Maths-
General
Easy

Question

If a2 + 4b2 = 12ab, then log (a + 2b) =

  1. fraction numerator 1 over denominator 2 end fraction (log a + log b – log 2)    
  2. log a/2 + log b/2 + log 2    
  3. fraction numerator 1 over denominator 2 end fraction (log a + log b + 4 log 2)    
  4. fraction numerator 1 over denominator 2 end fraction (log a – log b + 4log 2)    

The correct answer is: fraction numerator 1 over denominator 2 end fraction (log a + log b + 4 log 2)

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Given that logpx = α and logqx = β, then value of logp/q x equals-

Given that logpx = α and logqx = β, then value of logp/q x equals-

maths-General
General
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If f(x) is the primitive of fraction numerator sin invisible function application root index 3 of x end root log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of – 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of root index 3 of x end root end exponent – 1 right parenthesis end fraction (x not equal to 0), then stack l i m with x rightwards arrow 0 below f ' (x) is -

 f(x) = not stretchy integral fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of – 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of x to the power of 1 divided by 3 end exponent end exponent – 1 right parenthesis end fraction
stack l i m with x rightwards arrow 0 below f'(x)
stack l i m with x rightwards arrow 0 below fraction numerator open parentheses fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent over denominator x to the power of 1 divided by 3 end exponent end fraction close parentheses x to the power of 1 divided by 3 end exponent open square brackets fraction numerator cos invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator 3 x end fraction close square brackets left parenthesis 3 x right parenthesis over denominator open parentheses fraction numerator tan to the power of – 1 end exponent invisible function application square root of x over denominator square root of x end fraction close parentheses to the power of 2 end exponent. x. open parentheses fraction numerator e to the power of x to the power of 1 divided by 3 end exponent end exponent – 1 over denominator x to the power of 1 divided by 3 end exponent end fraction close parentheses. x to the power of 1 divided by 3 end exponent end fraction
= 3

If f(x) is the primitive of fraction numerator sin invisible function application root index 3 of x end root log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of – 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of root index 3 of x end root end exponent – 1 right parenthesis end fraction (x not equal to 0), then stack l i m with x rightwards arrow 0 below f ' (x) is -

Maths-General
 f(x) = not stretchy integral fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of – 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of x to the power of 1 divided by 3 end exponent end exponent – 1 right parenthesis end fraction
stack l i m with x rightwards arrow 0 below f'(x)
stack l i m with x rightwards arrow 0 below fraction numerator open parentheses fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent over denominator x to the power of 1 divided by 3 end exponent end fraction close parentheses x to the power of 1 divided by 3 end exponent open square brackets fraction numerator cos invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator 3 x end fraction close square brackets left parenthesis 3 x right parenthesis over denominator open parentheses fraction numerator tan to the power of – 1 end exponent invisible function application square root of x over denominator square root of x end fraction close parentheses to the power of 2 end exponent. x. open parentheses fraction numerator e to the power of x to the power of 1 divided by 3 end exponent end exponent – 1 over denominator x to the power of 1 divided by 3 end exponent end fraction close parentheses. x to the power of 1 divided by 3 end exponent end fraction
= 3
General
Maths-

If f(x) is the primitive of fraction numerator sin invisible function application root index 3 of x end root log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of negative 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of root index 3 of x end root end exponent minus 1 right parenthesis end fraction(x not equal to 0), then stack l i m with x rightwards arrow 0 below f ´ left parenthesis x right parenthesis is-

f(x) = not stretchy integral fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent log invisible function application left parenthesis 1 plus 3 x right parenthesis d x over denominator left parenthesis tan to the power of negative 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of 5 x to the power of 1 divided by 3 end exponent end exponent minus 1 right parenthesis end fraction
stack l i m with x rightwards arrow 0 below f ´ left parenthesis x right parenthesis = fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of negative 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of 5 x to the power of 1 divided by 3 end exponent end exponent minus 1 right parenthesis end fraction
= fraction numerator x to the power of 1 divided by 3 end exponent fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent over denominator x to the power of 1 divided by 3 end exponent end fraction. fraction numerator log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator 3 x end fraction.3 x over denominator x. open parentheses fraction numerator tan to the power of negative 1 end exponent invisible function application square root of x over denominator square root of x end fraction close parentheses to the power of 2 end exponent open parentheses fraction numerator e to the power of 5 x to the power of 1 divided by 3 end exponent end exponent minus 1 over denominator 5 x to the power of 1 divided by 3 end exponent end fraction close parentheses.5 x to the power of 1 divided by 3 end exponent end fraction
= fraction numerator 1.1.3 over denominator 1.1.5 end fraction equals 3/5

If f(x) is the primitive of fraction numerator sin invisible function application root index 3 of x end root log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of negative 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of root index 3 of x end root end exponent minus 1 right parenthesis end fraction(x not equal to 0), then stack l i m with x rightwards arrow 0 below f ´ left parenthesis x right parenthesis is-

Maths-General
f(x) = not stretchy integral fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent log invisible function application left parenthesis 1 plus 3 x right parenthesis d x over denominator left parenthesis tan to the power of negative 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of 5 x to the power of 1 divided by 3 end exponent end exponent minus 1 right parenthesis end fraction
stack l i m with x rightwards arrow 0 below f ´ left parenthesis x right parenthesis = fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of negative 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of 5 x to the power of 1 divided by 3 end exponent end exponent minus 1 right parenthesis end fraction
= fraction numerator x to the power of 1 divided by 3 end exponent fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent over denominator x to the power of 1 divided by 3 end exponent end fraction. fraction numerator log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator 3 x end fraction.3 x over denominator x. open parentheses fraction numerator tan to the power of negative 1 end exponent invisible function application square root of x over denominator square root of x end fraction close parentheses to the power of 2 end exponent open parentheses fraction numerator e to the power of 5 x to the power of 1 divided by 3 end exponent end exponent minus 1 over denominator 5 x to the power of 1 divided by 3 end exponent end fraction close parentheses.5 x to the power of 1 divided by 3 end exponent end fraction
= fraction numerator 1.1.3 over denominator 1.1.5 end fraction equals 3/5
parallel
General
maths-

If f left parenthesis x right parenthesis equals open vertical bar table row cell s i n invisible function application x end cell cell s i n invisible function application a end cell cell s i n invisible function application b end cell row cell c o s invisible function application x end cell cell c o s invisible function application a end cell cell c o s invisible function application b end cell row cell t a n invisible function application x end cell cell t a n invisible function application a end cell cell t a n invisible function application b end cell end table close vertical bar where 0 less than a less than b less than fraction numerator pi over denominator 2 end fraction hen the equation f(x) = 0 has, in the interval (a, b)

If f left parenthesis x right parenthesis equals open vertical bar table row cell s i n invisible function application x end cell cell s i n invisible function application a end cell cell s i n invisible function application b end cell row cell c o s invisible function application x end cell cell c o s invisible function application a end cell cell c o s invisible function application b end cell row cell t a n invisible function application x end cell cell t a n invisible function application a end cell cell t a n invisible function application b end cell end table close vertical bar where 0 less than a less than b less than fraction numerator pi over denominator 2 end fraction hen the equation f(x) = 0 has, in the interval (a, b)

maths-General
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The velocity-time graph of a particle moving along a straight line is shown in figure. The displacement of the body in 5s is

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equals fraction numerator 1 over denominator 2 end fraction open parentheses 3 cross times 2 minus fraction numerator 1 over denominator 2 end fraction cross times 1 cross times 2 plus 1 cross times 1 close parenthesesm=3m

The velocity-time graph of a particle moving along a straight line is shown in figure. The displacement of the body in 5s is

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equals fraction numerator 1 over denominator 2 end fraction open parentheses 3 cross times 2 minus fraction numerator 1 over denominator 2 end fraction cross times 1 cross times 2 plus 1 cross times 1 close parenthesesm=3m
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Let f x( ) and g x( ) are defined and differentiable for x greater or equal than x subscript 0 end subscript and f open parentheses x subscript 0 close parentheses equals g open parentheses x subscript 0 close parentheses comma f to the power of ´ left parenthesis x right parenthesis greater than straight g to the power of straight prime left parenthesis x right parenthesis text  for  end text x greater than x subscript 0 then

ϕ left parenthesis x right parenthesis equals f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis text  where  end text x element of open square brackets x subscript 0 end subscript comma b close square brackets by c element of open parentheses x subscript 0 end subscript comma b close parentheses contains as member
ϕ to the power of ´ end exponent left parenthesis c right parenthesis equals fraction numerator ϕ left parenthesis b right parenthesis minus ϕ open parentheses x subscript 0 end subscript close parentheses over denominator b minus x subscript 0 end subscript end fraction greater than 0
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Let f x( ) and g x( ) are defined and differentiable for x greater or equal than x subscript 0 end subscript and f open parentheses x subscript 0 close parentheses equals g open parentheses x subscript 0 close parentheses comma f to the power of ´ left parenthesis x right parenthesis greater than straight g to the power of straight prime left parenthesis x right parenthesis text  for  end text x greater than x subscript 0 then

maths-General
ϕ left parenthesis x right parenthesis equals f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis text  where  end text x element of open square brackets x subscript 0 end subscript comma b close square brackets by c element of open parentheses x subscript 0 end subscript comma b close parentheses contains as member
ϕ to the power of ´ end exponent left parenthesis c right parenthesis equals fraction numerator ϕ left parenthesis b right parenthesis minus ϕ open parentheses x subscript 0 end subscript close parentheses over denominator b minus x subscript 0 end subscript end fraction greater than 0
therefore f left parenthesis x right parenthesis greater than g left parenthesis x right parenthesis text  for  end text x equals b
parallel
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maths-

In the given figure, if POQ is a diameter of the circle and PR = QR, then RPQ is

In the given figure, if POQ is a diameter of the circle and PR = QR, then RPQ is

maths-General
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maths-

In the given figure, find the values of ‘x’ and ‘y’

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maths-General
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maths-

PQRS is a cyclic quadrilateral and PQ is the diameter of the circle. If QPR= ° 35 , then the value of PSR is

PQRS is a cyclic quadrilateral and PQ is the diameter of the circle. If QPR= ° 35 , then the value of PSR is

maths-General
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Find the value of ‘x’ in the given figure

Find the value of ‘x’ in the given figure

maths-General
General
maths-

In the given figure, find PR

In the given figure, find PR

maths-General
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maths-

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In the given figure PA, PB, EC, FB, FD and ED are tangents to the circle. If PA=13cm,CE = 4.5cm and EF = 9cm then PF is

maths-General
parallel
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If TP and TQ are the two tangents to a circle with centre ‘O’ such that thentext end text text PTO end text, is

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maths-General
General
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Consider a rubber ball freely falling from a height h equals 4.9 blank m onto a horizontal elastic plate. Assume that the duration of collision is negligible and the collision with the plate is totally elastic. Then the velocity as a function of time will be

h equals fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent comma blank(paraboli(c)
v equals negative g tand after the collision, v equals g t (straight line)
Collision is perfectly elastic then ball reaches to same height again and again with same velocity

Consider a rubber ball freely falling from a height h equals 4.9 blank m onto a horizontal elastic plate. Assume that the duration of collision is negligible and the collision with the plate is totally elastic. Then the velocity as a function of time will be

physics-General
h equals fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent comma blank(paraboli(c)
v equals negative g tand after the collision, v equals g t (straight line)
Collision is perfectly elastic then ball reaches to same height again and again with same velocity
General
physics-

A ball is thrown vertically upwards. Which of the following plots represents the speed-time graph of the ball during its flight if the air resistance is not ignored

For upward motion
Effective acceleration equals negative open parentheses g plus a close parentheses
And for downward motion
Effective acceleration equals left parenthesis g minus a right parenthesis
But both are constants. So the slope of speed-time graph will be constant

A ball is thrown vertically upwards. Which of the following plots represents the speed-time graph of the ball during its flight if the air resistance is not ignored

physics-General
For upward motion
Effective acceleration equals negative open parentheses g plus a close parentheses
And for downward motion
Effective acceleration equals left parenthesis g minus a right parenthesis
But both are constants. So the slope of speed-time graph will be constant
parallel

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