A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

If the equation $$a_n{x}^n+a_{n-1}{x}^{n-1}+...........+a_{1}x=0$$ $$a_1\ne 0,n\ge 2,$$ has a positive root $$x=\alpha$$, then the equation $$n{a}_n{x}^{n-1}+\left(n-1\right)a_{n-1}{x}^{n-2}+...........+a_1=0$$ has a positive root, which is

The normal to the curve $$x=a\left(\cos \theta +\theta \sin \theta \right),y=a\left(\sin \theta -\theta \cos \theta \right)$$ at any point $$\theta {}^{\prime }$$ is such that

A spherical iron ball $$10$$ cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $$50$$ cm$${}^3$$ /min. When the thickness of ice is $$5$$ cm, then the rate at which the thickness of ice decreases is

Let f be differentiable for all x. If f(1) = -2 and f'(x) $$\ge$$ 2 for x $$\in \left[1,6\right]$$, then

Area of the greatest rectangle that can be inscribed in the ellipse {{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1

A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceleration of $2\mathrm{cm}/{\mathrm{s}}^2$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20\mathrm{cm}/\mathrm{s}$. Then the lizard will catch the insect after :

The area enclosed between the curve $$y={\log }_e\left(x+e\right)$$ and the coordinate axes is :

The parabolas $$y^2=4x$$ and $$x^2=4y$$ divide the square region bounded by the lines $$x=4,$$ $$y=4$$ and the coordinate axes. If $$S_1,S_2,S_3$$ are respectively the areas of these parts numbered from top to bottom ; then $$S_1,S_2,S_3$$ is :

Let $$f(x)$$ be a non - negative continuous function such that the area bounded by the curve $$y=f(x),$$ $$x$$-axis and the ordinates x = {\pi \over 4} and x = \beta > {\pi \over 4} is \left( {\beta \sin \beta + {\pi \over 4}\cos \beta + \sqrt 2 \beta } \right). Then f\left( {{\pi \over 2}} \right) is

Out of the following pair, which one does NOT have identical dimensions is

A fish looking up through the water sees the outside world contained in a circular horizon. If the refractive index of water is {4 \over 3} and the fish is $$12$$ $$cm$$ below the surface, the radius of this circle in $$cm$$ is

A thin glass (refractive index $$1.5$$) lens has optical power of $$-5$$ $$D$$ in air. Its optical power in a liquid medium with refractive index $$1.6$$ will be

Two sources of equal $$emf$$ are connected to an external resistance $$R.$$ The internal resistance of the two sources are $$R_1$$ and $$R_2\left(R_1>R_1\right).$$ If the potential difference across the source having internal resistance $$R_2$$ is zero, then

An energy source will supply a constant current into the load if its internal resistance is

In a potentiometer experiment the balancing with a cell is at length $$240$$ $$cm.$$ On shunting the cell with a resistance of $$2\Omega ,$$ the balancing length becomes $$120$$ $$cm$$. The internal resistance of the cell is

The resistance of hot tungsten filament is about $$10$$ times the cold resistance. What will be resistance of $$100$$ $$W$$ and $$200$$ $$V$$ lamp when not in use ?

A moving coil galvanometer has $$150$$ equal divisions. Its current sensitivity is $$10$$- divisions per milliampere and voltage sensitivity is $$2$$ divisions per millivolt. In order that each division reads $$1$$ volt, the resistance in $$ohms$$ needed to be connected in series with the coil will be -

Two voltmeters, one of copper and another of silver, are joined in parallel. When a total charge $$q$$ flows through the voltmeters, equal amount of metals are deposited. If the electrochemical equivalents of copper and silver are $$Z_1$$ and $$Z_2$$ respectively the charge which flows through the silver voltmeter is

In the circuit, the galvanometer $$G$$ shows zero deflection. If the batteries $$A$$ and $$B$$ have negligible internal resistance, the value of the resistor $$R$$ will be -