If $$a\in R$$ and the equation $$-3{\left(x-\left[x\right]\right)}^2+2\left(x-\left[x\right]\right)+a^2=0$$ (where [$$x$$] denotes the greater integer $$\le x$$) has no integral solution, then all possible values of a lie in the interval :

The image of the line {{x - 1} \over 3} = {{y - 3} \over 1} = {{z - 4} \over { - 5}}\, in the plane $$2x-y+z+3=0$$ is the line :

The angle between the lines whose direction cosines satisfy the equations $$l+m+n=0$$ and $$l^2=m^2+n^2$$ is :

The variance of first 50 even natural numbers is

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is :

If $$(10{)}^9+2(11{)}^1(10^8)+3(11{)}^2(10{)}^7+.........+10(11{)}^9=k(10{)}^9,$$, then k is equal to :

If $$f$$ and $$g$$ are differentiable functions in $$\left[0,1\right]$$ satisfying $$f\left(0\right)=2=g\left(1\right),g\left(0\right)=0$$ and $$f\left(1\right)=6,$$ then for some $$c\in \left]0,1\right[$$

If $$x=-1$$ and $$x=2$$ are extreme points of $$f\left(x\right)=\alpha \log \left|x\right|+\beta x^2+x$$ then

The area of the region described by $$A=\{\left(x,y\right):x^2+y^2\le 1$$ and $$y^2\le 1-x\}$$ is :

The statement $$\sim \left(p\leftrightarrow \sim q\right)$$ is :

In a Young's double slit experiment, the distance between the two identical slits is 6.1 times larger than the slit width. Then the number of intensity maxima observed within the central maximum of the single slit diffraction pattern is :

A piece of wood from a recently cut tree shows 20 decays per minute. A wooden piece of same size placed in a museum (obtained from a tree cut many years back) shows 2 decays per minute. If half life of C14 is 5730 years, then age of the wooden piece placed in the museum is approximately :

Match List - I (Experiment performed) with List - II (Phenomena discovered/associated) and select the correct option from the options given below the lists : .tg {border-collapse:collapse;border-spacing:0;} .tg td{font-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;} .tg th{font-family:Arial, sans-serif;font-size:14px;font-weight:normal;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;} .tg .tg-bn4o{font-weight:bold;font-size:18px;text-align:center;vertical-align:top} .tg .tg-asmn{font-weight:bold;font-size:18px;text-align:center} .tg .tg-a0j4{font-size:18px;text-align:left} .tg .tg-nx8p{font-size:18px;text-align:left;vertical-align:top} List - I List - II (a) Davisson and Germer Experiment (i) Wave nature of electrons (b) Millikan’s oil drop experiment (ii) Charge of an electron (c) Rutherford experiment (iii) Quantisation of energy levels (d) Franck - Hertz experiment (iv) Existence of nucleus

The diameter of the objective lens of microscope makes an angle $$\beta$$ at the focus of the microscope. Further, the medium between the object and the lens is an oil of refractive index n. Then the resolving power of the microscope.

A ray of light is incident from a denser to a rarer medium. The critical angle for total internal reflection is $$\theta$$iC and the Brewster's angle of incidence is $$\theta$$iB, such that sin$$\theta$$iC/sin$$\theta$$iB = $$\eta$$ = 1.28. The relative refractive index of the two media is :

An example of a perfect diamagnet is a superconductor. This implies that when a superconductor is put in a magnetic field of intensity B, the magnetic field Bs inside the superconductor will be such that :

A student measured the length of a rod and wrote it as 3.50 cm. Which instrument did he use to measure it?

A green light is incident from the water to the air - water interface at the critical angle $$\left(\theta \right)$$. Select the correct statement.

A thin convex lens made from crown glass \left( {\mu = {3 \over 2}} \right) has focal length $$f$$. When it is measured in two different liquids having refractive indices {4 \over 3} and {5 \over 3}, it has the focal lengths $$f_1$$ and $$f_2$$ respectively. The correct relation between the focal lengths is :

In a large building, three are $$15$$ bulbs of $$40$$ $$W$$, $$5$$ bulbs of $$100$$ $$W$$, $$5$$ fans of $$80$$ $$W$$ and $$1$$ heater of $$1$$ $$kW.$$ The voltage of electric mains is $$220$$ $$V.$$ The minimum capacity of the main fuse of the building will be: