Let \[M=\int\limits_{0}^{\frac{\pi }{2}}{\frac{\cos x}{x+2}dx}\] and \[N=\int\limits_{0}^{\frac{\pi }{4}}{\frac{\sin x.\cos x}{{{(x+1)}^{2}}}dx}\] then the value of (M - N) equals
The intersection of the planes \[2x-y-3z=8\] and \[x+2y-4z=14\] is the line L. The value of 'a' for which the line L is perpendicular to the line through (a, 2, 2) and (6, 11, -1) is
A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is \[{{60}^{o}}\] and when he moves 40m away from the tree, the angle of elevation becomes \[{{30}^{o}}\]. The breadth of the river is
If \[\vec{a},\vec{b}\] are any two perpendicular vectors of equal magnitude and \[\left| 3\vec{a}+4\vec{b} \right|+\left| 4\vec{a}-3\vec{b} \right|=20\], then \[\left| {\vec{a}} \right|\] equals
In IITJEE-2016, a multiple choice question has four alternatives in which two or more than two are correct. Number of ways in which this question can be answered, is
If \[f(x)=\left\{ \begin{matrix} x\ln x, & x>0 \\ 0, & x=0 \\ \end{matrix} \right.\]and conclusion of LMVT holds at \[x=1\] in the interval [0, a] for \[f(x)\], then \[[{{a}^{2}}]\] is equal to [Note: [k] denotes the greatest integer less than or equal to k.]
A chord PQ is a normal to the parabola \[{{y}^{2}}=4ax\] at P and subtends a right angle at the vertex. If \[SQ=\lambda \,SP\] where S is the focus then the value of \[\lambda \] is
Let \[f(x)\left\{ \begin{matrix} \frac{{{(\sin x)}^{13}}-\ell n\left( 1+\sin x{{)}^{13}} \right)}{{{(\tan x)}^{26}}} & x\ne 0 \\ k, & x=0 \\ \end{matrix} \right.\]. If \[f(x)\] is continuous at \[x=0\] then the value of k is
A curve \[y=f(x)\] is passing through (0, 0). If the slope of the curve at any point \[(x,y)\] is equal to \[(x+xy)\], then the number of solution of the equation \[f(x)=1\], is
A plane P is perpendicular to the vector \[\vec{A}=2\hat{i}+3\hat{j}+6\hat{k}\] and contains the terminal point of the vector \[\vec{B}=\hat{i}+5\hat{j}+3\hat{k}\]. The distance from the origin to the plane P, is
If the value of y (greater than 1) satisfying the equation \[\int\limits_{1}^{y}{x\,\ell n\,x\,dx=\frac{1}{4}}\] can be expressed in the form of \[{{e}^{\frac{m}{n}}}\] , where m and n are relative prime then \[(m+n)\] is equal to [Note : e denotes Napier's constant]
Number of possible integral values of m in \[(-10,10)\] for which the quadratic equation \[{{x}^{2}}+(m+6)\left| x \right|+2m+8=0\] has two distinct real solutions are
For \[\lambda \in R\], let \[f(\lambda )=\] det \[(A-\lambda I)\] where \[A\left[ \begin{matrix} 1 & 2 \\ -1 & 3 \\ \end{matrix} \right]\]and I is an identity matrix of order 2. The minimum value of \[f(\lambda )\] is equal to
A circle passes through the points (2, 2) and (9, 9) and touches the x-axis. The absolute value of the difference of x-coordinate of the point of contact is
The focal chord of the parabola \[{{(y-2)}^{2}}=16(x-1)\] is a tangent to the circle \[{{x}^{2}}+{{y}^{2}}-14x-4y=51=0\], then slope of the focal chord can be
Consider two lines in space as \[{{L}_{i}}={{\vec{r}}_{1}}=\hat{j}+2\hat{k}+\lambda (3\hat{i}-\hat{j}-\hat{k})\] And \[{{L}_{2}}={{\vec{r}}_{2}}=4\hat{i}+3\hat{j}+6\hat{k}+\mu (\hat{i}+2\hat{k})\]. If the shortest distance between these lines is \[\sqrt{d}\] then d equals
A curve passing through M(1, 1) have the property that any tangent intersects the y-axis at the point which is equidistant from the point of tangency and the origin, can be
If \[\vec{p}\] and q are two diagonals of a quadrilateral such that \[\left| \vec{p}-\vec{q} \right|=\vec{p}.\,\vec{q}\], \[\left| {\vec{p}} \right|=1\], \[\left| {\vec{q}} \right|=\sqrt{2}\], then the area of quadrilateral is equal to
If the angle between the plane \[x-3y+2z=1\] and the line \[\frac{x-1}{2}=\frac{y-1}{1}=\frac{z-1}{-3}\] is \[\theta \], then the value of \[\cos ec\theta \] is
Let a circle of radius \[\frac{2}{\sqrt{3}}\] is touching the lines \[{{x}^{2}}-4xy+{{y}^{2}}=0\]in the first quadrant at points A and B. If area of triangle OAB (0 being the origin) is \[\Delta \] then \[{{\Delta }^{2}}\] equals
In Young's double slit experiment, first slit has width four times the width of the second slit. The ratio of the maximum intensity to the minimum intensity in the interference fringe system is :
For compound microscope\[{{f}_{0}}=1\,cm\], \[{{f}_{e}}=2.5\,cm\]. An object is placed at distance \[1.2\,cm\] from object lens. What should be length of microscope for normal adjustment?
A point source of electromagnetic radiation has an average power output of 1500 W. The maximum value of electric field at a distance of 3m from this source in \[V{{m}^{-1}}\] is
An LCR circuit of \[R=100\,\Omega \] is connected to an AC source 100 V, 50 Hz. the magnitude of phase difference between current and voltage is \[{{30}^{o}}\]. The power dissipated in the LCR circuit is :
A step - down transformer, transforms a supply line voltage of 2200 V into 220V. The primary coil has 5000 turns. The efficiency and power transmitted by the transformer are \[90%\] and 8 kW respectively. Then, the power supplied is:
At a temperature of \[{{30}^{o}}C\], the susceptibility of a ferromagnetic material is found to be \[\chi \] . Its susceptibility at \[{{333}^{o}}C\] is :
A circular coil of 20 turns and radius 10 cm is placed in uniform magnetic field of 0.10 T normal to the plane of the coil. If the current in coil is 5A, then the torque acting on the coil will be:
\[{{B}_{1}},{{B}_{2}}\] and \[{{B}_{2}}\] are the three identical bulbs connected to a battery of steady emf with key K closed. What happens to the brightness of the bulbs \[{{B}_{1}}\] and \[{{B}_{2}}\], when the key is opened.
A)
Brightness of the bulb \[{{B}_{1}}\] increase and that of \[{{B}_{2}}\] decreases.
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B)
Brightness of the bulb \[{{B}_{1}}\] and \[{{B}_{2}}\] increase
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C)
Brightness of the bulb \[{{B}_{1}}\] decrease and \[{{B}_{2}}\] increases.
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D)
Brightness of the bulb \[{{B}_{1}}\] and \[{{B}_{2}}\] decreases.
The resistance of a wire at 300 K is found to be\[0.3\] \[\Omega \]. If the temperature coefficient of resistance of wire is \[1.5\times {{10}^{-3}}{{K}^{-1}}\], the temperature at which the resistance becomes \[0.5\,\Omega \]. is
An air filled parallel plate capacitor has a capacity of 2pF. The separation of the plates is doubled and the interspace between the plates is filled with wax. If the capacity is increased to 6pF, the dielectric constant of wax is :
An electric dipole of length 1 cm is placed with axis making an angle of \[30{}^\circ \] to an electric field of strength \[{{10}^{4}}N{{C}^{-1}}\]. If it experiences a torque of \[10\sqrt{2}\] Nm, the potential energy of the dipole is:
In an experiment the angles are required to be measured using an instrument 29 deviation of the main scale exactly coincide with the 30 deviation of the vernier scale. If the smallest division of the main scale is half - a - degree \[(={{0.5}^{o}})\], then the least count of the instrument is:
When a ceiling fan is switched off, its angular velocity falls to half while it makes 36 rotations. how many rotation will it make before coming to rest?
A mass of M kg is suspended by a weightless string. The minimum force that is required to displace it until the string makes an angle of \[{{45}^{o}}\] with the initial vertical direction is:
A mass m moves with a velocity v and collides in elastically with another identical mass. After v collision the 1 st mass moves with velocity \[\frac{v}{\sqrt{3}}\] in a direction perpendicular to the initial direction of motion. Find the speed of the second mass after collision.
A uniform circular disc of mass 50 kg and radius \[0.4\] m is rotating with an angular velocity of 10 rad/s about its own axis, which is vertical. Two uniform circular rings, each of mass \[6.25\] kg and radius \[0.2\] m are gently placed symmetrically on the disc in such a manner that they are touching each other along the axis of the disc and are horizontal. Assume that the friction is large enough such that the rings are at rest relative to the disc and the system rotates about the original axis. The new angular velocity (in rad/s~1) of the system is :
Block A of mass 2 kg is placed over block B of mass, 8 kg. The combination is placed over a rough horizontal surface. Coefficient of friction between B and the floor is \[0.5\]. Coefficient of friction between A and B is \[0.4\]. A horizontal force of 10 N is applied on block B. The force of friction between A and B is:
Same quantity of ice is filled in each of the two metal containers P and Q having the same size, shape and wall thickness but made of different materials. The containers are kept in identical surroundings. The ice in P melts completes is Rune \[{{t}_{1}}\] whereas in Q takes a time \[{{t}_{2}}\]. The ratio of thermal conductivities of the materials of P and Q is :
A large open tank has two holes in its wall. One is a square hole of side a at a depth of \[x\] from the top and the other is a circular hole of radius r at a depth \[4x\] from the top. When the tank is completely filled with water, the quantities of water flowing out per second from both holes are the same. Then r is equal to :
A vessel whose bottom has round hole with diameter of 1 mm is filled with water. Assuming that surface tension acts only at hole, then the maximum height to which the water can be filled in vessel without leakage is (surface tension of water \[=7.5\times {{10}^{-2}}N{{m}^{-1}}\] and \[g=10\,m{{s}^{-2}}\])
A body is released from a point at distance r from the centre of earth. If R is the radius of the earth and \[r>R\], then the velocity of the body at the time of striking the earth will be:
A cart is moving horizontal along a straight line with constant speed \[30\,m{{s}^{-1}}\]. A projectile is to fired from the moving cart in such a way that it will return to the cart after the cart has moved 80 m. At what speed (relative to the cart) must the projectile be fired?
A long cylindrical tube carries a highly polished piston and has a side opening. A tunning fork of frequency v is sounded at the sound heard by the listener changes if the piston is moved in or out. At a particular position of the piston is moved through a distance of 9 cm, the intensity of sound becomes minimum, if the speed of sound is 360 m/s, the value of v is:
At some constant equilibrium temperature T K, the following gaseous equilibrium is set in a vessel fitted with a movable piston. \[{{N}_{2}}+3{{H}_{2}}\overset{{}}{leftrightarrows}2N{{H}_{3}}\] Now at this equilibrium state, total pressure is increased to some extent so as to attain a new equilibrium state. Which of the following will be true at the new equilibrium state?
A)
concentration of \[N{{H}_{3}}\] will increase but that of \[{{N}_{2}}\] and \[{{H}_{2}}\] will remain same,
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B)
concentration of \[N{{H}_{3}},\,{{N}_{2}}\] and \[{{H}_{2}}\] will increase
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C)
There shall be no change in the concentration of any of these gases
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D)
concentration of \[N{{H}_{3}}\] will increase but that of \[{{N}_{2}}\] and \[{{H}_{2}}\] will decrease
Two elements 'A' and 'B' form three covalent non-volatile solids with molecular formula \[A{{B}_{2}}\],\[A{{B}_{4}}\] and\[A{{B}_{8}}\]. One gram of each is dissolved in 20 g \[{{C}_{8}}{{H}_{6}}(\ell )\] separately For the solutions of \[A{{B}_{2}}\] and \[A{{B}_{4}}\] lowering in freezing points are found to \[\frac{5}{2}\] and \[\frac{5}{3}\] degree respectively. Assuming that here is no dissociation or association of any compound and \[{{K}_{f}}\] for \[{{C}_{6}}{{H}_{6}}(\ell )\] be 5 K.kg. \[mo{{l}^{-1}}\], what will be \[\Delta T\], for solution of \[A{{B}_{8}}\].
The rate constant of a first order reaction is \[4\times {{10}^{-3}}{{\sec }^{-1}}\] At a reactant concentration of \[0.02\] M. the rate of reaction would be :
A hydrogen gas electrode is made by dipping platinum wire in a solution of \[HC\,l\] of \[pH=10\] and by passing hydrogen gas around the platinum wire at one atm pressure The oxidation potential of electrode would be? \[(T=298\,K)\]
An acidic buffer solution has \[[HA]=1.0\]M and\[[NaA]=1.0M\]. To \[(10+x)\]mL of this buffer solution 9 mL of \[1.0M\,HCl\] is added so that pH changes by one unit. The value of \[x\] is