KCET 2020

A metal exists as an oxide with formula $$M_{0.96}$$ O. Metal, $$M$$ can exist as $$M^{2+}$$ and $$M^{3+}$$ in its oxide $$M_{0.96} \mathrm{O}$$. The percentage of $$\mathrm{M}^{3+}$$ in the oxide is nearly

A gas mixture contains $$25 \% \mathrm{~He}$$ and $$75 \% \mathrm{~CH}_4$$ by volume at a given temperature and pressure. The percentage by mass of methane in the mixture is approximately _________.

A cylindrical wire has a mass $$(0.3 \pm 0.003) \mathrm{g}$$, radius $$(0.5 \pm 0.005) \mathrm{mm}$$ and length $$(6 \pm 0.06) \mathrm{cm}$$. The maximum percentage error in the measurement of its density is

If the side of a cube is increased by $$5 \%$$, then the surface area of a cube is increased by

One end of a string of length $$l$$ is connected to a particle of mass $$m$$ and the other to a small peg on a smooth horizontal table. If the particle moves in a circle with speed $$v$$, the net force on the particle (directed towards the centre) is ($$T$$ is the tension in the string)

Rain is falling vertically with a speed of $$12 \mathrm{~ms}^{-1}$$. A woman rides a bicycles with a speed of $$12 \mathrm{~ms}^{-1}$$ in east to west direction. What is the direction in which she should hold her umbrella?

A body is initially at rest. It undergoes one-dimensional motion with constant acceleration. The power delivered to it at time $$t$$ is proportional to

A thin uniform rectangular plate of mass $$2 \mathrm{~kg}$$ is placed in $$x y$$-plane as shown in figure. The moment of inertial about $$x$$-axis is $$I_x=0.2 \mathrm{~kgm}^2$$ and the moment of inertia about $$Y$$-axis is $$I_y=0.3 \mathrm{~kgm}^2$$. The radius of gyration of the plate about the axis passing through $$O$$ and perpendicular to the plane of the plate is

At a metro station, a girl walks up a stationary escalator in $$20 \mathrm{~s}$$. If she remains stationary on the escalator, then the escalator take her up in $$30 \mathrm{~s}$$. The time taken by her to walk up on the moving escalator will be

A wheel starting from rest gains an angular velocity of $$10 \mathrm{~rad} / \mathrm{s}$$ after uniformly accelerated for $$5 \mathrm{~s}$$. The total angle through which it has turned is

A tray of mass $$12 \mathrm{~kg}$$ is supported by two identical springs as shown in figure. When the tray is pressed down slightly and then released, it executes SHM with a time period of $$1.5 \mathrm{~s}$$. The spring constant of each spring is

Young's modulus of a perfect rigid body is

The value of acceleration due to gravity at a height of $$10 \mathrm{~km}$$ from the surface of earth is $$x$$. At what depth inside the earth is the value of the acceleration due to gravity has the same value $$x$$ ?

Iceberg floats in water with part of it submerged. What is the fraction of the volume of iceberg submerged, if the density of ice is $$\rho_i=0.917 \mathrm{~g} \mathrm{~cm}^{-3}$$ ?

A train whistling at constant frequency $$n$$ is moving towards a station at a constant speed $$v$$. The train goes past a stationary observer on the station. The frequency $$n$$ of the sound as heard by the observer is plotted as a function of time $$t$$. Identify the correct curve.

A sphere, a cube and a thin circular plate all of same material and same mass initially heated to same high temperature are allowed to cool down under similar conditions. Then, the

In Young's double slit experiment, the distance between the slits and the screen is $$1.2 \mathrm{~m}$$ and the distance between the two slits is $$2.4 \mathrm{~mm}$$. If a thin transparent mica sheet of thickness $$1 \mu \mathrm{m}$$ and RI 1.5 is introduced between one of the interfering beams, the shift in the position of central bright fringe is

A certain amount of heat energy is supplied to a monoatomic ideal gas which expands at constant pressure. What fraction of the heat energy is converted into work?

In an adiabatic expansion of an ideal gas the product of pressure and volume

Three polaroid sheets $$P_1, P_2$$ and $$P_3$$ are kept parallel to each other such that the angle between pass axes of $$P_1$$ and $$P_2$$ is $$45^{\circ}$$ and that between $$P_2$$ and $$P_3$$ is $$45^{\circ}$$. If unpolarised beam of light of intensity $$128 \mathrm{~Wm}^{-2}$$ is incident on $$P_1$$. What is the intensity of light oming out of $$P_3$$ ?

An object approaches a convergent lens from the left of the lens with a uniform speed $$5 \mathrm{~m} / \mathrm{s}$$ and stops at the focus, the image

Two poles are separated by a distance of $$3.14 \mathrm{~m}$$. The resolving power of human eye is $$1 \mathrm{~min}$$ of an arc. The maximum distance from which he can identify the two poles distinctly is

A light beam of intensity $$20 \mathrm{~W} / \mathrm{cm}^2$$ is incident normally on a perfectly reflecting surface of sides $$25 \mathrm{~cm} \times 15 \mathrm{~cm}$$. The momentum imparted to the surface by the light per second is

A long cylindrical wire of radius $$R$$ carries a uniform current $$I$$ flowing through it. The variation of magnetic field with distance $$r$$ from the axis of the wire is shown by

The magnetic field at the origin due to a current element $$i d \mathbf{l}$$ placed at a point with vector position $$\mathrm{r}$$ is

In the given figure, the magnetic field at $$O$$.

The refracting angle of prism is $$A$$ and refractive index of material of prism is $$\cot \frac{A}{2}$$. The angle of minimum deviation is $$A$$

The following figure shows a beam of light converging at point $$P$$. When a concave lens of focal length $$16 \mathrm{~cm}$$ is introduced in the path of the beam at a place shown by dotted line such that $$O P$$ becomes the axis of the lens, the beam converges at a distance $$x$$ from the lens. The value of $$x$$ will be equal to

The ratio of magnetic field at the centre of a current carrying circular coil to its magnetic moment is $$x$$, if the current and the radius both are doubled. The new ratio will become

In a permanent magnet at room temperature

A rod of length $$2 \mathrm{~m}$$ slides with a speed of $$5 \mathrm{~ms}^{-1}$$ on a rectangular conducting frame as shown in figure. There exists a uniform magnetic filed of $$0.04 \mathrm{~T}$$ perpendicular to the plane of the figure. If the resistance of the rod is $$3 \Omega$$. The current through the rod is

A cyclotron is used to accelerate protons $$\left({ }_1^l \mathrm{H}\right)$$ deuterons $$\left({ }_1^2 \mathrm{H}\right)$$ and $$\alpha$$-particles $$\left({ }_2^4 \mathrm{He}\right)$$. While exiting under similar conditions, the minimum $$\mathrm{KE}$$ is gained by

Two long straight parallel wires are a distance $$d$$ part. They carry steady equal currents flowing out of the plane of the paper. The variation of magnetic field $B$ along the line $$x x^{\prime}$$ is given by

A paramagnetic sample shows a net magnetisation of $$8 \mathrm{Am}^{-1}$$ when placed in an external magnetic field of $$0.6 \mathrm{~T}$$ at a temperature of $$4 \mathrm{~K}$$. When the same sample is placed in an external magnetic field of $$0.2 \mathrm{~T}$$ at a temperature of $$16 \mathrm{~K}$$, the magnetisation will be

The electric field lines on the left have twice the separation on those on the right as shown in figure. If the magnitude of the field at $$A$$ is $$40 \mathrm{~Vm}^{-1}$$, what is the force on $$20 \mu \mathrm{C}$$ charge kept at $$B$$ ?

An infinitely long thin straight wire has uniform charge density of $$\frac{1}{4} \times 10^{-2} \mathrm{~cm}^{-1}$$. What is the magnitude of electric field at a distance $$20 \mathrm{~cm}$$ from the axis of the wire?

The current in a coil of inductance $$0.2 \mathrm{H}$$ changes from $$5 \mathrm{~A}$$ to $$2 \mathrm{~A}$$ in $$0.5 \mathrm{~s}$$. The magnitude of the average induced emf in the coil is

A dipole moment $$p$$ and moment of inertia $$I$$ is placed in a uniform electric field $$\mathbf{E}$$. If it is displaced slightly from its stable equilibrium position, the period of oscillation of dipole is

When a soap bubble is charged?

A point charge $$q$$ is placed at the corner of a cube of side $$a$$ as shown in the figure. What is the electric flux through the face $$A B C D$$ ?

The colour code for a carbon resistor of resistance $$0.2 \mathrm{k} \Omega \pm 10 \%$$ is

A metal rod of length $$10 \mathrm{~cm}$$ and a rectangular cross-section of $$1 \mathrm{~cm} \times \frac{1}{2} \mathrm{~cm}$$ is connected to a battery across opposite faces. The resistance will be

A potentiometer has a uniform wire of length $$5 \mathrm{~m}$$. A battery of emf $$10 \mathrm{~V}$$ and negligible internal resistance is connected between its ends. A secondary cell connected to the circuit gives balancing length at $$200 \mathrm{~cm}$$. The emf of the secondary cell is

Figure shows three points $$A, B$$ and $$C$$ in a region of uniform electric field $$\mathbf{E}$$. The line $$A B$$ is perpendicular and $$B C$$ is parallel to the field lines. Then, which of the following holds good ? ($$V_A, V_B$$ and $$V_C$$ represent the electric potential at points $$A, B$$ and $$C$$, respectively)

Each resistance in the given cubical network has resistance of $$1 \Omega$$ and equivalent resistance between $$A$$ and $$B$$ is

A car has a fresh storage battery of emf $$12 \mathrm{~V}$$ and internal resistance $$2 \times 10^{-2} \Omega$$. If the starter motor draws a current of $$80 \mathrm{~A}$$. Then, the terminal voltage when the starter is On is

In the given circuit, the resonant frequency is

$$I-V$$ characteristic of a copper wire of length $$L$$ and area of cross-section $$A$$ is shown in figure. The slope of the curve becomes

In the given circuit the peak voltage across $$C, L$$ and $$R$$ are $$30 \mathrm{~V}, 110 \mathrm{~V}$$ and $$60 \mathrm{~V}$$, respectively. The rms value of the applied voltage is

A $$220 \mathrm{~V}$$ AC supply is connected between points $$A$$ and $$B$$ as shown in figure, what will be the potential difference $$V$$ across the capacitor?

The power factor of $$R-L$$ circuit is $$\frac{1}{\sqrt{3}}$$. If the inductive reactance is $$2 \Omega$$. The value of resistance is

The difference between equivalent capacitances of two identical capacitors connected in parallel to that in series is $$6 \mu \mathrm{F}$$. The value of capacitance of each capacitor is

A radio-active elements has half-life of 15 years. What is the fraction that will decay in 30 years?

A beam of fast moving alpha particles were directed towards a thin film of gold. The parts $$A, B$$ and $$C$$ of the transmitted and reflected beams corresponding to the incident parts $$A, B$$ and $$C$$ of the beam are shown in the adjoining diagram. The number of alpha particles in

The period of revolution of an electron revolving in nth orbit of $$\mathrm{H}$$-atom is proportional to

Angular momentum of an electron in hydrogen atom is $$\frac{3 h}{2 \pi}$$ ($$h$$ is the Planck's constant). The KE of the electron is

Two protons are kept at a separation of $$10 \mathrm{~nm}$$. Let $$F_n$$ and $$F_e$$ the nuclear force and the electromagnetic force between them

During a $$\beta^{-}$$-decay

In the following circuit what are P and Q

The de-Broglie wavelength associated with electron of hydrogen atom in this ground state is

A positive hole in a semiconductor is

A hot filament liberates an electron with zero initial velocity. The anode potential is $$1200 \mathrm{~V}$$. The speed of the electron when it strikes the anode is

$$0.4 \mathrm{~g}$$ of dihydrogen is made to react with $$7.4 \mathrm{~g}$$ of dichlorine to form hydrogen chloride. The volume of hydrogen chloride formed at $$273 \mathrm{~K}$$ and $$1 \mathrm{~bar}$$ pressure is

The following graph represents the variation of photocurrent with anode potential for a metal surface. Here $$I_1, I_2$$ and $$I_3$$ represents intensities and $$\gamma_1, \gamma_2, \gamma_3$$ represent frequency for curves $$1,2$$ and $$3$$ respectively, then

With regard to photoelectric effect, identify the correct statement among the following.

How many moles of acidified $$\mathrm{K}_2 \mathrm{Cr}_2 \mathrm{O}_7$$ is required to liberate 6 moles of $$\mathrm{I}_2$$ from an aqueous solution of $$\mathrm{I}^{-}$$?

Which of the following pair of solutions is isotonic?

The conjugate base of $$\mathrm{NH}_3$$ is

Solute '$$X$$' dimerises in water to the extent of $$80\%$$. 2.5g of '$$X$$' in $$100 \mathrm{~g}$$ of water increases the boiling point by $$0.3^{\circ} \mathrm{C}$$. The molar mass of 'X' is $$[K_b=0.52 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}]$$

When the same quantity of heat is absorbed by a system at two different temperatures $$T_1$$ and $$T_2$$, such that $$T_1>T_2$$, change in entropies are $$\Delta S_1$$ and $$\Delta S_2$$ respectively. Then

In which of the following cases a chemical reaction is possible?

Silicon doped with gallium forms

The oxidation number of nitrogen atoms in $$\mathrm{NH}_4 \mathrm{NO}_3$$ are

A sol of $$\mathrm{AgI}$$ is prepared by mixing equal volumes of $$0.1 \mathrm{~M} \mathrm{~AgNO}_3$$ and $$0.2 \mathrm{~M} \mathrm{~KI}$$, which of the following statement is correct?

A metal crystallises in face centred cubic structure with metallic radius $$\sqrt{2} \mathop A\limits^o$$. The volume of the unit cell (in $$\mathrm{m}^3$$ ) is

During adsorption of a gas on a solid at low temperature

The rate constant of a reaction is given by $$k=P Z e^{-E_a / R T}$$ under standard notation. In order to speed up the reaction, which of the following factors has to be decreased?

The time required for $$60 \%$$ completion of a first order reaction is $$50 \mathrm{~min}$$. The time required for $$93.6 \%$$ completion of the same reaction will be

The pair of electrolytes that posses same value for the constant $$(A)$$ in the Debye-Huckel-Onsager equation, $$\Lambda_m=\Lambda_m^{\circ}-A \sqrt{C}$$ is

For an elementary reaction $$2 A+3 B \rightarrow 4 C+D$$ the rate of appearance of $C$ at time '$$T$$' is $$2.8 \times 10^{-3} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~S}^{-1}$$. Rate of disappearance of $$B$$ at '$$t$$' $t$ will be

Given $$E_{F{e^{3 + }}/F{e^{2 + }}}^o = + 0.76\,V$$ and $$E_{\mathrm{I}_2 / \mathrm{I}^{-}}^0=+0.55 \mathrm{~V}$$. The equilibrium constant for the reaction taking place in galvanic cell consisting of above two electrodes is $$\left[\frac{2303 R T}{F}=0.06\right]$$

If an aqueous solution of $$\mathrm{NaF}$$ is electrolysed between inert electrodes, the product obtained at anode is

Bond angle is $$\mathrm{PH}_4^{+}$$ is more than that of $$\mathrm{PH}_3$$. This is because

Which of the following has the lowest boiling point?

The formal charge on central oxygen atom in ozone is

The percentage of $$s$$-character in the hybrid orbitals of nitrogen in $$\mathrm{NO}_2^{+}, \mathrm{NO}_3^{-}$$ respectively are

Copper is extracted from copper pyrites by

Incorrectly matched pair is

Phosphorus pentachloride

The oxide of potassium that does not exist is

Sulphide ore on roasting gives a gas $$X$$. $$X$$ reacts with $$\mathrm{Cl}_2$$ in the presence of activated charcoal to give $$Y$$. $$Y$$ is

Aqueous solution of a salt $$(A)$$ forms a dense white precipitate with $$\mathrm{BaCl}_2$$ solution. The precipitate dissolves in dilute $$\mathrm{HCl}$$ to produce a gas $$(B)$$ which decolourises acidified $$\mathrm{KMnO}_4$$ solution. $$A$$ and $$B$$ respectively are

The metal that produce $$\mathrm{H}_2$$ with both dil. $$\mathrm{HCl}$$ and $$\mathrm{NaOH}(a q)$$ is

Function of potassium ethyl xanthate in froth floatation process is to make the ore

Number of stereoisomers exhibited by $$\left[\mathrm{Co}(\mathrm{en})_2 \mathrm{Cl}_2\right]^{+}$$ is

Identify the set of paramagnetic ions among the following.

A Lewis acid '$$X$$' reacts with $$\mathrm{LiAlH}_4$$ in ether medium to give a highly toxic gas. This gas when heated with $$\mathrm{NH}_3$$ gives a compound commonly known as inorganic benzene. The gas is

The last element of the $$p$$-block in 6 th period is represented by the outer most electronic configuration.

For $$\mathrm{Cu}_2 \mathrm{Cl}_2$$ and $$\mathrm{CuCl}_2$$ in aqueous medium, which of the following statement is correct?

The coordination number of $$\mathrm{Fe}$$ and $$\mathrm{Co}$$ in the complex ions, $$\left.\mathrm{Fe}\left(\mathrm{C}_2 \mathrm{O}_4\right)_3\right]^{3-}$$ and $$\left[\mathrm{Co}(\mathrm{SCN})_4\right]^{2-}$$ are respectively

Give the IUPAC name of $$\left[\mathrm{Pt}\left(\mathrm{NH}_3\right)_4\right]\left[\mathrm{Pt} \mathrm{Cl}_4\right]$$ is

Which of the following is not a green house gas?

Which of the following is the strongest base?

Which of the following is not a pair of functional isomers?

In the reaction,

The number of possible isomers for the organic compound X is

Identify 'X' in the following reaction.

The carbonyl compound that does not undergo aldol condensation is

The product 'P' is

The steps involved in the conversion of propan -2-ol to propan -1-ol are in the order

Which of the following halide shows highest reactivity towards $$\mathrm{S}_{\mathrm{N}} 1$$ reaction?

Which of the following on heating gives an ether as major products?

(P) $$\mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_2 \mathrm{Br}+\mathrm{CH}_3 \mathrm{ONa}$$

(Q) $$\mathrm{C}_6 \mathrm{H}_5 \mathrm{ONa}+\mathrm{CH}_3 \mathrm{Br}$$

(R) $$\left(\mathrm{CH}_3\right)_3 \mathrm{C}-\mathrm{Cl}+\mathrm{CH}_3 \mathrm{ONa}$$

(S) $$\mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}=\mathrm{CHCl}+\mathrm{CH}_3 \mathrm{ONa}$$

The final product for the reaction is

Which one of the following vitamins is not stored in adipose tissue?

Hypothyroidism is caused by the deficiency of

Which of the following monomers can undergo condensation polymerisation?

Which of the following polymer has strongest intermolecular forces of attraction?

Hinsberg's reagent is

$$\mathrm{C}_1-\mathrm{C}_4$$ glycosidic bond is not found in

The value of $${ }^{16} C_9+{ }^{16} C_{10}-{ }^{16} C_6-{ }^{16} C_7$$ is

Prolonged exposure of chloroform in humans may cause damage to liver. It is due to the formation of the following compound.

A die is thrown 10 times, the probability that an odd number will come up at least one time is

If the sum of $$n$$ terms of an AP is given by $$S_n=n^2+n$$, then the common difference of the $$\mathrm{AP}$$ is

Which of the following is not related to drug-enzyme interaction?

A food additive that acts as an antioxidant is

The probability of solving a problem by three persons $$A, B$$ and $$C$$ independently is $$\frac{1}{2}, \frac{1}{4}$$ and $$\frac{1}{3}$$ respectively. Then the probability of the problem is solved by any two of them is

If $$A=\{1,2,3,4,5,6\}$$, then the number of subsets of A which contain at least two elements is

If $$A$$ and $$B$$ are two events such that $$P(A)=\frac{1}{3}, P(B)=\frac{1}{2}$$ and $$P(A \cap B)=\frac{1}{6}$$, then $$P\left(A^{\prime} / B\right)$$ is

If $$n(A)=2$$ and total number of possible relations from Set A to set B is 1024, then $$n(B)$$ is

Events $$E_1$$ and $$E_2$$ from a partition of the sample space $$S$$. $$A$$ is any event such that $$P\left(E_1\right)=P\left(\dot{E}_2\right)=\frac{1}{2}, P\left(E_2 / A\right)=\frac{1}{2}$$ and $$P\left(A / E_2\right)=\frac{2}{3}$$, then $$P\left(E_1 / A\right)$$ is

If $$A, B, C$$ are three mutually exclusive and exhaustive events of an experiment such that $$P(A)=2 P(B)=3 P(C)$$, then $$P(B)$$ is equal to

If a relation $$R$$ on the set $$\{1,2,3\}$$ be defined by $$R=\{(1,1)\}$$, then $$R$$ is

If $$A=\{a, b, c\}$$, then the number of binary operations on $$A$$ is

The two vector $$\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$$ and $$\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$$ represent the two sides $$\overline{A B}$$ and $$\overline{A C}$$ respectively of a $$\triangle A B C$$. The length of the median through $$A$$ is

The number of terms in the expansion of $$(x+y+z)^{10}$$ is

If $$|\mathbf{a}+\mathbf{b}|^2+|\mathbf{a} \cdot \mathbf{b}|^2=144|\mathbf{a}|=6$$, then $$|\mathbf{b}|$$ is equal to

If $$\mathbf{a}$$ and $$\mathbf{b}$$ are unit vectors and $$\theta$$ is the angle between $$\mathbf{a}$$ and $$\mathbf{b}$$, then $$\sin \frac{\theta}{2}$$ is equal to

If a line makes an angle of with each of $$X$$ and $$Y$$-axis, then the acute angle made by Z-axis is

If the vectors $$2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$$ and $$\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$$ are coplanar, then the value of $$\lambda$$ is

The sine of the angle between the straight line $$\frac{x-2}{3}=\frac{3-y}{-4}=\frac{z-4}{5}$$ and the plane $$2 x-2 y+z=5$$ is

If $$A=\left(\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right)$$ then $$A^4$$ is equal to

The point $$(1,-3,4)$$ lies in the octant

The distance of the point $$(1,2,-4)$$ from the line $$\frac{x-3}{2}=\frac{y-3}{3}=\frac{z+5}{6}$$ is

If $$f(x)=\left|\begin{array}{ccc}x^3-x & a+x & b+x \\ x-a & x^2-x & c+x \\ x-b & x-c & 0\end{array}\right|$$, then

If $$\left(\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right) A=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$$ then the matrix $$a$$ is

If $$A$$ and $$B$$ are square matrices of same order and $$B$$ is a skew symmetric matrix, then $$A^{\prime} B A$$ is

The standard deviation of the data 6, 7, 8, 9, 10 is

If $$a_1 a_2 a_3 \ldots a_9$$ are in AP, then the value of $$\left|\begin{array}{lll}a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9\end{array}\right|$$ is

If $$A$$ is a square matrix of order 3 and $$|A|=5$$, then $$\mid A$$ adj. $$A \mid$$ is

If $$P(n): 2^n< n !$$ Then the smallest positive integer for which $$P(n)$$ is true if

Corner points of the feasible region determined by the system of linear constraints are $$(0,3),(1,1)$$ and $$(3,0)$$. Let $$z=p x=q y$$, where, $$p, q>0$$. Condition on $$p$$ and $$q$$, so that the minimum of $$z$$ occurs at $$(3,0)$$ and $$(1,1)$$ is

The feasible region of an LPP is shown in the figure. If $$z=11 x+7 y$$, then the maximum value of $$Z$$ occurs at

The negation of the statement "For all real numbers $$x$$ and $$y, x+y=y+x^{\prime \prime}$$ is

The value of $$\sin ^2 51^{\circ}+\sin ^2 39^{\circ}$$ is

If $$z=x+i y$$, then the equation $$|z+1|=|z-1|$$ represents

If $$\tan A+\cot A=2$$, then the value of $$\tan ^4 A+\cot ^4 A=$$

The area of the region bounded by the curve $$y^2=8 x$$ and the line $$y=2 x$$ is

If $$f(x)=\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$$, then $$f^{\prime}(\sqrt{3})$$ is

The domain of the function defined by $$f(x)=\cos ^{-1} \sqrt{x-1}$$ is

The area of the region bounded by the line $$y=2 x+1, X$$-axis and the ordinates $$x=-1$$ and $$x=1$$ is

The value of $$\cos \left(\sin ^{-1} \frac{\pi}{3}+\cos ^{-1} \frac{\pi}{3}\right)$$ is Does not exist

The right hand and left hand limit of the function are respectively.

$$f(x)=\left\{\begin{array}{cc} \frac{e^{1 / x}-1}{e^{1 / x}+1}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0 \end{array}\right.$$

Let $$f:[2, \infty) \rightarrow R$$ be the function defined $$f(x)=x^2-4 x+5$$, then the ranges of $$f$$ is

$$\lim _\limits{x \rightarrow 0}\left(\frac{\tan x}{\sqrt{2 x+4}-2}\right) \text { is equal to }$$

If $$f(x)=\left\{\begin{array}{cc}\frac{1-\cos K x}{x \sin x}, & \text { if } x \neq 0 \\ \frac{1}{2}, & \text { if } x=0\end{array}\right.$$ is continuous at $$x=0$$, then the value of $$K$$ is

The value of $$\int \frac{1+x^4}{1+x^6} d x$$ is

The value of $$\int e^{\sin x} \sin 2 x d x$$ is

The order of the differential equation obtained by eliminating arbitrary constants in the family of curves $$c_1 y=\left(c_2+c_3\right) e^{x+c_4}$$ is

If $$\int \frac{3 x+1}{(x-1)(x-2)(x-3)} d x A \log |x-1| B \log |x-2|+C \log |x-3|+C$$, then the values of $$A, B$$ and $$C$$ are respectively

If $$2^x+2^y=2^{x+y}$$, then $$\frac{d y}{d x}$$ is

The general solution of the differential equation $$x^2 d y-2 x y d x=x^4 \cos x d x$$ is

If $$y=2 x^{n+1}+\frac{3}{x^n}$$, then $$x^2 \frac{d^{2 y}}{d x^2}$$ is

If $$(x e)^y=e^y$$, then $$\frac{d y}{d x}$$ is

The value of $$\int_{-\pi / 2}^{\pi / 2} \frac{\cos x}{1+e^x} d x$$ is

If the curves $$2 x=y^2$$ and $$2 x y=K$$ intersect perpendicularly, then the value of $$K^2$$ is

The maximum value of $$\frac{\log _e x}{x}$$, if $$x>0$$ is

The curve passing through the point $$(1,2)$$ given that the slope of the tangent at any point $$(x, y)$$ is $$\frac{3 x}{y}$$ represents

Find the value of $$\int_0^1 \frac{\log (1+x)}{1+x^2} d x$$ is

The value of $$\int_{-1 / 2}^{1 / 2} \cos ^{-1} x d x$$ is

The two lines $$l x+m y=n$$ and $$l^{\prime} x+m^{\prime} y=n^{\prime}$$ are perpendicular if

If the parabola $$x^2=4$$ ay passes through the point $$(2,1)$$, then the length of the latus rectum is