KCET 2022

If voltage across a bulb rated $$220 \mathrm{~V}, 100 \mathrm{~W}$$ drops by $$2.5 \%$$ of its rated value, then the percentage of the rated value by which the power would decrease is

Vacant space in body centered cubic lattice unit cell is about

If $$3 x+i(4 x-y)=6-i$$ where $$x$$ and $$y$$ are real numbers, then the values of $$x$$ and $$y$$ are respectively,

If $$|\mathbf{a}|=2$$ and $$|\mathbf{b}|=3$$ and the angle between $$\mathbf{a}$$ and $$\mathbf{b}$$ is $$120^{\circ}$$, then the length of the vector $$\left|\frac{\mathbf{a}}{2}-\frac{\mathbf{b}}{3}\right|$$ is

The solution of the differential equation $$\frac{d y}{d x}=(x+y)^2$$ is

$$\int_0^1 \frac{x e^x}{(2+x)^3} d x$$ is equal to

If $$\frac{d y}{d x}+\frac{y}{x}=x^2$$, then $$2 y(2)-y(1)=$$

$$\int_0^{\pi / 2} \sqrt{\sin \theta} \cos ^3 \theta d \theta$$ is equal to

The function $$f(x)=4 \sin ^3 x-6 \sin ^2 x +12 \sin x+100$$ is strictly

If $$[x]$$ is the greatest integer function not greater than $$x$$, then $$\int_\limits0^8[x] d x$$ is equal to

Evaluate $$\int_\limits2^3 x^2 d x$$ as the limit of a sum

The Vernier scale of a travelling microscope has 50 divisions which coincides with 49 main scale divisions. If each main scale division is $$0.5 \mathrm{~mm}$$, then the least count of the microscope is

$$\int_0^{\pi / 2} \frac{\cos x \sin x}{1+\sin x} d x$$ is equal to

The displacement $$x$$ (in $$\mathrm{m}$$) of a particle of mass $$m$$ (in $$\mathrm{kg}$$) moving in one dimension under the action of a force, is related to time $$t$$ (in sec) by $$t=\sqrt{x}+3$$. The displacement of the particle when its velocity is zero, will be

Two masses of $$5 \mathrm{~kg}$$ and $$3 \mathrm{~kg}$$ are suspended with the help of massless inextensible strings as shown in figure below.

When whole system is going upwards with acceleration $$2 \mathrm{~m} / \mathrm{s}^2$$, the value of $$T_1$$ is (use, $$g=9.8 \mathrm{~m} / \mathrm{s}^2$$)

Two objects are projected at an angle $$\theta^{\circ}$$ and $$\left(90-\theta^{\circ}\right)$$, to the horizontal with the same speed. The ratio of their maximum vertical heights is

A smooth chain of length $$2 \mathrm{~m}$$ is kept on a table such that its length of $$60 \mathrm{~cm}$$ hangs freely from the edge of the table. The total mass of the chain is $$4 \mathrm{~kg}$$. The work done in pulling the entire chain on the table is (Take, $$g=10 \mathrm{~m} / \mathrm{s}^2$$)

A car is moving in a circular horizontal track of radius $$10 \mathrm{~m}$$ with a constant speed of $$10 \mathrm{~ms}^{-1}$$. A bob is suspended from the roof of the car by a light wire of length $$1.0 \mathrm{~m}$$. The angle made by the wire with the vertical is (in rad)

Electrical as well as gravitational affects can be thought to be caused by fields. Which of the following is true for an electrical or gravitational field?

The angular speed of a motor wheel is increased from $$1200 \mathrm{~rpm}$$ to $$3120 \mathrm{~rpm}$$ in $$16 \mathrm{~s}$$. The angular acceleration of the motor wheel is

The displacement of a particle executing SHM is given by $$x=3 \sin \left[2 \pi t+\frac{\pi}{4}\right]$$, where $$x$$ is in metre and $$t$$ is in seconds. The amplitude and maximum speed of the particles is

The centre of mass of an extended body on the surface of the earth and its centre of gravity

In case of Fraunhoffer diffraction at a single slit, the diffraction pattern on the screen is correct for which of the following statements?

When a compact disc (CD) is illuminated by small source of white light coloured bands are observed. This is due to

For light diverging from a finite point source,

A metallic rod breaks when strain produced is $$0.2 \%$$. The Young's modulus of the material of the $$\operatorname{rod} 7 \times 10^9 \mathrm{~N} / \mathrm{m}^2$$. The area of crosssection to support a load of $$10^4 \mathrm{~N}$$ is

The fringe width for red colour as compared to that for violet colour is approximately

"Heat cannot be flow itself from a body at lower temperature to a body at higher temperature". This statement corresponds to

Which of the following radiations of electromagnetic waves has the highest wavelength ?

A ray of light passes through an equilateral glass prism in such a manner that the angle of incidence is equal to the angle of emergence and each of these angles is equal to $$\frac{3}{4}$$ of the angle of the prism. The angle of deviation is

Consider a glass slab which is silvered at one side and the other side is transparent. Given the refractive index of the glass slab to be 1.5. If a ray of light is incident at an angle of $$45^{\circ}$$ on the transparent side, then the deviation of the ray of light from its initial path, when it comes out of the slab is

The power of a equi-concave lens is $$-4.5 \mathrm{D}$$ and is made of a material of refractive index 1.6, the radii of curvature of the lens is

Focal length of a convex lens will be maximum for

A convex lens of focal length $$f$$ is placed somewhere in between an object and a screen. The distance between the object and the screen is $$x$$. If the numerical value of the magnification produced by the lens is $$m$$, then the focal length of the lens is

Which of the following statements proves that Earth has a magnetic field ?

A solenoid of length $$50 \mathrm{~cm}$$ having 100 turns carries a current of $$2.5 \mathrm{~A}$$. The magnetic field at one end of the solenoid is

A circular coil of wire of radius $$r$$ has $$n$$ turns and carries a current $$I$$. The magnetic induction $$B$$ at a point on the axis of the coil at a distance $$\sqrt{3} r$$ from its centre is

A proton moves with a velocity of $$5 \times 10^6 \hat{\mathbf{\widehat j} m \mathrm{~m}^{-1}}$$ through the uniform electric field, $$\mathbf{\overrightarrow E}=4 \times 10^6[2 \hat{\mathbf{i}}+0.2 \hat{\mathbf{j}}+0.1 \hat{\mathbf{k}}] \mathrm{Vm}^{-1}$$ and the uniform magnetic field $$\mathbf{\overrightarrow B}=0.2[\hat{\mathbf{i}}+0.2 \hat{\mathbf{j}}+\hat{\mathbf{k}}] \mathrm{T}$$. The approximate net force acting on the proton is

A metallic rod of mass per unit length $$0.5 \mathrm{~kg} \mathrm{~m}^{-1}$$ is lying horizontally on a smooth inclined plane which makes an angle of $$30^{\circ}$$ with the horizontal. A magnetic field of strength $$0.25 \mathrm{~T}$$ is acting on it in the vertical direction. When a current $$I$$ is flowing through it, the rod is not allowed to slide down. The quantity of current required to keep the rod stationary is

A long solenoid has 500 turns, when a current of $$2 \mathrm{~A}$$ is passed through it, the resulting magnetic flux linked with each turn of the solenoid is $$4 \times 10^{-3} \mathrm{~Wb}$$, then self induction of the solenoid is

Four charges $$+q_1+2 q_1+q$$ and $$-2 q$$ are placed at the corners of a square $$A B C D$$ respectively. The force on a unit positive charge kept at the centre $$O$$ is

An electric dipole with dipole moment $$4 \times 10^{-9} \mathrm{C}-\mathrm{m}$$ is aligned at $$30^{\circ}$$ with the direction of a uniform electric field of magnitude $$5 \times 10^4 \mathrm{NC}^{-1}$$, the magnitude of the torque acting on the dipole is

A tiny spherical oil drop carrying a net charge $$q$$ is balanced in still air, with a vertical uniform electric field of strength $$\frac{81}{7} \pi \times 10^5 \mathrm{~V} / \mathrm{m}$$. When the field is switched OFF, the drop is observed to fall with terminal velocity $$2 \times 10^{-3} \mathrm{~ms}^{-1}$$. Here $$g=9.8 \mathrm{~m} / \mathrm{s}^2$$, viscosity of air is $$1.8 \times 10^{-5} \mathrm{Ns} / \mathrm{m}^2$$ and density of oil is $$900 \mathrm{~kg} \mathrm{~m}^{-3}$$. The magnitude of $$q$$ is

A magnetic field of flux densiity $$1.0 \mathrm{~Wb} \mathrm{~m}^{-2}$$ acts normal to a 80 turn coil of $$0.01 \mathrm{~m}^2$$ area. If this coil is removed from the field in $$0.2 \mathrm{~s}$$, then the emf induced in it is

A charged particle of mass $$m$$ and charge $$q$$ is released from rest in an uniform electric field E. Neglecting the effect of gravity, the kinetic energy of the charged particle after $$t$$ seconds is

Two tiny spheres carrying charges $$1.8 \mu \mathrm{C}$$ and $$2.8 \mu \mathrm{C}$$ are located at $$40 \mathrm{~cm}$$ apart. The potential at the mid - point of the line joining the two charges is

A charged particle is moving in an electric field of $$3 \times 10^{-10} \mathrm{Vm}^{-1}$$ with mobility $$2.5 \times 10^6 \mathrm{~m}^2 / \mathrm{V}-\mathrm{s}$$, its drift velocity is

10 identical cells each potential $$E$$ and internal resistance $$r$$ are connected in series to form a closed circuit.

Determine the potential difference across three cells using an ideal voltmeter.

The electric field and the potential of an electric dipole vary with distance $$r$$ as

Wire bound resistors are made by winding the wires of an alloy of

In an atom electrons revolve around the nucleus along a path of radius $$0.72\mathop A\limits^o$$ making $$9.4 \times 10^{18}$$ revolutions per second. The equivalent currents is [given, $$e=1.6 \times 10^{-19} \mathrm{C}$$]

A galvanometer of resistance $$50 \Omega$$ is connected to a battery $$3 \mathrm{~V}$$ along with a resistance $$2950 \Omega$$ in series. A full scale deflection of 30 divisions is obtained in the galvanometer. In order to reduce this deflection to 20 divisions, the resistance in series should be

A series resonant $$\mathrm{AC}$$ circuit contains a capacitance $$10^{-6} \mathrm{~F}$$ and an inductor of $$10^{-4} \mathrm{H}$$. The frequency of electrical oscillations will be

A parallel place capacitor is charged by connecting a $$2 \mathrm{~V}$$ battery across it. It is then disconnected from the battery and a glass slab is introduced between plates. Which of the following pairs of quantities decrease?

A wire of a certain material is stretched slowly by $$10 \%$$. Its new resistance and specific resistance becomes respectively

When a metal conductor connected to left gap of a meter bridge is heated, the balancing point

In a series $$L C R$$ circuit, $$R=300 \Omega, L=0.9 \mathrm{H}, C=2.0 \mu \mathrm{F}$$ and $$\omega=1000 \mathrm{~rad} / \mathrm{s}$$, then impedance of the circuit is

An alternating current is given by $$i=i_1 \sin \omega t+i_2 \cos \omega t$$. The rms current is given by

If an electron is revolving in its Bohr orbit having Bohr radius of 0.529$$\mathop A\limits^o $$, then the radius of third orbit is

A fully charged capacitor $$C$$ with initial charge $$q_0$$ is connected to a coil of self inductance $$L$$ at $$t=0$$. The time at which the energy is stored equally between the electric and the magnetic field is

Which of the following radiations is deflected by electric field?

In accordance with the Bohr's model, the quantum number that characterises the Earth's revolution around the Sun in an orbit of radius $$1.5 \times 10^{11} \mathrm{~m}$$ with orbital speed $$3 \times 10^4 \mathrm{~ms}^{-1}$$ is [given, mass of Earth $$=6 \times 10^{24} \mathrm{~kg}$$]

The radius of hydrogen atom in the ground state is 0.53$$\mathop A\limits^o $$. After collision with an electron, it is found to have a radius of 2.12$$\mathop A\limits^o $$, the principal quantum number $$n$$ of the final state of the atom is

A nuclear reactor delivers a power of $$10^9 \mathrm{~W}$$, the amount of fuel consumed by the reactor in one hour is

The resistivity of a semiconductor at room temperature is in between

Binding energy of a nitrogen nucleus $$\left[{ }_7^{14} \mathrm{~N}\right]$$, given $$m\left[{ }_7^{14} \mathrm{~N}\right]=14.00307 \mathrm{u}$$

The forbidden energy gap for Ge crystal at 0K is

The kinetic energy of the photoelectrons increases by $$0.52 \mathrm{~eV}$$ when the wavelength of incident light is changed from $$500 \mathrm{~nm}$$ to another wavelength which is approximately

The de-Broglie wavelength of a particle of kinetic energy $$K$$ is $$\lambda$$, the wavelength of the particle, if its kinetic energy $$\frac{K}{4}$$ is

In a photo electric experiment, if both the intensity and frequency of the incident light are doubled, then the saturation photoelectric current

Which property of $$\mathrm{CO}_2$$ makes it biologically and geo-chemically important?

If wavelength of photon is $$2.2 \times 10^{-11} \mathrm{~m}$$ and $$h=6.6 \times 10^{-34} \mathrm{~J} \mathrm{~s}$$, then momentum of photon

The work done when 2 moles of an ideal gas expands rèversibly and isothermally from a volume of $$1 \mathrm{~L}$$ to $$10 \mathrm{~L}$$ at $$300 \mathrm{~K}$$ is ($$R=0.0083 \mathrm{~kJ} \mathrm{~K} \mathrm{~mol}^{-1}$$)

The volume of $$2.8 \mathrm{~g}$$ of $$\mathrm{CO}$$ at $$27^{\circ} \mathrm{C}$$ and 0.821 atm pressure is ($$R=0.08210 \text { L. atm K} \mathrm{~mol}^{-1}$$)

Which logic gate is represented by the following combination of logic gates?

An aqueous solution of alcohol contains $$18 \mathrm{~g}$$ of water and $$414 \mathrm{~g}$$ of ethyl alcohol. The mole fraction of water is

If $$3 \mathrm{~g}$$ of glucose (molar mass $$=180 \mathrm{~g}$$) is dissolved in $$60 \mathrm{~g}$$ of water at $$15^{\circ} \mathrm{C}$$, the osmotic pressure of the solution will be

Which of the following colligative properties can provide molar mass of proteins, polymers and colloids with greater precision?

Solubility of a gas in a liquid increases with

The rise in boiling point of a solution containing $$1.8 \mathrm{~g}$$ of glucose in $$100 \mathrm{~g}$$ of solvent is $$0.1^{\circ} \mathrm{C}$$. The molal elevation constant of the liquid is

Which among the following has highest $$\mathrm{pH}$$ ?

1 mole of $$\mathrm{HI}$$ is heated in a closed container of capacity of $$2 \mathrm{~L}$$. At equilibrium half a mole of $$\mathrm{HI}$$ is dissociated. The equilibrium constant of the reaction is

Which can adsorb larger volume of hydrogen gas?

All Cu(II) halides are known, except the iodide, the reason for it is that

Colloidal solution commonly used in the treatment of skin disease is

How many number of atoms are there in a cube based unit cell, having one atom on each corner and 2 atoms on each body diagonal of cube?

Which of the following is not true about the amorphous solids?

In which of the following compounds, an element exhibits two different oxidation states?

The molar conductivity is maximum for the solution of concentration

Alkali halides do not show dislocation defect because

Specific conductance of $$0.1 \mathrm{~M} \mathrm{~HNO}_3$$ is $$6.3 \times 10^{-2} \mathrm{~ohm}^{-1} \mathrm{~cm}^{-1}$$. The molar conductance of the solution is

For spontaneity of a cell, which is correct?

In fuel cells _______ are used as catalysts.

Half-life of a reaction is found to be inversely proportional to the fifth power of its initial concentration, the order of reaction is

Amphoteric oxide among the following

The property of halogens which is not correctly matched is

For $$n$$th order of reaction, half-life period is directly proportional to

Elements $$X, Y$$ and $$Z$$ have atomic numbers 19, 37 and 55 respectively. Which of the following statements is true about them ?

A first order reaction is half completed in $$45 \mathrm{~min}$$. How long does it need $$99.9 \%$$ of the reaction to be completed?

The rate of the reaction, $$\mathrm{CH}_3 \mathrm{COOC}_2 \mathrm{H}_5+\mathrm{NaOH} \longrightarrow \mathrm{CH}_3 \mathrm{COONa}+\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}$$ is given by the equation, rate $$=k\left[\mathrm{CH}_3 \mathrm{COOC}_2 \mathrm{H}_5\right][\mathrm{NaOH}]$$. If concentration is expressed in $$\mathrm{mol} \mathrm{L}^{-1}$$, the unit of $$k$$ is

Which of the following hydrides is electron deficient?

What will be the value of $$x$$ in $$\mathrm{Fe}^{x+}$$, if the magnetic moment, $$\mu=\sqrt{24} \mathrm{~BM}$$ ?

The strong reducing property of hypophosphorus acid is due to

$$\left(\mathrm{NH}_4\right)_2 \mathrm{Cr}_2 \mathrm{O}_7$$ on heating liberates a gas. The same gas will be obtained by

In oxygen and carbon molecule the bonding is

Which noble gas has least tendency to form compounds?

A transition metal exists in its highest oxidation state. It is expected to behave as

The correct IUPAC name of cis-platin is

The complex hexamineplatinum(IV)chloride will give _____ number of ions on ionisation.

Crystal field splitting energy (CFSE) for $$\left[\mathrm{CoCl}_6\right]^{4-}$$ is $$18000 \mathrm{~cm}^{-1}$$. The crystal field splitting energy (CFSE) for $$\left[\mathrm{CoCl}_4\right]^{2-}$$ will be

The IUPAC name for

The only lanthanoid which is radioactive

The major product obtained when ethanol is heated with excess of conc. $$\mathrm{H}_2 \mathrm{SO}_4$$ at $$443 \mathrm{~K}$$ is

Identify, A and B in the following reaction

Which one of the following chlorohydrocarbon readily undergoes solvolysis?

In Kolbe's reaction the reacting substances are

An organic compound with molecular formula $$\mathrm{C}_7 \mathrm{H}_8 \mathrm{O}$$ dissolves in $$\mathrm{NaOH}$$ and gives a characteristic colour with $$\mathrm{FeCl}_3$$. On treatment with bromine, it gives a tribromo derivative $$\mathrm{C}_7 \mathrm{H}_5 \mathrm{OBr}_3$$. The compound is

In the following pairs of halogen compounds, which compound undergoes faster $$S_N 1$$ reaction?

Ethanoic acid undergoes Hell-Volhard Zelinsky reaction but methanoic acid does not, because of

Reaction by which benzaldehyde cannot be prepared is

Among the following, the products formed by the reaction of anisole with $$\mathrm{HI}$$ are

The general name of the compound formed by the reaction between aldehyde and alcohol is

Which is most VISCOUS?

The test to differentiate between pentan-2-one and pentan-3-one is

In carbylamine test for primary amines the resulting foul smelling product is

A nucleic acid, whether DNA or RNA gives on complete hydrolysis, two purine bases, two pyrimidine bases, a pentose sugar and phosphoric acid. Nucleotides which are intermediate products in the hydrolysis contain

Which of the following is correctly matched?

Which institute has approved the emergency use of 2-deoxy-D-glucose as additive therapy for COVID-19 patients?

A secondary amine is

Identify the products $$A$$ and $$B$$ in the reactions :

$$\begin{aligned} & R-X+\mathrm{AgCN} \longrightarrow A+\mathrm{Ag} X \\ & R-X+\mathrm{KCN} \longrightarrow B+\mathrm{KX} \end{aligned}$$

If $$A=\{1,2,3, \ldots, 10\}$$, then number of subsets of $$A$$ containing only odd numbers is

If $$a_1, a_2, a_3, \ldots, a_{10}$$ is a geometric progression and $$\frac{a_3}{a_1}=25$$, then $$\frac{a_9}{a_5}$$ equals

If $$A$$ and $$B$$ are two events such that $$P(A)=\frac{1}{2}, P(B)=\frac{1}{2}$$ and $$P(A \mid B)=\frac{1}{4}$$, then $$P\left(A^{\prime} \cap B^{\prime}\right)$$ is

If all permutations of the letters of the word MASK are arranged in the order as in dictionary with or without meaning, which one of the following is 19th word?

Find the mean number of heads in three tosses of a fair coin.

If the set $$x$$ contains 7 elements and set $$y$$ contains 8 elements, then the number of bijections from $$x$$ to $$y$$ is

Suppose that the number of elements in set $$A$$ is $$p$$, the number of elements in set $$B$$ is $$q$$ and the number of elements in $$A \times B$$ is 7, then $$p^2+q^2=$$

Let the relation $$R$$ is defined in $$N$$ by $$a R b$$, if $$3 a+2 b=27$$ then $$R$$ is

If $$|\mathbf{a} \times \mathbf{b}|^2+|\mathbf{a} \cdot \mathbf{b}|^2=36$$ and $$|\mathbf{a}|=3$$, then $$|\mathbf{a}|$$ is equal to

If $$\alpha=\hat{\mathbf{i}}-3 \hat{\mathbf{j}}, \beta=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$$, then express $$\beta$$ in the form $$\beta=\beta_1+\beta_2$$ where $$\beta_1$$ is parallel to $$\alpha$$ and $$\beta_2$$ is perpendicular to $$\alpha$$, then $$\beta_1$$ is given by

A pandemic has been spreading all over the world. The probabilities are 0.7 that there will be a lockdown, 0.8 that the pandemic is controlled in one month if there is a lockdown and 0.3 that it is controlled in one month if there is no lockdown. The probability that the pandemic will be controlled in one month is

If $$A$$ and $$B$$ are two independent events such that $$P(\bar{A})=0.75, P(A \cup B)=0.65$$ and $$P(B)=x$$, then find the value of $$x$$.

The octant in which the point (2, $$-$$4, $$-7$$) lies is

If $$A$$ is a skew symmetric matrix, then A$$^{2021}$$ is

The angle between the pair of lines $$\frac{x+3}{3}=\frac{y-1}{5}=\frac{z+3}{4}$$ and $$\frac{x+1}{1}=\frac{y-4}{4}=\frac{z-5}{2}$$ is

The coordinates of foot of the perpendicular drawn from the origin to the plane $$2 x-3 y+4 z=29$$ are

The distance of the point whose position vector is $$(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})$$ from the plane $$\mathbf{r} \cdot(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})=4$$ is

If $$A$$ is a matrix of order $$3 \times 3$$, then $$\left(A^2\right)^{-1}$$ is equal to

If $$A$$ is a $$3 \times 3$$ matrix such that $$|5 \cdot \operatorname{adj} A|=5$$, then $$|A|$$ is equal to

If there are two values of '$$a$$' which makes determinant

$$\Delta=\left|\begin{array}{ccc} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a \end{array}\right|=86$$

Then, the sum of these number is

If $$A=\left[\begin{array}{ll}2 & -1 \\ 3 & -2\end{array}\right]$$, then the inverse of the matrix $$A^3$$ is

If $$A_n=\left[\begin{array}{cc}1-n & n \\ n & 1-n\end{array}\right]$$, then $$\left|A_1\right|+\left|A_2\right|+\ldots .\left|A_{2021}\right|=$$

If the standard deviation of the numbers $$-1, 0,1, k$$ is $$\sqrt{5}$$ where $$k>0$$, then $$k$$ is equal to

If $$A=\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right]$$, then $$(a I+b A)^n$$ is (where $$I$$ is the identify matrix of order 2)

The degree measure of $$\frac{\pi}{32}$$ is equal to

The trigonometric function $$y=\tan x$$ in the II quadrant

The value of $$\sin \frac{5 \pi}{12} \sin \frac{\pi}{12}$$ is

A dietician has to develop a special diet using two foods $$X$$ and $$Y$$. Each packet (containing $$30 \mathrm{~g}$$ ) of food. $$X$$ contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Y contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240 units of calcium, atleast 460 units of iron and atmost 300 units of cholesterol. The corner points of the feasible region are

The corner points of the feasible region of an LPP are $$(0,2),(3,0),(6,0),(6,8)$$ and $$(0,5)$$, then the minimum value of $$z=4 x+6 y$$ occurs at

$$\sqrt{2+\sqrt{2+\sqrt{2+2 \cos 8 \theta}}}=$$

The domain of the function $$f(x)=\frac{1}{\log _{10}(1-x)}+\sqrt{x+2}$$ is

Area of the region bounded by the curve $$y=\tan x$$, the $$X$$-axis and line $$x=\frac{\pi}{3}$$ is

If $$f(x)=\left\{\begin{array}{cc}x^2-1, & 0< x<2 \\ 2 x+3, & 2 \leq x<3\end{array}\right.$$,

the quadratic equation whose roots are $$\lim _\limits{x \rightarrow 2^{-}} f(x)$$ and $$\lim _\limits{x \rightarrow 2^{+}} f(x)$$ is

$$\lim _\limits{y \rightarrow 0} \frac{\sqrt{3+y^3}-\sqrt{3}}{y^3}=$$

If $$f: R \rightarrow R$$ be defined by

$$f(x)=\left\{\begin{array}{llc} 2 x: & x>3 \\ x^2: & 1< x \leq 3 \\ 3 x: & x \leq 1 \end{array}\right.$$

then $$f(-1)+f(2)+f(4)$$ is

Domain $$\cos ^{-1}[x]$$ is, where [ ] denotes a greatest integer function

If $$\int \frac{d x}{(x+2)\left(x^2+1\right)}=a \log \left|1+x^2\right|+b \tan ^{-1} x +\frac{1}{5} \log |x+2|+c,$$ then

If $$y=\left(1+x^2\right) \tan ^{-1} x-x$$, then $$\frac{d y}{d x}$$ is

If $$f(1)=1, f^{\prime}(l)=3$$, then the derivative of $$f(f(f(x)))+(f(x))^2$$ at $$x=1$$ is

$$\int \frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha} d x$$ is equal to

If $$x=e^\theta \sin \theta, y=e^\theta \cos \theta$$ where $$\theta$$ is a parameter, then $$\frac{d y}{d x}$$ at $$(1,1)$$ is equal to

If $$y=e^{\sqrt{x \sqrt{x} \sqrt{x}}...,} x >1$$, then $$\frac{d^2 y}{d x^2}$$ at $$x=\log _e 3$$ is

The coordinates of the point on the $$\sqrt{x}+\sqrt{y}=6$$ at which the tangent is equally inclined to the axes is

If $$e^y+x y=e$$ the ordered pair $$\left(\frac{d y}{d x}, \frac{d^2 y}{d x^2}\right)$$ at $$x=0$$ is equal to

If $$y=x^{\sin x}+(\sin x)^x$$, then $$\frac{d y}{d x}$$ at $$x=\frac{\pi}{2}$$ is

The function $$f(x)=\log (1+x)-\frac{2 x}{2+x}$$ is increasing on

If $$y(x)$$ is the solution of differential equation $$x \log x \frac{d y}{d x}+y=2 x \log x, y(e)$$ is equal to

The sum of the degree and order of the differential equation $$\left(l+y_1^2\right)^{2 / 3}=y_2$$ is

If the straight line $$2 x-3 y+17=0$$ is perpendicular to the line passing through the points $$(7,17)$$ and $$(15, \beta)$$, then $$\beta$$ equals

If the vertices of a triangle are $$(-2,6),(3,-6)$$ and $$(1,5)$$, then the area of the triangle is