Consider the following relations $R=\{(x,y)\mid x,y$ are real numbers and $x=w y$ for some rational number $w\}$; $S=\{\left(\frac{m}{n},\frac{p}{q}\right)\mid m,n,p$ and $q$ are integers such that $n,q\ne 0$ and $qm=pm\}$. Then

Let $$f:R\to R$$ be a positive increasing function with \mathop {\lim }\limits_{x \to \infty } {{f(3x)} \over {f(x)}} = 1. Then \mathop {\lim }\limits_{x \to \infty } {{f(2x)} \over {f(x)}} =

There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

If $$\alpha$$ and $$\beta$$ are the roots of the equation $$x^2-x+1=0,$$ then $${\alpha }^{2009}+{\beta }^{2009}=$$

For a regular polygon, let $$r$$ and $$R$$ be the radii of the inscribed and the circumscribed circles. A $$false$$ statement among the following is :

A line $$AB$$ in three-dimensional space makes angles $$45^{\circ }$$ and $$120^{\circ }$$ with the positive $$x$$-axis and the positive $$y$$-axis respectively. If $$AB$$ makes an acute angle $$\theta$$ with the positive $$z$$-axis, then $$\theta$$ equals :

Statement-1 : The point $$A(3,1,6)$$ is the mirror image of the point $$B(1,3,4)$$ in the plane $$x-y+z=5.$$ Statement-2 : The plane $$x-y+z=5$$ bisects the line segment joining $$A(3,1,6)$$ and $$B(1,3,4).$$

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is

A person is to count 4500 currency notes. Let $$a_n$$ denote the number of notes he counts in the $$n^{th}$$ minute. If $$a_1$$ = $$a_2$$ = ....= $$a_{10}$$= 150 and $$a_{10}$$, $$a_{11}$$,.... are in an AP with common difference - 2, then the time taken by him to count all notes is

The equation of the tangent to the curve y = x + {4 \over {{x^2}}}, that is parallel to the $$x$$-axis, is

Let $$f:R\to R$$ be defined by $f\left( x \right) = \left\{ {\matrix{ {k - 2x,\,\,if} & {x \le - 1} \cr {2x + 3,\,\,if} & {x > - 1} \cr } } \right.$ If $$f$$has a local minimum at $$x=-1$$, then a possible value of $$k$$ is

Let $$f:R\to R$$ be a continuous function defined by $f\left( x \right) = {1 \over {{e^x} + 2{e^{ - x}}}}$ Statement - 1 : f\left( c \right) = {1 \over 3}, for some $$c\in R$$. Statement - 2 : 0 < f\left( x \right) \le {1 \over {2\sqrt 2 }}, for all $$x\in R$$

The area bounded by the curves $$y=\cos x$$ and $$y=\sin x$$ between the ordinates $$x=0$$ and x = {{3\pi } \over 2} is

Let S be a non-empty subset of R. Consider the following statement: P : There is a rational number x ∈ S such that x > 0. Which of the following statements is the negation of the statement P?

The respective number of significant figures for the numbers 23.023, 0.0003 and 2.1 $$\times$$ 10–3 are

An initially parallel cylindrical beam travels in a medium of refractive index $$\mu \left(I\right)={\mu }_0+{\mu }_{2}I,$$ where $${\mu }_0$$ and $${\mu }_2$$ are positive constants and $$I$$ is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius. As the beam enters the medium, it will

An initially parallel cylindrical beam travels in a medium of refractive index $$\mu \left(I\right)={\mu }_0+{\mu }_{2}I,$$ where $${\mu }_0$$ and $${\mu }_2$$ are positive constants and $$I$$ is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius. The speed of light in the medium is

Two conductors have the same resistance at $$0^{\circ }C$$ but their temperature coefficients of resistance are $${\alpha }_1$$ and $${\alpha }_2.$$ The respective temperature coefficients of their series and parallel combinations are nearly

Statement - 1 : Two particles moving in the same direction do not lose all their energy in a completely inelastic collision. Statement - 2 : Principle of conservation of momentum holds true for all kinds of collisions.

The figure shows the position$$-$$time $$(x-t)$$ graph of one-dimensional motion of body of mass $$0.4$$ $$kg.$$ The magnitude of each impulse is