# Statistics Test 3

Total Questions:50 Total Time: 75 Min

Remaining:

## Questions 1 of 50

Question:The resultant of two forces 3P and 2P is R, if the first force is doubled, the resultant is also doubled. The angle between the forces is [MNR 1985; UPSEAT 2000]

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

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

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

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

## Questions 2 of 50

Question:The resultant of two forces $$\overrightarrow P$$ and $$\overrightarrow Q$$ is of magnitude P. If the force $$\overrightarrow P$$ is doubled, $$\overrightarrow Q$$ remaining unaltered, the new resultant will be

Along $$\overrightarrow P$$

Along $$\overrightarrow Q$$

At $${60^o}$$ to $$\overrightarrow Q$$

At right angle to $$\overrightarrow Q$$

## Questions 3 of 50

Question:If the resultant of two forces 2P and $$\sqrt 2 P$$ is $$\sqrt {10} P$$ , then the angle between them will be

$$\pi$$

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

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

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

## Questions 4 of 50

Question:Two forces acting in opposite directions on a particle have a resultant of 34 Newton; if they acted at right angles to one another, their resultant would have a magnitude of 50 Newton. The magnitude of the forces are

48, 14

42, 8

40, 6

36, 2

## Questions 5 of 50

Question:Three forces of magnitude 30, 60 and P acting at a point are in equilibrium. If the angle between the first two is $${60^o}$$ , the value of P is

$$30\sqrt 7$$

$$30\sqrt 3$$

$$20\sqrt 6$$

$$25\sqrt 2$$

## Questions 6 of 50

Question:The resultant of two forces P and Q acting at an angle $$\theta$$ is equal to $$(2m + 1)$$ $$\sqrt {{P^2} + {Q^2}}$$ ; when they act at an angle $${90^o} - \theta$$ , the resultant is $$(2m - 1)$$ $$\sqrt {{P^2} + {Q^2}}$$ ; then $$\tan \theta$$ =

$$\frac{1}{m}$$

$$\frac{{m + 1}}{{m - 1}}$$

$$\frac{{m - 1}}{{m + 1}}$$

$$\sqrt {1 + {m^2}}$$

## Questions 7 of 50

Question:If forces of magnitude P, Q and R act at a point parallel to the sides BC, CA and AB respectively of a $$\Delta ABC$$ , then the magnitude of their resultant is

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

$$\sqrt {{P^2} + {Q^2} + {R^2} - 2PQ\cos C - 2QR\cos A - 2PR\cos B}$$

$$P + Q + R$$

None of these

## Questions 8 of 50

Question:Two forces of magnitudes $$P + Q$$ and $$P - Q$$ Newton are acting at an angle of $${135^o}$$ . If their resultant is a force of 2 Newton perpendicular to the line of action of the second force, then

$$P = (\sqrt 2 + 1),Q = (\sqrt 2 - 1)$$

$$P = (\sqrt 2 - 1),Q = (\sqrt 2 + 1)$$

$$P = (\sqrt 3 + 1),Q = (\sqrt 3 - 1)$$

$$P = (\sqrt 3 - 1),Q = (\sqrt 3 + 1)$$

## Questions 9 of 50

Question:Forces of 1, 2 unit act along the lines $$x = 0$$ and $$y = 0$$ . The equation of the line of action of the resultant is

$$y - 2x = 0$$

$$2y - x = 0$$

$$y + x = 0$$

$$y - x = 0$$

## Questions 10 of 50

Question:If N is resolved in two components such that first is twice of other, the components are

$$5N,5\sqrt 2 N$$

$$10N,10\sqrt 2 N$$

$$\frac{N}{{\sqrt 5 }},\frac{{2N}}{{\sqrt 5 }}$$

None of these

## Questions 11 of 50

Question:O is the circumcentre of $$\Delta ABC$$ . If the forces P, Q and R acting along OA, OB, and OC are in equilibrium then P : Q : R is

$$\sin A:\sin B:\sin C$$

$$\cos A:\cos B:\cos C$$

$$a\cos A:b\cos B:c\cos C$$

$$a\sec A:b\sec B:c\sec C$$

## Questions 12 of 50

Question:Three forces P, Q and R acting on a particle are in equilibrium. If the angle between P and Q is double the angle between P and R, then P is equal to

$$\frac{{{Q^2} + {R^2}}}{R}$$

$$\frac{{{Q^2} - {R^2}}}{Q}$$

$$\frac{{{Q^2} - {R^2}}}{R}$$

$$\frac{{{Q^2} + {R^2}}}{Q}$$

## Questions 13 of 50

Question:Three forces P, Q, R are acting at a point in a plane. The angle between P, Q and Q, R are $${150^o}$$ and $${120^o}$$ respectively, then for equilibrium; forces P, Q, R are in the ratio

1:02:03

1 : 2: 3$${1/2}$$

3:02:01

$${(3)^{1/2}}:2:1\ ## Questions 14 of 50 Question:Three forces keep a particle in equilibrium. One acts towards west, another acts towards north-east and the third towards south. If the first be 5N, then other two are ### Answers Choices: \(5\sqrt 2 N,\,5\sqrt 2 N$$

$$5\sqrt 2 N,\,5N$$

$$5N,\,5N$$

None of these

## Questions 15 of 50

Question:A particle is attracted to three points A, B and C by forces equal to $$\overrightarrow {PA} ,\overrightarrow {PB}$$ and $$\overrightarrow {PC}$$ respectively such that their resultant is $$\lambda \overrightarrow {PG} ,$$ where G is the centroid of $$\Delta ABC$$ . Then $$\lambda =$$

1

2

3

None of these

## Questions 16 of 50

Question:Three forces of magnitudes 8 N, 5N and 4N acting at a point are in equilibrium, then the angle between the two smaller forces is

$${\cos ^{ - 1}}\left( {\frac{{23}}{{40}}} \right)$$

$${\cos ^{ - 1}}\left( {\frac{{ - 23}}{{40}}} \right)$$

$${\sin ^{ - 1}}\left( {\frac{{23}}{{40}}} \right)$$

None of these

## Questions 17 of 50

Question:The resultant of forces P and Q acting at a point including a certain angle $$\alpha$$ is R, that of the forces $$2P$$ and Q acting at the same angle is $$2R$$ and that of $$P$$ and $$2Q$$ acting at the supplementary angle is $$2R$$ . Then $$P:Q:R =$$

1:02:03

$$\sqrt 6 :\sqrt 2 :\sqrt 5$$

$$\sqrt 2 :\sqrt 3 :\sqrt 5$$

None of these

## Questions 18 of 50

Question:The resultant of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the magnitude of the other. The ratio of the larger force to the smaller is

$$3:2\sqrt 2$$

$$3\sqrt 3 :2$$

$$3:2$$

$$4:3$$

## Questions 19 of 50

Question:$$ABCD$$ is a rigid square, on which forces 2, 3 and 5 kg-wt act along AB, AD and CA respectively. Then the magnitude of the resultant correct to one decimal place in kg-wt is

1

2

1.6

None of these

## Questions 20 of 50

Question:A horizontal force F is applied to a small object P of mass m on a smooth plane inclined to the horizon at an angle q. If F is just enough to keep P in equilibrium, then F =

$$mg{\cos ^2}\theta$$

$$mg{\sin ^2}\theta$$

$$mg\cos \theta$$

$$mg\tan \theta$$

## Questions 21 of 50

Question:Like parallel forces act at the vertices A, B and C of a triangle ABC and are proportional to the lengths BC, AC and AB respectively. The centre of the force is at the

Centroid

Circum-centre

Incentre

None of these

## Questions 22 of 50

Question:Three forces P, Q, R act along the sides BC, CA, AB of a $$\Delta ABC$$ taken in order, if their resultant passes through the centroid of $$\Delta ABC,$$ then

$$P + Q + R = 0$$

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

$$\frac{P}{{\cos A}} + \frac{Q}{{\cos B}} + \frac{R}{{\cos C}} = 0$$

None of these

## Questions 23 of 50

Question:P, Q, R are the points on the sides BC, CA, AB of the triangle ABC such that $$BP:PC = CQ:QA = AR:RB = m:n$$ . If $$\Delta$$ denotes the area of the $$\Delta ABC$$ , then the forces $$\overrightarrow {AP} ,\overrightarrow {BQ} ,\overrightarrow {CR}$$ reduce to a couple whose moment is

$$2\frac{{m + n}}{{m - n}}\Delta$$

$$2\frac{{n - m}}{{n + m}}\Delta$$

$$2({m^2} - {n^2})\Delta$$

$$2({m^2} + {n^2})\Delta$$

## Questions 24 of 50

Question:If the resultant of forces P,Q,R acting along the sides BC, CA, AB of a $$\Delta ABC$$ passes through its circumcentre, then

$$P\sin A + Q\sin B + R\sin C = 0$$

$$P\cos A + Q\cos B + R\cos C = 0$$

$$P\sec A + Q\sec B + R\sec C = 0$$

$$P\tan A + Q\tan B + R\tan C = 0$$

## Questions 25 of 50

Question:The resultant of two unlike parallel forces of magnitude P each acting at a distance of p is a

Force P

Couple of moment p.P

Force 2P

Force $$\frac{P}{2}$$

## Questions 26 of 50

Question:Force forming a couple are of magnitude 12N each and the arm of the couple is 8m. The force of an equivalent couple whose arm is 6m is of magnitude

8N

16N

12N

4N

## Questions 27 of 50

Question:The resultant of three equal like parallel forces acting at the vertices of a triangle act at its

Incentre

Circumcentre

Orthocentre

Centroid

## Questions 28 of 50

Question:If the force acting along the sides of a triangle, taken in order, are equivalent to a couple, then the forces are

Equal

Proportional to sides of triangle

In equilibrium

In arithmetic progression

## Questions 29 of 50

Question:If two like parallel forces of $$\frac{P}{Q}$$ Newton and $$\frac{Q}{P}$$ Newton have a resultant of 2 Newton, then

$$P = Q$$

$$P = 2Q$$

$$2P = Q$$

None of these

## Questions 30 of 50

Question:Two parallel forces not having the same line of action form a couple if they are

Like and unlike

Like and equal

Unequal and unlike

Equal and unlike

## Questions 31 of 50

Question:Two unlike parallel forces P and Q act at points 5m apart. If the resultant force is 9N and acts at a distance of 10m from the greater force P, then

$$P = 16N,Q = 7N$$

$$P = 15N,Q = 6N$$

$$P = 27N,Q = 18N$$

$$P = 18N,Q = 9N$$

## Questions 32 of 50

Question:A force $$\sqrt 5$$ unit act along the line $$\frac{{x - 3}}{2} = \frac{{y - 4}}{{ - 1}}$$ , the moment of the force about the point (4, 1) along z-axis is

0

$$5\sqrt 5$$

$$- \sqrt 5$$

5

## Questions 33 of 50

Question:The height from the base of a pillar must be end B of a rope AB of given length be fixed so that a man standing on the ground and pulling at its other end with a given force may have the greatest tendency to make the pillar overturn is

AB

AB/2

$$AB/\sqrt 2$$

None of these

## Questions 34 of 50

Question:If each of the two unlike parallel forces P and Q (P > Q) acting at a distance d apart be increased by S, then the point of application of the resultant is moved through a distance

$$\frac{d}{{P - Q}}$$

$$\frac{S}{{P - Q}}$$

$$\frac{{Sd}}{{P - Q}}$$

$$\frac{S}{{d(P - Q)}}$$

## Questions 35 of 50

Question:A couple is of moment G and the force forming the couple is P. If P is turned through a right angle, the moment of the couple thus formed is H. If instead, the force P are turned an angle $$\alpha$$ , then the moment of couple becomes

$$G\sin \alpha - H\cos \alpha$$

$$H\cos \alpha + G\sin \alpha$$

$$G\cos \alpha + H\,\sin \alpha$$

$$H\sin \alpha - G\cos \alpha$$

## Questions 36 of 50

Question:There is a system of coplanar forces acting on a rigid body represented in magnitude, direction and sense by the sides of a polygon taken in order, then the system is equivalent to

A single non-zero force

A zero force

A couple, where moment is equal to the area of polygon

A couple, where moment is twice the area of polygon

## Questions 37 of 50

Question:Three coplanar forces each equal to P, act at a point. The middle one makes an angle of $${60^o}$$ with each one of the remaining two forces. If by applying force Q at that point in a direction opposite to that of the middle force, equilibrium is achieved, then

$$P = Q$$

$$P = 2Q$$

$$2P = Q$$

None of these

## Questions 38 of 50

Question:A 2m long uniform rod ABC is resting against a smooth vertical wall at the end A and on a smooth peg at a point B. If distance of B from the wall is 0.3m, then

$$AB < 0.3m$$

$$AB < 1.0m$$

$$AB > 0.3m$$

$$AB > 1.0m$$

2 and 3 are correct

## Questions 39 of 50

Question:A uniform rod AB, 17m long whose mass is 120kg rests with one end against a smooth vertical wall and the other end on a smooth horizontal floor, this end being tied by a chord 8m long, to a peg at the bottom of the wall, then the tension of the chord is

32 kg wt

16 kg wt

64 kg wt

8 kg wt

## Questions 40 of 50

Question:A uniform rod AB of length a hangs with one end against a smooth vertical wall, being supported by a string of length l, attached to the other end of the rod and to a point of the rod vertically above B. If the rod rests inclined to the wall at an angle $$\theta$$ , then $${\cos ^2}\theta =$$

$$\frac{{({l^2} - {a^2})}}{{{a^2}}}$$

$$\frac{{({l^2} - {a^2})}}{{2{a^2}}}$$

$$\frac{{({l^2} - {a^2})}}{{3{a^2}}}$$

None of these

## Questions 41 of 50

Question:A body of weight W rests on a rough plane, whose coefficient of friction is $$\mu ( = \tan \lambda )$$ which is inclined at an angle $$\alpha$$ with the horizon. The least force required to pull the body up the plane is

$$2\,W\sin (\alpha + \lambda )$$

$$W\sin (\alpha + \lambda )$$

$$\,W\sin (\alpha - \lambda )$$

$$2\,W\sin (\alpha - \lambda )$$

## Questions 42 of 50

Question:The minimum force required to move a body of weight W placed on a rough horizontal plane surface is

$$W\sin \lambda$$

$$W\cos \lambda$$

$$W\tan \lambda$$

$$W\cot \lambda$$

## Questions 43 of 50

Question:A body of weight 4 kg is kept in a plane inclined at an angle of $${30^o}$$ to the horizontal. It is in limiting equilibrium. The coefficient of friction is then equal to

$$\frac{1}{{\sqrt 3 }}$$

$$\sqrt 3$$

$$\frac{1}{{4\sqrt 3 }}$$

$$\frac{{\sqrt 3 }}{4}$$

## Questions 44 of 50

Question:A weight W can be just supported on a rough inclined plane by a force P either acting along the plane or horizontally. The ratio$$\frac{P}{W}$$ , for the angle of friction $$\varphi$$ , is

$$\tan \varphi$$

$$\sec \varphi$$

$$\sin \varphi$$

None of these

## Questions 45 of 50

Question:A bar AB of weight W rests like a ladder, with upper end A against a smooth vertical wall and the lower end B on a rough horizontal plane. If the bar is just on the point of sliding, then the reaction at A is equal to $$(\mu$$ is the coefficient of friction)

$$\mu$$W

W

Normal reaction at B

W/$$\mu$$

## Questions 46 of 50

Question:A uniform ladder rests in limiting equilibrium, its lower end on a rough horizontal plane and its upper end against a smooth vertical wall. If q is the angle of inclination of the ladder to the vertical wall and $$\mu$$ is the coefficient of friction, then tanq is equal to

$$\mu$$

2 $$\mu$$

$$\frac{{3\mu }}{2}$$

$$\mu$$ + 1

## Questions 47 of 50

Question:A body of 6 kg rests in limiting equilibrium on an inclined plane whose slope is 30 $$^0$$. If the plane is raised to slope of 60 $$^0$$, the force in kg wt along the plane required to support it is

3

$$2\sqrt 3$$

$$\sqrt 3$$

$$3\sqrt 3$$

## Questions 48 of 50

Question:The coefficient of friction between the floor and a box weighing 1 ton if a minimum force of 600 kgf is required to start the box moving is

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

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

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

1

## Questions 49 of 50

Question:The C.G. of three particles placed at the vertices of a triangle is at its

Incentre

Centroid

Circumcentre

Orthocentre

## Questions 50 of 50

Question:In a circular disc of uniform metal sheet of radius 10 cm and centre O, two circular holes of radii 5cm and 2.5cm are punched. The centre G1 and G2 of the holes are on the same diameter of the circular disc. If G is the centre of gravity of the punched disc, then OG =

$$\frac{{22}}{{25}}cm$$
$$\frac{{55}}{{22}}cm$$
$$\frac{{25}}{{22}}cm$$