If the pairs of lines \[{{x}^{2}}+2xy+a{{y}^{2}}=0\] and \[a{{x}^{2}}+2xy+{{y}^{2}}=0\] have exactly one line in common then the joint equation of the other two lines can be-
If the ellipse \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{1}=1\] meet the ellipse \[\frac{{{x}^{2}}}{1}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\] in four distinct points and \[a={{b}^{2}}-10\,b+25\] then the value of b does not satisfy -
The mean of 63 children on an arithmetic test are respectively 27.6 and 7.1. To them are added a new group of 26 who had less training and whose mean is 19.2 and S.D. 6.2. The value of the combined group differ from the original as to the mean is
The function \[f(x)=0\]has eight distinct real solutions and \[f\] also satisfy \[f(4+x)=f(4-x).\]The sum of all the eight solutions of \[f(x)=0\] is -
Let the tangent to the \[\sqrt{x}+\sqrt{y}=\sqrt{a}\,\,(a>0)\] curve at any point on it cuts the coordinate axes at P and Q. Then \[OP\text{ }+\text{ }OQ,\] where 0 is origin is equal to-
Before a race, the chances of three runners A, B and C were estimated to be proportional to 5 : 3 : 2, but during the race, A meets with an accident which reduces his chance to 1/3. The respective chances of B and C now winning are
A triangle has base 10 cm long and the base angles of \[50{}^\circ \] and \[70{}^\circ .\] If the perimeter of the triangle is \[x+y\text{ }cos\text{ z}{}^\circ ,\] where \[z\in (0,90{}^\circ ),\] then the value of \[x\text{ }+\text{ }y\text{ }+\text{ }z\] equals -
Let \[\vec{a}={{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k},\] \[\vec{b}={{b}_{1}}\hat{i}+{{b}_{2}}\hat{j}+{{b}_{3}}\hat{k},\] and \[\vec{c}={{c}_{1}}\hat{i}+{{c}_{2}}\hat{j}+{{c}_{3}}\hat{k},\] be three nonzero vectors such that \[\left| \,\,\vec{c}\,\, \right|=1\] angle between \[\vec{a}\] and \[\vec{b}\] is \[\frac{\pi }{4}\]and \[\vec{c}\] perpendicular to \[\vec{a}\] and \[\vec{b}\] ,then
Sum of the real values of ?a? for which the \[({{a}^{2}}-3a+2)\,\,{{x}^{2}}+({{a}^{2}}-4a+3)\,\,x+({{a}^{2}}-6a+5)=0\] equation possess three distinct roots is given by -
A bimetallic strip is formed out of two identical strips one of copper and other of brass. The coefficients of linear expansion of the two metals are \[{{\alpha }_{C}}\] and \[{{\alpha }_{B}}\] . On heating the temperature of the strip goes up by \[\Delta T\] and the strip bends to form an arc of radius of R Then R is
A)
Proportional to \[\Delta T\]
doneclear
B)
Inversely proportion a to \[\Delta T\]
doneclear
C)
Proportional to \[\left| {{\alpha }_{B}}-{{\alpha }_{C}} \right|\]
The displacement of a particle starting from rest and moving with constant acceleration is calculated by the formula \[s=\frac{1}{2}a{{t}^{2}}\] . If there occurs an error of 10% in the measurement of time then the error in the calculation of \[s\]is:
A sliding wire of length 0.25 m and having a resistance of \[0.5\Omega \] moves along conducting guiding rails AB and CD with a uniform speed of 4 m/s. A magnetic field of 0.5 T exists normal to the plane of ABCD directed into the page. The guides are short -circuited with resistances of 4 and \[2\Omega \]. as shown. The current through the sliding wire is:
The energy of a particle executing simple harmonic motion is given by \[E=A{{x}^{2}}+\text{ }B{{v}^{2}}\] where \[x\] is the displacement from mean position \[~x=0\] and v is the velocity of the particle at x then choose the correct statement(s)
A)
Amplitude of SHM. is \[\sqrt{\frac{2E}{A}}\]
doneclear
B)
Maximum velocity of the particle during S.H.M. is \[\sqrt{EB}\]
doneclear
C)
Time period of motion is \[2\pi \sqrt{\frac{B}{A}}\]
doneclear
D)
Displacement of the particle is proportional to the velocity of the particle.
Two small identical metal balls of radius rare at a distance a from each other and are charged, one with a potential \[{{V}_{1}}\]and the other with a potential \[{{V}_{2}}\]. The charges on the balls are:
Two plane mirrors \[{{M}_{1}}\] and \[{{M}_{2}}\] each have length 2 m and are separated from one another by 1 cm. A ray of light is incident at one end of mirror \[{{M}_{1}}\] at angle\[45{}^\circ \]. How many reflections the ray will Suffer before going out from the other end? :
A horizontal uniform glass tube of 100 cm, length sealed at both ends contain 10 cm mercury column in the middle. The temperature and pressure of air on either side of mercury column are respectively \[81{}^\circ C\] and 76 cm of mercury. If the air column at one end is kept at \[0{}^\circ C\] and the other end at \[273{}^\circ C\], the pressure of air which is at \[0{}^\circ C\] is (in cm of Hg)
A parallel plate capacitor with air between the plates has a capacitance of \[9pF\]. The separation between the plates is \['d'\]. The space between the plates is now filled with two dielectrics. One of the dielectric has dielectric constant \[{{k}_{1}}=3\] and thickness \[d/3\] while the other one has dielectric constant \[{{k}_{2}}=6\] and thickness \[2d/3\] Capacitance of the capacitor is now
In Fresnel's \[\operatorname{biprism}\text{ }(\mu =1.5)\] experiment the distance between source and biprism is 0.3 m and that between biprism and screen is 0.7 m and angle of prism is \[1{}^\circ \]. The fringe width with light of wavelength \[6000\,\overset{\text{o}}{\mathop{\text{A}}}\,\] will be
There are 32 cells each having \[emf 1\text{ }V\] and internal resistance \[1\text{ }\Omega \]. All the cells are connected together in a closed loop. When terminal voltage of cells are measured they are found to be either \[0.5V\] for \[1.5V\]. How many cells are having their polarities reversed with respect to the majority of cells?
A bullet looses \[{{\left( \frac{1}{n} \right)}^{\operatorname{th}}}\] of its velocity passing through one plank. The number of such planks that are required to stop the bullet can be:
Ge and Si diodes conduct at \[0.3 V\]and \[0.7\,V\] respectively. In the following figure if Ge diode connection are reversed, the value of \[{{\operatorname{V}}_{0}}\] changes by
Radium nucleus was moving with constant velocity V. It disintegrates into He add Rn. He and Rn moves as shown in the figure. The CM of the system of the He and Rn will move along---A--- after the explosion of Ra. Here A refers to
Figure shows a circular wire loop of radius r, carrying current i, placed in a perpendicular magnetic field B. The radius of cross-section of the wire is 'a'. Find decrease in the radius of the loop if the magnetic field is switched off.
[Young's modulus of the material of the wire is Y]
In a hydrogen like atom electron makes transition from an energy level with quantum number n to another with quantum number \[(n-1)\]. For \[~n>>1\], the frequency of radiation emitted is :
A system consists of a cylinder, piston and a spring as shown. The initial volume of the cylinder is 100 l and its pressure is \[100 kPa\] so that it just balances the atmosphere pressure plus the piston weight. In this position, the spring connected to the piston exerts no force on it. Heat is now transferred to the system so as to expand air to double its volume, at which the pressure in the cylinder is \[300 kPa\]. The work done by the system is (in case of a spring \[F=-kx\]):
The half-life of a radioactive isotope is 3 hours. If the initial mass of the isotope were 256 gm, the mass of it remaining undecayed after 18 hours would be:
Coppercrystallisesinastructureoffacecenterdcubicunitcell.Theatomicradiusofcopperis \[1.28\overset{{}^\circ }{\mathop{A}}\,.\] What is axial length on an edge of copper.
Dinucleotide is obtained by joining two nucleotides together by phosphodiester linkage. Between which carbon atoms of pentose sugars of nucleotides are these linkages present?
The range of \['\alpha '\] for which the point \[(\alpha ,\alpha )\] lies inside the region bounded by the Curves \[y=\sqrt{1-{{x}^{2}}}\]and \[x+y=1\] is -
The solution of the differential equation \[{{x}^{2}}\frac{dx}{dx}.cos\left( \frac{1}{x} \right)-y\,\,\sin \left( \frac{1}{x} \right)=-\,1.\] Where \[y\to -\,1\,\,as\,x\to \infty \] is -
A variable chord PQ of parabola \[{{y}^{2}}=4ax\] subtends a right angle at the vertex. Find the locus of point of intersection of the tangents at P and Q.
If the straight line \[x\text{ }=\text{ }y\sqrt{3,}\] cuts the ellipse \[{{x}^{2}}+{{y}^{2}}+xy=3\] at points P and Q, then \[\left| \,OP\, \right|\,\,\,\left| \,OQ\, \right|\] is (where ' O ' is the origin)
A student uses a convex lens to determine the width of a slit. For this he fixes the positions of the object and the screen and moves the lens to get a real image on the screen. The images of the slit width are found to be 2.1 cm and 0.48 cm wide respectively when the lens is moved through 15 cm. Therefore, the slit width and the focal length of the lens respectively, are.
A circular loop wire with current \[{{i}_{1}}=({{i}_{0}}/\pi )\] and a V shaped wire with current;, are arranged in a plane as shown in diagram. If magnetic field at O is zero, value of \[{{\operatorname{i}}_{2}}\] is:
Let there be a spherically symmetric charge Distribution with charge density varying as \[\rho (r)={{\rho }_{0}}\left( \frac{5}{3}-\frac{r}{R} \right)\,\]Upto \[r=R,\] and \[\rho \left( r \right)=0\,\] \[\operatorname{for}\,r>R\]where r is the distance from the origin. The electric field at a distance \[ar(r<R)\]from the origin is given by
A man travelling in a car with a maximum constant speed of 20 m/s watches his friend start off at a distance of 100 m on motor cycle with constant acceleration \['a'.\] The man in the car will reach his friend when \['a'\] is
A helicopter is moving to the right at a constant horizontal velocity. It experiences three forces \[{{\vec{F}}_{gravitational}},\] and force on it caused by rotor \[{{\vec{F}}_{rotor.}}\]Which of the following diagrams can be a correct free-body diagram representing forces on the helicopter?
Two identical balls are set into motion simultaneously from an equal h. While ball A is thrown horizontally with velocity\[v\], B is just released to fall by itself. Choose the alternative that best represents the motion of A and B with respect to an observer who moves with velocity \[v/2\]with respect to the ground as shown in the figure.
A radioactive sample of \[{{\operatorname{U}}^{238}}\] decay to Pb through a process for which half-life is \[4.5\times {{10}^{9}}\] years. The ratio of number of nuclei of Pb to \[{{\operatorname{U}}^{238}}\]after a time of \[1.5\times {{10}^{9}}\] years (given \[{{2}^{1/3}}=1.26\])
Take a bicycle rim and extend its axle on both sides. Tie two strings at both ends A and B as shown in the figure. Hold both the strings together in one hand such that the rim is vertical. If you leave one string, the rim will tilt. Now, keeping the rim in vertical position with both the strings in one hand, put the wheel in fast rotation around the axle with the other hand. Then, leave one string, say\[B\], from your hand, and observe what happens.
What will happen, if we leave string B?
A)
The rim will stop rotating
doneclear
B)
The rim will rotate in a vertical plane and the plane of rotation will processes about string A
The cell \[Pt\,({{H}_{2}})\,\,(1\,atm)|{{H}^{+}}(pH=?)||{{I}^{-}}\,(a=1)|Agl\,(s),\] Ag has emf, \[{{E}_{298K}}=0.\] The standard electrode potential for the reaction \[Agl+{{e}^{-}}\to Ag+{{I}^{\Theta }}\]is \[-\,0.151\,\text{volt}\text{.}\] Calculate the pH value.
The freezing point of aqueous solution that contains 3% urea, 7.45% KCl and 9% of glucose is (given \[{{K}_{f}}\] of water = 1.86 and asume molarity = molality).
The activation energy is lowered by \[8.314\,\,KJ\,\,mo{{l}^{-1}}\] for the catalysed reaction. How many times the rate of this catalysed reaction greater than that of uncatalysed reaction? \[(Given\,\,{{e}^{3.33}}=28)\]