If the line \[y=ax+b\] is tangent to the curve \[f(x)=x-{{x}^{3}}\]at point \[(-1,0)\]then the eccentricity of the ellipse \[\frac{{{x}^{2}}}{{{a}^{4}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is
Median of \[^{2n}{{C}_{0,}}^{2n}{{C}_{1,}}^{2n}{{C}_{2,}}^{2n}{{C}_{3,}}{{.......}^{2n}}{{C}_{n,}}\] (where n is even) is \[[note:{{}^{n}}{{C}_{r}}=\frac{n!}{r!(n-r)!}]\]
If \[\alpha \] and \[\beta \] are the roots of the quadratic equation \[p{{x}^{2}}+qx+r=0\]where \[\alpha \beta =99\] and p, q, r (taken in that order) are in arithmetic progression, then \[(\alpha +\beta )\] equals
If \[\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\] are non-coplanar unit vectors such that \[\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\frac{1}{\sqrt{2}}(\overrightarrow{b}+\overrightarrow{c}),\] then the angle between the vectors \[\overrightarrow{a},\overrightarrow{b}\] is
The length of perpendiculars from the foci S and S' on any tangent to ellipse \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{9}=1\] are a and c respectively, then the value of \[\int\limits_{-ac}^{ac}{\{2x\}\,dx}\] equal to Note; {k} denotes fractional part of k.
Let \[f:\text{ }R\to R\] be a differentiable function and \[f(1)=10.\]Then the value of \[\underset{x\to I}{\mathop{\lim }}\,\int\limits_{10}^{f(x)}{\frac{2t}{x-1}}\] equals
If the system of equations \[2x-y+z\text{ }=\text{ }0;\text{ }x-2y+z=0;\] \[\lambda x-y+2z=0\]has infinitely many solution and \[f(x)\]be a continuous function such that \[f(x)+f(x+5)=100,\]then the value of \[\int\limits_{0}^{2\lambda }{f(x)}\,dx\]is equal to
Let \[f(x)={{\cot }^{-1}}\left( \frac{3x-{{x}^{3}}}{1-3{{x}^{2}}} \right)\] and \[g(x)={{\cot }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)\]then \[\underset{x\to c}{\mathop{\lim }}\,\frac{f(x)-f(c)}{g(x)-g(c)}\] where \[c\in \left( 0,\frac{1}{2} \right)\] is
A tangent to the circle \[{{x}^{2}}+{{y}^{2}}=4\]intersects the hyperbola \[{{x}^{2}}-2{{y}^{2}}=2\]at P and Q. If locus of mid- point of PQ is \[{{({{x}^{2}}-2{{y}^{2}})}^{2}}=\lambda ({{x}^{2}}+4{{y}^{2}}),\]then \[\lambda \] equals
John observes that the angle of elevation of first floor of building at a point A on the ground is \[{{30}^{o}}.\] The moves\[\sqrt{3}\] units towards the building to the point B and find that the angle of elevation of the second floor of building is \[{{60}^{o}}.\] If each floor has the same height, the height of seventh floor from the ground in units is
Let \[z\] be a complex number satisfying \[\frac{{{n}^{2}}-1}{4}.\]on the argon plane. If the locus of \[z\]is a conic C of area a7i and eccentricity e, then the value of \[|a-e|\] is equal to
Suppose y is a function of x that satisfies \[\frac{dy}{dx}=\frac{\sqrt{1-{{y}^{2}}}}{{{x}^{2}}}\]and \[y=0\]at \[x=\frac{2}{\pi }\]then \[y\left( \frac{3}{\pi } \right)\]is equal to
The equations of perpendicular bisectors of two sides AB and AC of a triangle ABC are \[x+y+1=0\]and \[x-y+1=0.\]respectively. If circumradius of \[\Delta \,ABC\] is 2 units and the locus of vertex A is \[{{x}^{2}}+{{y}^{2}}+gx+c=0,\]then \[({{g}^{2}}+{{c}^{2}}),\] is equal to
If \[|{{z}_{i}}|=2\] and \[(1-i){{z}_{2}}+(1-i){{\overrightarrow{z}}_{_{2}}}=8\sqrt{2},\] then the minimum value of \[|{{z}_{1}}-{{z}_{2}}|\] equals \[[Note:i=\sqrt{-1}.]\]
If P is the affix of \[z\] in the Argand diagram and P moves so that \[\frac{z-i}{z-1}\] is always purely imaginary, then the locus of z is \[(Note\text{ }:\text{ }i\text{ =}\sqrt{-1})\]
A)
circle with centre \[\left( \frac{1}{2},\frac{1}{2} \right)\] and radius \[\frac{1}{\sqrt{2}}\]
doneclear
B)
circle with centre \[\left( \frac{-1}{2},\frac{-1}{2} \right)\] and radius \[\frac{1}{\sqrt{2}}\]
doneclear
C)
circle with centre (2, 2) and radius \[\frac{1}{2}\]
doneclear
D)
circle with centre \[\left( \frac{-1}{2},\frac{-1}{2} \right)\] radius \[\frac{1}{2}\]
Statement-1: The Variance of first n even natural numbers is \[\frac{{{n}^{2}}-1}{4}.\]
Statement-2: The sum of first n natural numbers is \[\frac{n(n+1)}{2}\] and the sum of squares of first n natural numbers is \[\frac{n(n+1)(2n+1)}{6}.\]
A)
Statement-1 is true, statement-2 is true, statement-2 is correct explanation for statement-1.
doneclear
B)
Statement-1 is true, Statement-2 is true, Statement-2 is NOT a correct explanation for Statement-1
A wire has a mass \[0.3\pm 0.003\text{ }g,\] radius \[0.5\pm 0.005\text{ }mm\text{ }\] and length \[6\pm 0.06\text{ }cm\]. The maximum percentage error in measurement of its density is :
A particle moves in a projectile motion. At a point Q when its velocity vector makes an angle of \[{{30}^{o}}\]with horizontal, radius of curvature of path at Q is \[\frac{80}{3\sqrt{3}}\](in metres). If the particle is projected at \[{{60}^{o}},\] find the velocity of projection (m/sec)
A sphere B is in air sandwiched between blocks A and C which are compressed by two spring of spring constant \[{{k}_{1}}\] and \[{{k}_{2}}\] compressed by \[{{x}_{1}}\] and \[{{x}_{2}}\]respectively. The coefficient of friction between sphere and blocks is same and other surfaces are smooth. The sphere is at equlibrium then the ratio\[\frac{{{K}_{1}}}{{{K}_{2}}}\] will be:
A ring A of mass m is attached to a spring of constant K, which is fixed at C on a smooth circulate track of radius R. Points A and C are diametrically opposite. When the ring slips from rest on the track to point B, making an angle of \[{{30}^{o}}\] with \[(\angle ACB={{30}^{o}})\]spring becomes unstretched. Find the velocity of the ring at B.
A shell is fired from a cannon with a velocity V (m/s) at an angle 9 with the horizontal direction. At the highest point in its path, it explodes into two pieces of equal mass. One of the pieces retraces its path to the cannon. The speed of the other piece. immediately after the explosion is :
A Cubical block of mass m and side L rests on a rough horizontal surface with coefficient of friction\[\mu .\] A horizontal force F is applied on the block as shown. If the coefficient of friction is sufficiently high, so that block does not slide before toppling, the minimum force to topple the block is :
A thin wire of length L and uniform linear mass density \[\rho \] is bent into a circular loop with centre at O. The moment of inertia of the loop about the axis \[x-x'\]is :
The frequency of oscillation is \[\left( \frac{10}{\pi } \right)\] of a. particle of mass 0.1 kg which executes SHM along \[x-axis.\]The kinetic energy is 0.3J and potential energy is 0.2 J at position \[x.\]The potential energy is zero at mean position. Find the amplitude of oscillations (in metres):
Two identical thin rings each of radius R and placed coaxially at a distance R have a uniform mass distribution and mass \[{{m}_{1}}\] and \[{{m}_{2}}\] respectively. Work done in moving a particle of mass m from centre of one ring to the centre of other ring will be:
The end A is subjected to a tensile force F, while force applied at end B is \[\Delta \,F\] lesser than applied at end A and opposite to F. Tension at midpoint of the rod will be:
The graph as shown represents the variation of temperature \[(T)\] of the bodies \[X\] and \[Y\]having same surface area with time \[(t)\] due to emission of radiation. Find the correct relation between emissivity and absorptivity power of the two bodies.
A closed organ pipe and an open organ pipe of same length L when set into vibrations simultaneously in their fundamental mode produce 2 beats. When their lengths are changed they produced 7 beats. If new length of open organ pipe is \[{{L}_{0}}\] and new length of closed organ pipe is \[{{L}_{C}}\] then :
A fish in a lake (refractive index 4/3 for water) is viewed through a convex lens. From water surface, the lens is placed in air at half of the distance of the fish from the water surface, so that the image is formed at the fish itself. The focal length of the lens is how many times the depth of fish in water?
Two beam of light having intensities \[I\] and \[4I\] interfere to produce of fringe pattern on a screen. The phase difference between the beams is \[\pi /2\] at point A and \[\pi \] at point B. Then the difference between resultant intensities at A and B is :
Consider a system of three charges \[\frac{q}{3},\frac{q}{3}\] and \[(-)\frac{2q}{3}\]placed at points A, B and C respectively. ABC is equilateral triangle having 0 to be the centre of its circum circle of radius R .Choose correct option.
A)
the electric field at point O is \[\frac{q}{8\pi {{\varepsilon }_{0}}{{R}^{2}}}\] directed along the O to A.
doneclear
B)
the magnitude of the force between the charges at C and B is \[\frac{{{q}^{2}}}{64\pi {{\varepsilon }_{0}}{{R}^{2}}}\]
doneclear
C)
the potential at O is zero.
doneclear
D)
the potential at point O is \[\frac{q}{12\pi {{\varepsilon }_{0}}R}\]
Two very long straight parallel wires carry steady currents \[I\] and \[-I\]respectively. The distance between wires is d. At a certain instant of time, a point charge q is at a point equidistant from two wires in the plane of the wires. Its instantaneous velocity \[\overrightarrow{v}\] is perpendicular to this plane. The magnitude of the force due to the magnetic field acting on the charge at this instant is:
The network shown in figure is part of a complete circuit. If at a certain instant the current \[(i)\] is 5A and is decreasing at a rate of \[{{10}^{3}}\text{ }A/s,\] then \[{{V}_{B}}-{{V}_{A}}\] is
When an AC source of emf \[e={{\text{E}}_{0}}\text{ }sin\text{ (}100t\text{)}\]is connected across a circuit, the phase difference between the emf e and the current i in the circuit is observed to be \[\pi /4\] as shown in the diagram. If the circuit consists possibly only of \[R-C\] or \[R-L\] or \[L-C\] in series, find the relationship between the two elements.
A nucleus with mass number 220 initially at rest emits an \[\alpha -\]particle. If the Q value of the reaction is 5.5 MeV. Calculate the kinetic energy of \[\alpha -\]particle.
Energy required to remove both the electrons of helium is 79 eV. The energy required to remove one electron from the two electrons of neutral helium atom is
The electrical conductivity of a semi-conductor increases when radiation of wavelength shorter than 2480 mn is incident on it. The band gap (in eV) for semiconductor is:
A microwave telephone link operating at the central frequency 10 GHz has been established. If 2% of this is available for microwave communication channel, then maximum number of telephones channel each has band with 8 kHz simultaneously granted will be:
A bar magnet is oscillating in magnetic field of earth with a time period T. If its mass becomes 4 times of initial mass, what will be the effect on the time period and motion?
A)
New time period is 4T motion remains SHM
doneclear
B)
New time period is T/2 motion remains SHM
doneclear
C)
New time period is nearly T but motion is not SHM.
In a system, a piston caused an expansion against an external pressure of 1.2 atmgivinga change in volume of 32 It. for which \[\Delta \,E\text{ }=\text{ }-51\text{ }kJ.\]What was the value of heat involved ? (Take 1 It atm = 100 J)
In a simple cubic lattice of anions, the side length of the unit cell is \[2.88\overset{\text{o}}{\mathop{A}}\,\] The diameter of the void in the body centre is
In which reactions major product is carbonyl compound? (i) \[{{C}_{6}}{{H}_{6}}+C{{H}_{3}}COOL\xrightarrow{Anhy.AlC{{l}_{3}}}\] (ii) \[{{(PH-C{{H}_{2}})}_{2}}C{{d}_{2}}Cd+2C{{H}_{3}}COCl\xrightarrow{{}}\] (iii) \[C{{H}_{2}}-C\equiv CH\xrightarrow[{{H}_{2}}S{{O}_{4}}]{HgS{{O}_{4}}}\] (iv) \[PH-C{{H}_{3}}\xrightarrow[(2){{H}_{2}}{{O}^{+}}]{(1)Cr{{O}_{2}}C{{l}_{\,}}}\]