Let \[\overrightarrow{v}=\overrightarrow{i}\times (\overrightarrow{j}\times (\overrightarrow{k}\times (\overrightarrow{i}\times (\overrightarrow{j}\times (\overrightarrow{k}\times (\overrightarrow{i}\times \overrightarrow{j}\times (\overrightarrow{j}\times \overrightarrow{k})))))))).\] Then \[\left| |\overrightarrow{v}| \right|\] is equal to
In a tennis tournament, the odds that player A will be the champion is 4 to 3 and the odds that player B will be champion is 1 to 4. What are the odds that either A or B will become the champion?
The line \[y=3x-6\]is the tangent to the graph of \[f(x)\]at the point (2, 0). Then the value of \[\underset{x\to 2}{\mathop{\lim }}\,\frac{{{x}^{2}}(2x+1)f(x)}{(x-2)}\]is
Suppose, a population A has 100 observations 101, 102, ...., 200 and another population B has 100 observations 151, 152,...., 250. If \[{{V}_{A}}\] and \[{{V}_{B}}\]represent the variances of the two population respectively, then\[\frac{{{V}_{A}}}{{{V}_{B}}}\] is
A triangle has sides of length 13, 30 and 37. If the radius of the inscribed circle is \[\frac{p}{q}\] (where p and q are coprime), then the value of \[{{q}^{p+3}}\] is
If the vectors \[3\overrightarrow{p}+\overrightarrow{q};\text{ }5\overrightarrow{p}-3\overrightarrow{q};\text{ }2\overrightarrow{p}+\overrightarrow{q};\text{ }4\overrightarrow{p}-2\overrightarrow{q}\] are pairs of mutually perpendicular vectors then sin \[(\overrightarrow{p}\wedge \overrightarrow{p})\] is
'O' is the vertex of the parabola \[{{y}^{2}}\text{ }=\text{ }8x\] and L is the upper end of the latus rectum. If LH is drawn perpendicular to OL meeting OX in H, then the length of the double ordinate through H is \[\lambda \sqrt{5}\] where \[\lambda \] is equal to
Let \[a={{i}^{{{k}_{1}}}}+{{i}^{{{k}_{2}}}}+{{i}^{{{k}_{3}}}}+{{i}^{{{k}_{4}}}},(i=\sqrt{-1})\] where each \[{{k}_{a}}\] is randomly chosen from the set {1, 2, 3,4}. The probability that a = 0, is
If the equation \[co{{t}^{4}}x-2\,cos\,e{{c}^{2}}x+{{a}^{2}}\text{ }=0\]has at least one solution then, sum of all possible integral values of 'a' is equal to
If \[\left( \begin{matrix} \frac{1}{4} & 0 \\ x & \frac{1}{4} \\ \end{matrix} \right)={{\left[ \begin{matrix} 2 & 0 \\ -a & 2 \\ \end{matrix} \right]}^{-2}},\] then the value of \[\frac{a}{x}\]is equal to
Let circles \[{{C}_{1}}\] and \[{{C}_{2}}\] on Argand plane be given by \[|z+1|\,=3\] and \[|z-2|\,=7\] respectively. If a variable circle \[|z-{{z}_{ 0}}|\,=r\] be inside circle \[{{C}_{2}}\] such that is touches \[{{C}_{1}}\] externally and \[{{C}_{2}}\] internally then locus of \[{{z}_{0}}\]describes a conic E whose eccentricity is equal to
C is a curve defined by the parametric equation \[x\text{ }=\text{ }{{t}^{2}}\] and \[y={{t}^{3}}-3t.\]The equation of the tangent line to this curve at (4, 2) is
Let \[{{I}_{1}}=\int\limits_{0}^{x}{{{e}^{tx}}.{{e}^{-{{t}^{2}}}}dt}\]and \[{{I}_{2}}=\int\limits_{0}^{x}{{{e}^{\frac{-{{t}^{2}}}{4}}}dt}\] where \[x\text{ }>\text{ }0\] then the value of \[\frac{{{I}_{1}}}{{{I}_{2}}}\] is
If the curves \[{{x}^{\frac{2}{3}}}+{{y}^{\frac{2}{3}}}+={{c}^{\frac{2}{3}}}\] and \[\left( \frac{{{x}^{2}}}{{{a}^{2}}} \right)+\left( \frac{{{y}^{2}}}{{{b}^{2}}} \right)=1\]touches each other then
If a curve passing through (1, 1) is such that the tangent drawn at any point P on it intersects the \[x\] -axis at Q and the reciprocal of abscissa of point P is equal to twice \[x\] -intercept of tangent at P. Then the equation of the curve is
If the lines \[x=1\text{ }4-a,\text{ }y=-3-\lambda a,\text{ }z=1+\lambda a\] and \[x=\frac{b}{2},y=1+b,z=2-b\] are coplanar, then \[\lambda \] is equalto
Let \[{{C}_{n}}=\int\limits_{\frac{1}{n+1}}^{\frac{1}{n}}{\left( \frac{{{\tan }^{-1}}(nx)}{{{\sin }^{-1}}(nx)} \right)}\,dx,\] then \[\underset{n\to \infty }{\mathop{\lim }}\,{{n}^{2}}{{C}_{n}}\] equals to
A student performs an experiment for determination of \[g\left[ =\frac{4{{\pi }^{2}}l}{{{T}^{2}}} \right],\] where \[l=1\,m\]and he commits an error of \[\Delta \,L.\] For the experiment, he takes the time for n oscillations with the stopwatch for least count \[\Delta \,T\] and he commits a human error of 0.1 sec. For which of the following data, the measurement of g will be most accurate.
A particle is moving eastward with velocity of5m/ sec. In 10 seconds, the velocity changes to 5 m/sec northwards. The average acceleration in this time is:
A)
\[\frac{1}{\sqrt{2}}m/se{{c}^{2}}\] towards north - west
doneclear
B)
\[\frac{1}{\sqrt{2}}m/se{{c}^{2}}\] towards north-east
A bead of mass m is attached to a spring of natural length \[\left( \frac{d+x}{d-x} \right)W\] with other end of it fixed at O. The spring moves in a track of which part ABC is semicircular of radius R and path CDA varies from \[R\,to\frac{R}{2}\] and then again to R. The bead is in equilibrium at D and starts moving downward. Find the ratio of normal reaction on the bead to the centrifugal force in the bottom most position.
Acceleration time graph of a particle is shown. Work done by all the forces acting on the particle of mass m in time interval \[{{t}_{1}}\] and \[{{t}_{2}}\] while \[{{a}_{1}}\] is the acceleration at time \[{{t}_{1}}\] is given by:
A rod of weight W is supported by two parallel knife edges A and B and is in equilibrium in a horizontal position, the knife are at a distance d from each other. The centre of mass of the rod is at distance \[x\] from A. The normal reaction on A is:
Find the minimum time at which ratio of kinetic energy to potential energy of a particle is 3. The particle starts SHM from its equilibrium position. T is the time period of oscillation.
The magnitude of potential energy per unit mass of the object at the surface of the earth is E and the escape velocity of the object is \[{{V}_{e}}.\] Find the correct graph.
A sphere of mass M is suspended from two wires of same length L and cross - sectional area A with Young's modulus \[{{Y}_{1}}\] and \[{{Y}_{2}}\]. The elastic potential energy of the system will be :
Two metallic spheres \[{{S}_{1}}\] and \[{{S}_{2}}\] are made of same material and have got identical surface finish. The mass of \[{{S}_{1}}\] is thrice that of \[{{S}_{2}}\] Both of the spheres are heated to the same high temperature and placed in the same room having lower temperature but are thermally insulated from each other. The ratio of initial rate of cooling of \[{{S}_{1}}\] to that of \[{{S}_{2}}\] is:
A source of sound of frequency 600 Hz is placed inside water. The speed of sound in water is 1500 m/s and in air, it is 300 m/s. The frequency of sound recorded by an observer who is standing in air is :
Find the angle of prism of that rays of light incident normally on face AB of the prism of refractive index \[\mu ,\]come out of the prism along face AC.
A concave mirror is placed on a horizontal table with its axis directed vertically upwards. Let O be the pole of the mirror and C its centre of curvature. A point object is placed at C. It has a real image, also located at C. If the mirror is now filled with water, the image will be:
A)
real and will remain at C
doneclear
B)
real and located at a point between C and \[\alpha \]
Five identical capacitor plates, each of area A are arranged such that adjacent plates are at a distance d apart, the plates are connected to a source of emf V as shown. The charge on plate 4 is:
A rectangular loop carrying a current T is situated near a long straight wire, such that the wire is parallel to one of the sides of the loop and is in the plane of the loop. If steady current I is established in the wire as shown in the figure, the loop will:
A square coil ABCD is lying in horizontal plane. A time varying current pass through two vertical wires at B and D in downwards direction. The induced current in the coil is
A non conducting ring of radius r has charge per unit length \[\lambda .\] The magnetic field is perpendicular to the plane of the ring. The radius of the ring charges at the constant rate, K. Torque experienced by the ring is :
After some time number of radioactive nuclei of an isotope become constant at \[\frac{AB}{In(2)}\] where B is the half time and A is the rate at which radioactive isotope is produced then A is : (when no. of radioactive nuclei become constant)
The intensity of \[X-\]rays eomming out froma coolidge tube is related with wavelength \[\lambda \]. The minimum wavelength found is \[{{\lambda }_{C}}\] and the wavelength of \[{{K}_{\alpha }}\] line is \[{{\lambda }_{K}}\] As the accelerating voltage is increased.
The transition of the state n =4 ton =3 in a hydrogen like atom results in ultraviolet radiation. Infrared radiation will be obtained in the transition:
Frequencies higher than 10 MHz are foam not to be reflected by the ionosphere on a particular day at a place, than the maximum electron density of ionosphere will be:
An electron moving with a speed u along the positive\[x\] -axis at \[y=0\]enters a region of uniform magnetic field \[\overrightarrow{B}\text{ }=-{{B}_{0}}\hat{k}\] which exists to the right of \[y\] -axis. The electron exits from the region after some time with the speed v at coordinate y, then:
A brass pendulum is so adjusted at \[{{20}^{o}}C\] that its time period is 1 sec. The coefficient of linear expansion of brass is \[1.93\times \text{1}{{0}^{-15}}{{\text{ }}^{o}}{{C}^{-1}}.\] If the temperature becomes \[{{30}^{o}}C,\] then the clock will slowed down in one week by:
The intensity of x-ray beam becomes half when it cross through a led sheet of thickness t. The intensity becomes \[1/x\] when it cross through a led sheet of thickness 3t.Then the value of \[x\] will be :
AT \[{{27}^{o}}C\] the volume of a monoatomic ideal gas is V. The gas is expanded to 2V by adiabatic process. Find (i) final temperature of gas and (ii) change in internal energy [R=8.31J/mol-K]
In the following conversion \[C{{H}_{3}}-\overset{\begin{smallmatrix} O \\ || \end{smallmatrix}}{\mathop{C}}\,-C{{H}_{3}}\xrightarrow{X}C{{H}_{3}}-\overset{\begin{smallmatrix} OH \\ | \end{smallmatrix}}{\mathop{C-C{{H}_{3}}}}\,\xrightarrow{Y}C{{H}_{3}}-C=C{{H}_{2}}\]Identify \[X\] and \[Y.\]
\[Cl-\overset{\begin{smallmatrix} O \\ || \end{smallmatrix}}{\mathop{C}}\,-OE{{T}^{\xrightarrow[(3eq.)]{C{{H}_{\ MgBr}}}\xrightarrow{{{H}_{3}}{{O}^{+}}}}}\] Product is:
A)
\[C{{H}_{3}}-\overset{\begin{smallmatrix} O \\ || \end{smallmatrix}}{\mathop{C}}\,-OC{{H}_{3}}\]
doneclear
B)
\[C{{H}_{3}}-\overset{\begin{smallmatrix} O \\ || \end{smallmatrix}}{\mathop{C}}\,-C{{H}_{3}}\]
doneclear
C)
\[C{{H}_{3}}-\overset{\begin{smallmatrix} O \\ | \end{smallmatrix}}{\mathop{\underset{\begin{smallmatrix} | \\ OH \end{smallmatrix}}{\mathop{C}}\,}}\,-C{{H}_{3}}\]
doneclear
D)
\[C{{H}_{3}}-\overset{\begin{smallmatrix} O \\ | \end{smallmatrix}}{\mathop{\underset{\begin{smallmatrix} | \\ C{{H}_{\,\,\,3}} \end{smallmatrix}}{\mathop{C}}\,}}\,-C{{H}_{3}}\]
75.2g of \[{{C}_{6}}{{H}_{5}}OH\] (phenol) is dissolved in a 1 Kg solvent of \[{{K}_{f}}=14\]if the depression in freezing point is 7K then find the % of phenol that dimerises :
Calculate the pH at \[{{25}^{o}}C\] after 50 M of 0.1 M ammonia solution .is treated with 25 ml of 0.1 M HCl solution. (The dissociation constant of ammonia; \[{{K}_{b}}=2\times {{10}^{-5}})\]
A solid is made of two elements \[X\] and \[Z.\]The atoms \[Z\] are in \[CCP\] arrangement while the atom \[X\] occupy all the tetrahedral sites. What is the formula of the compound?
2 mol of \[{{N}_{2}}\]is mixed with 6 mol of \[{{H}_{2}}\] in a closed vessel of one litre capacity if 50% of \[{{N}_{2}}\] is converted into \[N{{H}_{3}}\] at equilibrium, the ratio of \[\frac{{{K}_{f}}}{{{K}_{b}}}\] (rate constants) value of \[{{K}_{c}}\] for the reaction\[{{K}_{2(g)}}+3{{H}_{2(g)}}2N{{H}_{3(g)}}\] is :
The change in potential of ferric to ferrous ion at 298K if ferric ion concentration is reduced ten times white ferrous ion concentration is kept same, will be:
If at \[{{27}^{o}}C\] rate of apperance of \[{{H}_{2}}\] (g) is \[7.2\times {{10}^{-8}}\] mole\[{{L}^{-1}}{{\sec }^{-1}}\] than rate of reaction in mole \[{{L}^{-1}}{{\sec }^{-1}}\] is \[N{{H}_{3}}(g)\xrightarrow{{}}\frac{1}{2}{{N}_{2}}(g)+\frac{3}{2}{{H}_{2}}(g)\]