An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of obtained on this is noted. If the toss of the Coin resluts in tail in tail then a card from a well-shuffled cards numbered 1, 2, 3, ..., 9 is randomly the number on the card is noted. The that the noted number is either 7 or 8 is:
The Plane passing through the point \[(4,\,-1,\,\,2)\] and parallel to the lines \[\frac{x+2}{3}=\frac{y-2}{-1}=\frac{z+1}{2}\] and \[\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-4}{3}\] also passes through the point:
Consider the quadratic equation \[(c-5){{x}^{2}}-2cs+(c-4)=0,\] \[c\,\ne \,5.\] Let S be the set of all integral values of g for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is:
In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is:
If the parabolas \[{{y}^{2}}=4b(x-c)\] and \[{{y}^{2}}=8ax\] have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c)?
For each\[t\in R,\]let \[[t]\] be the greatest integer less than or equal to \[t.\] Then, \[\underset{x\to 1+}{\mathop{\lim }}\,\frac{(1-|x|+\sin |1-x|)sin\left( \frac{\pi }{2}[1-x] \right)}{|1-x\left\| 1-x \right.|}\]
Let de\[d\in R,\] and \[A=\left[ \begin{matrix} -2 & 4+d & (sin\theta )-2 \\ 1 & (sin\theta )+2 & d \\ 5 & (2sin\theta )-d & (-sin\theta )+2+2d \\ \end{matrix} \right],\] \[\theta \in [0,2\pi ].\] If the minimum value of det [A is 8. then a value of d is:
Let \[{{z}_{1}}\] and \[{{z}_{2}}\] be any two non-zero complex numbers such that \[3\left| {{z}_{1}} \right|=4\left| {{z}_{2}} \right|.\]If \[z=\frac{3{{z}_{1}}}{2{{z}_{2}}}+\frac{2{{z}_{2}}}{3{{z}_{1}}}\] then:
A)
Re (z) = 0
doneclear
B)
\[\left| z \right|=\sqrt{\frac{5}{2}}\]
doneclear
C)
\[\left| z \right|=\frac{1}{2}\sqrt{\frac{17}{2}}\]
Let \[f(x)=\left\{ \begin{matrix} \max \left\{ \left| x \right|,{{x}^{2}} \right\} & \left| x \right|\le 2 \\ 8-2\left| x \right| & 2<\left| x \right|\le 4 \\ \end{matrix}. \right.\] Let S be the set of points in the interval \[(-4,4)\] at which differentiable. Then S:
If a circle C passing through the point (4, 0) touches the circle \[{{x}^{2}}+{{y}^{2}}+4x-6y=12\] externally at the point \[(1,-1),\] then the radius of C is:
Surface tension of a liquid at room temperature is \[6.99\times {{10}^{-2}}N{{m}^{-1}}\]. If vapour pressure of the liquid is \[2.33\times {{10}^{3}}\] Pa, then radius of smallest drop which can be formed without evaporating at room temperature is
A ball is dropped over an inclined plane of inclination \[45{}^\circ \] If the ball is dropped from a height of 5 m above the inclined plane and collision is elastic, then the length AB over the plane is \[(take,g=10ms)\]
A Michelson interferometer invented to form an interference pattern by splitting a light beam into two parts and recombining them after both parts covers unequal distances. This interferometer is used in LIGO appratus. This appratus detects
A capacitor is connected to a cell and after a long time, a dielectric slab is allowed to fall through the gap of capacitor plates without touching them.
As slab passes through the gap needle of a sensitive galvanometer shows
A)
no deflection at all
doneclear
B)
deflection towards left hand side
doneclear
C)
deflection towards right hand side
doneclear
D)
initial left then towards right hand side deflrction
A thin rod of length I, coefficient of linear expansion \[\alpha \] is heated, so that its temperature changes by AT. Fractional increase in moment of inertia of rod about an axis perpendicular to its length and passing through one of its end is
A transverse propagating wave is described by \[y=A\sin 2\pi \left( Bt-\frac{x}{c} \right)\]. If maximum particle speed is exactly equal to wave speed, then value of \[\frac{C}{A}\] is
A sound source emitting sound with frequency \[{{f}_{0}}\] is moving towards a fixed wall. Now, consider three observers \[{{O}_{1}},{{O}_{2}}\]and \[{{O}_{3}}\] as shown below.
Ratio of frequencies observed by observers \[{{f}_{1}} :{{f}_{2}}:{{f}_{3}}\] is (take, \[{{\upsilon }_{s}}=\] speed of source, \[\upsilon =\] speed of sound in air and assume that \[{{O}_{3}}\] receives only reflected sound and \[{{v}_{s}}=\frac{v}{2}\])
Let a radioactive sample undergoes \[\alpha \], \[\beta \] and \[\gamma \]decays (not necessarily in any particular order), then correct order of half-lives of these decays is
Assuming that \[{{O}_{2}}\] molecule is spherical in shape with radius \[2\overset{o}{\mathop{A}}\,\], the percentage of the volume of \[{{O}_{2}}\] molecules to the total volume of gas at S.T.P. is:
If a certain mass of ideal gas is made to undergo separately adiabatic and isothermal expansions reversibly to the same pressure, starting from the same initial conditions of temperature and pressure, then as compared to that of isothermal expansion, in the case of adiabatic expansion, the final:
A)
volume and temperature will be higher
doneclear
B)
volume and temperature will be lower
doneclear
C)
temperature will be lower but the final volume will be higher
doneclear
D)
volume will be lower but the final temperature will be higher
In a closed container He, \[{{O}_{2}}\] and \[{{O}_{3}}\] gases are placed in 1:2:3 weight ratio at constant temperature. Then find the correct statement (s):
A)
The ratio of their partial pressure is 1:2:3
doneclear
B)
The ratio of their mole fraction is 1:2:3
doneclear
C)
The ratio of their partial pressure is 6:1:1
doneclear
D)
Total pressure inside the container does not depend upon the nature of the gas
All the carbon in \[{{H}_{2}}C=\underset{\underset{H}{\mathop{|}}\,}{\mathop{C}}\,-C\equiv C-\underset{\underset{H}{\mathop{|}}\,}{\mathop{C}}\,=C{{H}_{2}}\] are in \[s{{p}^{2}}\] hybridisation.
doneclear
B)
In \[{{C}_{2}}{{H}_{2}}{{(CN)}_{2}}\] there are six \['\sigma '\] bonds.
doneclear
C)
In diamond 'C' is in \[s{{p}^{2}}\] hybridisation.
doneclear
D)
In \[{{C}_{3}}{{O}_{2}}\] ail the carbons are in sp hybridisation.
Let \[\vec{a}=2\hat{i}+{{\lambda }_{1}}\hat{j}+3\hat{k},\] \[\hat{b}=4\hat{i}+(3-{{\lambda }_{2}})\hat{j}+6\hat{k}\] and \[\vec{c}=3\hat{i}+6\hat{j}+({{\lambda }_{3}}-1)\hat{K}\] be three vectors such that \[\vec{b}=2\vec{a}\] and \[\vec{a}\] is perpendicular to \[\vec{c}.\] Then a possible value of \[({{\lambda }_{1}},{{\lambda }_{2}},{{\lambda }_{3}})\] is:
Let A be a point on the line \[\vec{r}=(1-3\mu )\hat{i}+(\mu -1)\hat{j}+(2+5\mu )\hat{k}\] and B (3, 2,6) be a point in the space. Then the value of \[\mu \] for which the vector \[\overset{\xrightarrow{{}}}{\mathop{AB}}\,\] is parallel to the plane \[x-4y+3z=1\] is:
Let \[n\ge 2\] be a natural number and \[0<\theta <\pi \text{/2}.\] Then \[\int{\frac{{{\left( {{\sin }^{n}}\theta -\sin \theta \right)}^{\frac{1}{n}}}\cos \theta }{{{\sin }^{n+1}}\theta }d\theta }\] is equal to: (where C is a constant of integration)
Consider a triangular plot ABC with sides AB = 7 m, BC = 5 m and CA = 6 m. A vertical lamp-post at the mid-point D of AC subtends an angle \[30{}^\circ \] at B. The height (in m) of the lamp-post is:
If the line \[3x+3y-24=0\] intersects the x-axis at point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is:
A metal bar (length L, mass m) can slides over two connected rails fitted over an inclined plane without friction. Rails are of very less resistance and metal bar is of resistance R. When a magnetic field B is switch ON perpendicular to ground, then terminal speed of the metal bar obtained is
Two identical cylindrical rods of radii 10 cm each and refractive index \[\sqrt{3}\] rests over a flat horizontal plane mirror.
A horizontal light ray is made incident over right side glass rod I, such that it leaves the rod at a height of 10 cm from the plane mirror. Final emergent ray is found leaving glass rod parallel to the mirror.
Five identical rods are connected between two large reservoirs of temperatures \[100{}^\circ C\] and \[0{}^\circ C\]. Each rod has thermal conductivity k, length l, and area A.
A free neutron decays into a proton and an electron as, \[_{0}^{1}n\xrightarrow{{}}_{1}^{1}p+_{-1}^{0}e+v\] neutron-hydrogen atom mass difference is \[840\mu u\]. then the maximum possible kinetic energy of electron will be\[\left( 1u=932MeV \right)\]
A chord ACB, 5m long is attached at points A and B to the vertical walls 3 m apart.
A pulley of negligible mass and negligible radius carries 200 N load is free to roll over chord without friction. Dimension x in figure, when pulley is in equilibrium is
Equal moles of \[BaC{{l}_{2}}\] and \[Mg{{(Cl{{O}_{4}})}_{2}}\] are present in two different solutions. As increasingly different masses of \[{{K}_{2}}S{{O}_{4}}\] are added lo these solutions in two independent experiments, following graph is obtained. Which of the following can be concluded from this graph? (Ba=137, K=39, Mg=24.3, S=32, O=16, N=14, Cl=35.5)
A)
\[BaS{{O}_{4}}\]and\[MgS{{O}_{4}}\], are precipitated in (1) and (2)
doneclear
B)
\[BaS{{O}_{4}}\] and \[KCl{{O}_{4}}\], are precipitated in (1) and (2)
doneclear
C)
\[KCl{{O}_{4}}\] and \[KCl\] are precipitated In (1) and (2)
doneclear
D)
\[KCl\] and \[MgS{{O}_{4}}\]. are precipitated in (1) and (2)
A racemic acid \[C{{H}_{3}}CHClCOOH\] is allowed to react with (S)-2-methylbutan-1-ol to form ester \[C{{H}_{3}}CHCl-\underset{\underset{O}{\mathop{||}}\,}{\mathop{C}}\,-OC{{H}_{2}}CH(C{{H}_{3}})C{{H}_{2}}C{{H}_{3}}\] and the reaction mixture is carefully distilled. The correct statement about the mixture distillate is:
\[E_{F{{e}^{3+}}/F{{e}^{+2}}}^{0}=+0.77\,V;E_{F{{e}^{+3}}/Fe}^{0}=-0.036V\]. What is \[E_{Fe/F{{e}^{+2}}}^{0}\] and is \[F{{e}^{+2}}\] stable to disproportionation in aqueous solution under standard conditions
Reaction \[A+B\to C+D\] follows rate law, \[r=k{{[A]}^{1/2}}{{[B]}^{1/2}}\] starting with 1 M of A and B each. What is the time taken for concentration of A become 0.1 M? Give \[k=2.303\times {{10}^{-2}}{{\sec }^{-1}}\].