KCET 2021

A proton moving with a momentum $$p_1$$ has a kinetic energy $$1 / 8$$th of its rest mass-energy. Another light photon having energy equal to the kinetic energy of the possesses a momentum $$p_2$$. Then, the ratio $$\frac{p_1-p_2}{p_1}$$ is equal to

The number of six membered and five membered rings in Buckminster fullerene respectively is

The $$K_{\mathrm{H}}$$ value ($$\mathrm{K}$$ bar) of argon (I), carbondioxide (II), formaldehyde (III) and methane (IV) are respectively $$40.3,167, 1.83 \times 0^{-5}$$ and 0.413 at $$298 \mathrm{~K}$$. The increasing order of solubility of gas in liquid is

The number of angular and radial nodes in $$3 p$$ orbital respectively are

The trajectory of projectile is

A particle starts from rest. Its acceleration $$a$$ versus time $$t$$ is shown in the figure. The maximum speed of the particle will be

For a projectile motion, the angle between the velocity and acceleration is minimum and acute at

For a body moving along a straight line, the following $$v$$-$$t$$ graph is obtained.

According to the graph, the displacement during

A particle starts form rest and its angular displacement (in radians) is given by $$\theta=\frac{t^2}{20}+\frac{t}{5}$$. If the angular velocity at the end of $$t=4$$ is $$k$$, then the value of $$5 k$$ is

If $$I_n=\int_0^{\frac{\pi}{4}} \tan ^n x d x$$, where $$n$$ is positive integer, then $$I_{10}+I_8$$ is equal to

In figure $$E$$ and $$v_{\mathrm{cm}}$$ represent the total energy and speed of centre of mass of an object of mass $$1 \mathrm{~kg}$$ in pure rolling. The object is

A particle starts from the origin at $$t=0$$ with a velocity of $$10 \hat{\mathbf{j}} \mathrm{ms}^{-1}$$ and move in the $$x$$-$$y$$ plane with a constant acceleration of $$(8 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}) \mathrm{ms}^{-2}$$. At an instant when the $$x$$-coordinate of the particle is $$16 \mathrm{~m}, y$$-coordinate of the particle is

The maximum range of a gun on horizontal plane is $$16 \mathrm{~km}$$. If $$\mathrm{g}=10 \mathrm{~ms}^{-2}$$, then muzzle velocity of a shell is

A ball hits the floor and rebounds after an inelastic collision. In this case

A $$1 \mathrm{~kg}$$ ball moving at $$12 \mathrm{~ms}^{-1}$$ collides with a $$2 \mathrm{~kg}$$ ball moving in opposite direction at $$24 \mathrm{~ms}^{-1}$$. If the coefficient of restitution is $$2 / 3$$, then their velocities after the collision are

A coin placed on a rotating turn table just slips if it is placed at a distance of $$4 \mathrm{~cm}$$ from the centre. If the angular velocity of the turn table is doubled it will just slip at a distance of

To propagate both longitudinal and transverse waves, a material must have

Two capillary tubes $$P$$ and $$Q$$ are dipped vertically in water. The height of water level in capillary tube $$P$$ is $$\frac{2}{3}$$ of the height in capillary tube $$Q$$. The ratio of their diameter is

Which of the following curves represent the variation of coefficient of volume expansion of an ideal gas at constant pressure?

A pendulum oscillates simple harmonically and only if

I. the sizer of the bob of pendulum is negligible in comparison with the length of the pendulum.

II. the angular amplitude is less than $$10^{\circ}$$.

Choose the correct option.

A number of Carnot engines are operated at identical cold reservoir temperatures $$(T_L)$$. However, their hot reservoir temperatures are kept different. A graph of the efficiency of the engines versus hot reservoir temperature $$(T_H)$$ is plotted. The correct graphical representation is

Two bodies of masses $$8 \mathrm{~kg}$$ are placed at the vertices $$A$$ and $$B$$ of an equilateral triangle $$A B C$$. A third body of mass $$2 \mathrm{~kg}$$ is placed at the centroid $$G$$ of the triangle. If $$A G=B G=C G=1 \mathrm{~m}$$, where should a fourth body of mass $$4 \mathrm{~kg}$$ be placed, so that the resultant force on the $$2 \mathrm{~kg}$$ body is zero?

Which of the following statements are correct with reference to single slit diffraction pattern?

(I) Fringes are of unequal width.

(II) Fringes are of equal width.

(III) Light energy is conserved.

(IV) Intensities of all bright fringes are equal.

In refraction, light waves are bent on passing from one medium to second medium because, in the second medium

A gas mixture contains monoatomic and diatomic molecules of 2 moles each. The mixture has a total internal energy of (symbols have usual meanings)

In the Young's double slit experiment a monochromatic source of wavelength $$\lambda$$ is used. The intensity of light passing through each slit is $$I_0$$. The intensity of light reaching the screen $$S_C$$ at a point $$P$$, a distance $$x$$ from $$O$$ is given by (Take, $$d<< D$$)

If the refractive index from air to glass is $$\frac{3}{2}$$ and that from air to water is $$\frac{4}{3}$$, then the ratio of focal lengths of a glass lens in water and in air is

A slit of width $$a$$ is illuminated by red light of wavelength $$6500 \mathop A\limits^o$$. If the first diffraction minimum falls at $$30^{\circ}$$, then the value of $$a$$ is

Suppose that the electric field amplitude of electromagnetic wave is $$E_0=120 \mathrm{~NC}^{-1}$$ and its frequency $$f=50 \mathrm{~MHz}$$. Then, which of the following value is incorrectly computed?

A strong magnetic field is applied on a stationary electron. Then, the electron

A copper rod $$A B$$ of length $$l$$ is rotated about end $$A$$ with a constant angular velocity $$\omega$$. The electric field at a distance $$x$$ from the axis of rotation is

The source of electromagnetic wave can be a charge

The size of the image of an object, which is at infinity, as formed by a convex lens of focal length $$30 \mathrm{~cm}$$ is $$2 \mathrm{~cm}$$. If a concave lens of focal length $$20 \mathrm{~cm}$$ is placed between the convex lens and the image at a distance of $$26 \mathrm{~cm}$$ from the lens, the new size of the image is

Two thin biconvex lenses have focal lengths $$f_1$$ and $$f_2$$. A third thin biconcave lens has focal length of $$f_3$$. If the two biconvex lenses are in contact, then the total power of the lenses is $$P_1$$. If the first convex lens is in contact with the third lens, then the total power is $$P_2$$. If the second lens is in contact with the third lens, the total power is $$P_3$$, then

The physical quantity which is measure in the unit of wb $$\mathrm{A}^{-1}$$ is

Two parallel wires in free space are $$10 \mathrm{~cm}$$ apart and each carries a current of $$10 \mathrm{~A}$$ in the same direction. The force exerted by one wire on the other [per unit length] is

A tightly wound long solenoid has $n$ turns per unit length, a radius $$r$$ and carries a current $$I$$. A particle having charge $$q$$ and mass $$m$$ is projected from a point on the axis in a direction perpendicular to the axis. The maximum speed of the particle for which the particle does not strike the solenoid is

Earth's magnetic field always has a horizontal component except at

A toroid with thick windings of $$N$$ turns has inner and outer radii $$R_1$$ and $$R_2$$, respectively. If it carries certain steady current $$I$$, the variation of the magnetic field due to the toroid with radial distance is correctly graphed in

Which of the field pattern given below is valid for electric field as well as for magnetic field?

The current following through an inductance coil of self-inductance $$6 \mathrm{~mH}$$ at different time instants is as shown. The emf induced between $$t=20 \mathrm{~s}$$ and $$t=40 \mathrm{~s}$$ is nearly

A $$2 \mathrm{~g}$$ object, located in a region of uniform electric field $$\mathrm{E}=\left(300 \mathrm{NC}^{-1}\right) \hat{\mathbf{i}}$$ carries a charge $$Q$$. The object released from rest at $$x=0$$, has a kinetic energy of $$0.12 \mathrm{~J}$$ at $$x=0.5 \mathrm{~m}$$. Then, $$Q$$ is

A wire of resistance $$3 \Omega$$ is stretched to twice its original length. The resistance of the new wire will be

Electric field due to infinite, straight uniformly charged wire varies with distance $$r$$ as

Which of the following statements is false in the case of polar molecules?

A copper wire of length $$1 \mathrm{~m}$$ and uniform cross-sectional area $$5 \times 10^{-7} \mathrm{~m}^2$$ carries a current of $$1 \mathrm{~A}$$. Assuming that, there are $$8 \times 10^{28}$$ free electrons per $$\mathrm{m}^3$$ in copper, how long will an electron take to drift from one end of the wire to the other?

Eight drops of mercury of equal radii combine to form a big drop. The capacitance of a bigger drop as compared to each smaller drop is

In figure, charge on the capacitor is plotted against potential difference across the capacitor. The capacitance and energy stored in the capacitor are respectively.

If a slab of insulating material (conceptual). $$4 \times 10^{-3} \mathrm{~m}$$ thick is introduced between the plates of a parallel plate capacitor, the separation between the plates has to be increased by $$3.5 \times 10^{-3} \mathrm{~m}$$ to restore the capacity to original value. The dielectric constant of the material will be

In the given arrangement of experiment on meter bridge, if $$A D$$ corresponding to null deflection of the galvanometer is $$X$$, what would be its value if the radius of the wire $$A B$$ is doubled?

Consider an electrical conductor connected across a potential difference $$V$$. Let $$\Delta q$$ be a small charge moving through it in time $$\Delta t$$. If $$I$$ is the electric current through it,

I. the kinetic energy of the charge increases by $$I V \Delta t$$.

II. the electric potential energy of the charge decreases by $$I V \Delta t$$.

III. the thermal energy of the conductor increases by $$I V \Delta t$$.

Choose the correct option.

An electrician requires a capacitance of $$6 \mu \mathrm{F}$$ in a circuit across a potential difference of $$1.5 \mathrm{~kV}$$. A large number of $$2 \mu \mathrm{F}$$ capacitors which can withstand a potential difference of not more than $$500 \mathrm{~V}$$ are available. The minimum number of capacitors required for the purpose is

What will be the reading in the voltmeter and ammeter of the circuit shown?

Energy of an electron in the second orbit of hydrogen atom is $$E_2$$. The energy of electron in the third orbit of $$\mathrm{He}^{+}$$ will be

Which graph in the following diagram correctly represents the potential energy of a pair of nucleons as a function of their separation?

The figure shows standing de-Broglie waves due to the revolution of electron in a certain orbit of hydrogen atom. Then, the expression for the orbit radius is (All notations have their usual meanings)

LC-oscillations are similar and analogous to the mechanical oscillations of a block attached to a spring. The electrical equivalent of the force constant of the spring is

In an oscillating $$L C$$-circuit, $$L=3 \mathrm{mH}$$ and $$C=2.7 \mu \mathrm{F}$$. At $$t=0$$, the charge on the capacitor is zero and the current is $$2 \mathrm{~A}$$. The maximum charge that will appear on the capacitor will be

An electron in an excited state of $$\mathrm{Li}^{2+}$$ ion has angular momentum $$\frac{3 h}{2 \pi}$$. The de-Broglie wavelength of electron in this state is $$p \pi a_0$$ (where, $$a_0=$$ Bohr radius). The value of $$p$$ is

According to Einstein's photoelectric equation to the graph between kinetic energy of photoelectrons ejected and the frequency of incident radiation is

In a nuclear reactor heavy nuclei is not used as moderators because

Identify the incorrect statement.

The work-function of a metal is 1 eV. Light of wavelength $$3000 \mathop A\limits^o$$ is incident on this metal surface. The velocity of emitted photoelectrons will be

The circuit given represents which of the logic operations?

$$K_a$$ values for acids $$\mathrm{H}_2 \mathrm{SO}_3, \mathrm{HNO}_2, \mathrm{CH}_3 \mathrm{COOH}$$ and $$\mathrm{HCN}$$ are respectively $$13 \times 10^{-2}, 4 \times 10^{-4}, 1.8 \times 10^{-5}$$ and $$4 \times 10^{-10}$$, which of the above acids produces stronger conjugate base in aqueous solution?

When the absolute temperature of ideal gas is doubled and pressure is halved, the volume of gas

Three photodiodes $$D_1, D_2$$ and $$D_3$$ are made of semiconductors having band gaps of $$2.5 \mathrm{~eV}, 2 \mathrm{~eV}$$ and $$3 \mathrm{~eV}$$, respectively. Which one will be able to detect light of wavelength $$600 \mathrm{~nm}$$ ?

Choose the correct statement.

A pure compound contains $$2.4 \mathrm{~g}$$ of $$\mathrm{C}, 1.2 \times 10^{23}$$ atoms of $$\mathrm{H}, 0.2$$ moles of oxygen atoms. Its empirical formula is

For the reaction,

$$A(g)+B(g) \rightleftharpoons C(g)+D(g) ; \Delta H=Q \mathrm{~kJ}$$

The equilibrium constant cannot be disturbed by

Which of the following is not true for oxidation?

Zeta potential is

In chrysoberyl, a compound containing beryllium, aluminium and oxygen, oxide ions form cubic close packed structure. Aluminium ions occupy $$\frac{1}{4}$$th of octahedral voids. The formula of the compound is

Henry's law constant for the solubility of $$\mathrm{N}_2$$ gas in water at $$298 \mathrm{~K}$$ is $$1.0 \times 10^5 \mathrm{~atm}$$. The mole fraction of $$\mathrm{N}_2$$ in air is 0.8 . The number of moles of $$\mathrm{N}_2$$ from air dissolved in 10 moles of water at $$298 \mathrm{~K}$$ and $$5 \mathrm{~atm}$$ pressure is

A colloidal solution is subjected to an electric field than colloidal particles more towards anode. The amount of electrolytes of $$\mathrm{BaCl}_2, \mathrm{AlCl}_3$$ and $$\mathrm{NaCl}$$ required to coagulate the given colloid is in the order

The vapour pressure of pure liquids $$A$$ and $$B$$ are 450 and $$700 \mathrm{~mm}$$ of $$\mathrm{Hg}$$ at $$350 \mathrm{~K}$$ respectively. If the total vapour pressure of the mixture is $$600 \mathrm{~mm}$$ of $$\mathrm{Hg}$$, the composition of the mixture in the solution is

The correct statement regarding defects in solid is

A metal crystallises in bcc lattice with unit cell edge length of $$300 \mathrm{~pm}$$ and density $$615 \mathrm{~g~cm}^{-3}$$. The molar mass of the metal is

The rate of a gaseous reaction is given by the expression $$k[A][B]^2$$. If the volume of vessel is reduced to one half of the initial volume, the reaction rate as compared to original rate is

The resistance of $$0.01 \mathrm{~m} \mathrm{~KCl}$$ solution at $$298 \mathrm{~K}$$ is $$1500 \Omega$$. If the conductivity of $$0.01 \mathrm{~m} \mathrm{~KCl}$$ solution at $$298 \mathrm{~K}$$ is $$0.1466 \times 10^{-3} \mathrm{~S} \mathrm{~cm}^{-1}$$. The cell constant of the conductivity cell in $$\mathrm{cm}^{-1}$$ is

Consider the following electrodes

$$\begin{aligned} & P=\mathrm{Zn}^{2+}(0.0001 \mathrm{M}) / \mathrm{Zn}, Q=\mathrm{Zn}^{2+}(0.1 \mathrm{M}) / \mathrm{Zn} \\ & R=\mathrm{Zn}^{2+}(0.01 \mathrm{M}) / \mathrm{Zn}, S=\mathrm{Zn}^{2+}(0.001 \mathrm{M}) / \mathrm{Zn} \end{aligned}$$

$$E^{\circ}(\mathrm{Zn} / \mathrm{Zn}^{2+})=-0.76 \mathrm{~V}$$ electrode potentials of the above electrodes in volts are in the order

$$\mathrm{H}_2(g)+2 \mathrm{AgCl}(s) \rightleftharpoons 2 \mathrm{Ag}(s)+2 \mathrm{HCl}(a q)$$

$$E_{\text {cell }}^{\circ}$$ at $$25^{\circ} \mathrm{C}$$ for the cell is $$0.22 \mathrm{~V}$$. The equilibrium constant at $$25^{\circ} \mathrm{C}$$ is

The correct order of boiling point in the following compounds is

Higher order $$(>3)$$ reactions are rare due to

Bond enthalpies of $$A_2, B_2$$ and $$A B$$ are in the ratio $$2: 1: 2$$. If bond enthalpy of formation of $$A B$$ is $$-100 \mathrm{~kJ} \mathrm{~mol}^{-1}$$. The bond enthalpy of $$B_2$$ is

Permanent hardness cannot be removed by

The third ionisation enthalpy is highest in

Which of the following is an incorrect statement?

A colourless, neutral, paramagnetic oxide of nitrogen '$$P$$' on oxidation gives reddish brown gas $$Q$$. $$Q$$ on cooling gives colourless gas $$R$$. $$R$$ on reaction with $$P$$ gives blue solid $$S$$. Identify $$P, Q, R, S$$ respectively

$$\mathrm{XeF}_6$$ on partial hydrolysis gives a compound $$X$$, which has square pyramidal geometry '$$X$$' is

The property of the alkaline earth metals that increases with their atomic number is

In the detection of II group acid radical, the salt containing chloride is treated with concentrated sulphuric acid, the colourless gas is liberated. The name of the gas is

Which of the following compound on heating gives $$\mathrm{N}_2 \mathrm{O}$$ ?

Which of the following property is true for the given sequence?

$$\mathrm{NH}_3>\mathrm{PH}_3>\mathrm{AsH}_3>\mathrm{SbH}_3>\mathrm{BiH}_3 \text { ? }$$

Which of the following pairs has both the ions coloured in aqueous solution? [Atomic numbers of

$$\mathrm{Sc}=21, \mathrm{Ti}=22, \mathrm{Ni}=28, \mathrm{Cu}=29, \mathrm{Mn}=25]$$

The IUPAC name of $$\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_5\left(\mathrm{CO}_3\right)\right] \mathrm{Cl}$$ is

Which one of the following is correct for all elements from Sc to Cu?

Which of the following does not represent property stated against it?

For the crystal field splitting in octahedral complexes,

Homoleptic complexes among the following are

(A) $$\mathrm{K}_3\left[\mathrm{Al}\left(\mathrm{C}_2 \mathrm{O}_4\right)_3\right]$$

(B) $$\left[\mathrm{CoCl}_2(\mathrm{en})_2\right]^{+}$$

(C) $$\mathrm{K}_2\left[\mathrm{Zn}(\mathrm{OH})_4\right]$$

Arrange benzene, $$n$$-hexane and ethyne in decreasing order of their acidic behaviour.

A compound $$'A' \left(\mathrm{C}_7 \mathrm{H}_8 \mathrm{O}\right)$$ is insoluble in $$\mathrm{NaHCO}_3$$ solution but dissolve in $$\mathrm{NaOH}$$ and give a characteristic colour with neutral $$\mathrm{FeCl}_3$$ solution. When treated with bromine water compound '$$A$$' forms the compound $$B$$ with the formula $$\mathrm{C}_7 \mathrm{H}_5 \mathrm{OBr}_3$$. '$$A$$' is

The correct order for wavelengths of light absorbed in the complex ions $$\left[\mathrm{CoCl}\left(\mathrm{NH}_3\right)_5\right]^{2+},\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_6\right]^{3+}$$ and $$\left[\mathrm{Co}(\mathrm{CN})_6\right]^{3-}$$ is

Peroxide effect is observed with the addition of $$\mathrm{HBr}$$ but not with the addition of HI to unsymmetrical alkene because

The correct IUPAC name of

The order of reactivity of the compounds $$\mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_2 \mathrm{Br}, \mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}\left(\mathrm{C}_6 \mathrm{H}_5\right) \mathrm{Br}, \mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}\left(\mathrm{CH}_3\right) \mathrm{Br}$$ and $$\mathrm{C}_6 \mathrm{H}_5 \mathrm{C}\left(\mathrm{CH}_3\right)\left(\mathrm{C}_6 \mathrm{H}_5\right) \mathrm{Br}$$ in $$\mathrm{S}_{\mathrm{N}} 2$$ reaction is

$$\begin{aligned} & \mathrm{CH}_3 \mathrm{CHO} \xrightarrow[\text { (ii) } \mathrm{H}_3 \mathrm{O}^{+}]{\text {(i) } \mathrm{CH}_3 \mathrm{MgBr}} A \xrightarrow[\Delta]{\text { Conc. } \mathrm{H}_2 \mathrm{SO}_4} \\ & B \xrightarrow[\text { (ii) } \mathrm{H}_2 \mathrm{O}, \mathrm{OH}^{-}]{\text {(i) } \mathrm{B}_2 \mathrm{H}_6} C \\ & \end{aligned}$$

$$A$$ and $$C$$ are

$$A, B$$ and $$C$$ respectively are

An organic compound $$X$$ on treatment with PCC in dichloromethane gives the compound $$Y$$. Compound $$Y$$ reacts with $$\mathrm{I}_2$$ and alkali to form yellow precipitate of triiodomethane. The compound $$X$$ is

The major product of the following reaction is

$$\mathrm{CH}_2=\mathrm{CH}-\mathrm{CH}_2-\mathrm{OH} \xrightarrow[\text { Excess }]{\mathrm{HBr}} \text { Product }$$

In set of reactions, identify $$D$$

$$\mathrm{CH}_3 \mathrm{COOH} \xrightarrow{\mathrm{SOCl}_2} A \xrightarrow[\text { Anhy. } \mathrm{AlCl}_3]{\text { Benzene }} B \xrightarrow{\mathrm{HCN}} C \xrightarrow{\mathrm{H}_2 \mathrm{O}} D$$

Which is the most suitable reagent for the following conversion?

The product '$$A$$' gives white precipitate when treated with bromine water. The product '$$B$$' is treated with barium hydroxide to give the product $$C$$. The compound $$C$$ is heated strongly to form product $$D$$. The product $$D$$ is

The reagent which can do the conversion $$\mathrm{CH}_3 \mathrm{COOH} \longrightarrow \mathrm{CH}_3-\mathrm{CH}_2-\mathrm{OH}$$ is

$$\mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_2 \mathrm{Cl} \xrightarrow{\text { Alc. } \mathrm{NH}_3} A \xrightarrow{2 \mathrm{CH}_3 \mathrm{Cl}} B.$$ The product $$B$$ is

The compound A (major product) is

The method by which aniline cannot be prepared is

A hydrocarbon $$A\left(\mathrm{C}_4 \mathrm{H}_8\right)$$ on reaction with $$\mathrm{HCl}$$ gives a compound $$\mathrm{B}\left(\mathrm{C}_4 \mathrm{H}_9 \mathrm{Cl}\right)$$ which on reaction with $$1 \mathrm{~mol}$$ of $$\mathrm{NH}_3$$ gives compound $$\mathrm{C}\left(\mathrm{C}_4 \mathrm{H}_{10} \mathrm{N}\right)$$. On reacting with $$\mathrm{NaNO}_2$$ and $$\mathrm{HCl}$$ followed by treatment with water, compound $$C$$ yields an optically active compound $$D$$. The compound $$D$$ is

If the middle term of the AP is 300, then the sum of its first 51 terms is

Primary structure in a nucleic acid contain 3 bases as GATGC ... The chain which is complementary to this chain is

RNA and DNA are chiral molecules, their chirality is due to the presence of

A student has to answer 10 questions, choosing at least 4 from each of the parts $$A$$ and $$B$$. If there are 6 questions in part $$A$$ and 7 in part $$B$$, then the number of ways can the student choose 10 questions is

$$A$$ and $$B$$ are non-singleton sets and $$n(A \times B)=35$$. If $$B \subset A$$, then $${ }^{n(A)} C_{n(B)}$$ is equal to

Given that, $$A$$ and $$B$$ are two events such that $$P(B)=\frac{3}{5}, P\left(\frac{A}{B}\right)=\frac{1}{2}$$ and $$P(A \cup B)=\frac{4}{5}$$, then $$P(A)$$ is equal to

If $$A, B$$ and $$C$$ are three independent events such that $$P(A)=P(B)=P(C)=P$$, then $$P$$ (at least two of $$A, B$$ and $$C$$ occur) is equal to

A car manufacturing factory has two plants $$X$$ and $$Y$$. Plant $$X$$ manufactures $$70 \%$$ of cars and plant $$Y$$ manufactures $$30 \%$$ of cars. $$80 \%$$ of cars at plant $$X$$ and $$90 %$$ of cars at plant $$Y$$ are rated as standard quality. A car is chosen at random and is found to be standard quality. The probability that it has come from plant $$X$$ is :

Two dice are thrown. If it is known that the sum of numbers on the dice was less than 6 the probability of getting a sum as 3 is

In a certain two $$65 \%$$ families own cell phones, 15000 families own scooter and $$15 \%$$ families own both. Taking into consideration that the families own at least one of the two, the total number of families in the town is

If $$P(A)=0.59, P(B)=0.30$$ and $$P(A \cap B)=0.21$$ then $$P\left(A^{\prime} \cap B^{\prime}\right)$$ is equal to

A vector a makes equal acute angles on the coordinate axis. Then the projection of vector $$\mathbf{b}=5 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+\hat{\mathbf{k}}$$ on $$\mathbf{a}$$ is

If the area of the parallelogram with $$\mathbf{a}$$ and $$\mathbf{b}$$ as two adjacent sides is 15 sq units, then the area of the parallelogram having $$\mathrm{3 a+2 b}$$ and $$\mathbf{a}+3 \mathbf{b}$$ as two adjacent sides in sq units is

The diagonals of a parallelogram are the vectors $$3 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$$. and $$-\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-8 \hat{\mathbf{k}}$$. Then the length of the shorter side of parallelogram is

If $$\mathbf{a} \cdot \mathbf{b}=0$$ and $$\mathbf{a}+\mathbf{b}$$ makes an angle $$60^{\circ}$$ with $$a$$, then

The angle between the lines whose direction cosines are $$\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\right)$$ and $$\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2}\right)$$

The area of the quadrilateral $$A B C D$$ when $$A(0,4,1), B(2,3,-1), C(4,5,0)$$ and $$D(2,6,2)$$ is equal to

The equation of the line joining the points $$(-3,4,11)$$ and $$(1,-2,7)$$ is

The mid points of the sides of triangle are $$(1,5,-1)(0,4,-2)$$ and $$(2,3,4)$$ then centroid of the triangle

If a plane meets the coordinate axes at $$A, B$$ and $$C$$ in such a way that the centroid of $$\triangle A B C$$ is at the point $$(1,2,3)$$, then the equation of the plane is

If $$A=\left[\begin{array}{ccc}1 & -2 & 1 \\ 2 & 1 & 3\end{array}\right]$$

$$ B=\left[\begin{array}{ll}2 & 1 \\ 3 & 2 \\ 1 & 1\end{array}\right]$$, then $$(A B)^{\prime}$$ is equal to

If $$A$$ and $$B$$ are invertible matrices then which of the following is not correct?

Let $$M$$ be $$2 \times 2$$ symmetric matrix with integer entries, then $$M$$ is invertible if

If $$A$$ and $$B$$ are matrices of order 3 and $$|A|=5,|B|=3$$, then $$|3 A B|$$ is

The value of $$\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}$$ is

If $$x^3-2 x^2-9 x+18=0$$ and $$A=\left|\begin{array}{lll}1 & 2 & 3 \\ 4 & x & 6 \\ 7 & 8 & 9\end{array}\right|$$ then the maximum value of $$A$$ is

The shaded region is the solution set of the inequalities

The value of $$\cos 1200^{\circ}+\tan 1485^{\circ}$$ is

If $$\left(\frac{1+i}{1-i}\right)^x=1$$, then

The standard deviation of the numbers $$31,32,33 \ldots 46,47$$ is

The area of the region bounded by $$y=-\sqrt{16-x^2}$$ and $$X$$-axis is

The function $$f(x)=\sqrt{3} \sin 2 x-\cos 2 x+4$$ is one-one in the interval

Domain of $$f(x)=\frac{x}{1-|x|}$$ is

$$\tan ^{-1}\left[\frac{1}{\sqrt{3}} \sin \frac{5 \pi}{2}\right] \sin ^{-1}\left[\cos \left(\sin ^{-} \frac{\sqrt{3}}{2}\right)\right]$$ is equal to

$$\cos \left[\cot ^{-1}(-\sqrt{3})+\frac{\pi}{6}\right]$$ is equal to

$$f: R \rightarrow R$$ defined by $$f(x)$$ is equal to $$\left\{\begin{array}{l}2 x, x> 3 \\ x^2, 1< x \leq 3, \text { then } f(-2)+f(3)+f(4) \text { is } \\ 3 x, x \leq 1\end{array}\right.$$

Let $$A=\{x: x \in R, x$$ is not a positive integer) Define $$f: A \rightarrow R$$ as $$f(x)=\frac{2 x}{x-1}$$, then $$f$$ is

If $$f(x)=\left|\begin{array}{ccc}\cos x & 1 & 0 \\ 0 & 2 \cos x & 3 \\ 0 & 1 & 2 \cos x\end{array}\right|$$, then $$\lim _\limits{x \rightarrow \pi} f(x)$$ is equal to

Domain of the function

$$f(x)=\frac{1}{\sqrt{\left[x^2\right]-[x]-6}},$$

where $$[x]$$ is greatest integer $$\leq x$$ is

$$\int \frac{x^3 \sin \left(\tan ^{-1}\left(x^4\right)\right)}{1+x^8} d x$$ is equal to

At $$x=1$$, the function

$$f(x)=\left\{\begin{array}{cc} x^3-1, & 1< x < \infty \\ x-1, & -\infty< x \leq 1 \end{array}\right. \text { is }$$

Consider the following statements

Statement 1 : $$\lim _\limits{x \rightarrow 1} \frac{a x^2+b x+c}{x^2+b x+a}$$ is 1

(where $$a+b+c \neq 0$$).

Statement 2 : $$\lim _\limits{x \rightarrow -2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2}$$ is $$\frac{1}{4}$$.

If $$a$$ and $$b$$ are fixed non-zero constants, then the derivative of $$\frac{a}{x^4}-\frac{b}{x^2}+\cos x$$ is $$m a+n b-p$$, where

The value of $$\int e^x\left[\frac{1+\sin x}{1+\cos x}\right] d x$$ is equal to

If $$y=\left(\cos x^2\right)^2$$, then $$\frac{d y}{d x}$$ is equal to

Consider the following statements

Statement 1 : If $$y=\log _{10} x+\log _e x$$, then $$\frac{d y}{d x}=\frac{\log _{10} e}{x}+\frac{1}{x}$$

Statement 2 : If $$\frac{d}{d x}\left(\log _{10} x\right)=\frac{\log x}{\log 10}$$ and $$\frac{d}{d x}\left(\log _e x\right)=\frac{\log x}{\log e}$$

The value of $$\int \frac{x^2 d x}{\sqrt{x^6+a^6}}$$ is equal to

The value of $$\int \frac{x e^x d x}{(1+x)^2}$$ is equal to

The cost and revenue functions of a product are given by $$c(x)=20 x+4000$$ and $$R(x)=60 x+2000$$ respectively, where $$\mathrm{x}$$ is the number of items produced and sold. The value of $$x$$ to earn profit is

For constant $$a, \frac{d}{d x}\left(x^x+x^a+a^x+a^a\right)$$ is

The number of solutions of $$\frac{d y}{d x}=\frac{y+1}{x-1}$$, when $$y(\mathrm{l})=2$$ is

Solution of differential equating $$x d y-y d x=0$$ represents

If the parametric equation of curve is given by $$x=\cos \theta+\log \tan \frac{\theta}{2}$$ and $$y=\sin \theta$$, then the points for which $$\frac{d y}{d x}=0$$ are given by

If $$y=(x-1)^2(x-2)^3(x-3)^5$$, then $$\frac{d y}{d x}$$ at $$x=4$$ is equal to

The value of $$\int_0^{4042} \frac{\sqrt{x} d x}{\sqrt{x}+\sqrt{4042-x}}$$ is equal to

If the area of the ellipse is $$\frac{x^2}{25}+\frac{y^2}{\lambda^2}=1$$ is $$20 \pi$$ sq units, then $$\lambda$$ is

The equation of straight line which passes through the point $$\left(a \cos ^3 \theta, a \sin ^3 \theta\right)$$ and perpendicular to $$x \sec \theta+y \operatorname{cosec} \theta=a$$ is

The maximum slope of the curve $$y=-x^3+3 x^2+2 x-27$$ is

If the parabola $$y=\alpha x^2-6 x+\beta$$ passes through the point $$(0,2)$$ and has its tangent at $$x=\frac{3}{2}$$ parallel to $$X$$-axis, then

The function $$f(x)=x^2-2 x$$ is strictly decreasing in the interval

For a reaction, $$A+2 B \rightarrow$$ Products, when concentration of $$B$$ alone is increased half-life remains the same. If concentration of $$A$$ alone is doubled, rate remains the same. The unit of rate constant for the reaction is

If the rate constant for a first order reaction is $$k$$, the time $$(t)$$ required for the completion of $$99 \%$$ of the reaction is given by