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Section-A

1. a) By vector method prove Cosine Law of Trigonometry. 5 ?? ????

b) Show that: (?a x b) x c = (?a . ?c) b –(b.c )a 5

? ??

? ??ct d2ã 2c

2.a) If ã = a Sin wt + b Coswt+ 2 Sin wt,then provethat + w2?ã = Coswt

wdt2 w ???

where a , b , c are constant vectors. 5 '

b) If f?(t)= 4 i^, f (t)=4^jand f (t) = 0 at t = 0 show that tip of the position vector f (t) describes a

_parabola. 5

n-1 ^

?

3. a) If ã=x i^+ y ^j+ –z k ^ , Show that

(rn)= n rã. 5 _

_ b) Prove the vector identity Curl (ô f ) = ô (Curl f ) – f x grade ô where ô is a constant function and f is a vector function. 5

Section-B

4. a) If three forces acting on particle are in equilibrium then magnitude of each force is proportional to the

sine of the angle between the other two. 5 b) Resolved part of resultant R of two forces P and Q in the direction of P is of magnitude Q. Show that

-1

Pangle between the two forces is 2Sin

2Q . 5

5. a) A uniform rod AB of length ‘a’ and weight w is freely hinged to a vertical wall at A and is maintained in equilibrium by a light string of length ‘a’ fastened to B and to point C at a distance b vertically above A.

a2 + 2b2

Prove that the reaction at hinge A is W . 5

2b b) Find the amount of least force to drag a particle down on an inclined plane. 5

6. a) A hemisphere rests on a rough inclined plane with angle of friction ë. Show that inclination of the plane

-1

base to horizontal cannot be greater than Sin(2 Sin ë). 5

b) The least force P which moves a weight w up an inclined plane. Show that the least force acting parallel to the plane which will move the weight upwards is P 1+ì2 , where ì is coefficient of friction. 5

  1. a) Find the C.M of a hollow right circular cone of semi vertical angle á and height h. 5 b) FindcentreofGravity ofthecurve y = sinx asu=x = ð. 5

  2. a) A uniform ladder with its upper end against a smooth vertical wall. And its foot on a rough horizontal 1

ground. Show that force of friction at the ground is 2 w tan è, where w is the weight of the ladder and è

is its inclination with the vertical. 5

b) Four uniform rods are freely joined at their extremities and form a parallelogram ABCD which is suspended from A and is kept in the shape of inextensible string AC. Prove that the tension of string is equal to half the whole weight. 5

*** B.A/B.Sc-I (09/A) xxv ***

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last updated on 28-04-2019