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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

  1. 15    
  2. 9    
  3. 7    
  4. 8    

    Answer:The correct answer is: 7q2–4pgreater 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.

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    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.
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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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    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
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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

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    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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    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:

    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:

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    Radius of a conductor increases uniformly from left end to right end as shown in fig  Material of the conductor is isotropic and its curved surface is thermally insulated from surrounding. Its ends are maintained at temperatures T subscript 1 end subscript and T subscript 2 end subscript left parenthesis T subscript 1 end subscript greater than T subscript 2 end subscript right parenthesis: If, in steady state, heat flow rate is equal to H, then which of the following graphs is correct

    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

    Radius of a conductor increases uniformly from left end to right end as shown in fig  Material of the conductor is isotropic and its curved surface is thermally insulated from surrounding. Its ends are maintained at temperatures T subscript 1 end subscript and T subscript 2 end subscript left parenthesis T subscript 1 end subscript greater than T subscript 2 end subscript right parenthesis: If, in steady state, heat flow rate is equal to H, then which of the following graphs is correct

    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

    As we know, Rate of cooling proportional to fraction numerator 1 over denominator s p e c i f i c blank h e a t left square bracket c right square bracket end fraction
    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
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    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
    As we know, Rate of cooling proportional to fraction numerator 1 over denominator s p e c i f i c blank h e a t left square bracket c right square bracket end fraction
    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
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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?

    lambda subscript m end subscript T equals c o n s t a n t
    From the graph T subscript 3 end subscript greater than T subscript 2 end subscript greater than T subscript 1 end subscript
    Temperature of sun will be maximum

    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?

    physics-General
    lambda subscript m end subscript T equals c o n s t a n t
    From the graph T subscript 3 end subscript greater than T subscript 2 end subscript greater than T subscript 1 end subscript
    Temperature of sun will be maximum
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    Six identical metallic rods are joined together in a pattern as shown in the figure. Points A and D are maintained at temperature 60 ℃ and 240 ℃. The temperature of the junction B will be

    R equals fraction numerator l over denominator K A end fraction
    fraction numerator T subscript A end subscript minus T subscript B end subscript over denominator R end fraction equals fraction numerator T subscript B end subscript minus T subscript C end subscript over denominator R end fraction equals fraction numerator T subscript C end subscript minus T subscript D end subscript over denominator R end fraction
    60 minus T subscript B end subscript equals T subscript B end subscript minus T subscript C end subscript(i)

    60 minus T subscript B end subscript equals T subscript C end subscript minus 240(ii)
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    fraction numerator T subscript A end subscript minus T subscript B end subscript over denominator R end fraction equals fraction numerator T subscript B end subscript minus T subscript C end subscript over denominator R end fraction equals fraction numerator T subscript C end subscript minus T subscript D end subscript over denominator R end fraction
    60 minus T subscript B end subscript equals T subscript B end subscript minus T subscript C end subscript(i)

    60 minus T subscript B end subscript equals T subscript C end subscript minus 240(ii)
    Solving (i) and (ii)
    T subscript B end subscript equals 120 ℃
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    Which curve shows the rise of temperature with the amount of heat supplied, for a piece of ice?

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    Which curve shows the rise of temperature with the amount of heat supplied, for a piece of ice?

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    The graph signifies

    The graph signifies

    physics-General
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    The figure given below shows the cooling curve of pure wax material after heating. It cools from A to B and solidifies along B D. If L and C are respective values of latent heat and the specific heat of the liquid wax, the ratio L divided by C is

    Let the quantity of heat supplied per minute be Q. Then quantity of heat supplied in 2 blank m i n equals m C left parenthesis 90 minus 80 right parenthesis
    In 4 blank m i n comma blankheat supplied equals 2 m C open parentheses 90 minus 80 close parentheses
    therefore 2 m blank C open parentheses 90 minus 80 close parentheses equals m L rightwards double arrow fraction numerator L over denominator C end fraction equals 20

    The figure given below shows the cooling curve of pure wax material after heating. It cools from A to B and solidifies along B D. If L and C are respective values of latent heat and the specific heat of the liquid wax, the ratio L divided by C is

    physics-General
    Let the quantity of heat supplied per minute be Q. Then quantity of heat supplied in 2 blank m i n equals m C left parenthesis 90 minus 80 right parenthesis
    In 4 blank m i n comma blankheat supplied equals 2 m C open parentheses 90 minus 80 close parentheses
    therefore 2 m blank C open parentheses 90 minus 80 close parentheses equals m L rightwards double arrow fraction numerator L over denominator C end fraction equals 20
    General
    physics-

    One end of a thermally insulated rod is kept at a temperature T subscript 1 end subscript and other at T subscript 2 end subscript. The rod is composed of two sections of lengths l subscript 1 end subscript and l subscript 2 end subscript and thermal conductivities K subscript 1 end subscript and K subscript 2 end subscript respectively. The temperature at the interface of the two sections is

    Let temperature at the interface is T.
    For part AB,

    fraction numerator Q subscript 1 end subscript over denominator t end fraction proportional to fraction numerator open parentheses T subscript 1 end subscript minus T close parentheses K subscript 1 end subscript over denominator l subscript 1 end subscript end fraction
    For part B C comma
    fraction numerator Q subscript 2 end subscript over denominator t end fraction proportional to fraction numerator open parentheses T minus T subscript 2 end subscript close parentheses K subscript 2 end subscript over denominator l subscript 2 end subscript end fraction
    At equilibrium, fraction numerator Q subscript 1 end subscript over denominator t end fraction equals fraction numerator Q subscript 2 end subscript over denominator t end fraction
    therefore fraction numerator open parentheses T subscript 1 end subscript minus T close parentheses K subscript 1 end subscript over denominator l subscript 1 end subscript end fraction equals fraction numerator open parentheses T minus T subscript 2 end subscript close parentheses K subscript 2 end subscript over denominator l subscript 2 end subscript end fraction
    rightwards double arrow T equals fraction numerator T subscript 1 end subscript K subscript 1 end subscript l subscript 2 end subscript plus T subscript 2 end subscript K subscript 2 end subscript l subscript 1 end subscript over denominator K subscript 1 end subscript l subscript 2 end subscript plus K subscript 2 end subscript l subscript 1 end subscript end fraction

    One end of a thermally insulated rod is kept at a temperature T subscript 1 end subscript and other at T subscript 2 end subscript. The rod is composed of two sections of lengths l subscript 1 end subscript and l subscript 2 end subscript and thermal conductivities K subscript 1 end subscript and K subscript 2 end subscript respectively. The temperature at the interface of the two sections is

    physics-General
    Let temperature at the interface is T.
    For part AB,

    fraction numerator Q subscript 1 end subscript over denominator t end fraction proportional to fraction numerator open parentheses T subscript 1 end subscript minus T close parentheses K subscript 1 end subscript over denominator l subscript 1 end subscript end fraction
    For part B C comma
    fraction numerator Q subscript 2 end subscript over denominator t end fraction proportional to fraction numerator open parentheses T minus T subscript 2 end subscript close parentheses K subscript 2 end subscript over denominator l subscript 2 end subscript end fraction
    At equilibrium, fraction numerator Q subscript 1 end subscript over denominator t end fraction equals fraction numerator Q subscript 2 end subscript over denominator t end fraction
    therefore fraction numerator open parentheses T subscript 1 end subscript minus T close parentheses K subscript 1 end subscript over denominator l subscript 1 end subscript end fraction equals fraction numerator open parentheses T minus T subscript 2 end subscript close parentheses K subscript 2 end subscript over denominator l subscript 2 end subscript end fraction
    rightwards double arrow T equals fraction numerator T subscript 1 end subscript K subscript 1 end subscript l subscript 2 end subscript plus T subscript 2 end subscript K subscript 2 end subscript l subscript 1 end subscript over denominator K subscript 1 end subscript l subscript 2 end subscript plus K subscript 2 end subscript l subscript 1 end subscript end fraction
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    maths-

    If n element of N comma and theperiod of fraction numerator c o s invisible function application n x over denominator s i n invisible function application open parentheses fraction numerator x over denominator n end fraction close parentheses end fraction is 4 pi, then n equals

    text L.C.M. of  end text open parentheses fraction numerator 2 pi over denominator n end fraction comma 2 pi n close parentheses equals 4 pi

    If n element of N comma and theperiod of fraction numerator c o s invisible function application n x over denominator s i n invisible function application open parentheses fraction numerator x over denominator n end fraction close parentheses end fraction is 4 pi, then n equals

    maths-General
    text L.C.M. of  end text open parentheses fraction numerator 2 pi over denominator n end fraction comma 2 pi n close parentheses equals 4 pi
    General
    physics-

    The adjoining diagram shows the spectral energy density distribution E subscript lambda end subscript of a black body at two different temperatures. If the areas under the curves are in the ratio 16 : 1, the value of temperature T is

    fraction numerator A subscript T end subscript over denominator A subscript 2000 end subscript end fraction equals fraction numerator 16 over denominator 1 end fraction [Given]
    Area under e subscript lambda end subscript minus lambda curve represents the emissive power of body and emissive power proportional to T to the power of 4 end exponent
    rightwards double arrow fraction numerator A subscript T end subscript over denominator A subscript 2000 end subscript end fraction equals open parentheses fraction numerator T over denominator 2000 end fraction close parentheses to the power of 4 end exponent rightwards double arrow fraction numerator 16 over denominator 1 end fraction equals open parentheses fraction numerator T over denominator 2000 end fraction close parentheses to the power of 4 end exponent rightwards double arrow T equals 4000 K

    The adjoining diagram shows the spectral energy density distribution E subscript lambda end subscript of a black body at two different temperatures. If the areas under the curves are in the ratio 16 : 1, the value of temperature T is

    physics-General
    fraction numerator A subscript T end subscript over denominator A subscript 2000 end subscript end fraction equals fraction numerator 16 over denominator 1 end fraction [Given]
    Area under e subscript lambda end subscript minus lambda curve represents the emissive power of body and emissive power proportional to T to the power of 4 end exponent
    rightwards double arrow fraction numerator A subscript T end subscript over denominator A subscript 2000 end subscript end fraction equals open parentheses fraction numerator T over denominator 2000 end fraction close parentheses to the power of 4 end exponent rightwards double arrow fraction numerator 16 over denominator 1 end fraction equals open parentheses fraction numerator T over denominator 2000 end fraction close parentheses to the power of 4 end exponent rightwards double arrow T equals 4000 K
    General
    physics-

    Five identical rods are joined as shown in figure. Point A and C are maintained at temperature 120 ℃ and 20 ℃ respectively. The temperature of junction B will be

    If thermal resistance of each rod is considered R then, the given combination can be redrawn as follows

    open parentheses H e a t blank c u r r e n t close parentheses subscript A C end subscript equals open parentheses H e a t blank c u r r e n t close parentheses subscript A B end subscript
    fraction numerator left parenthesis 120 minus 20 right parenthesis over denominator 2 R end fraction equals fraction numerator left parenthesis 120 minus theta right parenthesis over denominator R end fraction rightwards double arrow theta equals 70 ℃

    Five identical rods are joined as shown in figure. Point A and C are maintained at temperature 120 ℃ and 20 ℃ respectively. The temperature of junction B will be

    physics-General
    If thermal resistance of each rod is considered R then, the given combination can be redrawn as follows

    open parentheses H e a t blank c u r r e n t close parentheses subscript A C end subscript equals open parentheses H e a t blank c u r r e n t close parentheses subscript A B end subscript
    fraction numerator left parenthesis 120 minus 20 right parenthesis over denominator 2 R end fraction equals fraction numerator left parenthesis 120 minus theta right parenthesis over denominator R end fraction rightwards double arrow theta equals 70 ℃