In the xy-plane, three distinct lines \[{{\ell }_{1}},{{\ell }_{2}},{{\ell }_{3}}\] are concurrent at \[M(\lambda ,0)\]. Also the lines\[{{\ell }_{1}},{{\ell }_{2}},{{\ell }_{3}}\], are normal's to the parabola \[{{y}^{2}}=6x\]at the points\[A\text{ }\left( {{x}_{1}},\text{ }{{y}_{1}} \right),B\left( {{x}_{2}},\text{ }{{y}_{2}} \right),C\left( {{x}_{3}},\text{ }{{y}_{3}} \right)\]respectively. Then
The minimum value of \[|{{z}_{1}}-{{z}_{2}}|\] as \[{{z}_{1}}\] and \[{{z}_{2}}\] vary over the curve \[|\sqrt{3}(1-2z)+2i|=2\sqrt{7}\]and \[|\sqrt{3}(-1-z)-2i|=\sqrt{3}(9-z)+18i|\] respectively, is [Note: \[i=\sqrt{-1}\]]
Two aero planes I and II bomb a target in succession. The probabilities of I and II scoring hit correctly are \[\frac{3}{10}\] and \[\frac{1}{5}\] respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is
If the straight lines \[\frac{x-1}{k}=\frac{y-2}{2}=\frac{z-3}{3}\], and \[\frac{x-2}{3}=\frac{y-3}{k}=\frac{z-1}{2}\] intersect at a point, then the integer k is equal to
Let P(x) be a polynomial of degree 4 having extremumat 1,2 and Lim \[\underset{x\to 0}{\mathop{Lim}}\,\left( 1+\frac{P(x)}{{{x}^{2}}} \right)=2\] then Pis equal to
The graph of quadratic polynomial\[f\left( x \right)=\left( x-a \right)\left( x-b \right)\]where \[a,\text{ }b>0\]and \[a\ne b\], then the graph does not pass through
If \[\overrightarrow{x}\] and \[\overrightarrow{y}\] be unit vectors and \[|\overrightarrow{z}|=\frac{2}{7}\] such that \[\overrightarrow{z}+(\overrightarrow{z}\times \overrightarrow{x})=\overrightarrow{y}\] and \[\theta \] is the angle between \[\overrightarrow{x}\] and \[\overrightarrow{z}\] , then the value of sin \[\theta \] is
The total number of polynomials of the form\[{{x}^{3}}+a{{x}^{2}}+bx+c\] that are divisible by \[{{x}^{2}}+1\], where \[a,b,c\text{ }\in \text{ }\left\{ 1,2,3..........9,10 \right\}\]is equal to
In a set of 2n observations, half of them are equal to \[\alpha \] and the remaining half are equal to\[-\alpha \]. If the standard deviation of all the observations is 2, then the value of \[|\alpha |\] is equal to
Let \[\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k},\,\,\,\overrightarrow{b}=\hat{i}+2\hat{j}-\hat{k}\] and \[\,\,\overrightarrow{c}=\hat{i}+\hat{j}-2\hat{k}\] be three vectors. A vector of the type \[\overrightarrow{b}+\lambda \overrightarrow{c}\] for some scalar \[\lambda \], whose projection on \[\overrightarrow{a}\] is of magnitude \[\sqrt{\frac{2}{3}}\] is
If the events A and B are such that\[P(A)=\frac{1}{4},\,P\left( \frac{A}{B} \right)=\frac{1}{2}\] and\[\,P\left( \frac{B}{A} \right)=\frac{2}{3}\], then P equal to
A variable line L is drawn through \[O\left( 0,0 \right)\]to meet the lines \[{{L}_{1}}:y-x-10=0\]and \[{{L}_{2}}:\text{ }y-\text{ }x-20=0\] at the points A and B respectively. A point P is taken on L such that -\[\frac{2}{OP}=\frac{1}{OA}+\frac{1}{OB}\]and P, A, B lies on same side of origin 0. The locus of P is
An observer on the top of a tree finds the angle of depression of a car moving toward the tree to be \[{{30}^{o}}\]. After 3 minutes, this angle becomes\[{{60}^{o}}\]. After how much more time, will the car reach the tree?
Normal is drawn to the ellipse \[\frac{{{x}^{2}}}{27}+{{y}^{2}}=1\] at a point \[\left( 3\sqrt{3}cos\theta ,\sin \theta \right)\]where \[0<\theta <\frac{\pi }{2}\]. The value of \[\theta \] such that the area of triangle formed by normal and coordinate axes is maximum, is
If \[i=\sqrt{-1}\] and\[\left( 1+\frac{1}{{{1}^{2}}} \right).\left( 1+\frac{1}{{{(1+i)}^{2}}} \right)\left( 1+\frac{1}{{{(1+2i)}^{2}}} \right)\]\[...........\left( 1+\frac{1}{{{(1+(n-1)i)}^{2}}} \right)=\frac{10+8i}{1+8i},\]then n is equal to
If the equation of plane containing the line\[\frac{-x-1}{3}=\frac{y-1}{2}=\frac{z+1}{-1}\]and passing through the point \[\left( 1,-1,0 \right)\]is \[ax+y+bz+c=0\], then (a + b + c) is equal to
A particle is projected with \[10m{{s}^{-1}}\]at \[{{60}^{o}}\] from horizontal ground. Find magnitude at its average velocity for total time of flight.
A particle of mass 1 kg is projected from ground with velocity \[20\text{ }m{{s}^{-1}}\]at \[{{45}^{o}}\] from horizontal. Magnitude of its angular momentum just before it hits the ground about the point of projection will be \[\left( g=10m{{s}^{-2}} \right)\]
A 60 kg man running on horizontal road and increases his speed from \[2\text{ }m{{s}^{-1}}\]to \[\text{6 }m{{s}^{-1}}\] During this period work done by:
If infinite number of particle each of mass m are placed on x-axis at x = r, x = 2r, x = 4r..... and so on then magnitude of net gravitational field intensity at the origin will be:
Equation of a longitudinal wave is given as \[s={{10}^{2}}\,\sin 2\pi \left( 1000t+\frac{50x}{17} \right)\] (all SI units),At \[t=0\], change in pressure is maximum (in modulus) at x = ?
The charge per unit length of the four quadrants of the ring of radius R are \[2\lambda ,-2\lambda ,\lambda \] and \[-\lambda \], respectively as shown in figure. The magnitude of electric field at the centre is:
Between two infinitely long wires having linear charge densities\[\lambda \], and\[-\lambda \], there are two points A and B as shown in figure. The amount of work done by the electric field in moving a point charge \[{{q}_{0}}\] from A to B is equal to:
A bulb \[{{B}_{1}}\] marked (100W, 250W) and two bulbs \[{{B}_{2}}\] and \[{{B}_{3}}\] each have mark (60W, 350V) are connected to 250V source as shown in figure. Now \[{{W}_{1}},{{W}_{2}}\] and \[{{W}_{3}}\] are the output powers of the bulbs \[{{B}_{3}}\]off' and B, respectively. Then
Given\[{{R}_{1}}=1\Omega \,\,=2\Omega ,\,\,\,\,{{C}_{1}}=2\mu F,\] \[{{C}_{2}}=4\mu F\], the time constant (in \[\mu s\]) for the circuit I, II, III are respectively.
Two magnets placed one above the other oscillate with a period of 6 sec. If one of them is reversed, the time period becomes 2 sec. The ratio of their magnetic moment is nearest to:
Two thick wires and two thin wires, all of the same materials and same length, form a square in three different ways P, Q and R as shown in figure. The magnetic field at the centre of the square is zero in case of:
A metal wire PQ having a mass of 10 g and length 4.9 cm lies at rest on the metal rails as shown in figure. A vertical downward magnetic field of magnitude 0.8T exists in the space. The resistance of the circuit is slowly decreased and it is found that when the resistance goes below 200, the wire starts sliding on the rails. The coefficient of fraction is:
Two coaxial solenoids are made by winding thin insulated wire over a pipe of cross sectional area \[A=10\text{ }c{{m}^{2}}\] and length = 20 cm. If one of the solenoids has 300 turns and the other 400 turns, their mutual inductance is \[({{\mu }_{0}}=4\pi \times {{10}^{-7}}Tm{{A}^{-1}})\]
A converging lens of focal length 15 cm and a converging mirror of focal length 20 cm are placed with their principal axes coinciding. A point source S is placed on the principal axis at a distance of 12 cm from the lens as shown in figure. It is found that the final beam comes out parallel to the principal axis. Find the separation between the mirror and the lens in cm.
Monochromatic light of wavelength \[5000\overset{o}{\mathop{\text{A}}}\,\] is incident on two slits separated by a distance of \[5\times {{10}^{-4}}\text{ }m\]. Interference pattern is seen on the screen placed at a distance of 1 m from the slits. A thin glass plate of thickness \[1.5\times {{10}^{-6}}m\] and refractive index 1.5 is placed between one of the slits and the screen. If intensity in the absence of plate was \[{{I}_{0}}\], then new intensity at the centre of the screen will be:
In an n - p - n transistor amplifier circuit, when working in common emitter configuration, when the input signal changes by 0.02 V, the base current changes by \[10\mu A\] and the collector current by 1mA. the load in the collector circuit is \[10k\Omega ,\], the power gain is
The following figure shown a logic gate circuit with two inputs A and B and output C. The voltage waveforms of A, B and C are as shown in second figure below. The logic circuit gate is:
Light of wavelength \[\lambda \] strikes a photosensitive surface and electrons are ejected with kinetic energy E. If the KE is to be increased to 2E, the wavelength must be changed to \[\lambda '\] where
A reversible engine converts 1/6 part of the heat absorbed from source into work. When the temperature of sink is reduced by\[{{62}^{o}}C\], then its efficiency is doubled. Temperature of source is:
A circular steel strip of 1.20 m diameters is to be shrunk to a cart wheel whose average diameter is 0.36 cm greater than the inner diameter of the strip. If the coefficient of linear expansion of steel is \[1.2\times {{10}^{-5}}\,\,per\,{{\,}^{o}}C\], then the temperature to which the strip must be heated in order to just slip on the wheel is:
Two rods of a substance with same length, having radii in ratio \[{{r}_{1}}:{{r}_{2}}=1:3\] are twisted with same couple. Find ratio of torsion produced in rods.
The equation of progressive wave is: \[y=4\sin \left\{ \pi \left( \frac{t}{5}-\frac{x}{9} \right)+\frac{\pi }{6} \right\}\]where x and y are in cm. Which of the following statement is true?
A magnet oscillates in a magnetic field with periodic time T. If its mass is made four times without changing other variables then what will be the effect on its periodic time and velocity.
A)
it will oscillate with 4T periodic time
doneclear
B)
it will oscillate with 2T periodic time
doneclear
C)
it will not oscillate but will perform periodic motion with same time period
A damped oscillator attains the first amplitude of 500 mm after starting from rest. After 100 oscillations the amplitude remains 50 mm. The time period of motion is 2.3 sec. The damping coefficient well be:
An excited hydrogen atom emits a photon of wavelength K in returning to ground state, if R is the Rydberg constant, then the quantum number n of the excited state is
The equilibrium constant of the reaction\[2N{{H}_{3}}+\frac{5}{2}{{O}_{2}}2NO+3{{H}_{2}}O\] in term of \[{{K}_{1}}\] \[{{K}_{2}}\] and \[{{K}_{3}}\] is
A first order reaction at 298 K can be expressed by generalisation log \[K(In\,{{S}^{-1}})=17.64-\frac{2.5\times {{10}^{4}}}{T},\] then activation energy \[({{K}_{a}})\] of the reaction will be [K =Rate constant of reaction; T= Kelvin temperature]