# Statistics Test 2

Total Questions:50 Total Time: 75 Min

Remaining:

## Questions 1 of 50

Question:When two equal forces are inclined at an angle $$2\alpha$$ , their resultant is twice as great as when they act at an angle $$2\beta$$ , then

$$\cos \alpha = 2\sin \beta$$

$$\cos \alpha = 2\cos \beta$$

$$\cos \beta = 2\cos \alpha$$

$$\sin \beta = 2\cos \alpha$$

## Questions 2 of 50

Question:Two forces of 13 N and $$3\sqrt 3$$ N act on a particle at an angle $$\theta$$ and are equal to a resultant force of 14N, the angle between the forces is

$${30^o}$$

$${60^o}$$

$${45^o}$$

$${90^o}$$

## Questions 3 of 50

Question:If two forces $$P + Q$$ and $$P - Q$$ make an angle $$2\alpha$$ with each other and their resultant makes an angle $$\theta$$ with the bisector of the angle between the two forces, then $$\frac{P}{Q}$$ is equal to

$$\frac{{\tan \theta }}{{\tan \alpha }}$$

$$\frac{{\tan \alpha }}{{\tan \theta }}$$

$$\frac{{\sin \theta }}{{\sin \alpha }}$$

$$\frac{{\sin \alpha }}{{\sin \theta }}$$

## Questions 4 of 50

Question:The direction of three forces 1N, 2N and 3N acting at a point are parallel to the sides of an equilateral triangle taken in order, The magnitude of their resultant is

$$\frac{{\sqrt 3 }}{2}N$$

3N

$$\sqrt 3 N$$

$$\frac{3}{2}N$$

## Questions 5 of 50

Question:Forces of magnitudes 5, 10, 15 and 20 Newton act on a particle in the directions of North, South, East and West respectively. The magnitude of their resultant is

$$15\sqrt 2 N$$

$$10N$$

$$25\sqrt 2 N$$

$$5\sqrt 2 N$$

## Questions 6 of 50

Question:Forces of magnitudes $$P - Q,P$$ and $$P + Q$$ act at a point parallel to the sides of an equilateral triangle taken in order. The resultant of these forces, is

$$\sqrt 3 P$$

$$\sqrt 3 Q$$

$$3\sqrt 3 P$$

$$3P$$

## Questions 7 of 50

Question:If the resultant of two forces P and Q is $$\sqrt 3 Q$$ and makes an angle 30o with the direction of P, then

$$P = 2Q$$

$$Q = 2P$$

$$P = 3Q$$

None of these

## Questions 8 of 50

Question:The resolved part of a force of 16 Newton in a direction is $$8\sqrt 3$$ Newton. The inclination of the direction of the resolved part with the direction of the force is

$${30^o}$$

$${60^o}$$

$${120^o}$$

$${150^o}$$

## Questions 9 of 50

Question:Let P, 2P and 3P be the forces acting along AB, BC, CA of an equilateral $$\Delta ABC$$ . Suppose R is the magnitude of their resultant and $$\theta$$ the angle made by the resultant with the side BC, then

$$R = P\sqrt 3 ,\theta = \frac{\pi }{2}$$

$$R = 2P\sqrt 3 ,\theta = \frac{\pi }{2}$$

$$R = P\sqrt 3 ,\theta = \frac{\pi }{6}$$

$$R = 2P\sqrt 3 ,\theta = \frac{\pi }{6}$$

## Questions 10 of 50

Question:When a particle be kept at rest under the action of the following forces

$$\uparrow 8N, \uparrow 5N, \downarrow 13N$$

$$\uparrow 7N, \uparrow 4N, \downarrow 12N$$

$$\uparrow 5N, \uparrow 8N, \downarrow 10N$$

$$\uparrow 4N, \uparrow 2\sqrt 5 N, \downarrow 6N$$

## Questions 11 of 50

Question:In a triangle ABC three forces of magnitudes $$3\overrightarrow {AB} ,\;2\overrightarrow {AC}$$ and $$6\overrightarrow {CB}$$ are acting along the sides AB, AC and CB respectively. If the resultant meets AC at $$D$$ , then the ratio DC : AD will be equal to

1:1

1:2

1:3

1:4

## Questions 12 of 50

Question:If the forces of 12, 5 and 13 unit-wt balance at a point, two of them are inclined at

$${30^o}$$

$${45^o}$$

$${90^o}$$

$${60^o}$$

## Questions 13 of 50

Question:A particle is acted upon by three forces P, Q and R. It cannot be in equilibrium, if $$P:Q:R$$ =

$$1:3:5$$

$$3:5:7$$

$$5:7:9$$

$$7:9:11$$

## Questions 14 of 50

Question:Forces of 7 N, 5N and 3N acting on a particle are in equilibrium, the angle between the pair of forces 5 and 3 is

$${30^o}$$

$${60^o}$$

$${90^o}$$

$${120^o}$$

## Questions 15 of 50

Question:With two forces acting at a point, the maximum effect is obtained when their resultant is 4N. If they act at right angles, then their resultant is 3N. Then the forces are

$$\left( {2 + \frac{1}{2}\sqrt 3 } \right)N$$ and $$\left( {2 - \frac{1}{2}\sqrt 3 } \right)N$$

$$(2 + \sqrt 3 )N$$ and $$(2 - \sqrt 3 )N$$

$$\left( {2 + \frac{1}{2}\sqrt 2 } \right)N$$ and $$\left( {2 - \frac{1}{2}\sqrt 2 } \right)N$$

$$(2 + \sqrt 2 )N$$ and $$(2 - \sqrt 2 )N$$

## Questions 16 of 50

Question:The resultant of two forces P and Q is equal to $$\sqrt 3 Q$$ and makes an angle of $${30^o}$$ with the direction of P, then $$\frac{P}{Q} =$$

1 or 2

3 or 5

3 or 4

4 or 5

## Questions 17 of 50

Question:Two men carry a weight of 240 Newton between them by means of two ropes fixed to the weight. One rope is inclined at $${60^o}$$ to the vertical and the other at $${30^o}$$ . The tensions in the ropes are

$$120N,\,120N$$

$$120N,120\sqrt 3 N$$

$$120\sqrt 3 N,120\sqrt 3 N$$

None of these

## Questions 18 of 50

Question:A string ABC has its extremities tied to two fixed points A and B in the same horizontal line. If a weight W is knotted at a given point C, then the tension in the portion CA is (where a, b, c are the sides and $$\Delta$$ is the area of triangle ABC)

$$\frac{{Wb}}{{4c\Delta }}({a^2} + {b^2} + {c^2})$$

$$\frac{{Wb}}{{4c\Delta }}({b^2} + {c^2} - {a^2})$$

$$\frac{{Wb}}{{4c\Delta }}({c^2} + {a^2} - {b^2})$$

$$\frac{{Wb}}{{4c\Delta }}({a^2} + {b^2} - {c^2})$$

## Questions 19 of 50

Question:Forces M and N acting at a point O make an angle 150$$^0$$. Their resultant acts at O has magnitude 2 unit and is perpendicular to M. Then, in the same unit, the magnitudes of M and N are

$$2\sqrt 3 ,4$$

$$\sqrt {\frac{3}{2}} ,2$$

3, 4

4. 5

## Questions 20 of 50

Question:If the resultant of two forces of magnitude P and 2P is perpendicular to P, then the angle between the forces is

2$$\pi$$/3

3$$\pi$$/4

4$$\pi$$/5

5$$\pi$$/6

## Questions 21 of 50

Question:Two forces X and Y have a resultant F and the resolved part of F in the direction of X is of magnitude Y. Then the angle between the forces is

$${\sin ^{ - 1}}\sqrt {\frac{X}{{2Y}}}$$

$$2{\sin ^{ - 1}}\sqrt {\frac{X}{{2Y}}}$$

$$4{\sin ^{ - 1}}\sqrt {\frac{X}{{2Y}}}$$

None of these

## Questions 22 of 50

Question:The greatest and least magnitude of the resultant of two forces of constant magnitude are F and G. When the forces act an angle 2$$\alpha$$, the resultant in magnitudes is equal to

$$\sqrt {{F^2}{{\cos }^2}\alpha + {G^2}{{\sin }^2}\alpha }$$

$$\sqrt {{F^2}\sin \alpha + {G^2}{{\cos }^2}\alpha }$$

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

$$\sqrt {{F^2} - {G^2}}$$

## Questions 23 of 50

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

$$P + Q + R = 0$$

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

$$aP + bQ + cR = 0$$

None of these

## Questions 24 of 50

Question:If the resultant of two unlike parallel forces of magnitudes 10 N and 16 N act along a line at a distance of 24 cm from the line of action of the smaller force, then the distance between the lines of action of the forces is

12 cm

8 cm

9 cm

18 cm

## Questions 25 of 50

Question:If the position of the resultant of two like parallel forces P and Q is unaltered, when the positions of P and Q are interchanged, then

$$P = Q$$

$$P = 2Q$$

$$2P = Q$$

None of these

## Questions 26 of 50

Question:Three parallel forces P,Q, R act at three points A,B, C of a rod at distances of 2m, 8m and 6m respectively from one end. If the rod be in equilibrium, then $$P:Q:R$$=

$$1:2:3$$

$$2:3:1$$

$$3:2:1$$

None of these

## Questions 27 of 50

Question:The resultant of two like parallel forces is 12N. The distance between the forces is 18m. If one of the force is 4N, then the distance of the resultant from the smaller force is

4m

8m

12m

None of these

## Questions 28 of 50

Question:In a right angle $$\Delta ABC,$$ $$\angle A = {90^o}$$ and sides a, b, c are respectively 5 cm, 4 cm and 3 cm. If a force $$\overrightarrow F$$ has moments 0, 9 and 16 in N-cm respectively about vertices A, B and C, then magnitude of $$\overrightarrow F$$ is

9

4

5

3

## Questions 29 of 50

Question:If the forces 6W, 5W acting at a point (2, 3) in cartesian rectangular co-ordinates are parallel to the positive X and Y axis respectively, then the moment of the resultant force about the origin is

8W

3W

3W

8W

## Questions 30 of 50

Question:A man carries a hammer on his shoulder and holds it at the other end of its light handle in his hand. If he changes the point of support of the handle at the shoulder and if x is the distance between his hand and the point of support, then the pressure on his shoulder is proportional to

x

$${x^2}$$

$$1/x$$

$$1/{x^2}$$

## Questions 31 of 50

Question:If a couple is acting on 2 particles of mass 1 kg attached with a rigid rod of length 4m, fixed at centre, acting at the end and the angular acceleration of system about centre is 1 rad/s2 , then magnitude of force is

2N

4N

1N

None of these

## Questions 32 of 50

Question:Two like parallel forces P and Q act on a rigid body at A and B respectively. If P and Q be interchanged in positions, show that the point of application of the resultant will be displaced through a distance along AB, where d =

$$\frac{{P + Q}}{{P - Q}}AB$$

$$\frac{{2P + Q}}{{2P - Q}}AB$$

$$\frac{{P - Q}}{{P + Q}}AB$$

$$\frac{{P - Q}}{{2P + Q}}AB$$

## Questions 33 of 50

Question:A body of 6.5 kg is suspended by two strings of lengths 5 and 12 metre attached to two points in the same horizontal line whose distance apart is 13 m. The tension of the strings in kg wt are

3,5

2.5, 6

4, 5

3, 4

## Questions 34 of 50

Question:A sphere of radius r and weight W rests against a smooth vertical wall, to which is attached a string of length l where one end is fastened to a point on the wall and the other to the surface of the sphere. Then the tension in the string is

$$\frac{{W(l - r)}}{{\sqrt {({l^2} + 2lr)} }}$$

$$\frac{{W(l - r)}}{{l + r}}$$

$$\frac{{W(l + r)}}{{\sqrt {({l^2} + 2lr)} }}$$

None of these

## Questions 35 of 50

Question:A system of five forces whose directions and non-zero magnitudes can be chosen arbitrarily, will never be in equilibrium if n of the forces are concurrent, where

$$n = 2$$

$$n = 3$$

$$n = 4$$

$$n = 5$$

## Questions 36 of 50

Question:A uniform rod of weight W and length 2l is resting in a smooth spherical bowl of radius r. The rod is inclined to the horizontal at an angle of

$${0^o}$$

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

$${\tan ^{ - 1}}\left( {\frac{l}{r}} \right)$$

$$\frac{l}{{\sqrt {({r^2} - {l^2})} }}$$

## Questions 37 of 50

Question:There are three coplanar forces acting on a rigid body. If these are in equilibrium, then they are

Parallel

Concurrent

Concurrent or parallel

All of these

## Questions 38 of 50

Question:A uniform beam of length 2a rests in equilibrium against a smooth vertical plane and over a smooth peg at a distance h from the plane. If $$\theta$$ be the inclination of the beam to the vertical, then $${\rm{si}}{{\rm{n}}^{\rm{3}}}\theta$$is

$$\frac{h}{a}$$

$$\frac{{{h^2}}}{{{a^2}}}$$

$$\frac{a}{h}$$

$$\frac{{{a^2}}}{{{h^2}}}$$

## Questions 39 of 50

Question:A rough plane is inclined at an angle $$\alpha$$ to the horizon. A body is just to slide due to its own weight. The angle of friction would be

$${\tan ^{ - 1}}\alpha$$

$$\alpha$$

$$\tan \alpha$$

$$2\alpha$$

## Questions 40 of 50

Question:A particle is resting on a rough inclined plane with inclination $$\alpha$$. The angle of friction is $$\lambda$$ , the particle will be at rest if and only if,

$$\alpha > \lambda$$

$$\alpha \ge \lambda$$

$$\alpha \le \lambda$$

$$\alpha < \lambda$$

## Questions 41 of 50

Question:The relation between the coefficient of friction $$(\mu )$$ and the angle of friction $$(\lambda )$$ is given by

$$\mu = \cos \lambda$$

$$\mu = \sin \lambda$$

$$\mu = \tan \lambda$$

$$\mu = \cot \lambda$$

## Questions 42 of 50

Question:A rough inclined plane has its angle of inclination equal to $${45^o}$$ and $$\mu = 0.5$$ .The magnitude of the least force in kg wt, parallel to the plane required to move a body of 4kg up the plane is

$$3\sqrt 2$$

$$2\sqrt 2$$

$$\sqrt 2$$

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

## Questions 43 of 50

Question:A solid cone of semi- vertical angle $$\theta$$ is placed on a rough inclined plane. If the inclination of the plane is increased slowly and $$\mu < 4$$ $$\tan \theta$$ , then

Cone will slide down before toppling

Cone will topple before sliding down

Cone will slide and topple simultaneously

Cone will rest in limiting equilibrium

## Questions 44 of 50

Question:A circular cylinder of radius r and height h rests on a rough horizontal plane with one of its flat ends on the plane. A gradually increasing horizontal force is applied through the centre of the upper end. If the coefficient of friction is $$\mu$$ , the cylinder will topple before sliding, if

$$r < \mu h$$

$$r \ge \mu h$$

$$r \ge 2\mu h$$

$$r = 2\mu h$$

## Questions 45 of 50

Question:A uniform beam AB of weight W is standing with the end B on a horizontal floor and end A leaning against a vertical wall. The beam stands in a vertical plane perpendicular to the wall inclined at $${45^o}$$ to the vertical, and is in the position of limiting equilibrium. If the two points of contact are equally rough, then the coefficient of friction at each of them is

$$\sqrt 2 - 1$$

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

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

None of these

## Questions 46 of 50

Question:A body is pulled up an inclined rough plane. Let $$\lambda$$ be the angle of friction. The required force is least when it makes an angle $$k\lambda$$ with the inclined plane, where $$k =$$

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

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

1

2

## Questions 47 of 50

Question:A force of 35 kg is required to pull a block of wood weighing 140 kg on a rough horizontal surface. The coefficient of friction is

1

0

4

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

## Questions 48 of 50

Question:Weights 2, 3, 4 and 5 lbs are suspended from a uniform lever 6 ft long at distances of 1, 2, 3 and 4 ft from one end. If the weight of the lever is 11 lbs, then the distance of the point at which it will balance from this end is

$$\frac{{53}}{{25}}$$

$$\frac{{63}}{{25}}$$

$$\frac{{73}}{{25}}$$

None of these

## Questions 49 of 50

Question:On the same base AB, and on opposite side of it, isosceles triangles CAB and DAB are described whose altitudes are 12 cm and 6 cm respectively. The distance of the centre of gravity of the quadrilateral CADB from AB, is

0.5 cm

1cm

1.5cm

2cm

## Questions 50 of 50

Question:In a triangle ABC right angled at C, the lengths of sides AC and BC are 3 cm and 4 cm respectively. Parallel forces each equal to P act at the vertices A,B,C and parallel forces each equal to 2P act at the middle points of all the sides of the triangle. The distance of C.G. from the vertex C is

$$\frac{2}{3}$$ cm
$$\frac{4}{3}$$ cm
$$\frac{5}{3}$$ cm