The mean of 20 observations is 15. On checking it was found that the two observations were wrongly copied as 3 & 6. The correct values are 8 and 4, then correct mean will be given by:
If \[\alpha \] is a real root of the equation \[{{x}^{2}}+3x-\tan \left( \frac{1}{2} \right)=0,\] then \[{{\cot }^{-1}}\alpha +{{\cot }^{-1}}\frac{1}{\alpha }-\frac{\pi }{2}\] is equal to-
If \[\alpha \] and \[\beta \] are the roots of the equation \[a{{x}^{2}}+bx+c=0,\]\[(a,b,c\in \mathbf{R})\] then\[(1+\alpha +{{\alpha }^{2}})\,\,(1+\beta +{{\beta }^{2}}),\] is -
If the fraction \[\frac{{{x}^{3}}+(\alpha -10){{x}^{2}}-x-\alpha -6}{{{x}^{3}}+(\alpha -6){{x}^{2}}-x+\alpha -10}\] reduces to a quotient of two linear functions then \[\alpha \]-
If \[x\in \left( \frac{3\pi }{2},2\pi \right)\] then value of the expression \[{{\sin }^{-\,1}}\,(\cos \,({{\cos }^{-\,1}}(\cos x)+{{\sin }^{-\,1}}\,(\sin x)))\] equals-
Let ABCD is a convex quadrilateral in which \[\angle BAC=50{}^\circ ,\] \[\angle CAD=60{}^\circ ,\] \[\angle CBD=30{}^\circ \] & \[\angle BDC=25{}^\circ .\] If E is the point of intersection of AC & BD then \[\angle AEB\] equals -
Equation of the image of the line \[x+y={{\sin }^{-1}}({{a}^{3}}+1)+co{{s}^{-1}}({{a}^{2}}+1)-{{\tan }^{-1}}(a+1),\]\[a\in R\] about y-axis is given by-
Let \[\vec{b}=4\hat{i}+3\hat{j}\] and \[\vec{c}\] be two veetors perpendicular to each other in the ay-plane. Then a vector in the same plane have projection 1 and 2 along \[\vec{b}\] and \[\vec{c}\] respectively, is -
Let \[\vec{a},\,\] \[\vec{b},\] \[\vec{c}\] are three non-coplanar vectors such that \[{{\vec{r}}_{1}}=\vec{a}-\vec{b}+\vec{c},\] \[{{\vec{r}}_{2}}=\vec{b}+\vec{c}-\vec{a},\] \[{{\vec{r}}_{3}}=\vec{c}+\vec{a}-\vec{b},\]\[\vec{r}=2\vec{a}-3\vec{b}+4\vec{c},\] if \[\vec{r}={{\lambda }_{1}}{{\vec{r}}_{1}}+{{\lambda }_{2}}{{\vec{r}}_{2}}+{{\lambda }_{3}}{{\vec{r}}_{3}}\] then-
A U-tube contains water and oil separated mercury. The mercury columns in the two arms are at the same level with 10 cm of water in arm and 12.5 cm of oil in the other, as shown figure. What is the relative density of oil?
Let \[P(r)=\frac{Q}{\pi {{R}^{4}}}r\] be the charge density distribution for a solid sphere of radius \[R\] and total charge Q. For a point 'p' inside the sphere at distance r, from the centre of the sphere, the magnitude of electric field is:
A student measures the thickness of a human hair by looking at it through a microscope of magnification 100. He makes 20 observations and finds that the average width of the hair in the field of view of the microscope is 3.5 mm. What is the estimate on the thickness of hair?
A ray of light moving in air strikes at origin at grazing angle and then follows a path \[2y={{x}^{2}}\]for \[x\ge 0.\] The correct variation of refractive index with x co-ordinate is:
Two identical rods of mass M and length \[l\] are lying in a horizontal plane at an angle a. The MI of the system of two rods about an axis passing through \['O'\] and perpendicular to the plane of the rods is
Two vessels \[A\]and \[B\]of different shapes have the same base area and are filled with water upto the same height h figure. The force exerted by water on the base is \[{{F}_{A}}\] for vessel \[A\]and \[{{F}_{B}}\] for vessel\[B\]. The respective weights of the vessels are \[{{W}_{A}}\] and \[{{W}_{B}}\]. Then,
When the diffraction pattern from a certain slit illuminated with laser light \[\text{(}\lambda =6330A{}^\circ \text{)}\]is projected on a screen 150 cm from the slit, the second minima on each side are separated by 8 cm. This tells us that:
A)
The slit is approximately 0.005 cm wide
doneclear
B)
The slit is approximately 0.05 cm wide
doneclear
C)
\[\operatorname{a}/\lambda \] is approximately 7.5 (a is the slit width)
doneclear
D)
\[\operatorname{a}/\lambda \] is approximately 750
Three batteries of negligible internal resistance and three resistors of 4, 8 and 12 ohm are connected as shown in figure here. The current through the 12 ohm resistor is :
In figure, a crate slides down an inclined right angled through. The coefficient of kinetic friction between the crate and the trough is \[{{\mu }_{k}}.\] The acceleration of the crate is:
Hydrogen (H), deuterium [D], singly ionized helium \[\left( H{{e}^{+}} \right)\]and doubly ionized lithium \[\left( L{{i}^{++}} \right)\]all have one electron around the nucleus. Consider \[n=\text{2}\] to \[n=1\] transition. The wavelengths of emitted radiations are \[{{\lambda }_{1}},{{\lambda }_{2}},{{\lambda }_{3}}\]and \[{{\lambda }_{4}}\] respectively. Then approximately:
The moment of inertia of a thin uniform rod of length L and mass M about an axis passing through a point at a distance of 1/3 from one of its ends and perpendicular to the rod is
A charge \[(+Q)\] is uniformly distributed on a circular disc of radius R. The disc rotates with constant angular speed \[\omega \] about its own axis. The magnetic field at the centre of the disc is [Assume that the disc is in the XY - plane and rotates in anticlockwise direction]
Refer to the common emitter amplifier circuit shown below, using a transistor with \[\beta =80\] and \[{{V}_{BE}}=0.7volt\]. The value of resistance\[{{\operatorname{R}}_{B}}\] is
A square loop with 2.0 m sides is perpendicular to a uniform magnetic field, with half the area of the loop in the field is shown in figure. The loop contains a 20.0 V battery with negligible interval resistance. If the magnitude of the field varies with time according to\[B=0.042-0.871t,\], with \[B\] in tesla and t in second. The net emf of the circuit is:
A car accelerates from rest at a constant rate \[\alpha \] for some time, after which it decelerates at s constant rate \[\beta \] to come to rest. If total time a motion is t, then its displacement:
(i) The slag obtained during the extraction is lighter and has lower melting point than the metal (Fe or Cu) (ii) Froth floatation process may be used to increase the concentration of mineral chalcopyrites.
An inorganic compound [a] is formed on passing iodine gas through a concentrated liquor containing sodium sulphide and sodium sulphite. On adding a solution of [a] into the solution of cupric chloride, a white precipitate is formed which dissolves in excess of [a], forming a compound [b]. Hence, compound [b] is:
what will be that value of overall half-life of A in minutes?\[\left[ \text{Given}\,\,\text{thet}\,\,\frac{{{[B]}_{t}}}{{{[C]}_{t}}}=\frac{16}{9} \right]\]
\[6{{I}^{-}}\,(aq)+BrO_{3}^{-}(aq)+6{{H}^{+}}\,(aq)\to 3{{I}_{2}}\,(aq)+B{{r}^{-}}\,(aq)\]\[+3{{H}_{2}}O\,(\ell )\] These data were obtained when this reaction was studied.
It is an experiment fact that: \[DMG+Ni\,(II)\,salt+N{{H}_{4}}OH\xrightarrow{{}}\,\,\operatorname{Red}\,\,ppt.\] Which of the following is wrong about this red ppt:
When the complex \[{{K}_{6}}\,[{{(CN)}_{5}}Co-O-O-Co\,{{(CN)}_{5}}].\] is oxidised by bromine into \[{{K}_{5}}\,[{{(CN)}_{5}}Co-O-O-Co\,{{(CN)}_{5}}].\] Then which of the following statements will be true about this change:
Electrolysis of a solution of \[HSO_{4}^{\,-}\] ions produces \[{{S}_{2}}{{O}_{8}}^{-\,-}.\] Assuming 75% current efficiency, what current should be employed to achieve a production rate of 1 mole of \[{{S}_{2}}{{O}_{8}}^{-\,-}\] per hour?
You are given the following cell at 298 K, \[Zn\,\,\left| \begin{matrix} Z{{n}^{++}}_{(aq.)} \\ 0.01\,\,M \\ \end{matrix} \right|\,\,\left| \begin{matrix} HC{{l}_{(aq.)}} \\ 1.0\,\,\text{lit} \\ \end{matrix} \right|\,\,\left| \begin{matrix} {{H}_{2}}\,(g) \\ 1.0\,\,\text{atm} \\ \end{matrix} \right|\,\,Pt\] with \[{{E}_{\text{cell}}}=0.701\]and\[E_{Z{{n}^{2\,+}}/Zn}^{0}=-\,0.76\,V.\] Which of the following amounts of NaOH (equivalent weight = 40) will just make the pH of cathodic compartment to be equal to 7.0:
In a species, the weight of newborn ranges from 2 to 5 kg. 97% of the newborn with an average weight between 3 to 3.3 kg survive whereas 99% of the infants born with weight from 2 to 2.5 kg or 4.5 to 5 kg die. Which type of selection process is taking place?
It takes very long time for pineapple plants to produce flowers. Which combination of hormones can be applied to artificially induce flowering in pineapple plants throughout the year to increase yield?
What would be the heart rate of a person if the cardiac output is 5 L, blood volume in the ventricles at the end of diastole is 100 mL and at the end of ventricular systole is 50 mL?
Equation of a straight line meeting the circle \[{{x}^{2}}+{{y}^{2}}=100\] in two points, each point at a distance of 4 from the point (8, 6) on the circle, is-
If in a rectangle ABCD with BC = 3AB. Points P & Q are on BC such that \[\angle DBC={{\tan }^{-1}}(1/3);\] & \[\angle DPC={{\tan }^{-1}}(1/2)\] \[\angle DBC=\angle DQC-\angle DPC,\]then-
Messages are conveyed by arranging 4 white, 1 blue and 3 red flags on a pole. Flags of the same colour are alike. If a message is transmitted by the order in which the colours are arranged then the total number of messages that can be transmitted if exactly 6 flags are used is -
Four balls each of radius 10 cm and mass 1 kg, 2 kg, 3 kg and 4 kg are attached to the periphery of massless plate of radius 1 m. What is moment of inertia of the system about the centre of plate?
A vertical wire of length t has its both ends fixed. Space around the wire has a horizontal magnetic field of strength B. If current; is passed through the wire such that its mid-point is deflected by amount x, tension in the wire is
A ball\[A\], moving with kinetic energy\[k\], makes a head on elastic collision with a stationary ball with mass n times that of\[A\]. The maximum potential energy stored in the system during the collision is
Four moles of an ideal gas undergoes the cyclic process ABCDA as shown in figure. If \[3{{T}_{c}}=4{{T}_{B}}=12{{T}_{A}}=2400K,\] determine the work done by, the gas during the entire process.
A ray parallel to principal axis is incident at \[30{}^\circ \] from normal on concave mirror having radius of curvature R. The point on principal axis where rays are focussed is \[Q\]such that PQ is
One mole of an ideal monatomic gas has initial temperature \[{{\operatorname{T}}_{0}}\], is made to go through the cycle \[abca\] shown in the given figure. If U denotes the internal energy, then choose the correct alternative.
Let V and I be the readings of the voltmeter and the ammeter respectively as shown in the figure. Let \[{{\operatorname{R}}_{v}}\], and \[{{\operatorname{R}}_{A}}\] be their corresponding resistance Therefore,
From a newly formed radioactive substance (half-life 2 hours), the intensity of radiation is 64 times the permissible safe level. The minimum time after which work can be done safely from this source is
A large cylindrical rod of length L is made by joining two identical rods of copper and steel of length \[\left( \frac{L}{2} \right)\] each. The rods are completely insulated from the surroundings. If the free end of copper rod is maintained at \[100{}^\circ C\] and that of steel at \[0{}^\circ C\]then the temperature of junction is (Thermal conductivity of copper is 9 times that of steel)
Two liquids 'A' and 'B' are mixed in the molar ratio of 1 : 2 and the vapour pressure of the solution is 24 torr. When the two liquids are mixed in the reverse ratio, the vapour pressure of the solution increases by a fraction of \[\frac{1}{4}.\] The vapour pressures of pure 'A' and 'B' are respectively.
Three moles of an ideal gas \[[{{C}_{p}}=7/2\,\,R]\] at pressure 'P' and temperature 'T' is isothermally expanded to twice its initial volume. It is then compressed at constant pressure to its original volume. Finally the gas is brought at constant volume to its original pressure P. Consider the following diagrams. \[P-V\]and \[P-T\] diagrams for the processes
An acid-base indicator which is a weak acid has a \[p{{K}_{a}}\] \[\text{value}=5.45.\]At what concentration ratio of sodium acetate to acetic acid would the indicator show a colour half-way between those of its acid and conjugate base forms? \[p{{K}_{a}}\] of acetic \[\text{acid}=4.75.\]\[[\text{log }2=0.3]\]
Assuming the formation of an ideal solution, determine the boiling point of a mixture containing 1560 g benzene (molar mass = 78) and 1125 g chlorobenzene (molar mass = 112.54) using the following against an external pressure of 1000 Torr.
A student finds a crystal of NaCl having mass 58.5 mg. He uses some of it to prepare 0.05 M, 10 mL NaCl solution. How many unit cells are remaing in the crystal? \[({{N}_{A}}=6\times {{10}^{23}})\]
The product Z in the following reactions sequnce is \[C{{H}_{3}}-\overset{O}{\mathop{\overset{\parallel }{\mathop{C}}\,}}\,-C{{H}_{3}}\xrightarrow[(ii)\,\,{{H}_{2}}O]{(i)\,\,C{{H}_{3}}MgBr}\,(X)\xrightarrow[\text{Ether}]{\text{Na}\,\,\text{Metal}}\]\[(Y)\xrightarrow{C{{H}_{3}}-Br}\,(Z)\]
After the zinc distillation of Phenol it reacts with propene in acidic medium and then on air oxidation followed by acid hydrolysis the final product is/are-
Tidal Volume and Expiratory Reserve Volume of an athlete is 500 mL and 1000 mL, respectively. What will be his Expiratory Capacity if the Residual Volume is 1200 mL?