The position vectors of three points A, B, C are \[\hat{i}+2\hat{j}+3\hat{k}\cdot 2\hat{i}+3\hat{j}+\hat{k}\And 3\hat{i}+\hat{j}+2\hat{k}.\] A unit vector perpendicular to the plane of the triangle ABC is:
Let \[{{(1+{{x}^{2}})}^{2}}\,{{(1+x)}^{n}}={{A}_{0}}+{{A}_{1}}x+{{A}_{2}}{{x}^{2}}+.....\] If \[{{A}_{0}},\]\[{{A}_{1}},\]\[{{A}_{2}}\] are in A. P. then the value of n is:
If \[{{z}_{1}}\] and \[{{z}_{2}}\] are two complex numbers such that \[\operatorname{Re}\,({{z}_{2}})\ne 0,\]\[\operatorname{Re}\,({{z}_{1}}+{{z}_{2}})=0\] and \[\operatorname{Im}\,({{z}_{1}}{{z}_{2}})=0\] then
\[y=f\,(x)\] is a continuous function such that its graph passes through (a, 0). Then \[\underset{x\,\to \,a}{\mathop{\text{Limit}}}\,\frac{{{\log }_{e}}\,(1+3f\,(x))}{2f\,(x)}\] is
If the matrix \[\left[ \begin{matrix} 1 & 2a & 2 \\ 0 & -1 & 1 \\ a & 1 & 3 \\ \end{matrix} \right]\] is not equivalent to \[{{I}_{3}}\] then a is, \[a\in R.\]
If \[f\,(x)\] is a differentiable function satisfying \[f'\,(x)<2\] for all \[x\in R\] and \[f\,(1)=2,\] then greatest possible integral value of \[f\,(3)\]is
Tangents are drawn from a point on the directrix to the parabola \[{{y}^{2}}=4ax.\]The locus of foot of perpendicular drawn from this point to its chord of contact is a:
A missile is fired at a plane on which there are two targets I & II. The probability of hiting target I is \[{{P}_{1}}\] & that of hiting the II is \[{{P}_{2}}.\] If it is known that target I is not hit, then the probability that the target II is hit is:
If A.M. between \[{{p}^{th}}\] and \[{{q}^{th}}\] terms of an A. P. be equal to the A.M. between \[{{r}^{th}}\] and \[{{s}^{th}}\] term of the A. P., then \[p+q\] is equal to
The lines \[y=-\frac{3}{2}x\] and \[y=-\frac{2}{5}x\] intersect the curve \[3{{x}^{2}}+4xy+5{{y}^{2}}-4=0\] at the points P and Q respectively. The tangents drawn to the curve at P and Q:
The lengths of the diagonals of a parallelogram constructed on the vectors \[\vec{p}=2\vec{a}+\vec{b}\] & \[\vec{q}=\vec{a}-2\vec{b}\] where \[\vec{a}\] & \[\vec{b}\] are unit vectors forming an angle of \[60{}^\circ \] are:
Let \[{{z}_{1}},\]\[{{z}_{2}},\]\[{{z}_{3}},\]\[{{z}_{4}}\] be the vertices A, B, C, D respectively of a square on the argand plane taken in anticlockwise direction, then
Two bar magnets of the same mass, same length and breadth but having magnetic moments M and 2M are joined together pole for pole and suspended by a string. The time period of assembly in a magnetic field of strength H is 3 seconds. If now the polarity of one of the magnets is reversed and the combination is again made to oscillate in the same field, the time period of oscillation is :
Two magnets A and B are identical and these are arranged as shown. Their lengths are negligible in comparison to separation between them. A magnetic needle is placed between the magnets at point P and it gets deflected through an angle \[\theta .\] The ratio of distances and will be
The tangent galvanometers having coils of the same radius are connected in series. A current flowing in them produces deflections of \[60{}^\circ \] and \[45{}^\circ \] respectively. The ratio of the number of turns in the coil is:
An electron moves along the line AB which lies in the same plane as a circular loop of conducting wire as shown in figure. What will be the direction of the current induced if any in the loop?
A)
No current will be induced
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B)
The current will be clockwise
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C)
The current, will be anticlockwise
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D)
The current will change direction as the electron passes by
A solenoid has 2000 turns wound over a length of 0.3m, The area of its cross- section is \[1.2\text{ }\times \text{ 1}{{0}^{-3}}\text{ }{{m}^{2}}.\] Around its central portion a coil of 300 turns is wound. If an initial current of 2 amp in the solenoid is reversed in 0.25 sec., the emf induced in the coil is equal to -
In the circuit shown in the following figure E = 10 V, \[{{R}_{1}}=2\,ohm,\] \[{{R}_{2}}=3\,ohm\] and \[{{R}_{3}}=\text{6}\,ohm\]and L = 5 henry. The current \[{{I}_{1}}\] just after pressing the switch S is
In figure shown, find the magnitude of acceleration of m, given that string is inextensible and mass less and the acceleration of M is \[2\text{ }m/{{s}^{2}}\] towards left -
A square of mass M and sides of length L has a moment of inertia Io when rotated about an axis perpendicular to its surface and passing through its center, as shown. Now a lump of clay, also of mass M is attached to one corner of the square as shown. What is the new moment of inertia of the masses about the same axis of rotation?
A block of mass m is pushed across a rough surface by an applied force F, directed at an angle \[\theta \] relative to the horizontal as shown. The block experiences a friction force \[f\] in the opposite direction. What is the coefficient of friction between the block and the surface?
The free-body diagram shows all forces acting on a box supported by a stationary horizontal surface, where the length of each force vector is proportional to its magnitude. Which statement below is correct?
A)
The box must be moving to the left, due to the force of friction acting in that direction
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B)
The box must be accelerating to the right, as indicated by the force of friction in the opposite direction
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C)
The box must be moving to the right, as indicated by the force of friction in the opposite direction
A mass m in three-dimensional space is subjected to three forces: \[\theta \] \[{{F}_{2}}\] and \[{{F}_{3}},\] \[{{F}_{1}}\] and \[{{F}_{2}}\] have the same magnitude, with \[{{F}_{1}}\] in the positive-x direction, and \[{{F}_{2}}\] in the positive-y direction. If the mass has an acceleration of 0, which of the following statements is false?
A)
The magnitude of \[{{F}_{3}}\] is the same as that of \[{{F}_{1}}\]
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B)
The object is in equilibrium and could be stationary
A cube of mass 30 g wettable by water floats on the surface of water. Each face of the cube is 4 cm long. Surface tension of water = 70 dynes/cm. The distance of the lower face of the cube from the surface of water is \[(g=980\,\,cm\,\,{{s}^{-2}}):\]
An object is placed 21 cm in front of a concave mirror of radius of curvature 10 cm. A glass slab of thickness 3 cm and refractive index 1.5 is then placed close to the mirror in the space between the object and the mirror. The distance of the Bear surface of the slab from the mirror is 1 cm. The final image from the mirror will be formed at:
A simple pendulum has a period T on the earth. What is the period T of this same pendulum on the moon, where the acceleration due w gravity is 1/6 that of the earth?
An aeroplane is flying in a horizontal circle at a speed of 540 km/h. Banked for this turn, the wings of the plane are tilted at an angle \[45{}^\circ \] from the horizontal. Assume that a lift force acting perpendicular to the wings holds the aircraft in the sky. The radius of the circle in which the plane is flying is \[(Take\,\,\,g=10m/{{s}^{2}})\]
\[\wedge _{m}^{o}\] for \[NaCl,\text{ }HCl\] and \[NaA\] are 126.4, 425.9 and 100.5 \[S\,c{{m}^{2}}mo{{l}^{-1}},\] respectively. If the conductive of \[0.001\text{ }MHA\] is \[5\times {{10}^{-5}}S\,c{{m}^{-1}},\] degree of dissociation of HA is:
Molecules of benzoic acid \[({{C}_{6}}{{H}_{5}}COOH)\]dimerise in benzene, 'w' g of the acid dissolved in 30 g of benzene shows a depression in freezing point equal to 2 K. If the percentage association of the acid to form dimer in the solution is 80, then w is: (Given that \[{{k}_{f}}=5K\,kg\,mo{{l}^{-1}},\]Molar mass of benzoic acid \[=122\,g\,mo{{l}^{-1}}\])
During some surgical operations the drug curare. Which has a similar shape to acetylcholine is' injected into the muscles to relax them. Why do the muscles remain relaxed?
A)
Calcium ions cannot be taken up by membrane vesicless
Origin of life as a result of chemical evolution has been properly explained by or the most logical biochemical theory of origin of life has been given by
If \[ax+by-5=0\]is the equation of the shortest chord of the circle \[{{(x-3)}^{2}}+{{(y-4)}^{2}}=4\] passing through the point (2, 3), then \[|a+b|\] is
If the quadratic equation \[f\,(x)=p{{x}^{2}}-qx+r=0\] has two distinct roots in (0, 2) where p, q, \[r\in N\] and \[f\,(1)=-1\] then the minimum value of p is
If the line \[y=\sqrt{3}\,x\] intersects the curve \[{{x}^{3}}+{{y}^{3}}+3xy+5{{x}^{2}}+3{{y}^{2}}+4x+5y-1=0\] at the points A, B, C then OA. OB. OC is (Here 'O' is origin)
A semicircular loop of radius R is rotated about its straight edge which divides the space into two regions one having a uniform magnetic field B and the other having no field. If initially the plane of loop is perpendicular to \[\vec{B}\] (as shown), and if current flowing from O to A be taken as positive, the correct plot of induced current v/s time for one time period is
A ray hits the y-axis making an angle \[\theta \] with y-axis as shown in the figure. The variation of refractive index with x-coordinate is \[\mu ={{\mu }_{0}}\left( 1-\frac{x}{d} \right)\] for \[0\le x\le d,\]\[\left( 1-\frac{1}{{{\mu }_{0}}} \right)\] and \[\text{ }\mu \text{=}{{\mu }_{0}}\] for x < 0, where d is a positive constant. The maximum x-coordinate of the path traced by the ray is -
A cobalt (atomic no. = 27) target is bombarded with electrons, and the wavelengths of its characteristic x-rays spectrum are measured. A second weak characteristic spectrum is also found, due to an impurity in the target. The wavelength of the \[{{K}_{a}}\] lines are 225.0 pm (cobalt) and 100.0 pm (impurity). Atomic number of the impurity is (take b=1)
The conducting spheres of radii r and 3r initially have charges 3q & q respectively. Their separation is much larger than their radii. If they are joined by a conductor of high resistance, the force between them will -
In an LRC series circuit at resonance, current in the circuit is \[10\sqrt{2}A\]. If now frequency of the source is changed such that now current lags by \[45{}^\circ \] than applied voltage in the circuit. Which of the following is correct is -
A)
Frequency must be increased and current after the change is 10 A
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B)
Frequency must be decreased and current after the change is 10 A
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C)
Frequency must be decreased and current is same as that of initial value
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D)
The given information is insufficient to conclude anything
The oscillations represented by curve 1 in the graph are expressed by equation \[x\text{ }=\text{ }A\text{ }sin\omega t.\]The equation for the oscillations represented by curve 2 is expressed as -
A certain gas is taken to the five states represented by dots in the graph. The plotted lines are isotherms. Order of the most probable speed \[{{V}_{p}}\]of the molecules at these five states is -
An electron of the kinetic energy 10 eV collides with a hydrogen atom in 1st excited state. Assuming loss of kinetic energy in the collision to be quantized which of the following statements is incorrect.
A uniform stick of mass M is placed in a frictionless well as shown. The stick makes angle \[\theta \] was the horizontal. The force which the vertical wall exerts on right end of stick is -
For a certain reaction consider the plot of \[\ell nk\] versus 1/T given in the figure. If the rate constant of this reaction at 400 K is \[{{10}^{-5}}{{s}^{-1}},\]then the rate constant at 500 K is:
An open vessel at \[27{}^\circ C\] is heated until two fifth of the air (assumed as an ideal gas) in it has escaped from the vessel. Assuming that the volume of the vessel remains constant, the temperature at which the vessel has been heated is:
If the de Brogile wavelength of the electro in nth Bohr orbit in a hydrogenic atom is equal to \[1.5\pi {{a}_{0}}\](\[a{{ }_{0}}\] is Bohr radius), then the value of n/z. is;
The restriction fragment shown below contains a gene whose recessive allele is lethal. The normal allele has restriction sites for restriction enzyme pst I at sites I and II. The recessive allele lacks restriction sites I. An individual who had a sister with the lethal trait is being tested to determine if he is a carrier of that lethal trait.
Which of the band patterns would be produced on a gel if he is a carrier?
The Galapagos islands are a group of volcanic islands in the Eastern Pacific Ocean about 1000 km from South America. Thirteen species of finch are found on the islands they resemble each other closely but differ in their feeding habit and in the shape of their beaks. Assuming that an ancestral stock of finches came from the mainland what is the most likely explanation for the existence of similar but distinct species of Galapagos finches?
A)
Finches developed different kinds of beak in order to feed on different kinds of food
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B)
Finches evolved separately according to the habitat in which they settled
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C)
Finches from the mainland bred with a resident population of a related species and produced new genotypes
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D)
Finches underwent convergent evolution to produce very similar species
The diagram below represents the proportions of a population of newborn deer calves falling into various birth weight classes. The graph superimposed on the diagram represents mortality in relation to birth weight.
From the information given which one of the following interpretations is correct?
A)
Birth weight is undergoing stabilising selection
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B)
Birth weight is an example of discontinuous variation
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C)
Birth weight is inversely proportional to mortality
ATP can be formed by oxidative phosphorylation in the electron transport system and by glycolysis. In the complete oxidation of one molecule of glucose, approximately what percentage of ATP is formed by oxidative phosphorylation?
The diagram represents a sequence of reactions taking place in a bacterium in which amino acids are produced from one another by the action of specific enzymes. Number 1 to 6 represent different amino acids, letters V to Z represent different enzymes. All the amino acids are essential for survival.
The original strain of the bacterium required only amino acid 1. A mutant strain of this bacterium could only survive when produced with amino acids 1, 2 and 5 in its culture medium.
The table shows the result of an analysis of percentage concentration of three bases in nucleic acids from four sources. Three of the sources are DNA and one is RNA. Which source is RNA?
The graph shows the amount of product formed by a standard concentration of enzyme and a standard concentration of substrate at a temperature of \[20{}^\circ C.\]
Which graph shows the effect on the activity of the enzyme of decreasing the temperature to \[15{}^\circ C?\]