An ellipse is drawn by taking a diameter of thec circle $${\left(x-1\right)}^2+y^2=1$$ as its semi-minor axis and a diameter of the circle $$x^2+{\left(y-2\right)}^2=4$$ is semi-major axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is :

Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can be formed such that Y $$\subseteq$$ X, Z $$\subseteq$$ X and Y $$\cap$$ Z is empty, is :

If $$f:R\to R$$ is a function defined by f\left( x \right) = \left[ x \right]\cos \left( {{{2x - 1} \over 2}} \right)\pi, where [x] denotes the greatest integer function, then $$f$$ is

Consider the function, $$f\left(x\right)=\left|x-2\right|+\left|x-5\right|,x\in R$$ Statement - 1 : $$f^{\prime }\left(4\right)=0$$ Statement - 2 : $$f$$ is continuous in [2, 5], differentiable in (2, 5) and $$f$$(2) = $$f$$(5)

Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is:

The equation $$e^{\sin x}-e^{-\sin x}-4=0$$ has:

In a $$\Delta PQR,$$ If $$3\sin P+4\cos Q=6$$ and $$4\sin Q+3\cos P=1,$$ then the angle R is equal to :

A equation of a plane parallel to the plane $$x-2y+2z-5=0$$ and at a unit distance from the origin is :

If the line {{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over 4} and {{x - 3} \over 1} = {{y - k} \over 2} = {z \over 1} intersect, then $$k$$ is equal to :

Let x1, x2,........., xn be n observations, and let $$\overline{x}$$ be their arithematic mean and $${\sigma }^2$$ be their variance. Statement 1 : Variance of 2x1, 2x2,......., 2xn is 4$${\sigma }^2$$. Statement 2 : : Arithmetic mean of 2x1, 2x2,......, 2xn is 4$$\overline{x}$$.

Statement-1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) +.....+ (361 + 380 + 400) is 8000. Statement-2: $$\sum _{k=1}^n\left(k^3-{(k-1)}^3\right)=n^3$$, for any natural number n.

Let $$a,b\in R$$ be such that the function $$f$$ given by $$f\left(x\right)=In\left|x\right|+b{x}^2+ax,x\ne 0$$ has extreme values at $$x=-1$$ and $$x=2$$ Statement-1 : $$f$$ has local maximum at $$x=-1$$ and at $$x=2$$. Statement-2 : a = {1 \over 2} and b = {-1 \over 4}

A spherical balloon is filled with $$4500\pi$$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $$72\pi$$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases $$49$$ minutes after the leakage began is :

A line is drawn through the point $$(1,2)$$ to meet the coordinate axes at $$P$$ and $$Q$$ such that it forms a triangle $$OPQ,$$ where $$O$$ is the origin. If the area of the triangle $$OPQ$$ is least, then the slope of the line $$PQ$$ is :

The area between the parabolas {x^2} = {y \over 4} and $$x^2=9y$$ and the straight line $$y=2$$ is :

The negation of the statement “If I become a teacher, then I will open a school” is :

A spectrometer gives the following reading when used to measure the angle of a prism. Main scale reading: 58.5 degree Vernier scale reading : 09 divisions Given that 1 division on main scale corresponds to 0.5 degree. Total divisions on the vernier scale is 30 and match with 29 divisions of the main scale. The angle of the prism from the above data

Resistance of a given wire is obtained by measuring the current flowing in it and the voltage difference applied across it. If the percentage errors in the measurement of the current and the voltage difference are 3% each, then error in the value of resistance of the wire is

An object $$2.4$$ $$m$$ in front of a lens forms a sharp image on a film $$12$$ $$cm$$ behind the lens. A glass plate $$1$$ $$cm$$ thick, of refractive index $$1.50$$ is interposed between lens and film with its plane faces parallel to film. At what distance (from lens) should object shifted to be in sharp focus of film?

Two electric bulbs marked $$25W$$ $$-$$ $$220$$ $$V$$ and $$100W$$ $$-$$ $$220V$$ are connected in series to a $$440$$ $$V$$ supply. Which of the bulbs will fuse?