Maths-
General
Easy

Question

If ϕ is the angle between the diameter through any point on a standard ellipse and the normal at the point, then the greatest value of tan ϕ is–

  1. fraction numerator 2 a b over denominator a to the power of 2 end exponent plus b to the power of 2 end exponent end fraction    
  2. fraction numerator a to the power of 2 end exponent plus b to the power of 2 end exponent over denominator a b end fraction    
  3. fraction numerator a to the power of 2 end exponent minus b to the power of 2 end exponent over denominator 2 a b end fraction    
  4. fraction numerator b to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction    

The correct answer is: fraction numerator a to the power of 2 end exponent minus b to the power of 2 end exponent over denominator 2 a b end fraction


    Any point P on ellipse is (a cos theta, b sin theta)
     Equation of the diameter CP is y = open parentheses fraction numerator b over denominator a end fraction tan invisible function application theta close parenthesesx
    The normal to ellipse at P is
    ax sec theta – by cosec theta = a2e2
    Slopes of the lines CP and the normal GP are fraction numerator b over denominator a end fractiontan theta andfraction numerator a over denominator b end fractiontan theta

     tan ϕ = fraction numerator fraction numerator a over denominator b end fraction tan invisible function application theta minus fraction numerator b over denominator a end fraction tan invisible function application theta over denominator 1 plus fraction numerator a over denominator b end fraction tan invisible function application theta. fraction numerator b over denominator a end fraction tan invisible function application theta end fraction=fraction numerator tan invisible function application theta over denominator sec to the power of 2 end exponent invisible function application theta end fraction
    =fraction numerator a to the power of 2 end exponent minus b to the power of 2 end exponent over denominator a b end fractionsin  cos  = fraction numerator a to the power of 2 end exponent minus b to the power of 2 end exponent over denominator 2 a b end fractionsin 2
     The greatest value of tan ϕ = fraction numerator a to the power of 2 end exponent minus b to the power of 2 end exponent over denominator 2 a b end fraction.1 = fraction numerator a to the power of 2 end exponent minus b to the power of 2 end exponent over denominator 2 a b end fraction .

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    when source is fixed and observer is moving towards it
    upsilon ´ equals fraction numerator c plus a over denominator c end fraction. upsilon
    when source is moving towards observer at rest
    upsilon " equals fraction numerator c over denominator c minus a end fraction v ´ equals fraction numerator c plus a over denominator c minus a end fraction. upsilon equals c open square brackets fraction numerator 1 plus fraction numerator a over denominator c end fraction over denominator 1 minus fraction numerator a over denominator c end fraction end fraction close square brackets upsilon
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    Four identical metal butterflies are hanging from a light string of length 5 l at equally placed points as shown in the figure . The ends of the string are attached to a horizontal fixed support. The middle section of the string is horizontal. The relation between the angle theta subscript 1 end subscript and theta subscript 2 end subscript is given by

    table row cell T subscript 1 end subscript s i n invisible function application theta subscript 1 end subscript equals 2 m g end cell row cell T subscript 2 end subscript s i n invisible function application theta subscript 2 end subscript equals m g end cell row cell T subscript 1 end subscript c o s invisible function application 0 subscript 1 end subscript identical to T subscript 2 end subscript c o s invisible function application 0 subscript 2 end subscript end cell row cell 2 m g c o t invisible function application theta subscript 1 end subscript equals m g c o t invisible function application theta subscript 2 end subscript end cell row cell rightwards double arrow t a n invisible function application theta subscript 1 end subscript equals 2 t a n invisible function application theta subscript 2 end subscript end cell end table

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    table row cell T subscript 1 end subscript s i n invisible function application theta subscript 1 end subscript equals 2 m g end cell row cell T subscript 2 end subscript s i n invisible function application theta subscript 2 end subscript equals m g end cell row cell T subscript 1 end subscript c o s invisible function application 0 subscript 1 end subscript identical to T subscript 2 end subscript c o s invisible function application 0 subscript 2 end subscript end cell row cell 2 m g c o t invisible function application theta subscript 1 end subscript equals m g c o t invisible function application theta subscript 2 end subscript end cell row cell rightwards double arrow t a n invisible function application theta subscript 1 end subscript equals 2 t a n invisible function application theta subscript 2 end subscript end cell end table
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    physics-General
    The FBD of blocks is as shown From Newton's second law 4mg – 2T cosq = 4 mA .... (1)
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    General
    maths-

    If S and straight S to the power of straight prime are two foci of an ellipse fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction blank+blank fraction numerator blank y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction= 1 left parenthesis a less than b right parenthesis and P open parentheses x subscript 1 end subscript comma y subscript 1 end subscript close parentheses a point on it, then SP + straight S to the power of straight prime P is equal to-

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    maths-General
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    A cylinder rests in a supporting carriage as shown. The side AB of carriage makes an angle 30 to the power of ring operator end exponent with the horizontal and side BC is vertical. The carriage lies on a fixed horizontal surface and is being pulled towards left with an horizontal acceleration 'a'. The magnitude of normal reactions exerted by sides AB and BC of carriage on the cylinder be N subscript A B end subscript text  and  end text N subscript B C end subscriptrespectively. Neglect friction everywhere. Then as the magnitude of acceleration 'a ' of the carriage is increased, pick up the correct statement:

    The free body diagram of cylinder is as shown. Since net acceleration of cylinder is horizontal

    N subscript A B end subscript cos invisible function application 30 to the power of ring operator end exponent equals m g text  or  end text N subscript A B end subscript equals fraction numerator 2 over denominator square root of 3 end fraction m g
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    A cylinder rests in a supporting carriage as shown. The side AB of carriage makes an angle 30 to the power of ring operator end exponent with the horizontal and side BC is vertical. The carriage lies on a fixed horizontal surface and is being pulled towards left with an horizontal acceleration 'a'. The magnitude of normal reactions exerted by sides AB and BC of carriage on the cylinder be N subscript A B end subscript text  and  end text N subscript B C end subscriptrespectively. Neglect friction everywhere. Then as the magnitude of acceleration 'a ' of the carriage is increased, pick up the correct statement:

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    The free body diagram of cylinder is as shown. Since net acceleration of cylinder is horizontal

    N subscript A B end subscript cos invisible function application 30 to the power of ring operator end exponent equals m g text  or  end text N subscript A B end subscript equals fraction numerator 2 over denominator square root of 3 end fraction m g
    and NBC – NAB sin30° = ma or NBC = ma + NAB sin 30° .... (2)
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    In the figure shown, a person wants to raise a block lying on the ground to a height h. In both the cases if time required is same then in which case he has to exert more force. Assume pulleys and strings light.

    Since,blank h equals fraction numerator 1 over denominator 2 end fraction at2 Þ a should be same in both cases, because h and t are same in both cases as given.
    table row cell I n invisible function application left parenthesis i right parenthesis F subscript 1 end subscript minus m g equals m a. rightwards double arrow F subscript 1 end subscript equals m g plus m a end cell row cell text end text text ( end text text i end text text i end text text ) end text text end text 2 F subscript 2 end subscript minus m g equals m a rightwards double arrow F subscript 2 end subscript equals fraction numerator m g plus m a over denominator 2 end fraction end cell row cell text end text text I end text text n end text text end text end cell row cell therefore F subscript 1 end subscript greater than F subscript 2 end subscript end cell end table

    In the figure shown, a person wants to raise a block lying on the ground to a height h. In both the cases if time required is same then in which case he has to exert more force. Assume pulleys and strings light.

    physics-General
    Since,blank h equals fraction numerator 1 over denominator 2 end fraction at2 Þ a should be same in both cases, because h and t are same in both cases as given.
    table row cell I n invisible function application left parenthesis i right parenthesis F subscript 1 end subscript minus m g equals m a. rightwards double arrow F subscript 1 end subscript equals m g plus m a end cell row cell text end text text ( end text text i end text text i end text text ) end text text end text 2 F subscript 2 end subscript minus m g equals m a rightwards double arrow F subscript 2 end subscript equals fraction numerator m g plus m a over denominator 2 end fraction end cell row cell text end text text I end text text n end text text end text end cell row cell therefore F subscript 1 end subscript greater than F subscript 2 end subscript end cell end table
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    physics-

    A rod of length 2 l is moving such that its ends A and B move in contact with the horizontal floor and vertical wall respectively as shown in figure. O is the intersection point of the vertical wall and horizontal floor. The velocity vector of the centre of rod C is always directed along tangent drawn at C to the –

    At any instant of time the rod makes an angle q with horizontal, the x & y coordinates of centre of rod are

    table row cell x equals l c o s invisible function application theta y equals l s i n invisible function application theta end cell row cell therefore x to the power of 2 end exponent plus y to the power of 2 end exponent equals l to the power of 2 end exponent end cell end table
    Hence the centre C moves along a circle of radius lwith centre at O. \ velocity vector of C is always directed along the tangent drawn at C to the circle of radius l whose centre lies at O.

    A rod of length 2 l is moving such that its ends A and B move in contact with the horizontal floor and vertical wall respectively as shown in figure. O is the intersection point of the vertical wall and horizontal floor. The velocity vector of the centre of rod C is always directed along tangent drawn at C to the –

    physics-General
    At any instant of time the rod makes an angle q with horizontal, the x & y coordinates of centre of rod are

    table row cell x equals l c o s invisible function application theta y equals l s i n invisible function application theta end cell row cell therefore x to the power of 2 end exponent plus y to the power of 2 end exponent equals l to the power of 2 end exponent end cell end table
    Hence the centre C moves along a circle of radius lwith centre at O. \ velocity vector of C is always directed along the tangent drawn at C to the circle of radius l whose centre lies at O.