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text  If  end text y equals e to the power of 4 x end exponent plus 2 e to the power of negative x end exponent satisfies the relation fraction numerator d cubed y over denominator d x cubed end fraction plus A fraction numerator d y over denominator d x end fraction plus B y equals 0 then value of A and B respectively are:

Maths-General

  1. –13, 12
  2. –13, 14
  3. –13, –12
  4. 12, –13

    Answer:The correct answer is: –13, –12

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    According to Newton’s law of cooling, the rate of cooling is proportional to open parentheses increment theta close parentheses to the power of n end exponent, where increment theta is the temperature differences between the body and the surroundings andn is equal to

    According to Newton’s law of cooling the rate of loss of heat of a body is directly proportional to the difference in temperature of the body, i e comma
    negative fraction numerator d Q over denominator d t end fraction proportional to left parenthesis increment theta right parenthesis (i)
    Given, negative fraction numerator d Q over denominator d t end fraction proportional to open parentheses increment theta close parentheses to the power of n end exponent (ii)
    Comparing Eqs. (i) and (ii), we get
    n=1

    According to Newton’s law of cooling, the rate of cooling is proportional to open parentheses increment theta close parentheses to the power of n end exponent, where increment theta is the temperature differences between the body and the surroundings andn is equal to

    physics-General
    According to Newton’s law of cooling the rate of loss of heat of a body is directly proportional to the difference in temperature of the body, i e comma
    negative fraction numerator d Q over denominator d t end fraction proportional to left parenthesis increment theta right parenthesis (i)
    Given, negative fraction numerator d Q over denominator d t end fraction proportional to open parentheses increment theta close parentheses to the power of n end exponent (ii)
    Comparing Eqs. (i) and (ii), we get
    n=1
    General
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    Statement-1- The number blank to the power of 1000 end exponent C subscript 500 end subscript is not divisible by 11.Because
    Statement-2- If p is a prime, the exponent of p in n! is open square brackets fraction numerator n over denominator p end fraction close square brackets+ open square brackets fraction numerator n over denominator p to the power of 2 end exponent end fraction close square brackets+ open square brackets fraction numerator n over denominator p to the power of 3 end exponent end fraction close square brackets+……Where [x] denotes the greatest integer less or equal than x.

    Statement-1- The number blank to the power of 1000 end exponent C subscript 500 end subscript is not divisible by 11.Because
    Statement-2- If p is a prime, the exponent of p in n! is open square brackets fraction numerator n over denominator p end fraction close square brackets+ open square brackets fraction numerator n over denominator p to the power of 2 end exponent end fraction close square brackets+ open square brackets fraction numerator n over denominator p to the power of 3 end exponent end fraction close square brackets+……Where [x] denotes the greatest integer less or equal than x.

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    maths-

    Number of values of 'p' for which the equationopen parentheses p squared minus 3 p plus 2 close parentheses x squared minus open parentheses p squared minus 5 p plus 4 close parentheses x plus p minus p squared equals 0 possess more  than two roots ,is:

    For (p2– 3p+2)x2–(p2–5p+4)x+p–p2=0tobe anidentity
    p2–3p+2 =0 rightwards double arrow p = 1, 2...(i)
    p2 – 5p + 4 = 0 rightwards double arrow p = 1, 4...(ii)
    p – p2 = 0 rightwards double arrow p = 0, 1...(iii)
    For (i), (ii) & (iii) to hold simultaneously p = 1.

    Number of values of 'p' for which the equationopen parentheses p squared minus 3 p plus 2 close parentheses x squared minus open parentheses p squared minus 5 p plus 4 close parentheses x plus p minus p squared equals 0 possess more  than two roots ,is:

    maths-General
    For (p2– 3p+2)x2–(p2–5p+4)x+p–p2=0tobe anidentity
    p2–3p+2 =0 rightwards double arrow p = 1, 2...(i)
    p2 – 5p + 4 = 0 rightwards double arrow p = 1, 4...(ii)
    p – p2 = 0 rightwards double arrow p = 0, 1...(iii)
    For (i), (ii) & (iii) to hold simultaneously p = 1.
    General
    maths-

    Let p, qelement of{1,2,3,4}.Then number of equation of the form px2+qx+1=0,having real roots ,is

    q2–4pgreater or equal than0
    q=2 rightwards double arrow p=1
    q=3rightwards double arrow p=1,2
    q=4rightwards double arrow p=1,2,3,4
    Hence 7 values of (p, q)7equationsarepossible.

    Let p, qelement of{1,2,3,4}.Then number of equation of the form px2+qx+1=0,having real roots ,is

    maths-General
    q2–4pgreater or equal than0
    q=2 rightwards double arrow p=1
    q=3rightwards double arrow p=1,2
    q=4rightwards double arrow p=1,2,3,4
    Hence 7 values of (p, q)7equationsarepossible.
    General
    maths-

    The differential equation of all circles which pass through the origin and whose centres lie on y-axis is

    The differential equation of all circles which pass through the origin and whose centres lie on y-axis is

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    General
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    A simple magnifying lens is used in such a way that an image is formed at 25 cm away from the eye. In order to have 10 times magnification, the focal length of the lens should be

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    A simple magnifying lens is used in such a way that an image is formed at 25 cm away from the eye. In order to have 10 times magnification, the focal length of the lens should be

    physics-General
    fraction numerator D over denominator F end fraction or fraction numerator 25 over denominator F end fraction
    General
    maths-

    If x fraction numerator d y over denominator d x end fraction equals y left parenthesis log space y minus log space x plus 1 right parenthesis then the solution of the equation is :

    If x fraction numerator d y over denominator d x end fraction equals y left parenthesis log space y minus log space x plus 1 right parenthesis then the solution of the equation is :

    maths-General
    General
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    physics-General
    Since the curved surface of the conductor is thermally insulated, therefore, in steady state, the rate of flow of heat at every section will be the same. Hence the curve between H and x will be straight line parallel to x-axis
    General
    physics-

    Water and turpentine oil (specific heat less than that of water) are both heated to same temperature. Equal amounts of these placed in identical calorimeters are then left in air

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    because c subscript o i l end subscript less than c subscript W a t e r end subscript
    rightwards double arrow open parentheses R a t e blank o f blank c o o l i n g close parentheses subscript o i l end subscript greater than open parentheses R a t e blank o f blank c o o l i n g close parentheses subscript W a t e r end subscript

    It is clear that, at a particular time after start cooling, temperature of oil will be less than that of water
    So graph B represents the cooling curve of oil and A represents the cooling curve of water

    Water and turpentine oil (specific heat less than that of water) are both heated to same temperature. Equal amounts of these placed in identical calorimeters are then left in air

    physics-General
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    because c subscript o i l end subscript less than c subscript W a t e r end subscript
    rightwards double arrow open parentheses R a t e blank o f blank c o o l i n g close parentheses subscript o i l end subscript greater than open parentheses R a t e blank o f blank c o o l i n g close parentheses subscript W a t e r end subscript

    It is clear that, at a particular time after start cooling, temperature of oil will be less than that of water
    So graph B represents the cooling curve of oil and A represents the cooling curve of water
    General
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    Variation of radiant energy emitted by sun, filament of tungsten lamp and welding are as a function of its wavelength is shown in figure. Which of the following option is the correct match?

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    physics-General
    lambda subscript m end subscript T equals c o n s t a n t
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    If f(x) is a differentiable function satisfying f(x+ y)= f(x)f(y) straight for all x, y element of R and f'(0)=2 then f(x)=

    table row cell f left parenthesis x plus y right parenthesis equals f left parenthesis x right parenthesis f left parenthesis y right parenthesis         rightwards double arrow         f left parenthesis x right parenthesis equals k e to the power of x f left parenthesis 0 right parenthesis end exponent text end text text w end text text h end text text e end text text r end text text e end text text end text k text end text text i end text text s end text text end text text a end text text end text text c end text text o end text text n end text text s end text text t end text text a end text text n end text text t end text text end text end cell row cell text end text text A end text text s end text text end text f to the power of ´ end exponent left parenthesis 0 right parenthesis equals 2         rightwards double arrow         k equals 1 rightwards double arrow f left parenthesis x right parenthesis equals e to the power of 2 x end exponent end cell end table rightwards double arrow f left parenthesis x right parenthesis equals k e to the power of 2 x end exponent

    If f(x) is a differentiable function satisfying f(x+ y)= f(x)f(y) straight for all x, y element of R and f'(0)=2 then f(x)=

    maths-General
    table row cell f left parenthesis x plus y right parenthesis equals f left parenthesis x right parenthesis f left parenthesis y right parenthesis         rightwards double arrow         f left parenthesis x right parenthesis equals k e to the power of x f left parenthesis 0 right parenthesis end exponent text end text text w end text text h end text text e end text text r end text text e end text text end text k text end text text i end text text s end text text end text text a end text text end text text c end text text o end text text n end text text s end text text t end text text a end text text n end text text t end text text end text end cell row cell text end text text A end text text s end text text end text f to the power of ´ end exponent left parenthesis 0 right parenthesis equals 2         rightwards double arrow         k equals 1 rightwards double arrow f left parenthesis x right parenthesis equals e to the power of 2 x end exponent end cell end table rightwards double arrow f left parenthesis x right parenthesis equals k e to the power of 2 x end exponent
    General
    maths-

    If f(x) = ax2+ bx + c such that f(p) + f(q) = 0 where anot equal to0 ; p , qelement ofR then number of real roots of equation f(x) = 0in interval [p, q] is

    f(p)=–f(q) rightwards double arrow eitherf(p)f(q)<0 or f(p) =0=f(q)
    rightwards double arrow exactlyonerootin(p,q) orrootsarepandq

    If f(x) = ax2+ bx + c such that f(p) + f(q) = 0 where anot equal to0 ; p , qelement ofR then number of real roots of equation f(x) = 0in interval [p, q] is

    maths-General
    f(p)=–f(q) rightwards double arrow eitherf(p)f(q)<0 or f(p) =0=f(q)
    rightwards double arrow exactlyonerootin(p,q) orrootsarepandq
    General
    maths-

    If f(x) is a polynomial of degree five with leading coefficient one such that  f(1)=12,f(2)=22,f(3)=32,f(4)=42,f(5)=52 then

    f left parenthesis x right parenthesis minus x to the power of 2 end exponent equals left parenthesis x minus 1 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis x minus 3 right parenthesis left parenthesis x minus 4 right parenthesis left parenthesis x minus 5

    If f(x) is a polynomial of degree five with leading coefficient one such that  f(1)=12,f(2)=22,f(3)=32,f(4)=42,f(5)=52 then

    maths-General
    f left parenthesis x right parenthesis minus x to the power of 2 end exponent equals left parenthesis x minus 1 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis x minus 3 right parenthesis left parenthesis x minus 4 right parenthesis left parenthesis x minus 5