JEE MAINS FULL SYLLABUS PAPER-1 DEMO

If $$\vec{a}=2 \hat{i}+3 \hat{j}+6 \hat{k}$$ and $$\vec{b}=3 \hat{i}+4 \hat{j}$$ then

A student is experimenting with resonance tube apparatus in physics lab to find the speed of sound at room temperature. He got resonating lengths of air column as 17 cm and 51 cm, using tuning fork of frequency 512 Hz. Find the speed of sound at room temperature and specify , whether the side water reservoir was moved upward or downward to obtain the second resonance (51cm) ?

A small body is released from rest on a smooth inclined plane at $$t=0$$ Velocity versus time graph of the body as shown in the figure. The angle of inclination of the plane with horizontal is $$\left(g=10 \mathrm{~ms}^{-2}\right)$$

In a circular motion of a particle, the tangential acceleration of the particle is given by P. The radius of the circle is 4m. The particle was initially at rest. Time after which total acceleration of the particle makes an angle of $$45^{\circ}$$with the radial acceleration is

The potential energy$$\phi$$ in joule of a particle of mass 1 kg moving in $$x-y$$ plane obeys the law$$\phi=3 x+4 y$$.Here $$x$$ and $$y$$ are in meters. If the particle is at rest at (6m, 8m) at time 0, then the work done by conservative force on the particle from the initial position to the instant when it crosses the x-axis is

A particle of mass $$m_{0}$$ travelling at speed $$v_{0}$$ strikes a stationary particle of mass $$2 m_{0}$$ As a result, the particle of mass $$m_{0}$$ is deflected through $$45^{\circ}$$ and has a final speed of $$\frac{v_{0}}{\sqrt{2}}$$ Then the speed of the particle of mass $$2 m_{0}$$ after this collision is

A thin hollow sphere of mass m is completely filled with non viscous liquid of mass $$m$$ When the sphere roll-on horizontal ground such that centre moves with velocity $$v$$ kinetic energy of the system is equal to

Three identical particles each of mass M move along a common circular path of radius R under the mutual interaction of each other. The velocity of each particle is

Two simple harmonic motions are represented by the following equations $$y=40 \sin \omega t$$ and $$y_{2}=10(\sin \omega t+c \cos \omega t)$$ If their displacement amplitudes are equal, then the value of $$c$$ (in appropriate units) is

A bob of mass 10 kg is attached to a wire 0.3 m long. Its breaking stress is $$4.8 \times 10^{7} \mathrm{~N} / \mathrm{m}^{2}$$ Then area of cross-section of the wire is $$10^{-6} \mathrm{~m}^{2}$$ What is the maximum angular velocity with which it can be rotated in a horizontal circle ?

A body of density $$\rho$$ is dropped from rest from a height $$h$$ into a lake of density $$\sigma(\sigma>\rho)$$ The maximum depth the body sinks inside the liquid is (neglect viscous effect of liquid)

A string of length ‘ L’ is fixed at both ends. It is vibrating in its $$3{ }^{\text {rd }}$$ overtone with maximum amplitude ‘a’ The amplitude at a distance $$\frac{L}{3}$$ from one end is

A source is moving with constant speed $$v_{s}=20 \mathrm{~m} / \mathrm{s}$$ towards a stationary observer due east of the source. Wind is blowing at the speed of $$20 \mathrm{~m} / \mathrm{s}$$ at $$60^{\circ}$$ north of east. The source has frequency 500 Hz. Speed of sound $$=300 \mathrm{~m} / \mathrm{s}$$ The frequency registered by the observer is approximately

Switch S is closed at time $$t=0$$ Which one of the following statements is correct ?

A source of emf $$E=10 \mathrm{~V}$$ and having negligible internal resistance is connected to a variable resistance. The resistance varies as shown in figure. The total charge that has passed through the resistor R during the interval from $$t_{1}$$ to $$t_{2}$$ is

Six charges are placed at the vertices of a regular hexagon as shown in the figure. The electric field on the line passing through point O and perpendicular to the plane of the figure as a function of distance $$x$$ from point O is $$(x>>a)$$

The figure shows a graph of the current in a discharging circuit of a capacitor through a resistor of resistance $$10 \Omega$$

A capacitor of capacitance C is charged to a potential difference V from a cell and then disconnected from it. A charge$$+Q$$ is now given to its positive plate. The potential difference across the capacitor is now

Equal currents are flowing in three infinitely long wires along positive $$x, y$$ and $$z$$ directions. The magnetic field at a point $$(0,0,-a)$$ would be ( $$i=$$ current in each wire)

A uniformly charged ring of radius R is rotated about its axis with constant linear speed $$v$$ of each of its particles. The ratio of electric field to magnetic field at a point P on the axis of the ring at a distance $$x=R$$ from centre of ring is ($$c$$ is speed of light

Two parallel long straight conductors lie on a smooth plane surface. Two other parallel conductors rest on them at right angles so as to form a square of side $$a$$ A uniform magnetic filed B exists at right angles to the plane containing the conductors. Now conductors start moving outward with a constant velocity $$v_{0}$$ at $$t=0$$ Then induced current in the loop at any time$$t$$ is ($$\lambda$$ is resistance per unit length of the conductors)

When an alternating voltage of 220 V is applied across a device P, a current of 0.25 A flows through the circuit and it leads the applied voltage by an angle $$\pi / 2$$ radian. When the same voltage source is connected across another device Q, the same current is observed in the circuit but in phase with the applied voltage. What is the current when the same source is connected across a series combination of P and Q ?

Two identical glass $$\left(\mu_{g}=\frac{3}{2}\right)$$ equi convex lenses of focal length $$f$$ are kept in contact. The space between the two lenses is filled with water $$\left(\mu_{w}=\frac{4}{3}\right)$$ The focal length of the combination

A car is fitted with a convex mirror of focal length 20cm. A second car 2m broad and 1.6m high is 20m away from the first car. The position of the second car as seen from the mirror of the first car is approximately

Young’s double slit experiment is made in a liquid. The $$10^{\text {th }}$$ bright fringe in liquid lies where $$6^{\text {th }}$$ dark fringe lies in vaccum. The refractive index of the liquid is approximately

When a surface is illuminated with light of wavelength $$\lambda$$ the stopping potential is V. When the same surface is illuminated by light of wavelength $$2 \lambda$$ the stopping potential is $$\frac{V}{3}$$ Threshold wavelength for the metallic surface is

A stationary hydrogen atom of mass $$M$$ emits a photon corresponding to the first line of Lyman series. It $$R$$ is the Rydberg’s constant, the velocity that the atom acquires is

When a block of iron floats in mercury at $$0^{\circ} \mathrm{C}$$ a fraction $$K_{1}$$ of its volume is submerged, while at the temperature of $$60^{\circ} \mathrm{C}$$ a fraction $$K_{2}$$ is seen to be submerged. If the coefficient of volume expansion of iron is $$\gamma_{F e}$$ and that of mercury is $$\gamma_{H g}$$ then the ratio $$\frac{K_{1}}{K_{2}}$$ can be

One mole of a monoatomic ideal gas undergoes the process $$A \rightarrow B$$ in the given $$P-V$$ diagram. The molar heat capacity for this process is

A long horizontal gas filled tube of length 2L closed at both ends is rotated about a vertical axis through its centre normal to its length with an angular velocity $$\omega$$ If $$P_{0}$$ is the pressure at the centre, T is temperature in Kelvin scale, R is gas constant per unit mass, the pressure $$P$$ at the end $$B$$ will be

Which of the following compounds show geometrical isomerism
i)$$\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{Cl}_{2}\right]^{+}$$
ii) $$\left[\mathrm{Co}(\text { en })_{2} \mathrm{Cl}_{2}\right]^{+}$$

25ml of $$\mathrm{H}_{2} \mathrm{O}_{2}$$ on treatment with excess of acidified KI solution liberates iodine. Liberated iodine requires 40ml of $$0.4 \mathrm{~N} \mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}$$ solution. The volume strength of $$\mathrm{H}_{2} \mathrm{O}_{2}$$ solution is

The osmotic pressure of 0.1M solution of $$\mathrm{MgCl}_{2}$$ at 300K is 4.92atm. The percentage ionization of the salt is

The Gibb’s Energy for the decomposition of $$\mathrm{Al}_{2} \mathrm{O}_{3}$$ at $$500^{\circ} \mathrm{C}$$ is as follows $$\frac{2}{3} \mathrm{Al}_{2} \mathrm{O}_{3} \longrightarrow \frac{4}{3} \mathrm{Al}+\mathrm{O}_{2} ; \Delta_{r} G=+965 \mathrm{KJ} /$$ mole The potential difference needed for electrolytic reduction of $$\mathrm{Al}_{2} \mathrm{O}_{3}$$ at $$500^{\circ} \mathrm{C}$$ is at least

An impure sample of NaCl weighing 2 grams was dissolved in water and excess of $$\mathrm{AgNO}_{3}$$ solution was added. The weight of precipitate formed was 0.287g. The percentage purity of the sample is

The density of an ionic crystal is $$2.75 \mathrm{~g} \mathrm{~cm}^{-3}$$ and edge length of unit cell is 660 pm. The formula weight of ionic compound is 119. The number of unit cells present in 23.8 g of the ionic compound is

An organic compound has $$14 \%$$ of nitrogen. It is subjected to kjeldahl’s method and the ammonia evolved is absorbed in 100ml of normal sulphuric acid. About 400ml of water was added. And 20 ml of this new solution now required $$16 \mathrm{ml} \frac{\mathrm{N}}{5} \mathrm{NaOH}$$ for complete neutralization. Calculate the original weight of the organic compound taken

The number of monochloro derivatives that the compound ‘D’ can give

If $$H, D, H e$$ He atoms are travelling with same velocity, the ratio of their deBroglie wavelength is

For the decomposition of $$\mathrm{PCl}_{5}(g)$$ in a closed vessel, the degree of dissociation is $$x$$ at a total pressure P $$
P C l_{5}(\mathrm{~g}) \rightleftharpoons \mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g}) ; \mathrm{Kp}
$$ Which among the following relations is correct?

Observe the following statements
a) The almost identical radii of Zr and Hf is consequence of Lanthanide contraction
b) $$\mathrm{Eu}^{2+}$$ act as a strong reducing agent
c) An alloy of lanthanoid metal called misch metal is used in making bullets
d) Actinoid contraction is greater from element to element than lanthanide contraction
The correct statements are

. X and Y in this reaction is

In which of the following pair of compounds, the first compound have more dipolemoment than the second one

The pH of a sample of KOH solution is 12.6990. The weight of solute required to prepare 5 lit of the solution is

Which of the following fibre is of polyamides?

Which among the following are correct statements
a) Order of $$\mathrm{O}-\mathrm{O}$$ bond length is $$\mathrm{O}_{2}\left[\mathrm{AsF}_{6}\right]<\mathrm{O}_{2}N_{2}^{+}>N_{2}^{2-}$$
d) Ice has more density than water

The correct order of coagulation power of different ions is

A salt ‘ X ‘ on heating liberate one coloured gas and another coloureless gas. Both the gases are paramagnetic. The salt ‘ X ‘ may be

Compressibility factor for $$\mathrm{H}_{2}$$ behaving as real gas is

In which of the following arrangements the order is not according to the property indicated against it ?

The composition of a sample of Ion oxide is $$\mathrm{Fe}_{0.93} \mathrm{O}_{1.0}$$ The percentage of iron present in the form of $$\mathrm{Fe}^{3+}$$ is

$$93.75 \%$$ of a I order reaction is completed in 120 minutes. If initial concentration of the reactant is $$2 \mathrm{~mole} / \mathrm{lit}$$ the rate of reaction at the end of 120 min is

Which of the following is wrong statement

Which of the following is wrong

An organic compound (A) with molecular formula $$\mathrm{C}_{8} \mathrm{H}_{8} \mathrm{O}$$ forms an orange-red precipitate with $$2,4-D N P$$ and gives yellow precipitate on heating with iodine in presence of NaoH It neither reduces tollen’s or Fehling’s reagent nor does it decolourise bromine water or Bayer’s reagent. On drastic oxidation with chromic acid it gives carboxylic acid (B) having molecular formula $$\mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{2}$$ Then A and B are

An aldehyde, (P) which does not undergo self aldol condensation gives benzaldehyde and two moles of $$(\mathrm{Q})$$ on ozonolysis. Compound ( Q), on oxidation with silver ions gives oxalic acid. The structure of $$(\mathrm{P})$$ is given as

The product of the following I and II sequence are related as

Which of the following sequence of reagent is the good means to furnish the conversion ?$$\mathrm{R}-\mathrm{CH}_{2} \mathrm{OH} \longrightarrow \mathrm{R}-\mathrm{CH}_{2} \mathrm{NH}_{2}$$
(a) $$\mathrm{KMnO}_{4}, \mathrm{SOCl}_{2}, \mathrm{NH}_{3}, \Delta, \mathrm{NaOBr}$$
(b) $$\mathrm{SOCl}_{2}, \mathrm{NaCN}, \mathrm{H}_{2} / \mathrm{Ni}$$
(c) $$\mathrm{CrO}_{3}$$ in dilute acetone $$\mathrm{NH}_{3}, \mathrm{H}_{2}, \mathrm{Ni}$$
(d) $$\mathrm{Cu}, 300{ }^{0} \mathrm{C}, \mathrm{NH}_{3}, \mathrm{LiAlH}_{4}$$

Consider the following statements:
a) Keratin & Myosin are fibrous proteins
b) Deficiency of vitamin C causes disease bleeding gums (scurvy)
c) D-glucose and D- mannose are C-2 epimers
d) Fructose does not reduse fehling solution
Which of the given statements are correct

The correct order of stability of following carbanions

Let $$S$$ be the set of real numbers. For $$a, b \in S$$ relation $$R$$ is defined by $$a R b \Leftrightarrow|a-b|<1$$ then $$R$$ is

Given that matrix $$A=\left[\begin{array}{lll}x & 3 & 2 \\ 1 & y & 4 \\ 2 & 2 & z\end{array}\right]$$ If $$x y z=2013$$ and $$8 x+4 y+3 z=2012$$ then A (adj A) is equal to

In a $$\triangle P Q R$$ if $$3 \sin P+4 \cos Q=6$$ and $$4 \sin Q+3 \cos P=1$$ then angle ‘ R ‘ is

$$\theta \in[0,2 \pi]$$ and $$z_{1}, z_{2}, z_{3}$$ are three complex numbers such that they are collinear and $$(1+|\sin \theta|) z_{1}+(|\cos \theta|-1) z_{2}-\sqrt{2} z_{3}=0$$ If at least one of the complex numbers $$z_{1}, z_{2}, z_{3}$$ is non- zero then number of possible values of $$\theta$$ is

The number of real negative terms in the binomial expansion of $$(1+i x)^{4 n-2}, n \in N, x>0$$ is

Let $$f(x)=2 x^{n}+\lambda, \lambda \in$$ R and $$n \in N f(4)=133$$ and $$f(5)=255$$ then sum of all the positive integral divisors of $$(f(3)-f(2))$$ is

In a sequence of 21 terms the first 11 terms are in A.P. with common difference 2 and the last 11 terms are in G.P. with common ratio 2. If the middle term of the A.P. is equal to the middle term of the G.P., then the middle term of the entire sequence is

Value of where [ ] denotes step function is

If $$f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}, \forall x, y \in R$$ and $$f^{\prime}(0)=-1, f(0)=1$$ then $$f(2)=$$

The equation of the plane containing the line $$\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{4} \quad$$ and perpendicular to the plane $$x+2 y+z=12$$ is

If the lines $$\mathrm{x}=\mathrm{k}: \mathrm{k}=1,2,3 \ldots \ldots, n$$ meet the line $$\mathrm{y}=3 \mathrm{x}+4$$ at the points $$A_{k}\left(x_{k}, y_{k}\right): k=1,2,3 \ldots \ldots \ldots, n$$ then the ordinate of the point which is the mean position of the points $$A_{k}, k=1,2,3 \ldots, n$$ is

The chord of contact of the pair of tangents to the circle $$x^{2}+y^{2}=1$$ drawn from any point on the line $$2 x+y=4$$ always passes through the point

Total number of points of discontinuity of $$f(x)=[4+3 \sin x]$$ in $$(0, \pi)\{[$$.$$] denotes$$ greatest integer function $$\}$$

‘ A ‘ is a square matrix, if all elements in $$A^{n}$$ become zero, as $$n \rightarrow \infty$$ we say $$\lim _{n \rightarrow \infty} A^{n}=O$$ now $$A=\left(\begin{array}{ll}x & x \\ x & x\end{array}\right)(x>0)$$ then the complete set of values of $$x$$ such that $$\lim _{n \rightarrow \infty} A^{n}=O$$ is

The number of real common normals of $$y^{2}=4 x$$ and $$x^{2}=4 y$$ that can be drawn is equal to

A variable triangle ABC is inscribed in a circle of diameter $$x$$ units. At a particular instant the rate of change of side ‘a’ is $$x / 2$$ times the rate of change of the opposite angle $$A$$ then $$A=$$

The line $$3 x+2 y=24$$ meets $$x$$ -axis at $$A$$ and $$y$$ -axis at $$B$$ The perpendicular bisector of $$\overline{A B}$$ meets the line passing through $$(0,-1)$$ and parallel to x-axis at ‘C’ Then the area of $$\triangle A B C$$ is (in square units)

The area bounded by $$y=x e^{|\mathrm{x}|}$$ and the lines $$|\mathrm{x}|=1, \mathrm{y}=0$$ is ( in sq.units)

The solution of the differential equation when $$y^{1}(0)=1$$ and $$y(0)=0$$ is

A particle acted by constant forces $$4 \hat{i}+\hat{j}-3 \hat{k}$$ and $$3 \hat{i}+\hat{j}-\hat{k}$$ is displaced from the point $$\hat{i}+2 \hat{j}+3 \hat{k}$$ to the point $$5 \hat{i}+4 \hat{j}+\hat{k}$$. The total work done by the forces is

A point $$(a, b)$$ is called a good point if both ‘$$a$$’ and ‘$$b$$’ are integers. The number of good points on the curve $$\mathrm{xy}=225$$ are

The point ‘ B ‘ is the image of ‘ A ‘ in the line $$x+y+4=0$$ and ‘ C ‘ is the image of ‘ B ‘ in the line $$2 x-y+7=0$$ if $$A=(1,6)$$ then the circumcentre of the triangle ABC is

If $$n \in N \& n>1$$ then the digit in unit’s place of the number $$\sum_{k=0}^{100} \underline{k}+2^{2^{n}}$$

The number of non zero solutions of the system of equations $$x=\frac{2 z^{2}}{1+z^{2}}, y=\frac{2 x^{2}}{1+x^{2}}, z=\frac{2 y^{2}}{1+y^{2}}$$ is

The probability that $$\sin ^{-1}(\sin x)+\cos ^{-1}(\cos y)$$ is an integer where $$x, y \in\{1,2,3,4\}$$ is

The area bounded by the circles $$x^{2}+y^{2}=1 \& x^{2}+y^{2}=4$$ and the pair of lines $$2 \mathrm{x}^{2}-3 \mathrm{xy}-2 \mathrm{y}^{2}=0 \quad(\mathrm{y}>0)$$ is $$\qquad$$ ( in sq.units)

A random variable $$X$$ has its range $$\{0,1,2\}$$ and $$P(X=0)=3 c^{3}, P(X=1)=4 c-10 c^{2}$, $\mathrm{P}(\mathrm{X}=2)=5 \mathrm{c}-1$$ then $$\mathrm{P}(0<\mathrm{X}<3)=$ $\qquad$$

A tower ABCD stands on a level ground with foot A. At a point P on the ground, the portion $$\mathrm{AB}, \mathrm{AC}$$ and AD subtends angles $$\alpha, \beta$$ and $$\gamma$$ respectively. If $$\mathrm{AB}=\mathrm{a}, \mathrm{AC}=\mathrm{b}, \mathrm{AD}=\mathrm{c}, \mathrm{AP}=\mathrm{x}$$ and $$(\mathrm{a}+\mathrm{b}+\mathrm{c}) \mathrm{x}^{2}=$$

$$x=n \pi-\tan ^{-1} 3$$ is a solution of the equation $$12 \tan 2 x+\frac{\sqrt{10}}{\cos x}+1=0$$ for

Suppose a population ‘A’ has observations $$101,102, \ldots \ldots .200$$ and another population ‘B’ has observations $$151,152, \ldots \ldots . .250$$ If $$V_{A}$$ and $$V_{B}$$ represent variances of two populations respectively then $$\frac{V_{A}}{V_{B}}=$$