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Maths-

General

Easy

Question

$f colon R to the power of plus not stretchy rightwards arrow R text defined by end text f left parenthesis x right parenthesis equals log subscript e invisible function application x comma x element of left parenthesis 0 comma 1 right parenthesis comma f left parenthesis x right parenthesis equals 2 log subscript e invisible function application x comma x element of left square bracket 1 comma straight infinity right parenthesis text is end text$

onto
one-one
not one-one
a bijection

The correct answer is: a bijection

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$f colon R to the power of plus not stretchy rightwards arrow R text defined by end text f left parenthesis x right parenthesis equals 2 to the power of x comma x element of left parenthesis 0 comma 1 right parenthesis comma f left parenthesis x right parenthesis equals 3 to the power of x comma x element of left square bracket 1 comma straight infinity right parenthesis text is end text$

$f colon R to the power of plus not stretchy rightwards arrow R text defined by end text f left parenthesis x right parenthesis equals 2 to the power of x comma x element of left parenthesis 0 comma 1 right parenthesis comma f left parenthesis x right parenthesis equals 3 to the power of x comma x element of left square bracket 1 comma straight infinity right parenthesis text is end text$

A small spherical solid ball is dropped from a great height in a viscous liquid. Its journey in the liquid is best described in the diagram given below by the

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Two communicating vessels contain mercury. The diameter of one vessel is n times larger than the diameter of the other. A column of water of height h is poured into the left vessel. The mercury level will rise in the right-hand vessel (s = relative density of mercury and $rho$ = density of water) by

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There are two identical small holes of area of cross-section a on the opposite sides of a tank containing a liquid of density $rho$ . The difference in height between the holes is h. Tank is resting on a smooth horizontal surface. Horizontal force which will has to be applied on the tank to keep it in equilibrium is

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A cylinder containing water up to a height of 25 cm has a hole of cross-section $fraction numerator 1 over denominator 4 end fraction c m to the power of 2 end exponent$ in its bottom. It is counterpoised in a balance. What is the initial change in the balancing weight when water begins to flow out

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Water is filled in a cylindrical container to a height of 3m. The ratio of the cross-sectional area of the orifice and the beaker is 0.1. The square of the speed of the liquid coming out from the orifice is (g = 10 m/s²)

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A body floats in a liquid contained in a beaker. The whole system as shown falls freely under gravity. The up thrust on the body due to the liquid is

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A wooden block, with a coin placed on its top, floats in water as shown in fig. the distance l and h are shown there. After some time the coin falls into the water. Then

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A homogeneous solid cylinder of length L $left parenthesis L less than H divided by 2 right parenthesis$ . Cross-sectional area $A divided by 5$ is immersed such that it floats with its axis vertical at the liquid-liquid interface with length $L divided by 4$ in the denser liquid as shown in the fig. The lower density liquid is open to atmosphere having pressure $P subscript 0 end subscript$ . Then density D of solid is given by

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A U-tube in which the cross-sectional area of the limb on the left is one quarter, the limb on the right contains mercury (density 13.6 g/cm³). The level of mercury in the narrow limb is at a distance of 36 cm from the upper end of the tube. What will be the rise in the level of mercury in the right limb if the left limb is filled to the top with water

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An iron block and a wooden block are positioned in a vessel containing water as shown in the figure. The iron block (I) hangs from a massless string with a rigid support from the top while the wooden block floats being tied to the bottom through a massless string. If now the vessel starts accelerating upwards, the incorrect option is

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Two hollow hemispheres of metal are fitted together to form a sphere of radius R. Air from the shall is pumped out and pressure remained left inside is $fraction numerator 1 over denominator 4 end fraction$ th the outside atmospheric pressure P₀. The force required to separate the hemisphers is

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$f colon R to the power of plus not stretchy rightwards arrow R text defined by end text f left parenthesis x right parenthesis equals 2 to the power of x comma x element of left parenthesis 0 comma 1 right parenthesis comma f left parenthesis x right parenthesis equals 3 to the power of x comma x element of left square bracket 1 comma straight infinity right parenthesis text is end text$

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A particle is thrown in upward direction with Velocity V₀ It passes through a point p of height h at time t₁ and t2 so t₁+t₁

A particle is thrown in upward direction with Velocity V₀ It passes through a point p of height h at time t₁ and t2 so t₁+t₁

The function $f colon R not stretchy rightwards arrow R text defined by end text f left parenthesis x right parenthesis equals x minus left square bracket x right square bracket comma straight for all x element of R text is end text$

The function $f colon R not stretchy rightwards arrow R text defined by end text f left parenthesis x right parenthesis equals x minus left square bracket x right square bracket comma straight for all x element of R text is end text$

parallel

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