# Statistics Test 1

Total Questions:50 Total Time: 75 Min

Remaining:

## Questions 1 of 50

Question:The maximum resultant of two forces is P and the minimum resultant is $$Q$$ , the two forces are at right angles, the resultant is

$$P + Q$$

$$P - Q$$

$$\frac{1}{2}\sqrt {{P^2} + {Q^2}}$$

$$\sqrt {\frac{{{P^2} + {Q^2}}}{2}}$$

## Questions 2 of 50

Question:Two equal forces act at a point. If the square of the magnitude of their resultant is three times the product of their magnitudes, the angle between the forces is

$${30^o}$$

$${45^o}$$

$${90^o}$$

$${60^o}$$

## Questions 3 of 50

Question:A force is resolved into components P and Q equally inclined to it. Then

$$P = 2Q$$

$$2P = Q$$

$$P = Q$$

None of these

## Questions 4 of 50

Question:If the square of the resultant of two equal forces is equal to $$(2 - \sqrt 3 )$$ times their product, then the angle between the forces is

$${60^o}$$

$${150^o}$$

$${120^o}$$

$${30^o}$$

## Questions 5 of 50

Question:The resultant of two equal forces is equal to either of these forces. The angle between them is

$$\frac{\pi }{4}$$

$$\frac{\pi }{3}$$

$$\frac{\pi }{2}$$

$$\frac{{2\pi }}{3}$$

## Questions 6 of 50

Question:Let R be the resultant of P and Q and if $$\frac{P}{3} = \frac{Q}{7} = \frac{R}{5}$$ , then the angle between P and R is

$${\cos ^{ - 1}}\left( {\frac{{11}}{{14}}} \right)$$

$${\cos ^{ - 1}}\left( {\frac{{ - 11}}{{14}}} \right)$$

$$\frac{{2\pi }}{3}$$

$$\frac{{5\pi }}{6}$$

## Questions 7 of 50

Question:The resultant of two forces P and Q is at right angles to P, the resultant of P and $$Q'$$ acting at the same angle $$\alpha$$is at right angles to $$Q'$$ . Then,

$$P,Q,Q'$$ are in G.P.

$$Q,P,Q'$$ are in G.P.

$$P,Q',Q$$ are in G.P.

None of these

## Questions 8 of 50

Question:The resultant R of two forces P and Q act at right angles to P. Then the angle between the forces is

$${\cos ^{ - 1}}\left( {\frac{P}{Q}} \right)$$

$${\cos ^{ - 1}}\left( { - \frac{P}{Q}} \right)$$

$${\sin ^{ - 1}}\left( {\frac{P}{Q}} \right)$$

$${\sin ^{ - 1}}\left( { - \frac{P}{Q}} \right)$$

## Questions 9 of 50

Question:The sum of the two forces is 18 and their resultant perpendicular to the lesser of the forces is 12, then the lesser force is

5

3

7

15

## Questions 10 of 50

Question:The magnitudes of two forces are 3, 5 and the direction of the resultant is at right angles to that of the smaller force. The ratio of the magnitude of the larger force and of the resultant is

5:3

5:4

4:5

4:3

## Questions 11 of 50

Question:If A, B ,C are three forces in equilibrium acting at a point and if $${60^o}$$ , $${150^o}$$ and $${150^o}$$ respectively denote the angles between A and B, B and C and C and A, then the forces are in proportion of

$$\sqrt 3 :1:1$$

$$1:1:\sqrt 3$$

$$1:\sqrt 3 :1$$

$$1:2.5:2.5$$

## Questions 12 of 50

Question:If the angle $$\alpha$$ between two forces of equal magnitude is reduced to $$\alpha - \frac{\pi }{3}$$ , then the magnitude of their resultant becomes $$\sqrt 3$$ times of the earlier one. The angle $$\alpha$$ is

$$\pi /2$$

$$2\pi /3$$

$$\pi /4$$

$$4\pi /5$$

## Questions 13 of 50

Question:The resultant of two forces P and Q is R. If one of the forces is reversed in direction, the resultant becomes $$R'$$ , then

$${R'^2} = {P^2} + {Q^2} + 2PQ\cos \alpha$$

$${R'^2} = {P^2} - {Q^2} - 2PQ\cos \alpha$$

$${R'^2} + {R^2} = 2({P^2} + {Q^2})$$

$${R'^2} + {R^2} = 2({P^2} - {Q^2})$$

## Questions 14 of 50

Question:Forces proportional to AB, BC and 2CA act along the sides of triangle ABC in order, their resultants represented in magnitude and direction is

$$\overrightarrow {CA}$$

$$\overrightarrow {AC}$$

$$\frac{H}{{A + B}}$$

$$\overrightarrow {CB}$$

## Questions 15 of 50

Question:$$ABCD$$ is a parallelogram. A particle P is attracted towards A and C by forces proportional to PA and PC respectively and repelled from B and D by forces proportional to $$PB$$ and $$PD$$ . The resultant of these forces is

$$2\overrightarrow {PA}$$

$$2\overrightarrow {PB}$$

$$2\overrightarrow {PC}$$

None of these

## Questions 16 of 50

Question:Two forces $$P + Q,P - Q$$ make an angle $$2\alpha$$ with one another and their resultant make an angle $$\theta$$ with the bisector of the angle between them. Then

$$P\tan \theta = Q\tan \alpha$$

$$P\cot \alpha = Q\cot \theta$$

$$P\tan \alpha = Q\tan \theta$$

None of these

1 and 2 are correct

## Questions 17 of 50

Question:If two forces P and Q act on such an angle that their resultant force R is equal to force P, then if P is doubled then the angle between new resultant force and Q will be

$${30^o}$$

$${60^o}$$

$${45^o}$$

$${90^o}$$

## Questions 18 of 50

Question:A bead of weight W can slide on a smooth circular wire in a vertical plane, the bead is attached by a light thread to the highest point of the wire, and in equilibrium the thread is taut. Then the tension of the thread and the reaction of the wire on the bead, if the length of the string is equal to the radius of the wire, are

$$W,2W$$

$$W,W$$

$$W,3W$$

None of these

## Questions 19 of 50

Question:The resultant of two forces P and Q is R. If the direction of P is reversed keeping the direction Q same, the resultant remains unaltered. The angle between P and Q is

90$$^0$$

60$$^0$$

45$$^0$$

30$$^0$$

## Questions 20 of 50

Question:Two forces P and Q acting parallel to the length and base of an inclined plane respectively would each of them singly support a weight W, on the plane , then $$\frac{1}{{{P^2}}} - \frac{1}{{{Q^2}}}$$=

$$1/{W^2}$$

$$2/{W^2}$$

$$3/{W^2}$$

None of these

## Questions 21 of 50

Question:The moment of a system of coplanar forces (not in equilibrium) about three collinear points A,B,C in the plane are $${G_1},{G_2},{G_3}$$ then

$${G_1}.AB + {G_2}.BC + {G_3}.AC = 0$$

$${G_1}.BC + {G_2}.CA + {G_3}.AB = 0$$

$${G_1}.CA + {G_2}.AB + {G_3}.BC = 0$$

None of these

## Questions 22 of 50

Question:The resultant of two like parallel forces P, Q passes through a point O. If the resultant also passes through O when Q and R replace P and Q respectively, then

$$P,Q,R$$ are in G.P.

$$Q,P,R$$ are in G.P.

$$R,P,Q$$ are in G.P.

$$P,Q,R$$ are in A.P.

## Questions 23 of 50

Question:Any two coplanar couples of equal moments

Balance each other

Are equivalent

Need not be equivalent

None of these

## Questions 24 of 50

Question:Two like parallel forces P and 3P act on a rigid body at points A and B respectively. If the forces are interchanged in position, the resultant will be displaced through a distance of

$$\frac{1}{2}AB$$

$$\frac{1}{3}AB$$

$$\frac{1}{4}AB$$

$$\frac{3}{4}AB$$

## Questions 25 of 50

Question:Three like parallel forces P,Q, R act at the corners A,B,C of a $$\Delta ABC$$ . If their resultant passes through the incentre of $$\Delta ABC$$ , then

$$\frac{P}{a} + \frac{Q}{b} + \frac{R}{c} = 0$$

$$Pa + Qb + Rc = 0$$

$$\frac{P}{a} = \frac{Q}{b} = \frac{R}{c}$$

$$Pa = Qb = Rc$$

## Questions 26 of 50

Question:Two like parallel forces P and 3P are 40 cm apart. If the direction of P is reversed, then their resultant shifts through a distance of

30 cm

40 cm

50 cm

60 cm

## Questions 27 of 50

Question:Let a force P be represented by the straight line AB and O is any point. Then the moment of P about O is represented in magnitude by

$$\Delta AOB$$

$$2\Delta AOB$$

$$3\Delta AOB$$

$$(1/2)\Delta AOB$$

## Questions 28 of 50

Question:The resultant of two parallel forces P,Q acting at A and B respectively acts at C when like and at D when unlike. If P>Q, then CD =

$$\frac{{PQ}}{{{P^2} - {Q^2}}}AB$$

$$\frac{{2PQ}}{{{P^2} - {Q^2}}}AB$$

$$\frac{{2PQ}}{{{P^2} + {Q^2}}}AB$$

None of these

## Questions 29 of 50

Question:Two like parallel forces of 5N and 15 N, act on a light rod at two points A and B respectively 6m apart. The resultant force and the distance of its point of a application from the point A are

$$10N,4.5m$$

$$20N,4.5m$$

$$20N,1.5m$$

$$10N,15m$$

## Questions 30 of 50

Question:Two weights of 10gms and 2 gms hang from the ends of a uniform lever one metre long and weighing 4gms. The point in the lever about which it will balance is from the weight of 10gms at a distance of

5 cm

25 cm

45 cm

65 cm

## Questions 31 of 50

Question:Three like parallel forces P, Q, R act at the corner points of a triangle ABC. Their resultant passes through the circumcentre, if

$$\frac{P}{a} = \frac{Q}{b} = \frac{R}{c}$$

$$P = Q = R$$

$$P + Q + R = 0$$

None of these

## Questions 32 of 50

Question:Two unlike parallel forces acting at points A and B form a couple of moment G. If their lines of action are turned through a right angle, they form a couple of moment H. Show that when both act at right angles to AB, they form a couple of moment

GH

G$$^2$$ + H$$^2$$

$$\sqrt {{G^2} + {H^2}}$$

None of these

## Questions 33 of 50

Question:A and B are two like parallel forces. A couple of moment H lies in the plane of A and B and is contained with them. The resultant of A and B after combining is displaced through a distance

$$\frac{H}{{2(A + B)}}$$

$$\frac{H}{{A - B}}$$

$$\frac{{2H}}{{A - B}}$$

$$\frac{H}{{A + B}}$$

## Questions 34 of 50

Question:A heavy uniform rod, 15cm long, is suspended from a fixed point by strings fastened to its ends, their lengths being 9 and 12 cm. If $$\theta$$ be the angle at which the rod is inclined to the vertical, then $$\sin \theta =$$

$$\frac{4}{5}$$

$$\frac{8}{9}$$

$$\frac{{19}}{{20}}$$

$$\frac{{24}}{{25}}$$

## Questions 35 of 50

Question:A light string of length l is fastened to two points A and B at the same level at a distance 'a' apart. A ring of weight W can slide on the string, and a horizontal force P is applied to it such that the ring is in equilibrium vertically below B. The tension in the string is equal to

$$\frac{{aW}}{l}$$

laW

$$\frac{{W({l^2} + {a^2})}}{{2{l^2}}}$$

$$\frac{{2W({l^2} + {a^2})}}{{2{a^2}}}$$

## Questions 36 of 50

Question:Three coplanar forces each of weight 10 kilogram are acting at a particle. If their line of actions make same angle, then their resultant force will be

Zero

$$5\sqrt 2$$

$$10\sqrt 2$$

20

## Questions 37 of 50

Question:A circular ring having radius 3 cm hanged 4 cm above the centre of ring by four strings on equal interval of circumference. If angle between two consecutive strings is $$\theta$$ , then $$\cos \theta$$ will be

$$\frac{4}{5}$$

$$\frac{4}{{25}}$$

$$\frac{{16}}{{25}}$$

None of these

## Questions 38 of 50

Question:A uniform triangular lamina whose sides are of lengths 3 cm, 4 cm and 5 cm, is suspended by a string tied at the middle point of the largest side. In equilibrium position the inclination of this side to the vertical is

$${\sin ^{ - 1}}\left( {\frac{{24}}{{25}}} \right)$$

$${\sin ^{ - 1}}\left( {\frac{{12}}{{25}}} \right)$$

$${\cos ^{ - 1}}\left( {\frac{7}{{25}}} \right)$$

None of these

1 and 3 are correct

## Questions 39 of 50

Question:If the sum of the resolved parts of a system of coplanar forces along two mutually perpendicular direction is zero, then the sum of the moment of the forces about a given point

Is zero always

Is positive always

Is negative always

May have any value

## Questions 40 of 50

Question:In a right angled triangle one side is thrice the other side in length. The triangle is suspended by a string attached at the right angle. The angle that the hypotenuse of the triangle will make with the vertical is

$${\sin ^{ - 1}}(3/5)$$

$${\sin ^{ - 1}}(4/5)$$

$${60^o}$$

None of these

## Questions 41 of 50

Question:A body of weight 40 kg rests on a rough horizontal plane, whose coefficient of friction is 0.25. The least force which acting horizontally would move the body is

10 kg wt

20 kg wt

30 kg wt

40 kg wt

## Questions 42 of 50

Question:A hemi-spherical shell rests on a rough inclined plane, whose angle of friction is $$\lambda$$, the inclination of the plane base of the rim to the horizon cannot be greater than

$${\sin ^{ - 1}}(2\sin \lambda )$$

$${\cos ^{ - 1}}(2\cos \lambda )$$

$${\tan ^{ - 1}}(2\tan \lambda )$$

$${\cot ^{ - 1}}(2\cot \lambda )$$

## Questions 43 of 50

Question:A uniform ladder of length 70m and weight W rests against a vertical wall at an angle of $${45^o}$$with the wall. The coefficient of friction of the ladder with the ground and the wall are $$\frac{1}{2}$$ and $$\frac{1}{3}$$respectively. A man of weight $$\frac{W}{2}$$ climbs the ladder without slipping. The height in metre to which he can climb is

30

40

50

60

## Questions 44 of 50

Question:A body is on the point of sliding down an inclined plane under its own weight. If the inclination of the plane to the horizon be $${30^o}$$, the angle of friction will be

$${30^o}$$

$${60^o}$$

$${45^o}$$

$${15^o}$$

## Questions 45 of 50

Question:The end of a heavy uniform rod AB can slide along a rough horizontal rod AC to which it is attached by a ring. B and C are joined by a string. If $$\angle ABC$$be a right angle, when the rod is on the point of sliding,$$\mu$$ the coefficient of friction and $$\alpha$$ the angle between AB and the vertical, then

$$\mu = \frac{{2\tan \alpha }}{{(2 + {{\tan }^2}\alpha )}}$$

$$\mu = \frac{{\tan \alpha }}{{(2 + {{\tan }^2}\alpha )}}$$

$$\mu = \frac{{2\cot \alpha }}{{(1 + {{\cot }^2}\alpha )}}$$

$$\mu = \frac{{\cot \alpha }}{{(2 + {{\cot }^2}\alpha )}}$$

## Questions 46 of 50

Question:AB is a straight line of length 150cm. Two particles of masses 1kg and 3kg are placed at a distance of 15cm form A and 50 cm from B respectively. The distance of the third particle of mass 2kg from A, so that the C.G. of the system is at the middle point of AB is

40 cm

50 cm

67.5 cm

None of these

## Questions 47 of 50

Question:A square hole is punched out of a circular lamina of diameter 4cm, the diagonal of the square being a radius of the circle. Centroid of the remainder from the centre of the circle is at a distance

$$\frac{1}{{2\pi + 1}}$$

$$\frac{1}{{2\pi - 1}}$$

$$\frac{1}{{\pi + 1}}$$

$$\frac{1}{{\pi - 1}}$$

## Questions 48 of 50

Question:The centre of gravity of the surface of a hollow cone lies on the axis and divides it in the ratio

1:2

1:3

2:3

1:1

## Questions 49 of 50

Question:A body consists of a solid cylinder with radius a and height a together with a solid hemisphere of radius a placed on the base of the cylinder. The centre of gravity of the complete body is

Inside the cylinder

Inside the hemisphere

On the interface between the two

Outside both

## Questions 50 of 50

Question:The centre of gravity G of three particles of equal mass placed at the three vertices of a right angled isosceles triangle whose hypotenuse is equal to 8 cm is on the median through A such that AG is

$$\frac{4}{3}$$
$$\frac{5}{3}$$
$$\frac{8}{3}$$
$$\frac{{10}}{3}$$