

(nJ sP
-/
u B*y
a
J. (a) Examine the continuitY of the function / defined bY

x-lrl r+0
x
f(x) =
x x=0

at the point r = 0.

(b) If Y = e'' .si n-rx, then

(1 - x2)y,,-z - (2n + l)x !n,t - (n2 + m2)y,,
: 0'

.r+y
r f- ' Drove
(c)Ifu:tan-r r/x+1y ' that

du0ul
.\--f l-=-slnzu
0x 'Ay 4
A
+. (a) Find the equation of the tangent to the parabola
4x * 5, which is parallel to the line v - 2-r +3
: 0'
'2:
P.T.O.

4478 4

(b) Show that the position and nature of the double points
on the following curve

,rr - y3 - 7x2 + 4y + 15;r - 13 : 0.

(c) Trace the curve

v2(2a - x) : x3.

(a) Verify the Lagrange's mean value theorem for the
f-unction

(.') : r(.r - l)(.r - 2) in
[t :]
(b) Define Maclaurin's infinite series expansion with
Taylor's remainder and hence find the expansion of
r.' : sin x.

re' - log(1+ r)
(c) Evaluate lim,-o

xr +4x
(a) Evaluate J

(b) Find the area common to the parabolas : 4ax and
12 : 4by.

(c) Find the volume of the solid generated by rotating the
ellipse 4x2 * !) : 4 about the x-axis.
(1 400)