Let \[f(x)=\left\{ \begin{align} & {{\cot }^{-1}}x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ & \frac{1}{2}\left| x \right|+\frac{\pi }{4}-\frac{1}{2},\,\,\, \\ \end{align} \right.\begin{matrix} \left| x \right|\ge 1 \\ \left| x \right|<1' \\ \end{matrix}\] the number of points which domain of \[f\left( x \right)\]does not contain is
From a fixed point \[A\] on the circumference of a circle of radius \[r,\] the perpendicular \[AY\] is let fall on the tangent at \[P.\] the maximum area of the triangle \[APY\]is
\[let\,f:\mathbb{R}\to \mathbb{R},g:\mathbb{R}\to \mathbb{R}\] and h:\[\mathbb{R}\to \mathbb{R}\,\]be differentiable functions such that \[f(x)\,={{x}^{3}}+3x+2,\] g(f(x))=x and h (g(g(x)))=x for all x\[\in \mathbb{R}.\] then, which of the following is correct.
Two points \[P\]and \[Q\] are taken on the line joining the points \[A\](0, 0) and \[B\](3a, 0) such that \[AP=PQ=QB.\]circle are drawn on \[AP,PQ\,\,and\,\,QB\] as diameters. The locus of the points \[S,\]the sum of the squares of the tangents from which to the three circle is equal to \[{{b}^{2}},\]is
Let us define a region \[R\]in \[xy-\]plane as a set of points \[(x,y)\]satisfying \[\left[ {{x}^{2}} \right]\]=\[\left[ y \right]\](where \[\left[ x \right]\]denotes greatest integer \[\le x),\]then the region \[R\]defines
The change that doctor A will diagnose disease X correctly is \[60%\]the change that a patient of doctor A dies after correct treatment is\[~75%\]while it is \[80%\]after wrong Diagnosis. A patient of doctor A having disease X dies. The probability that his disease is correctly diagnosed is
Let \[f:\left\{ x,y,z \right\}\to \left\{ 1,2,3 \right\}\]be a one -one mapping such that only one of the following three statements is true and remaining two are false \[f(x)\ne 2,f(y)=3,f(z)\ne 1,then\]
For hyperbola \[\frac{{{x}^{2}}}{{{\cos }^{2}}\alpha }-\frac{{{y}^{2}}}{{{\sin }^{2}}\alpha }=1\]which of the following remains constant with change in \['\alpha '\]
If \[f\]and \[g\]are two continuous functions being even and odd respectively, then\[\int\limits_{-a}^{a}{\frac{f(x)}{{{b}^{g(x)}}+1}dx}\]is equal to (a being any non-zero number and \[b\] is positive real number,\[b\ne 1\])
Let \[f(x)={{a}^{x}}(a>0)\] be written as \[f(x)=g(x)\,+h(x)\] where \[g(x)\,\] is an even function and \[h(x)\] is an odd function. Then the value of \[g(x+y)+g(x-y)\] is
A determinant of the second order is made with the elements 0and 1. If \[\frac{m}{n}\]be the probability that the determinant made is non-negative, where m and n are relative primes, then the value of m-n is
A fixed container is filled with a piston which is attached to a spring of spring constant k. The other end of the spring is attached to a rigid wall. Initially the spring is in its natural length and the length of container between the piston and its side wall.is L. Now an ideal diatomic gas is slowly filled in the container so that the piston moves quasistatically. If pushed the piston by x so that the spring now is compressed by x. The total rotation kinetic energy of the gas molecules in terms of the displacement x of the piston is (there is vacuum outside the container)
A battery of internal resistance \[2\,\Omega \] is connected to a variable resistor whose value can vary from \[1\,\Omega \] to \[10\,\Omega \] the resistance is initially set at \[4\,\Omega \,.\] If the resistance is now increased then
Two identical spheres of same mass and specific gravity (which is the ratio of density of a substance and density of water) 2.4 have different charges of Q and\[-\,3\,Q\]. They are suspended from two string of same length \[\ell \] fixed to points at the same horizontal level, but distant \[\ell \] from each other. When the entire set up is transferred inside a liquid of specific gravity 0.8, it is observed that the inclination of each string in equilibrium remains unchanged. Then the dielectric constant of the liquid is
The element which has a \[{{k}_{\alpha }}\] x-rays line of wavelength \[18\,\,\overset{{}^\circ }{\mathop{A}}\,\] is \[(R={{1.110}^{7}}{{m}^{-1}},\,\,b=1\,\,\text{and}\,\,\sqrt{5/33}=0.39)\]
\[{{M}_{1}}\] and \[{{M}_{2}}\] are plane mirrors and kept parallel to each other. At point O there will be a maxima for wavelength. Light from monochromatic source S of wavelength \[\lambda \] is not reaching directly on the screen. Then \[\lambda \] is:
A parallel plate capacitor without any dielectric has capacitance \[{{C}_{0}}.\] A dielectric slab is made up of two dielectric slabs of dielectric constants K and 2K and is of same dimensions as that of capacitor plates and both the parts are of equal dimensions arranged serially as shown. If this dielectric slab is introduced (dielectric K enters first) in between the plates at constant speed, then variation of capacitance with time will be best represented by:
A uniform rod of length L is charged uniformly with charge q and is rotating about an axis passing through its centre and perpendicular to rod. Magnetic moment of the rod is:
A object is moving with velocity \[\nu .\] (w. r. t. earth) parallel to plane mirror \[{{M}_{1}}.\]Another plane mirror \[{{M}_{2}}\] makes an angle \[\beta \] with the vertical as shown in figure. Then velocity of image in mirror \[{{M}_{2}}\] w.r.t. the image in \[{{M}_{1}}\] is -
The graph between photo electric current and cathode potential when the anode is kept at zero potential, for light of two different intensities out of the same frequency looks like the one:
There are three concentric thin spheres of radius a, b, c (a > b > c). The total surface charge densities on their surfaces are \[\sigma ,\]\[-\,\sigma ,\]\[\sigma \] respectively. The magnitude: of electric field at r (distance from centre) such that a > r > b is:
The direction of field B at a point P symmetric to P with respect to the vertex i.e., along the axis and the same distance d, but inside the V is along:
A thin prism of glass is placed in air and water successively. If \[{}_{\alpha }{{\mu }_{g}}=3/2\] and \[{}_{\alpha }{{\mu }_{W}}=4/3,\] then the ratio of deviation produced by the prism for a small angle of incidence when placed in air and water is
A wire of fixed length is wound is such a way that it form a solenoid of length \['\ell '\] and radius 'r'. Its self inductance is fund to be L. Now if same wire is wound in such a way that it forms a solenoid of length \[\frac{\ell }{2}\] and radius \[\frac{r}{2},\] then the self inductance will be:
Two identical samples (same material and same amount) P and Q of a radioactive substance having mean life T are observed to have activities \[{{A}_{P}}\] & \[{{A}_{Q}}\] respectively at the time of observation. If P is older than Q, then the difference in their ager is:
The polymerization of ethylene to linear polyethylene is represented by the reaction: \[nC{{H}_{2}}=C{{H}_{2}}\to {{[-\,C{{H}_{2}}-C{{H}_{2}}-]}_{n}}\] where n has large integral value. Given that the average enthalpies of bond dissociation for C = C and \[C-C\] at 298K are 590 and \[331\,\,kJ\,\,mo{{l}^{-1}}\]respectively, calculate the enthalpy of polymerization per mol of ethylene at 298 K.
Element A bums in nitrogen to give an ionic compound B. Compound B reacts with water to give C and D. A solution of C becomes 'milky' on bubbling carbon dioxide. Identify A, B, C and D.
Compound A of molecular formula \[{{C}_{9}}{{H}_{7}}{{O}_{2}}Cl\] exists in ketoform and predominatly in enolic form B. On oxidation with \[KMn{{O}_{4}},\] A gives m-chlorobenzoic acid. Identify A and B.
Name the products m the following reaction: \[{{C}_{2}}{{H}_{5}}CHO+C{{H}_{3}}COO{{C}_{2}}{{H}_{5}}\xrightarrow[{{C}_{2}}{{H}_{5}}OH\,\,and\,\,heat]{NaO{{C}_{2}}{{H}_{5}}\,\,in\,\,absolute}?\]
Name the products in the following reaction: \[{{C}_{6}}{{H}_{5}}C{{H}_{2}}C{{O}_{2}}C{{H}_{3}}\xrightarrow[(ii){{H}^{+}}]{(i)C{{H}_{3}}MgBr(excess)}?\]
\[2NOBr\,(g)\,\,\,\,2NO\,(g)+B{{r}_{2}}\,(g).\] If nitrosyl bromide (NOBr) is 40% dissociated at certain temperature and a total equilibrium pressure of 0.30 atm. \[{{K}_{P}}\] for the reaction \[2NOB{{r}_{2}}\,(g)\,\,\,\,2\,NOBr\,(g)\]
\[{{H}_{2}}A\] is a weak triprotic acid \[({{K}_{{{a}_{1}}}}={{10}^{-\,5}},{{K}_{{{a}_{2}}}}={{10}^{-\,9}},{{K}_{{{a}_{3}}}}={{10}^{-\,13}})\]; What is the value of \[pX\]of 0.1 M \[{{H}_{3}}A\](aq.) solution ? where \[pX=-\,\log X\] and \[X=\frac{[{{A}^{3-}}]}{[H{{A}^{2-}}]}\]
A 0.60 g sample consisting of only \[Ca{{C}_{2}}{{O}_{4}}\]and \[Mg{{C}_{2}}{{O}_{4}}\] is heated at \[500{}^\circ C,\]converting the two salts of \[CaC{{O}_{3}}\] and \[MgC{{O}_{3}}.\] The sample then weighs 0.465 g. If the sample had been heated to \[900{}^\circ C,\]where the products are \[CaO\] and \[MgO,\]what would the mixtures of oxides have weighed?
150 mL of \[{{H}_{2}}{{O}_{2}}\] sample was divided into two parts. First part was treated with KI and formed KOH required 200 mL of M/2 \[{{H}_{2}}S{{O}_{4}}\] for neutralization. Other part was treated with \[KMn{{O}_{4}}\] yielding 6.74 litre of \[{{O}_{2}}\] at STP. Using % yield indicated find volume strength of \[{{H}_{2}}{{O}_{2}}\]sample used.
Which one of the following plants shows a very close relationship with a species of moth, where none of the two can complete its life cycle without the other?
Let \[f:\left[ 0,\infty )\to \left[ 0,\infty ) \right. \right.\]and \[g:\left[ 0,\infty )\to \left[ 0,\infty ) \right. \right.\]be non-increasing and non-decreasing functions respectively and \[h(x)\,=g(f(x)).\]if \[f\]and \[g\]are differentiable for all points in their respective domains and \[h(0)=0,\]than
The number of such points \[(a+1,\sqrt{3}a),\] where a is any integer, lying inside the region bounded by the circles\[{{x}^{2}}+{{y}^{2}}-2x-3=0\] and \[{{x}^{2}}+{{y}^{2}}-2x-15=0,\]
Tangents are draw to the circle \[{{x}^{2}}+{{y}^{2}}=50\] from a point \['p'\] lying on the x-axis. These tangents meet the y-axis at point \['{{P}_{1}}'\]and \['{{P}_{2}}'\]possible coordinates of \['P'\]so that area of triangle \[P{{P}_{1}}{{P}_{2}}\] is minimum, are
The focal length of a convex lens of refractive index 1, 5 is 2 cm. The focal length of the lens when immerged in liquid spirit of refractive index 1.25 is
In the figure shown \[{{v}_{1}},\]\[{{v}_{2}},\]\[{{v}_{3}}\] are AC voltmeters and A is AC ammeter. The readings of \[{{v}_{1}},\]\[{{v}_{2}},\]\[{{v}_{3}}\] and A are 10 V, 20 V, 20 V, 2A respectively. Find the values of R, C, L and the source voltage \[{{v}_{s}}.\] If the inductor is short circuit then what will be the readings of \[{{v}_{1}},\]\[{{v}_{2}}.\] and A.
A segment of angle \[\theta \] is cut from a half disc with symmetry of symmetrically as shown. If the centre of mass of the remaining. part is at a distance 'a' from O and the centre of mass of the original disc was at distance d then it can be definitely said that
A)
\[a=d\]
doneclear
B)
a > d
doneclear
C)
a < d
doneclear
D)
A, B, C depends on the angle of segment cut from disc.
A large open tank has two small holes in its vertical wall as shown in figure. One is a square hole of side 'L' at a depth '4y' from the top and the other is a circular hole of radius 'R' at a depth y 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.
If a charged particle of charge to mass ratio \[\frac{q}{m}=\alpha \] is entering in a magnetic field of strength B at a speed \[v=(2\,\alpha \,d)\,\,(B),\] then which of the following is correct:
A)
angle subtended by charged particle at the centre of circular path is \[2\pi .\]
doneclear
B)
the charge will move on a circular path and will come out from magnetic field at a distance 4d from the point of insertion.
doneclear
C)
the time for which particle will be in the magnetic field is \[\frac{2\pi }{\alpha B}.\]
doneclear
D)
the charged particle will subtend an angle of \[90{}^\circ \] at the centre of circular path
Two identical rectangular rods of metal are welded end to end in series between temperature \[0{}^\circ C\] and \[100{}^\circ C\] and 10 J of heat is conducted (in steady state process) through the rod in 2.00 min. If 5 such rods are taken and joined as shown in figure maintaining the same temperature difference between A and B, then the. time in which 20 J heat will flow through the rods is:
Two points A & B on a disc have velocities \[{{v}_{1}}\] & \[{{v}_{2}}\] at some moment. Their directions make angles \[60{}^\circ \] and \[30{}^\circ \] respectively with the line of separation as shown in figure. The angular velocity of disc is:
A chain of length L is placed on a horizontal surface as shown in figure. At any instant x is the length of chain on rough surface and the remaining portion lies on smooth surface. Initially \[x=0.\] A horizontal force P is applied to tin: chain (as shown in figure). In the duration x changes from \[x=0\] to \[x=L,\] for chain to move with constant speed.
A)
the magnitude of P should increase with time
doneclear
B)
the magnitude of P should decrease with time
doneclear
C)
the magnitude of P should increase first and then decrease with time
doneclear
D)
the magnitude of P should decrease first and then increase with time
An object approaches a fixed diverging lens with a constant velocity from infinity along the principal axis. The relative velocity between object and its image will be:
A ring of radius R is placed in the plane with its centre at origin and its axis along the x-axis and having uniformly distributed positive charge. A ring of radius \[r\left( <<R \right)\] and coaxial with the larger ring is moving along the axis with constant velocity then the variation of electrical flux \[\left( \phi \right)\] passing through the smaller ring with position will be best represented by :
Electrode potential for \[Zn\] electrode varies according to the equation. \[{{E}_{Z{{n}^{2+}}|Zn}}=E{{{}^\circ }_{Z{{n}^{2+}}|Zn}}-\frac{0.059}{2}\log \frac{1}{[Z{{n}^{2+}}]}.\] The graph of \[{{E}_{Z{{n}^{2+}}|Zn}}\] vs log \[[Z{{n}^{2+}}]\] is-
Graph between log K and \[\frac{1}{T}\] [Where K is rate constant \[({{S}^{-1}})\] and T is temperature (K)] is a straight line with OX = 5, \[\theta ={{\tan }^{-1}}\left[ -\frac{1}{2.303} \right].\] Hence \[Ea\]and log A respectively will be:
(I) \[[Mn{{(C{{l}_{6}}]}^{3\,-}},\] \[{{[Fe{{F}_{5}}]}^{3\,-}}\] and \[{{[Co{{F}_{5}}]}^{3\,-}}\] are paramagnetic having four, five and four unpaired electrons respectively.
(II) Valence bond theory gives a quantitative interpretation of the thermodynamics stabilities of coordination compounds.
(III) The crystal field splitting \[{{\Delta }_{0}},\] depends upon the field produced by the ligand and charge on the metal ion.
1 mol \[C{{H}_{3}}COOH\] is added in 250 g benzene Acetic acid dimerises in benzene due to hydrogen bond. \[{{K}_{b}}\] of benzene is 2K \[kg\,mo{{l}^{-\,1}}\]. The boiling point has increased by 6.4 K. % dimerisation of acetic acid is -