Mathematics BS Mathematics/MS Mathematics PU — University of the Punjab 2016

UNIVERSITY OF THE PUNJAB

A/2016

Examination:- B.A./B.Sc.

Roll NoL!038

Subject: Mathematics General

PAPER: B

TIME ALLOWED: 3 hrs.

MAX. MARKS: 100 -

NOTE: Attempt SIX questions in all selecting TWO questions from Section - 1,

TWO questions from Section - II, ONE question from Section - III and

ONE question from Section-IV.

Section - I

(a) Find angle between the pair of lines given by the equation

X2+4xy +y2-6x-3=0

(b) Show that the point P (a sec 0, b tan

0) lies on the hyperbola

a? 62 1 for all values of

0. Find

equation of normal at

0.

9, 8

Q.2

(a) Find equations of tangent and normal to the curves

× = 2a cos 0 - a cos 20, y = 2a sin 0 - a sin 20 at 0 = 2

(b) Show that the curves r = am cos me and r = am sin me cut each other orthogonally.

9, 8

Q.3

(a) By using vectors prove that cos (a +

B) = cos a cos B - sin a sin B.

(b) Find area of the parallelogram with adjacent sides

9, 8

a =i-j+ kand b = 2j - 3k

Q.4

(a) If (t) = (t - sin t) + (1 - cos t) | + (t sin t + cos t) k, then find f

(I) at t = 2

(b) A particle moves along the curve x = 4 cos t, y = 4 sin t, z = 6t. Find velocity and acceleration at

† =¡

9, 8

Section - II

Q.5

(a) A straight line makes angles of measure c, B, y, & with the four diagonals of a cube.

Prove that cos? a + cos? B + cos' y + cos? 8 =

3.

(b) Find measure of the acute angle between the planes

9, 8

2x+y-z-5=0 and x-Y,=

-2z+5=0.

Q.6

(a) Find equation of the plane contain

g the line × - y - z = 0 = 2x - y + 3z - 5 and perpendicular

to yz-plane.

(b) Show that the straight line

=K*1

- and the plane 3x + 4y - 2z = 22 have a unique point

of intersection.

9, 8

Q.7

(a) Find direction of Qibla at Islamabad with latitude 33° 40'N and longitude 78° 8' E.

(b) Show that

-3-4i=56（元+旬n・1当）

9, 8

Q,8

(a) If 2 cos 0 =x+x, 2 cos $ = y+

y

, 2 cos y =z+½ then prove that

2 cos (0 + 4 + y) = xyZ + xyz

(b) Express cost 0 in the series of sines or cosines of multiple of

0.

9,8