Maths-

General

Easy

Question

The area of an ellipse is 8π sq. units dist. between the foci is $4 square root of 3$ then e=

$s i n 30^{\circ}$
$s i n 45^{\circ}$
$s i n 60^{\circ}$
$s i n 75^{\circ}$

Hint:

Compare the two different formulas of eccentricity.

The correct answer is: $s i n space 60 to the power of ring operator$

$A r e a space o f space a n space e l l i p s e space equals space pi a b equals 8 pi space s q. space u n i t s therefore a b equals 8 space space space space..... open parentheses 1 close parentheses F o c i equals 2 a e equals 4 square root of 3 therefore a e equals fraction numerator 4 square root of 3 over denominator 2 end fraction equals 2 square root of 3 therefore e equals fraction numerator 2 square root of 3 over denominator space a end fraction space space space..... open parentheses 2 close parentheses e squared equals 1 minus b squared over a squared F r o m space e q n space 1 space a n d space 2 comma open parentheses fraction numerator 2 square root of 3 over denominator space a end fraction close parentheses squared equals 1 minus open parentheses 8 over a squared close parentheses squared open parentheses fraction numerator 2 square root of 3 over denominator space a end fraction close parentheses squared equals 1 minus open parentheses 64 over a to the power of 4 close parentheses open parentheses fraction numerator 12 over denominator space a squared end fraction close parentheses equals open parentheses fraction numerator a to the power of 4 minus 64 over denominator a to the power of 4 end fraction close parentheses 12 a squared equals a to the power of 4 minus 64 a to the power of 4 minus 12 a squared minus 64 equals 0 a to the power of 4 minus 16 a squared plus 4 a squared minus 64 equals 0 a squared equals 16 therefore a equals plus-or-minus 4 e equals fraction numerator 2 square root of 3 over denominator a end fraction equals fraction numerator 2 square root of 3 over denominator 4 end fraction equals fraction numerator square root of 3 over denominator 2 end fraction equals sin 60 degree$

A point $P$ moves in counter-clockwise direction on a circular path as shown in the figure. The movement of $´ ´ P ´ ´$ is such that it sweeps out a length $s equals t to the power of 3 end exponent plus 5$ , where $s$ is in metres and $t$ is in seconds. The radius of the path is $20 blank m$ . The acceleration of $´ ´ P ´ ´$ when $t equals 2 s$ is nearly

S equals t to the power of 3 end exponent plus 5

fraction numerator d s over denominator d t end fraction equals 3 t to the power of 2 end exponent equals v

therefore a subscript t end subscript equals fraction numerator d v over denominator d t end fraction equals 6 t

t equals 2 blank s e c

open vertical bar stack a with rightwards arrow on top close vertical bar equals square root of a subscript c end subscript superscript 2 end superscript plus a subscript t end subscript superscript 2 end superscript end root

equals square root of open parentheses fraction numerator v to the power of 2 end exponent over denominator R end fraction close parentheses to the power of 2 end exponent plus a subscript t end subscript superscript 2 end superscript end root equals square root of open parentheses fraction numerator 4 t to the power of 4 end exponent over denominator R end fraction close parentheses to the power of 2 end exponent plus open parentheses fraction numerator d v over denominator d t end fraction close parentheses to the power of 2 end exponent end root

equals square root of open parentheses 7.2 close parentheses to the power of 2 end exponent plus 144 end root

open vertical bar stack a with rightwards arrow on top close vertical bar equals 14 m divided by s to the power of 2 end exponent

A point $P$ moves in counter-clockwise direction on a circular path as shown in the figure. The movement of $´ ´ P ´ ´$ is such that it sweeps out a length $s equals t to the power of 3 end exponent plus 5$ , where $s$ is in metres and $t$ is in seconds. The radius of the path is $20 blank m$ . The acceleration of $´ ´ P ´ ´$ when $t equals 2 s$ is nearly

Physics-General

S equals t to the power of 3 end exponent plus 5

fraction numerator d s over denominator d t end fraction equals 3 t to the power of 2 end exponent equals v

therefore a subscript t end subscript equals fraction numerator d v over denominator d t end fraction equals 6 t

t equals 2 blank s e c

open vertical bar stack a with rightwards arrow on top close vertical bar equals square root of a subscript c end subscript superscript 2 end superscript plus a subscript t end subscript superscript 2 end superscript end root

equals square root of open parentheses fraction numerator v to the power of 2 end exponent over denominator R end fraction close parentheses to the power of 2 end exponent plus a subscript t end subscript superscript 2 end superscript end root equals square root of open parentheses fraction numerator 4 t to the power of 4 end exponent over denominator R end fraction close parentheses to the power of 2 end exponent plus open parentheses fraction numerator d v over denominator d t end fraction close parentheses to the power of 2 end exponent end root

equals square root of open parentheses 7.2 close parentheses to the power of 2 end exponent plus 144 end root

open vertical bar stack a with rightwards arrow on top close vertical bar equals 14 m divided by s to the power of 2 end exponent

General

physics-

In a two dimensional motion of a particle, the particle moves from point $A$ , position vector $stack r with rightwards arrow on top subscript 1 end subscript$ . If the magnitudes of these vectors are respectively, $r subscript 1 end subscript$ =3 and $r subscript 2 end subscript equals 4$ and the angles they make with the $x$ -axis are $theta subscript 1 end subscript equals 75 degree$ and 15 $degree$ , respectively, then find the magnitude of the displacement vector

Displacement

equals A B

angle between

stack r subscript 1 end subscript with rightwards arrow on top

and

stack r subscript 2 end subscript with rightwards arrow on top

theta equals 75 degree minus 15 degree equals 60 degree

From figure

A B to the power of 2 end exponent equals r subscript 1 end subscript superscript 2 end superscript plus r subscript 2 end subscript superscript 2 end superscript minus 2 r subscript 1 end subscript r subscript 2 end subscript c o s theta

equals 3 to the power of 2 end exponent plus 4 to the power of 2 end exponent minus 2 cross times 3 cross times 4

cos

60 degree

equals 13

A B equals square root of 13

In a two dimensional motion of a particle, the particle moves from point $A$ , position vector $stack r with rightwards arrow on top subscript 1 end subscript$ . If the magnitudes of these vectors are respectively, $r subscript 1 end subscript$ =3 and $r subscript 2 end subscript equals 4$ and the angles they make with the $x$ -axis are $theta subscript 1 end subscript equals 75 degree$ and 15 $degree$ , respectively, then find the magnitude of the displacement vector

physics-General

Displacement

equals A B

angle between

stack r subscript 1 end subscript with rightwards arrow on top

and

stack r subscript 2 end subscript with rightwards arrow on top

theta equals 75 degree minus 15 degree equals 60 degree

From figure

A B to the power of 2 end exponent equals r subscript 1 end subscript superscript 2 end superscript plus r subscript 2 end subscript superscript 2 end superscript minus 2 r subscript 1 end subscript r subscript 2 end subscript c o s theta

equals 3 to the power of 2 end exponent plus 4 to the power of 2 end exponent minus 2 cross times 3 cross times 4

cos

60 degree

equals 13

A B equals square root of 13

General

physics-

A block(B) is attached to two unstretched springs $S subscript 1 end subscript a n d blank S subscript 2 end subscript$ with springs constants $k blank a n d blank 4 k comma$ representively (see Fig. I) The other ends are attached to identical supports $M subscript 1 end subscript$ and $M subscript 2 end subscript$ not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere. The block $B$ is displaced towards wall I by small distance $x left parenthesis F i g blank I I right parenthesis$ and released. The block returns and moves a maximum distance y towards wall 2.Displacements $x blank a n d blank y$ are measured with respect to the equilibrium position of the block $B$ The ratio $fraction numerator y over denominator x end fraction$ is

From energy conservation,

fraction numerator 1 over denominator 2 end fraction k x to the power of 2 end exponent equals fraction numerator 1 over denominator 2 end fraction open parentheses 4 k close parentheses y to the power of 2 end exponent

fraction numerator y over denominator x end fraction equals fraction numerator 1 over denominator 2 end fraction

A block(B) is attached to two unstretched springs $S subscript 1 end subscript a n d blank S subscript 2 end subscript$ with springs constants $k blank a n d blank 4 k comma$ representively (see Fig. I) The other ends are attached to identical supports $M subscript 1 end subscript$ and $M subscript 2 end subscript$ not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere. The block $B$ is displaced towards wall I by small distance $x left parenthesis F i g blank I I right parenthesis$ and released. The block returns and moves a maximum distance y towards wall 2.Displacements $x blank a n d blank y$ are measured with respect to the equilibrium position of the block $B$ The ratio $fraction numerator y over denominator x end fraction$ is

physics-General

From energy conservation,

fraction numerator 1 over denominator 2 end fraction k x to the power of 2 end exponent equals fraction numerator 1 over denominator 2 end fraction open parentheses 4 k close parentheses y to the power of 2 end exponent

fraction numerator y over denominator x end fraction equals fraction numerator 1 over denominator 2 end fraction

General

physics-

A small roller coaster starts at point A with a speed $u blank$ on a curved track as shown in the figure.
The friction between the roller coaster and the track is negligible and it always remains in contact with the track. The speed of roller coaster at point $D blank$ on the track will be

There is no loss of energy. Therefore, the final velocity is the same as the initial velocity.
Hence, The speed of roller coaster at point D is

u

A small roller coaster starts at point A with a speed $u blank$ on a curved track as shown in the figure.
The friction between the roller coaster and the track is negligible and it always remains in contact with the track. The speed of roller coaster at point $D blank$ on the track will be

physics-General

There is no loss of energy. Therefore, the final velocity is the same as the initial velocity.
Hence, The speed of roller coaster at point D is

u

General

Physics-

A body of mass $m$ is moving with a uniform speed $v$ along a circle of radius $r$ , what is the average acceleration in going from $A$ to $B$ ?

Here,

T equals fraction numerator 2 pi r over denominator 4 v end fraction equals fraction numerator pi r over denominator 2 v end fraction

Change in velocity is going from

A

B

v square root of 2

Average acceleration

equals fraction numerator v square root of 2 over denominator pi r divided by 2 v end fraction equals fraction numerator 2 square root of 2 v to the power of 2 end exponent over denominator pi r end fraction

A body of mass $m$ is moving with a uniform speed $v$ along a circle of radius $r$ , what is the average acceleration in going from $A$ to $B$ ?

Physics-General

Here,

T equals fraction numerator 2 pi r over denominator 4 v end fraction equals fraction numerator pi r over denominator 2 v end fraction

Change in velocity is going from

A

B

v square root of 2

Average acceleration

equals fraction numerator v square root of 2 over denominator pi r divided by 2 v end fraction equals fraction numerator 2 square root of 2 v to the power of 2 end exponent over denominator pi r end fraction

General

physics-

A body of mass $M$ is moving with a uniform speed of $10 blank m divided by s$ on friction less surface under the influence of two forces $F subscript 1 end subscript$ and $F subscript 2 end subscript$ . The net power of the system is

because

Speed is constant

therefore

Work done by forces = 0

therefore

Power

equals fraction numerator W o r k over denominator T i m e end fraction equals 0

A body of mass $M$ is moving with a uniform speed of $10 blank m divided by s$ on friction less surface under the influence of two forces $F subscript 1 end subscript$ and $F subscript 2 end subscript$ . The net power of the system is

physics-General

because

Speed is constant

therefore

Work done by forces = 0

therefore

Power

equals fraction numerator W o r k over denominator T i m e end fraction equals 0

General

physics-

From an inclined plane two particles are projected with same speed at same angle $theta$ , one up and other down the plane as shown in figure, which of the following statements is/are correct?

H e r e comma blank alpha equals 2 theta comma blank beta equals theta

T i m e blank o f blank f l i g h t blank o f blank A blank i s comma

T subscript 1 end subscript equals fraction numerator 2 u sin invisible function application open parentheses alpha minus beta close parentheses over denominator g cos invisible function application beta end fraction

equals fraction numerator 2 u sin invisible function application open parentheses 2 theta minus theta close parentheses over denominator g cos invisible function application theta end fraction

equals fraction numerator 2 u over denominator g end fraction tan invisible function application theta

T i m e blank o f blank f l i g h t blank o f blank B blank i s comma blank T subscript 2 end subscript equals fraction numerator 2 u sin invisible function application theta over denominator g cos invisible function application theta end fraction

equals fraction numerator 2 u over denominator g end fraction tan invisible function application theta

So,

T subscript 1 end subscript equals T subscript 2 end subscript

. The acceleration of both the particles is

g

downwards. Therefore, relative acceleration between the two I zero or relative motion between the two is uniform. The relative velocity of

A

w.r.t.

B

is towards

A B

, therefore collision will take place between the two in mid air.

From an inclined plane two particles are projected with same speed at same angle $theta$ , one up and other down the plane as shown in figure, which of the following statements is/are correct?

physics-General

H e r e comma blank alpha equals 2 theta comma blank beta equals theta

T i m e blank o f blank f l i g h t blank o f blank A blank i s comma

T subscript 1 end subscript equals fraction numerator 2 u sin invisible function application open parentheses alpha minus beta close parentheses over denominator g cos invisible function application beta end fraction

equals fraction numerator 2 u sin invisible function application open parentheses 2 theta minus theta close parentheses over denominator g cos invisible function application theta end fraction

equals fraction numerator 2 u over denominator g end fraction tan invisible function application theta

T i m e blank o f blank f l i g h t blank o f blank B blank i s comma blank T subscript 2 end subscript equals fraction numerator 2 u sin invisible function application theta over denominator g cos invisible function application theta end fraction

equals fraction numerator 2 u over denominator g end fraction tan invisible function application theta

So,

T subscript 1 end subscript equals T subscript 2 end subscript

. The acceleration of both the particles is

g

downwards. Therefore, relative acceleration between the two I zero or relative motion between the two is uniform. The relative velocity of

A

w.r.t.

B

is towards

A B

, therefore collision will take place between the two in mid air.

General

physics-

A particle of a mass $0.1 blank k g$ is subjected to a force which varies with distance as shown in fig. If it starts its journey from rest at $x equals 0$ , its velocity at $x equals 12 blank m$ is

Area between curve and displacement axis

equals fraction numerator 1 over denominator 2 end fraction cross times open parentheses 12 plus 4 close parentheses cross times 10 equals 80 blank J

In this time body acquire kinetic energy

equals fraction numerator 1 over denominator 2 end fraction blank m v to the power of 2 end exponent

By the law of conservation of energy

fraction numerator 1 over denominator 2 end fraction m v to the power of 2 end exponent equals 80 blank J

rightwards double arrow fraction numerator 1 over denominator 2 end fraction blank cross times 0.1 cross times v to the power of 2 end exponent equals 80 rightwards double arrow v to the power of 2 end exponent equals 1600 rightwards double arrow v equals 40 blank m divided by s

A particle of a mass $0.1 blank k g$ is subjected to a force which varies with distance as shown in fig. If it starts its journey from rest at $x equals 0$ , its velocity at $x equals 12 blank m$ is

physics-General

Area between curve and displacement axis

equals fraction numerator 1 over denominator 2 end fraction cross times open parentheses 12 plus 4 close parentheses cross times 10 equals 80 blank J

In this time body acquire kinetic energy

equals fraction numerator 1 over denominator 2 end fraction blank m v to the power of 2 end exponent

By the law of conservation of energy

fraction numerator 1 over denominator 2 end fraction m v to the power of 2 end exponent equals 80 blank J

rightwards double arrow fraction numerator 1 over denominator 2 end fraction blank cross times 0.1 cross times v to the power of 2 end exponent equals 80 rightwards double arrow v to the power of 2 end exponent equals 1600 rightwards double arrow v equals 40 blank m divided by s

General

maths-

P is a point on the ellipse $fraction numerator x to the power of 2 end exponent over denominator 36 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 9 end fraction equals 1$ , S and $S to the power of 1 end exponent$ are the focii of the ellipse then $S P plus S to the power of l end exponent P equals$

maths-General

General

physics-

As per given future to complete the circular loop what should be the radius if initial height is $5 blank m$

h equals fraction numerator 5 over denominator 2 end fraction r rightwards double arrow r equals fraction numerator 2 over denominator 5 end fraction cross times h equals fraction numerator 2 over denominator 5 end fraction cross times 5 equals 2 blank m e t r e

As per given future to complete the circular loop what should be the radius if initial height is $5 blank m$

physics-General

h equals fraction numerator 5 over denominator 2 end fraction r rightwards double arrow r equals fraction numerator 2 over denominator 5 end fraction cross times h equals fraction numerator 2 over denominator 5 end fraction cross times 5 equals 2 blank m e t r e

General

physics-

Figure shows four paths for a kicked football. Ignoring the effects of air on the flight, rank the paths according to initial horizontal velocity component, highest first

R equals fraction numerator u to the power of 2 end exponent sin invisible function application 2 theta over denominator g end fraction equals fraction numerator 2 u subscript x end subscript v subscript y end subscript over denominator g end fraction

therefore

Range

proportional to

horizontal initial velocity

left parenthesis u subscript x end subscript right parenthesis

In path 4 range is maximum so football possess maximum horizontal velocity in the path

Figure shows four paths for a kicked football. Ignoring the effects of air on the flight, rank the paths according to initial horizontal velocity component, highest first

physics-General

R equals fraction numerator u to the power of 2 end exponent sin invisible function application 2 theta over denominator g end fraction equals fraction numerator 2 u subscript x end subscript v subscript y end subscript over denominator g end fraction

therefore

Range

proportional to

horizontal initial velocity

left parenthesis u subscript x end subscript right parenthesis

In path 4 range is maximum so football possess maximum horizontal velocity in the path

General

maths-

The area of the region bounded by $blank y equals open vertical bar x minus 1 close vertical bar blank$ and y=1 in sq. units is

maths-General

General

maths-

Area of the region bounded by $blank y equals open vertical bar x close vertical bar blank$ and y=2 is

maths-General

General

maths-

The area bounded by $blank y equals c o s blank x comma blank y equals x plus 1 blank$ and y=0 in the second quadrant is

maths-General

General

maths-

The area bounded by the two curves $blank y equals square root of x blank a n d blank x equals square root of y blank$ is

maths-General

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The area of an ellipse is 8π sq. units dist. between the foci is then e=

Compare the two different formulas of eccentricity.

The correct answer is:

Related Questions to study

A point moves in counter-clockwise direction on a circular path as shown in the figure. The movement of is such that it sweeps out a length , where is in metres and is in seconds. The radius of the path is . The acceleration of when is nearly

A point moves in counter-clockwise direction on a circular path as shown in the figure. The movement of is such that it sweeps out a length , where is in metres and is in seconds. The radius of the path is . The acceleration of when is nearly

In a two dimensional motion of a particle, the particle moves from point , position vector . If the magnitudes of these vectors are respectively, =3 and and the angles they make with the -axis are and 15, respectively, then find the magnitude of the displacement vector

In a two dimensional motion of a particle, the particle moves from point , position vector . If the magnitudes of these vectors are respectively, =3 and and the angles they make with the -axis are and 15, respectively, then find the magnitude of the displacement vector

A small roller coaster starts at point A with a speed on a curved track as shown in the figure.The friction between the roller coaster and the track is negligible and it always remains in contact with the track. The speed of roller coaster at point on the track will be

A small roller coaster starts at point A with a speed on a curved track as shown in the figure.The friction between the roller coaster and the track is negligible and it always remains in contact with the track. The speed of roller coaster at point on the track will be

A body of mass is moving with a uniform speed along a circle of radius , what is the average acceleration in going from to ?

A body of mass is moving with a uniform speed along a circle of radius , what is the average acceleration in going from to ?

A body of mass is moving with a uniform speed of on friction less surface under the influence of two forces and . The net power of the system is

A body of mass is moving with a uniform speed of on friction less surface under the influence of two forces and . The net power of the system is

From an inclined plane two particles are projected with same speed at same angle , one up and other down the plane as shown in figure, which of the following statements is/are correct?

From an inclined plane two particles are projected with same speed at same angle , one up and other down the plane as shown in figure, which of the following statements is/are correct?

A particle of a mass is subjected to a force which varies with distance as shown in fig. If it starts its journey from rest at , its velocity at is

A particle of a mass is subjected to a force which varies with distance as shown in fig. If it starts its journey from rest at , its velocity at is

P is a point on the ellipse , S and are the focii of the ellipse then

P is a point on the ellipse , S and are the focii of the ellipse then

As per given future to complete the circular loop what should be the radius if initial height is

As per given future to complete the circular loop what should be the radius if initial height is

Figure shows four paths for a kicked football. Ignoring the effects of air on the flight, rank the paths according to initial horizontal velocity component, highest first

Figure shows four paths for a kicked football. Ignoring the effects of air on the flight, rank the paths according to initial horizontal velocity component, highest first

The area of the region bounded by and y=1 in sq. units is

The area of the region bounded by and y=1 in sq. units is

Area of the region bounded byand y=2 is

Area of the region bounded byand y=2 is

The area bounded by and y=0 in the second quadrant is

The area bounded by and y=0 in the second quadrant is

The area bounded by the two curves is

The area bounded by the two curves is

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The area of an ellipse is 8π sq. units dist. between the foci is $4 square root of 3$ then e=

The correct answer is: $s i n space 60 to the power of ring operator$

A small roller coaster starts at point A with a speed $u blank$ on a curved track as shown in the figure.
The friction between the roller coaster and the track is negligible and it always remains in contact with the track. The speed of roller coaster at point $D blank$ on the track will be

A small roller coaster starts at point A with a speed $u blank$ on a curved track as shown in the figure.
The friction between the roller coaster and the track is negligible and it always remains in contact with the track. The speed of roller coaster at point $D blank$ on the track will be

A body of mass $m$ is moving with a uniform speed $v$ along a circle of radius $r$ , what is the average acceleration in going from $A$ to $B$ ?

A body of mass $m$ is moving with a uniform speed $v$ along a circle of radius $r$ , what is the average acceleration in going from $A$ to $B$ ?

A body of mass $M$ is moving with a uniform speed of $10 blank m divided by s$ on friction less surface under the influence of two forces $F subscript 1 end subscript$ and $F subscript 2 end subscript$ . The net power of the system is

A body of mass $M$ is moving with a uniform speed of $10 blank m divided by s$ on friction less surface under the influence of two forces $F subscript 1 end subscript$ and $F subscript 2 end subscript$ . The net power of the system is

From an inclined plane two particles are projected with same speed at same angle $theta$ , one up and other down the plane as shown in figure, which of the following statements is/are correct?

From an inclined plane two particles are projected with same speed at same angle $theta$ , one up and other down the plane as shown in figure, which of the following statements is/are correct?

A particle of a mass $0.1 blank k g$ is subjected to a force which varies with distance as shown in fig. If it starts its journey from rest at $x equals 0$ , its velocity at $x equals 12 blank m$ is

A particle of a mass $0.1 blank k g$ is subjected to a force which varies with distance as shown in fig. If it starts its journey from rest at $x equals 0$ , its velocity at $x equals 12 blank m$ is

As per given future to complete the circular loop what should be the radius if initial height is $5 blank m$

As per given future to complete the circular loop what should be the radius if initial height is $5 blank m$

The area of the region bounded by $blank y equals open vertical bar x minus 1 close vertical bar blank$ and y=1 in sq. units is

The area of the region bounded by $blank y equals open vertical bar x minus 1 close vertical bar blank$ and y=1 in sq. units is

Area of the region bounded by $blank y equals open vertical bar x close vertical bar blank$ and y=2 is

Area of the region bounded by $blank y equals open vertical bar x close vertical bar blank$ and y=2 is

The area bounded by $blank y equals c o s blank x comma blank y equals x plus 1 blank$ and y=0 in the second quadrant is

The area bounded by $blank y equals c o s blank x comma blank y equals x plus 1 blank$ and y=0 in the second quadrant is

The area bounded by the two curves $blank y equals square root of x blank a n d blank x equals square root of y blank$ is

The area bounded by the two curves $blank y equals square root of x blank a n d blank x equals square root of y blank$ is