JEE Advanced 2018 Question Paper with Solutions

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is $2.0\,N{m^{ - 1}}$ and the mass of the block is $2.0$ $kg.$ Ignore the mass of the spring. Initially the spring is an unstretched condition. Another block of mass $1.0$ $kg$ moving with a speed of $2.0$ $m{s^{ - 1}}$ collides elastically with the first block. The collision is such that the $2.0$ $kg$ block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is __________.

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $z = x/y.$ If the errors in $x,y$ and $z$ are $\Delta x,\Delta y$ and $\Delta z,$ respectively, then $ z ± Δ z = x ± Δ x y ± Δ y = x y ( 1 ± Δ x x ) ( 1 ± Δ y y ) − 1 . $ The series expansion for ${\left( {1 \pm {{\Delta y} \over y}} \right)^{ - 1}},$ to first power in $\Delta y/y.$ is $1 \pm \left( {\Delta y/y} \right).$ The relative errors in independent variables are always added. So the error in $z$ will be $ Δ z = z ( Δ x x + Δ y y ) . $ The above derivation makes the assumption that $\Delta x/x < < 1,$ $\Delta y/y < < 1.$ Therefore, the higher powers of these quantities are neglected. In an experiment the initial number of radioactive nuclei is $3000.$ It is found that $1000 \pm 40$ nuclei decayed in the first $1.0s.$ For $\left| x \right| < < 1.$ $\ln \left( {1 + x} \right) = x$ up to first power in $x.$ The error $\Delta \lambda ,$ in the determination of the decay constant $\lambda ,$ in ${s^{ - 1}},$ is

The potential energy of a particle of mass $m$ at a distance $r$ from a fixed point $O$ is given by $V\left( r \right) = k{r^2}/2,$ where $k$ is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius $R$ about the point $O$ . If $v$ is the speed of the particle and $L$ is the magnitude of its angular momentum about $O,$ which of the following statements is (are) true?

One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure (where $V$ is the volume and $T$ is the temperature). Which of the statements below is (are) true?

A uniform capillary tube of inner radius $r$ is dipped vertically into a beaker filled with water. The water rises to a height $h$ in the capillary tube above the water surface in the beaker. The surface tension of water is $\sigma .$ The angle of contact between water and the wall of the capillary tube is $\theta .$ Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

A reversible cyclic process for an ideal gas is shown below. Here, P , V , and T are pressure, volume and temperature, respectively. The thermodynamic parameters q , w , H and U are heat, work, enthalpy and internal energy, respectively. The correct option(s) is (are)

The solubility of a salt of weak acid $(AB)$ at $pH\,$ $3$ is $Y \times {10^{ - 3}}$ $mol\,{L^{ - 1}}.$ The value of $Y$ is ________________. (Given that the value of solubility product of $AB$ $\left( {{K_{sp}}} \right) = 2 \times {10^{ - 10}}$ and the value of ionization constant of $HB$ $\left( {{K_a}} \right) = 1 \times {10^{ - 8}}$ )

To measure the quantity of M n C l 2 dissolved in an aqueous solution, it was completely converted to K M n O 4 using the reaction, M n C l 2 + K 2 S 2 O 8 + H 2 O → K M n O 4 + H 2 S O 4 + H C l (equation not balanced). Few drops of concentrated H C l were added to this solution and gently warmed. Further, oxalic acid ( 225 m g ) was added in portions till the colour of the permanganate ion disappeared. The quantity of M n C l 2 (in mg) present in the initial solution is ____________. (Atomic weights in g m o l − 1 : M n = 55 , C l = 35.5 )

For a reaction, $A\,\,\rightleftharpoons\,\,P,$ the plots of $\left[ A \right]$ and $\left[ P \right]$ with time at temperature ${T_1}$ and ${T_2}$ are given below. If ${T_2} > {T_1},$ the correct statement(s) is (are) (Assume $\Delta {H^ \circ }$ and $\Delta {S^ \circ }$ are independent of temperature and ratio of $lnK$ at ${T_1}$ to $lnK$ at ${T_2}$ is greater than ${{{T_2}} \over {{T_1}}}.$ Here $H,$ $S,G$ and $K$ are enthalpy, entropy, Gibbs energy and equilibrium constant, respectively.)

For the given compound X , the total number of optically active stereoisomers is ____________.

The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the repetition of digits is allowed, is .................

Let S be the circle in the XY-plane defined the equation x 2 + y 2 = 4. Let E 1 E 2 and F 1 F 2 be the chords of S passing through the point P 0 (1, 1) and parallel to the X-axis and the Y-axis, respectively. Let G 1 G 2 be the chord of S passing through P 0 and having slope $-$ 1. Let the tangents to S at E 1 and E 2 meet at E 3 , then tangents to S at F 1 and F 2 meet at F 3 , and the tangents to S at G 1 and G 2 meet at G 3 . Then, the points E 3 , F 3 and G 3 lie on the curve

In a $\Delta$ PQR = 30 $^\circ$ and the sides PQ and QR have lengths 10 $\sqrt 3$ and 10, respectively. Then, which of the following statement(s) is(are) TRUE?

For a non-zero complex number z, let arg(z) denote the principal argument with $-$ $\pi$ < arg(z) $\le$ $\pi$ . Then, which of the following statement(s) is (are) FALSE?

Let a, b, c three non-zero real numbers such that the equation $\sqrt 3 a\cos x + 2b\sin x = c,x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right]$ , has two distinct real roots $\alpha$ and $\beta$ with $\alpha + \beta = {\pi \over 3}$ . Then, the value of ${b \over a}$ is ............

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, [ E ] and [ B ] stand for dimensions of electric and magnetic fields respectively, while [ ε 0 ] and [ μ 0 ] stand for dimensions of the permittivity and permeability of free space respectively. [ L ] and [ T ] are dimensions of length and time respectively. All the quantities are given in S I units. The relation between [ ε 0 ] and [ μ 0 ] is

Sunlight of intensity $1.3$ $kW\,{m^{ - 2}}$ is incident normally on a thin convex lens of focal length $20$ cm. Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light, in $kW\,{m^{ - 2}},$ at a distance $22$ cm from the lens on the other side is ___________.

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $[E]$ and $[B]$ stand for dimensions of electric and magnetic fields respectively, while $\left[ {{\varepsilon _0}} \right]$ and $\left[ {{\mu _0}} \right]$ stand for dimensions of the permittivity and permeability of free space respectively. $\left[ L \right]$ and $\left[ T \right]$ are dimensions of length and time respectively. All the quantities are given in $SI$ units. The relation between $[E]$ and $[B]$ is

In the $xy$ -plane, the region $y > 0$ has a uniform magnetic field ${B_1}\widehat k$ and the region $y < 0$ has another uniform magnetic field ${B_2}\widehat k.$ A positively charged particle is projected from the origin along the positive $y$ -axis with speed ${v_0} = \pi \,m{s^{ - 1}}$ at $t=0,$ as shown in the figure. Neglect gravity in this problem. Let $t=T$ be the time when the particle crosses the $x$ -axis from below for the first time. If ${B_2} = 4{B_1},$ the average speed of the particle, in $m{s^{ - 1}},$ along the $x$ -axis in the time interval $T$ is ___________.

Three identical capacitors ${C_1},{C_2}$ and ${C_3}$ have a capacitance of $1.0\,\mu F$ each and they are unchanged initially. They are connected in a circuit as shown in the figure and ${C_1}$ is then filled completely with a dielectric material of relative permittivity ${\varepsilon _r}.$ The cell electromotive force $\left( {emf} \right)\,\,{V_0} = 8V.$ First the switch ${S_1}$ is closed while the switch ${S_2}$ is kept open. When the capacitor ${C_3}$ is fully charged, ${S_1}$ is opened and ${S_2}$ is closed simultaneously. When all the capacitors reach equilibrium, the charge on ${C_3}$ is found to be $5\,\mu C.$ The value of ${\varepsilon _r} =$ _________________.

There are five students S 1 , S 2 , S 3 , S 4 and S 5 in a music class and for them there are five seats R 1 , R 2 , R 3 , R 4 and R 5 arranged in a row, where initially the seat R i is allotted to the student S i , i = 1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats. (There are two questions based on Paragraph "A", the question given below is one of them) For i = 1, 2, 3, 4, let T i denote the event that the students S i and S i+1 do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event ${T_1} \cap {T_2} \cap {T_3} \cap {T_4}$ is

There are five students S 1 , S 2 , S 3 , S 4 and S 5 in a music class and for them there are five seats R 1 , R 2 , R 3 , R 4 and R 5 arranged in a row, where initially the seat R i is allotted to the student S i , i = 1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats. (There are two questions based on Paragraph "A", the question given below is one of them) The probability that, on the examination day, the student S 1 gets the previously allotted seat R 1 , and NONE of the remaining students gets the seat previously allotted to him/her is

The value of ${({({\log _2}9)^2})^{{1 \over {{{\log }_2}({{\log }_2}9)}}}} \times {(\sqrt 7 )^{{1 \over {{{\log }_4}7}}}}$ is ....................

For every twice differentiable function $f:R \to [ - 2,2]$ with ${(f(0))^2} + {(f'(0))^2} = 85$ , which of the following statement(s) is(are) TRUE?

The number of real solutions of the equation sin − 1 ( ∑ i = 1 ∞ x i + 1 − x ∑ i = 1 ∞ ( x 2 ) i ) = π 2 − cos 1 ( ∑ i = 1 ∞ ( − x 2 ) i − ∑ i = 1 ∞ ( − x ) i ) lying in the interval $\left( { - {1 \over 2},{1 \over 2}} \right)$ is ........... . (Here, the inverse trigonometric functions sin $-$ 1 x and cos $-$ 1 x assume values in ${\left[ { - {\pi \over 2},{\pi \over 2}} \right]}$ and ${\left[ {0,\pi } \right]}$ , respectively.)

Based on the compounds of group $15$ elements, the correct statement(s) is (are)

Consider an ionic solid $MX$ with $NaCl$ structure. Construct a new structure $(Z)$ whose unit cell is constructed from the unit cell of $MX$ following the sequential instructions given below. Neglect the charge balance. $(i)\,\,\,\,\,$ Remove all the anions $(X)$ except the central one $(ii)\,\,\,\,$ Replace all the face centered cations $(M)$ by anions $(X)$ $(iii)\,\,$ Remove all the corner cations $(M)$ $(iv)\,\,\,\,$ Replace the central anion $(X)$ with cation $(M)$ The value of $\,\,\left( {{{number\,\,of\,\,anions} \over {number\,\,of\,\,cations}}} \right)\,\,$ in $Z$ is ___________.

In the following reaction sequence, the correct structure(s) of X is (are)

The compound Y is

The correct statement(s) regarding the binary transition metal carbonyl compounds is(are) (Atomic numbers : $Fe = 26,Ni = 28$ )