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Solution
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Q.1 Correct
Q.1 In-correct
Q.1 Unattempt
For any positive integer n, define

fn:(0,∞)→R as

fn=∑j=1ntan−1(11+(x+j)(x+j−1))

for all x∈(0, ∞). (Here, the inverse trigonometric function tan−1 x assumes values in (−π2,π2)). Then, which of the following statement(s) is (are) TRUE?
For any positive integer n, define

fn:(0,∞)→R as

fn=∑j=1ntan−1(11+(x+j)(x+j−1))

for all x∈(0, ∞). (Here, the inverse trigonometric function tan−1 x assumes values in (−π2,π2)). Then, which of the following statement(s) is (are) TRUE?
Q.2 Correct
Q.2 In-correct
Q.2 Unattempt
Let T be the line passing through the points P(−2, 7) and Q(2, −5). Let F1 be the set of al pairs of circles (S1, S2) such that T is tangent to S1 at P and tangent to S2 at Q, and also such that S1 and S2 touch each other at a point, say M. Let E1 be the set representing the locus of M as the pair (S1, S2) varies in F1. Let the set of all straight line segments joining a pair of distinct points of E1 and passing through the point R(1, 1) be F2. Let E2 be the set of the mid-points of the line segments in the set F2. Then, which of the following statement(s) is (are) TRUE?
Let T be the line passing through the points P(−2, 7) and Q(2, −5). Let F1 be the set of al pairs of circles (S1, S2) such that T is tangent to S1 at P and tangent to S2 at Q, and also such that S1 and S2 touch each other at a point, say M. Let E1 be the set representing the locus of M as the pair (S1, S2) varies in F1. Let the set of all straight line segments joining a pair of distinct points of E1 and passing through the point R(1, 1) be F2. Let E2 be the set of the mid-points of the line segments in the set F2. Then, which of the following statement(s) is (are) TRUE?
Q.3 Correct
Q.3 In-correct
Q.3 Unattempt
Let S be the set of all column matrices [b1b2b3] such that b1,b2,b3∈R and the system of equations (in real variables)

−x+2y+5z=b12x−4y+3z=b2x−2y+2z=b3

has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each [b1b2b3]∈S?
Let S be the set of all column matrices [b1b2b3] such that b1,b2,b3∈R and the system of equations (in real variables)

−x+2y+5z=b12x−4y+3z=b2x−2y+2z=b3

has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each [b1b2b3]∈S?
Q.4 Correct
Q.4 In-correct
Q.4 Unattempt
Consider two straight lines, each of which is tangent to both the circle x2 + y2 = (1/2) and the parabola y2 = 4x. Let these lines intersect at the point Q. Consider the ellipse whose centre is at the origin O(0, 0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is 2, then which of the following statement(s) is (are) TRUE?
Consider two straight lines, each of which is tangent to both the circle x2 + y2 = (1/2) and the parabola y2 = 4x. Let these lines intersect at the point Q. Consider the ellipse whose centre is at the origin O(0, 0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is 2, then which of the following statement(s) is (are) TRUE?
Q.5 Correct
Q.5 In-correct
Q.5 Unattempt
Let s, t, r be non-zero complex numbers and L be the set of solutions z=x+iy(x,y∈R,i=−1) of the equation sz+tz―+r=0 where z― = x − iy. Then, which of the following statement(s) is(are) TRUE?
Let s, t, r be non-zero complex numbers and L be the set of solutions z=x+iy(x,y∈R,i=−1) of the equation sz+tz―+r=0 where z― = x − iy. Then, which of the following statement(s) is(are) TRUE?
Q.6 Correct
Q.6 In-correct
Q.6 Unattempt
Let f : (0, π) → R be a twice differentiable function such that limt→x⁡f(x)sin⁡t−f(t)sin⁡xt−x=sin2x for all x∈ (0, π).

If f(π6)=−π12, then which of the following statement(s) is (are) TRUE?
Let f : (0, π) → R be a twice differentiable function such that limt→x⁡f(x)sin⁡t−f(t)sin⁡xt−x=sin2x for all x∈ (0, π).

If f(π6)=−π12, then which of the following statement(s) is (are) TRUE?
Q.7 Correct
Q.7 In-correct
Q.7 Unattempt
The value of the integral

∫01/213((x+1)2(1−x)6)1/4dx is ........
The value of the integral

∫01/213((x+1)2(1−x)6)1/4dx is ........
Q.8 Correct
Q.8 In-correct
Q.8 Unattempt
Let P be a matrix of order 3 × 3 such that all the entries in P are from the set {−1, 0, 1}. Then, the maximum possible value of the determinant of P is ............ .
Let P be a matrix of order 3 × 3 such that all the entries in P are from the set {−1, 0, 1}. Then, the maximum possible value of the determinant of P is ............ .
Q.9 Correct
Q.9 In-correct
Q.9 Unattempt
Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If α is the number of one-one functions from X to Y and β is the number of onto functions from Y to X, then the value of 15!(β−α) is ..................
Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If α is the number of one-one functions from X to Y and β is the number of onto functions from Y to X, then the value of 15!(β−α) is ..................
Q.10 Correct
Q.10 In-correct
Q.10 Unattempt
Let f : R → R be a differentiable function with f(0) = 0. If y = f(x) satisfies the differential equation dydx=(2+5y)(5y−2), then the value of limn→−∞⁡f(x) is ...........
Let f : R → R be a differentiable function with f(0) = 0. If y = f(x) satisfies the differential equation dydx=(2+5y)(5y−2), then the value of limn→−∞⁡f(x) is ...........
Q.11 Correct
Q.11 In-correct
Q.11 Unattempt
Let f : R → R be a differentiable function with f(0) = 1 and satisfying the equation f(x + y) = f(x) f'(y) + f'(x) f(y) for all x, y∈ R.

Then, the value of loge(f(4)) is ...........
Let f : R → R be a differentiable function with f(0) = 1 and satisfying the equation f(x + y) = f(x) f'(y) + f'(x) f(y) for all x, y∈ R.

Then, the value of loge(f(4)) is ...........
Q.12 Correct
Q.12 In-correct
Q.12 Unattempt
Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicular to the plane x + y = 3 and the mid-point of PQ lies in the plane x + y = 3) lies on the Z-axis. Let the distance of P from the X-axis be 5. If R is the image of P in the XY-plane, then the length of PR is ...............
Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicular to the plane x + y = 3 and the mid-point of PQ lies in the plane x + y = 3) lies on the Z-axis. Let the distance of P from the X-axis be 5. If R is the image of P in the XY-plane, then the length of PR is ...............
Q.13 Correct
Q.13 In-correct
Q.13 Unattempt
Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the X-axis, Y-axis and Z-axis, respectively, where O(0, 0, 0) is the origin. Let S(12,12,12) be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If p = SP, q = SQ, r = SR and t = ST, then the value of |(p × q) × (r × t)| is ............
Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the X-axis, Y-axis and Z-axis, respectively, where O(0, 0, 0) is the origin. Let S(12,12,12) be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If p = SP, q = SQ, r = SR and t = ST, then the value of |(p × q) × (r × t)| is ............
Q.14 Correct
Q.14 In-correct
Q.14 Unattempt
Let X=(10C1)2+2(10C2)2+3(10C3)2+...+10(10C10)2,

where 10Cr, r ∈{1, 2, ..., 10} denote binomial coefficients. Then, the value of 11430X is ..........
Let X=(10C1)2+2(10C2)2+3(10C3)2+...+10(10C10)2,

where 10Cr, r ∈{1, 2, ..., 10} denote binomial coefficients. Then, the value of 11430X is ..........
Q.15 Correct
Q.15 In-correct
Q.15 Unattempt
Let E1={x∈R:x≠1andxx−1>0} and


E2={x∈E1:sin−1(loge(xx−1))isarealnumber}

(Here, the inverse trigonometric function sin−1 x assumes values in [−π2,π2].).

Let f : E1 → R be the function defined by f(x) = loge(xx−1) and g : E2 → R be the function defined by g(x) = sin−1(loge(xx−1)).


The correct option is :
Let E1={x∈R:x≠1andxx−1>0} and


E2={x∈E1:sin−1(loge(xx−1))isarealnumber}

(Here, the inverse trigonometric function sin−1 x assumes values in [−π2,π2].).

Let f : E1 → R be the function defined by f(x) = loge(xx−1) and g : E2 → R be the function defined by g(x) = sin−1(loge(xx−1)).


The correct option is :
Q.16 Correct
Q.16 In-correct
Q.16 Unattempt
In a high school, a committee has to be formed from a group of 6 boys M1, M2, M3, M4, M5, M6 and 5 girls G1, G2, G3, G4, G5.

(i) Let α1 be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.

(ii) Let α2 be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.

i) Let α3 be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.

(iv) Let α4 be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls such that both M1 and G1 are NOT in the committee together.


The correct option is
In a high school, a committee has to be formed from a group of 6 boys M1, M2, M3, M4, M5, M6 and 5 girls G1, G2, G3, G4, G5.

(i) Let α1 be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.

(ii) Let α2 be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.

i) Let α3 be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.

(iv) Let α4 be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls such that both M1 and G1 are NOT in the committee together.


The correct option is
Q.17 Correct
Q.17 In-correct
Q.17 Unattempt
Let H:x2a2−y2b2=1, where a > b > 0, be a hyperbola in the XY-plane whose conjugate axis LM subtends an angle of 60∘ at one of its vertices N. Let the area of the ΔLMN be 43.
 
  List - I   List - II
P. The length of the conjugate axis of H is 1. 8
Q. The eccentricity of H is 2. 43
R. The distance between the foci of H is 3. 23
S. The length of the latus rectum of H is 4. 4
       
       
Let H:x2a2−y2b2=1, where a > b > 0, be a hyperbola in the XY-plane whose conjugate axis LM subtends an angle of 60∘ at one of its vertices N. Let the area of the ΔLMN be 43.
 
  List - I   List - II
P. The length of the conjugate axis of H is 1. 8
Q. The eccentricity of H is 2. 43
R. The distance between the foci of H is 3. 23
S. The length of the latus rectum of H is 4. 4
       
       
Q.18 Correct
Q.18 In-correct
Q.18 Unattempt
Let f1:R→R,f2:(−π2,π2)→R,f3:(−1,eπ/2−2)→R and f4:R→R be functions defined by

(i) f1(x)=sin⁡(1−e−x2),

(ii) f2(x)={|sin⁡x|tan−1xifx≠0,where1ifx=0

the inverse trigonometric function tan−1x assumes values in (−π2,π2),

(iii) f3(x)=[sin⁡(loge(x+2))], where for t∈R,[t] denotes the greatest integer less than or equal to t,

(iv)
Let f1:R→R,f2:(−π2,π2)→R,f3:(−1,eπ/2−2)→R and f4:R→R be functions defined by

(i) f1(x)=sin⁡(1−e−x2),

(ii) f2(x)={|sin⁡x|tan−1xifx≠0,where1ifx=0

the inverse trigonometric function tan−1x assumes values in (−π2,π2),

(iii) f3(x)=[sin⁡(loge(x+2))], where for t∈R,[t] denotes the greatest integer less than or equal to t,

(iv)
Q.1 Correct
Q.1 In-correct
Q.1 Unattempt
Galena (an ore) is partially oxidized by passing air through it at high temperature. After some time, the passage of air is stopped, but the heating is continued in a closed furnace such that the contents undergo self-reduction. The weight (in kg) of Pb produced per kg of O2 consumed is ___________.

(Atomic weights in gmol−1:O=16,S=32,Pb=207)
Galena (an ore) is partially oxidized by passing air through it at high temperature. After some time, the passage of air is stopped, but the heating is continued in a closed furnace such that the contents undergo self-reduction. The weight (in kg) of Pb produced per kg of O2 consumed is ___________.

(Atomic weights in gmol−1:O=16,S=32,Pb=207)
Q.2 Correct
Q.2 In-correct
Q.2 Unattempt
The correct option(s) to distinguish nitrate salts of Mn2+ and Cu2+ taken separately is (are)
The correct option(s) to distinguish nitrate salts of Mn2+ and Cu2+ taken separately is (are)
Q.3 Correct
Q.3 In-correct
Q.3 Unattempt
The total number of compounds having at least one bridging oxo group among the molecules given below is _________.

N2O3,N2O5, P4O6,P4O7, H4P2O5,H5P3O10, H2S2O3,H2S2O5
The total number of compounds having at least one bridging oxo group among the molecules given below is _________.

N2O3,N2O5, P4O6,P4O7, H4P2O5,H5P3O10, H2S2O3,H2S2O5
Q.4 Correct
Q.4 In-correct
Q.4 Unattempt
The correct option(s) regarding the complex

[Co(en)(NH3)3(H2O)]3+ (en=H2NCH2CH2NH2) is (are)
The correct option(s) regarding the complex

[Co(en)(NH3)3(H2O)]3+ (en=H2NCH2CH2NH2) is (are)
Q.5 Correct
Q.5 In-correct
Q.5 Unattempt
To measure the quantity of MnCl2 dissolved in an aqueous solution, it was completely converted to KMnO4 using the reaction,

MnCl2+K2S2O8+H2O→KMnO4+H2SO4+HCl (equation not balanced).

Few drops of concentrated HCl were added to this solution and gently warmed. Further, oxalic acid (225 mg) was added in portions till the colour of the permanganate ion disappeared. The quantity of MnCl2 (in mg) present in the initial solution is ____________.

(Atomic weights in gmol−1:Mn=55,Cl=35.5 )
To measure the quantity of MnCl2 dissolved in an aqueous solution, it was completely converted to KMnO4 using the reaction,

MnCl2+K2S2O8+H2O→KMnO4+H2SO4+HCl (equation not balanced).

Few drops of concentrated HCl were added to this solution and gently warmed. Further, oxalic acid (225 mg) was added in portions till the colour of the permanganate ion disappeared. The quantity of MnCl2 (in mg) present in the initial solution is ____________.

(Atomic weights in gmol−1:Mn=55,Cl=35.5 )
Q.6 Correct
Q.6 In-correct
Q.6 Unattempt
The surface of copper gets tarnished by the formation of copper oxide. N2 gas was passed to prevent the oxide formation during heating of copper at 1250 K. However, the N2 gas contains 1 mole % of water vapor as impurity. The water vapor oxidises copper as per the reaction given below : 2Cu(s)+H2O(g)→Cu2O(s)+H2(g)

PH2 is the minimum partial pressure of H2 (in bar) needed to prevent the oxidation at 1250 K. The value of ln⁡(PH2) is ________.

Given: total pressure =1 bar, R (universal gas constant ) = 8JK−1mol−1, ln⁡(10)=2.3. Cu(s) and Cu2O(s) are naturally immiscible.

At 1250
The surface of copper gets tarnished by the formation of copper oxide. N2 gas was passed to prevent the oxide formation during heating of copper at 1250 K. However, the N2 gas contains 1 mole % of water vapor as impurity. The water vapor oxidises copper as per the reaction given below : 2Cu(s)+H2O(g)→Cu2O(s)+H2(g)

PH2 is the minimum partial pressure of H2 (in bar) needed to prevent the oxidation at 1250 K. The value of ln⁡(PH2) is ________.

Given: total pressure =1 bar, R (universal gas constant ) = 8JK−1mol−1, ln⁡(10)=2.3. Cu(s) and Cu2O(s) are naturally immiscible.

At 1250
Q.7 Correct
Q.7 In-correct
Q.7 Unattempt
The Fischer presentation of D-glucose is given below.


The correct structure(s) of β-L-glucopyranose is (are)
The Fischer presentation of D-glucose is given below.


The correct structure(s) of β-L-glucopyranose is (are)
Q.8 Correct
Q.8 In-correct
Q.8 Unattempt
In the following reaction sequence, the amount of D (in g) formed from 10 moles of acetophenone is ___________.

(Atomic weights in gmol−1:H=1,C=12, N=14,O=16, Br=80.. The yield (%) corresponding to the product in each step is given in the parenthesis)

In the following reaction sequence, the amount of D (in g) formed from 10 moles of acetophenone is ___________.

(Atomic weights in gmol−1:H=1,C=12, N=14,O=16, Br=80.. The yield (%) corresponding to the product in each step is given in the parenthesis)
Q.9 Correct
Q.9 In-correct
Q.9 Unattempt
Aniline reacts with mixed acid (conc. HNO3 and conc. H2SO4) at 288 K to give P (51%), Q (47%) and R (2%). The major product(s) of the following reaction sequence is (are)

Aniline reacts with mixed acid (conc. HNO3 and conc. H2SO4) at 288 K to give P (51%), Q (47%) and R (2%). The major product(s) of the following reaction sequence is (are)
Q.10 Correct
Q.10 In-correct
Q.10 Unattempt
For the given compound X, the total number of optically active stereoisomers is ____________.



For the given compound X, the total number of optically active stereoisomers is ____________.
Q.11 Correct
Q.11 In-correct
Q.11 Unattempt
For a first order reaction A(g)→2B(g)+C(g) at constant volume and 300K, the total pressure at the beginning (t=0) and at time t are P0 and P1, respectively. Initially, only A is present with concentration [A]0, and t1/3 is the time required for the partial pressure of A to reach 1/3rd of its initial value. The correct option(s) is (are) (Assume that all these gases behave as ideal gases)
For a first order reaction A(g)→2B(g)+C(g) at constant volume and 300K, the total pressure at the beginning (t=0) and at time t are P0 and P1, respectively. Initially, only A is present with concentration [A]0, and t1/3 is the time required for the partial pressure of A to reach 1/3rd of its initial value. The correct option(s) is (are) (Assume that all these gases behave as ideal gases)
Q.12 Correct
Q.12 In-correct
Q.12 Unattempt
Match each set of hybrid orbitals from LIST - A with complex(es) given in LIST - B

  List - A     List - B
P. dsp2   1. [FeF6]4-
Q. sp3   2. [Ti(H2O)3Cl3]
R. sp3d2   3. [Cr(NH3)6]3+
S. d2sp3   4. [FeCl4]2-
      5. Ni(CO)4
      6. [Ni(CN)4]2-


The correct option is
Match each set of hybrid orbitals from LIST - A with complex(es) given in LIST - B

  List - A     List - B
P. dsp2   1. [FeF6]4-
Q. sp3   2. [Ti(H2O)3Cl3]
R. sp3d2   3. [Cr(NH3)6]3+
S. d2sp3   4. [FeCl4]2-
      5. Ni(CO)4
      6. [Ni(CN)4]2-


The correct option is
Q.13 Correct
Q.13 In-correct
Q.13 Unattempt
Consider the following reversible reaction, A(g)+B(g)→AB(g).

The activation energy of the backward reaction exceeds that of the forward reaction by 2RT (in Jmol−1). If the pre-exponential factor of the forward reaction is 4 times that of the reverse reaction, the absolute value of ΔG∘ (in Jmol−1 ) for the reaction at 300 K is ____________.

(Given; ln⁡(2)=0.7,RT=2500 Jmol−1 at 300 K and G is the Gibbs energy)
Consider the following reversible reaction, A(g)+B(g)→AB(g).

The activation energy of the backward reaction exceeds that of the forward reaction by 2RT (in Jmol−1). If the pre-exponential factor of the forward reaction is 4 times that of the reverse reaction, the absolute value of ΔG∘ (in Jmol−1 ) for the reaction at 300 K is ____________.

(Given; ln⁡(2)=0.7,RT=2500 Jmol−1 at 300 K and G is the Gibbs energy)
Q.14 Correct
Q.14 In-correct
Q.14 Unattempt
Consider an electrochemical cell:

A(s)|An+(aq,2M)|B2n+(aq,1M)|B(s).

The value of ΔH∘ for the cell reaction is twice that of ΔG∘ at 300 K. If the emf of the cell is zero, the ΔS∘ (in JK−1mol−1) of the cell reaction per mole of B formed at 300 K is ___________.

(Given: ln⁡(2)=0.7,R (universal gas constant) =8.3JK−1mol−1. H,S and G are enthalpy, entropy and Gibbs energy, respectively.)
Consider an electrochemical cell:

A(s)|An+(aq,2M)|B2n+(aq,1M)|B(s).

The value of ΔH∘ for the cell reaction is twice that of ΔG∘ at 300 K. If the emf of the cell is zero, the ΔS∘ (in JK−1mol−1) of the cell reaction per mole of B formed at 300 K is ___________.

(Given: ln⁡(2)=0.7,R (universal gas constant) =8.3JK−1mol−1. H,S and G are enthalpy, entropy and Gibbs energy, respectively.)
Q.15 Correct
Q.15 In-correct
Q.15 Unattempt
Dilution processes of different aqueous solutions, with water, are given in LIST - I. The effects of dilution of the solutions on [H+] are given in LIST - II

(Note: Degree of dissociation (a) of weak acid and weak base is <<1; degree of hydrolysis of salt <<1; [H+] represents the concentration of H+ ions)

  LIST-I   LIST-II
P. (10 mL of 0.1 M NaOH + 20 mL of
0.1 M acetic acid) diluted to 60 mL		 1. the value of [H+] does not change
on dilution
Q. (20 mL of 0.1 M NaOH + 20 mL of
0.1 M acetic acid) diluted to 80 mL		 2. the value of [H+] changes to half
of its initial value on dilution
R. (20 mL of 0.1 M HCL + 20 mL of
0.1 M ammonia solution) diluted to
80 mL		 3. the value of [H+] changes to two
times of its initial value on dilution
S. 10 mL saturated solution of Ni(OH)2
in equilibrium with excess solid
Ni(OH)2 is diluted to 20 mL (solid
Ni(OH)2 is still present after dilution).		 4. the value of [H+] changes to 1/root2
times of its initial value on dilution
    5. the value of [H+] changes to root2
times of its initial value on dilution


Match each process given in LIST-I with one or more effect(s) in LIST-II. The correct option is
Dilution processes of different aqueous solutions, with water, are given in LIST - I. The effects of dilution of the solutions on [H+] are given in LIST - II

(Note: Degree of dissociation (a) of weak acid and weak base is <<1; degree of hydrolysis of salt <<1; [H+] represents the concentration of H+ ions)

  LIST-I   LIST-II
P. (10 mL of 0.1 M NaOH + 20 mL of
0.1 M acetic acid) diluted to 60 mL		 1. the value of [H+] does not change
on dilution
Q. (20 mL of 0.1 M NaOH + 20 mL of
0.1 M acetic acid) diluted to 80 mL		 2. the value of [H+] changes to half
of its initial value on dilution
R. (20 mL of 0.1 M HCL + 20 mL of
0.1 M ammonia solution) diluted to
80 mL		 3. the value of [H+] changes to two
times of its initial value on dilution
S. 10 mL saturated solution of Ni(OH)2
in equilibrium with excess solid
Ni(OH)2 is diluted to 20 mL (solid
Ni(OH)2 is still present after dilution).		 4. the value of [H+] changes to 1/root2
times of its initial value on dilution
    5. the value of [H+] changes to root2
times of its initial value on dilution


Match each process given in LIST-I with one or more effect(s) in LIST-II. The correct option is
Q.16 Correct
Q.16 In-correct
Q.16 Unattempt
The desired product X can be prepared by reacting the major product of the reactions in LIST-I with one or more appropriate reagents in LIST-II (given, order of migratory aptitude: aryl > alkyl > hydrogen)



The correct option is
The desired product X can be prepared by reacting the major product of the reactions in LIST-I with one or more appropriate reagents in LIST-II (given, order of migratory aptitude: aryl > alkyl > hydrogen)



The correct option is
Q.17 Correct
Q.17 In-correct
Q.17 Unattempt
LIST-I contains reactions and LIST-II contains major products.



Match the reaction in LIST-I with one or more products in LIST-II and choose the correct option.
LIST-I contains reactions and LIST-II contains major products.



Match the reaction in LIST-I with one or more products in LIST-II and choose the correct option.
Q.18 Correct
Q.18 In-correct
Q.18 Unattempt
For a reaction, A⇌P, the plots of [A] and [P] with time at temperature T1 and T2 are given below.



If T2>T1, the correct statement(s) is (are) (Assume ΔH∘ and ΔS∘ are independent of temperature and ratio of lnK at T1 to lnK at T2 is greater than T2T1. Here H, S,G and K are enthalpy, entropy, Gibbs energy and equilibrium constant, respectively.)
For a reaction, A⇌P, the plots of [A] and [P] with time at temperature T1 and T2 are given below.



If T2>T1, the correct statement(s) is (are) (Assume ΔH∘ and ΔS∘ are independent of temperature and ratio of lnK at T1 to lnK at T2 is greater than T2T1. Here H, S,G and K are enthalpy, entropy, Gibbs energy and equilibrium constant, respectively.)
Q.1 Correct
Q.1 In-correct
Q.1 Unattempt
A particle of mass m is initially at rest at the origin. It is subjected to a force and starts moving along the x-axis. Its kinetic energy K changes with time as dK/dt=γt, where γ is a positive constant of appropriate dimensions. Which of a positive constant of appropriate dimensions. Which of the following statement is (are) true?
A particle of mass m is initially at rest at the origin. It is subjected to a force and starts moving along the x-axis. Its kinetic energy K changes with time as dK/dt=γt, where γ is a positive constant of appropriate dimensions. Which of a positive constant of appropriate dimensions. Which of the following statement is (are) true?
Q.2 Correct
Q.2 In-correct
Q.2 Unattempt
Consider a thin square plate floating on a viscous liquid in a large tank. The height h of the liquid in the tank is much less than the width of the tank. The floating place is pulled horizontally with a constant velocity μ0. Which of the following statements is (are) true?
Consider a thin square plate floating on a viscous liquid in a large tank. The height h of the liquid in the tank is much less than the width of the tank. The floating place is pulled horizontally with a constant velocity μ0. Which of the following statements is (are) true?
Q.3 Correct
Q.3 In-correct
Q.3 Unattempt
An infinitely long thin non-conducting wire is parallel to the z-axis and carries a uniform line charge density λ. It pierces a thin non-conducting spherical shell of radius R in such a way that the arc PQ subtends an angle 120∘ at the center O of the spherical shell, as shown in the figure. The permittivity of free space is ∈0. Which of the following statement is (are) true?

An infinitely long thin non-conducting wire is parallel to the z-axis and carries a uniform line charge density λ. It pierces a thin non-conducting spherical shell of radius R in such a way that the arc PQ subtends an angle 120∘ at the center O of the spherical shell, as shown in the figure. The permittivity of free space is ∈0. Which of the following statement is (are) true?
Q.4 Correct
Q.4 In-correct
Q.4 Unattempt
In a radioactive decay chain, 90232Th nucleus decays to 82212Pb nucleus. Let Nα and Nβ be the number of α and β− particles, respectively, emitted in this decay process. Which of the following statements is (are) true?
In a radioactive decay chain, 90232Th nucleus decays to 82212Pb nucleus. Let Nα and Nβ be the number of α and β− particles, respectively, emitted in this decay process. Which of the following statements is (are) true?
Q.5 Correct
Q.5 In-correct
Q.5 Unattempt
A wire is bent in the shape of a right angled triangle and is placed in front of a concave mirror of focal length f, as shown in the figure. Which of the figures shown in the four options qualitatively represent(s) the shape of the image of the bent wire? (These figures are not to scale.)

A wire is bent in the shape of a right angled triangle and is placed in front of a concave mirror of focal length f, as shown in the figure. Which of the figures shown in the four options qualitatively represent(s) the shape of the image of the bent wire? (These figures are not to scale.)
Q.6 Correct
Q.6 In-correct
Q.6 Unattempt
In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency 500 Hz is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length 50.7 cm and 83.9 cm. Which of the following statements is (are) true?
In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency 500 Hz is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length 50.7 cm and 83.9 cm. Which of the following statements is (are) true?
Q.7 Correct
Q.7 In-correct
Q.7 Unattempt
A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass m=0.4 kg is at rest on this surface. An impulse of 1.0 Ns is applied to the block at time t=0 so that it starts moving along the x-axis with a velocity v(t)=v0e−t/τ, where v0 is a constant and τ=4s. The displacement of the block, in metres, at t=τ is ______________ Take e−1=0.37.
A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass m=0.4 kg is at rest on this surface. An impulse of 1.0 Ns is applied to the block at time t=0 so that it starts moving along the x-axis with a velocity v(t)=v0e−t/τ, where v0 is a constant and τ=4s. The displacement of the block, in metres, at t=τ is ______________ Take e−1=0.37.
Q.8 Correct
Q.8 In-correct
Q.8 Unattempt
A moving coil galvanometer has 50 turns and each turn has an area 2×10−4m2. The magnetic field produced by the magnet inside the galvanometer is 0.02 T. The torsional constant of the suspension wire is 10−4Nmrad−1. When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by 0.2 rad. The resistance of the coil of the galvanometer is 50Ω. This galvanometer is to be converted into an ammeter capable of measuring current in the range 0−1.0 A. For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ohms, is _____________.
A moving coil galvanometer has 50 turns and each turn has an area 2×10−4m2. The magnetic field produced by the magnet inside the galvanometer is 0.02 T. The torsional constant of the suspension wire is 10−4Nmrad−1. When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by 0.2 rad. The resistance of the coil of the galvanometer is 50Ω. This galvanometer is to be converted into an ammeter capable of measuring current in the range 0−1.0 A. For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ohms, is _____________.
Q.9 Correct
Q.9 In-correct
Q.9 Unattempt
A ball is projected from the ground at an angle of 45o with the horizontal surface. It reaches a maximum height of 120 m and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of 30o with the horizontal surface. The maximium height it reaches after the bounce, in metres, is ______________.
A ball is projected from the ground at an angle of 45o with the horizontal surface. It reaches a maximum height of 120 m and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of 30o with the horizontal surface. The maximium height it reaches after the bounce, in metres, is ______________.
Q.10 Correct
Q.10 In-correct
Q.10 Unattempt
A particle, of mass 10−3 kg and charge 1.0 C, is initially at rest. At time t=0, the particle comes under the influence of an electric field E→(t)=E0sin ωti^, where E0=1.0NC−1 and ω=103rads−1. Consider the effect of only the electrical force on the particle. Then the maximum speed, in ms−1, attained by the particle at subsequent times is _______________.
A particle, of mass 10−3 kg and charge 1.0 C, is initially at rest. At time t=0, the particle comes under the influence of an electric field E→(t)=E0sin ωti^, where E0=1.0NC−1 and ω=103rads−1. Consider the effect of only the electrical force on the particle. Then the maximum speed, in ms−1, attained by the particle at subsequent times is _______________.
Q.11 Correct
Q.11 In-correct
Q.11 Unattempt
One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is 100K and the universal gas constant R=8.0 Jmol−1K−1, the decrease in its internal energy, in Joule, is ____________.
One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is 100K and the universal gas constant R=8.0 Jmol−1K−1, the decrease in its internal energy, in Joule, is ____________.
Q.12 Correct
Q.12 In-correct
Q.12 Unattempt
A steel wire of diameter 0.5 mm and Young's modulus 2×1011Nm−2 carries a load of mass M. The length of the wire with the load is 1.0 m.A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count 1.0 mm , is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by 1.2 kg, the vernier scale division which coincides with a main scale division is _____________. Take g=10ms−2. and π=3.2.
A steel wire of diameter 0.5 mm and Young's modulus 2×1011Nm−2 carries a load of mass M. The length of the wire with the load is 1.0 m.A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count 1.0 mm , is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by 1.2 kg, the vernier scale division which coincides with a main scale division is _____________. Take g=10ms−2. and π=3.2.
Q.13 Correct
Q.13 In-correct
Q.13 Unattempt
Consider a hydrogen-like ionized atom with atomic number Z with a single electron. In the emission spectrum of this atom, the photon emitted in the n=2 to n=1 transition has energy 74.8eV higher than the photon emitted in the n=3 to n=2 transition. The ionization energy of the hydrogen atom is 13.6 eV. The value of Z is ____________.
Consider a hydrogen-like ionized atom with atomic number Z with a single electron. In the emission spectrum of this atom, the photon emitted in the n=2 to n=1 transition has energy 74.8eV higher than the photon emitted in the n=3 to n=2 transition. The ionization energy of the hydrogen atom is 13.6 eV. The value of Z is ____________.
Q.14 Correct
Q.14 In-correct
Q.14 Unattempt
In a photoelectric experiment a parallel beam of monochromatic light with power of 200 W is incident on a perfectly absorbing cathode of work function 6.25 ev. The frequency of light is just above the threshold frequency so that the photoelectrons are emitted with negligible kinetic energy. Assume that the photoelectron emission efficiency is 100%. A potential difference of 500 V is applied between the cathode and the anode. All the emitted electrons are incident normally on the anode and are absorbed. The anode experiences a force F=n×10−4 N due to the impact of the electrons. The value of n is ______________. Mass of the electron Me=9×10−31kg and 1.0eV=1.6×10−19J.
In a photoelectric experiment a parallel beam of monochromatic light with power of 200 W is incident on a perfectly absorbing cathode of work function 6.25 ev. The frequency of light is just above the threshold frequency so that the photoelectrons are emitted with negligible kinetic energy. Assume that the photoelectron emission efficiency is 100%. A potential difference of 500 V is applied between the cathode and the anode. All the emitted electrons are incident normally on the anode and are absorbed. The anode experiences a force F=n×10−4 N due to the impact of the electrons. The value of n is ______________. Mass of the electron Me=9×10−31kg and 1.0eV=1.6×10−19J.
Q.15 Correct
Q.15 In-correct
Q.15 Unattempt
The electric field E is measured at a point P(0,0,d) generated due to various charge distributions and the dependence of E on d is found to be different for different charge distributions. List-I contains different relations between E and d. List-II describes different electric charge distributions, along with their locations. Match the functions in List-I with the related charge distributions in List-II.

LIST - I LIST - II
P. E is independent of d 1. A point charge Q at the origin
Q. E∝1/d 2. A small dipole with point charges
Q at (0,0,l) and −Q at
(0,0,−l). Take 2l<<d
R. E∝1/d2 3. An infinite line charge coincident
with the x-axis, with uniform linear charge density λ
S. E∝1/d3 4. Two infinite wires carrying
uniform linear charge density
parallel to the x-axis. The one
along (y=0,z=l) has
a charge density +λ and the one
along (y=0,z=−l) has a
charge density Take
    5. Infinite plane charge coincident
with the xy-plane with uniform surface charge density
The electric field E is measured at a point P(0,0,d) generated due to various charge distributions and the dependence of E on d is found to be different for different charge distributions. List-I contains different relations between E and d. List-II describes different electric charge distributions, along with their locations. Match the functions in List-I with the related charge distributions in List-II.

LIST - I LIST - II
P. E is independent of d 1. A point charge Q at the origin
Q. E∝1/d 2. A small dipole with point charges
Q at (0,0,l) and −Q at
(0,0,−l). Take 2l<<d
R. E∝1/d2 3. An infinite line charge coincident
with the x-axis, with uniform linear charge density λ
S. E∝1/d3 4. Two infinite wires carrying
uniform linear charge density
parallel to the x-axis. The one
along (y=0,z=l) has
a charge density +λ and the one
along (y=0,z=−l) has a
charge density Take
    5. Infinite plane charge coincident
with the xy-plane with uniform surface charge density
Q.16 Correct
Q.16 In-correct
Q.16 Unattempt
A planet of mass M, has two natural satellites with masses m1 and m2. The radii of their circular orbits are R1 and R2 respectively, Ignore the gravitational force between the satellites. Define v1,L1,K1 and T1 to be , respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite 1; and v2,L2,K2, and T2 to be the corresponding quantities of satellite 2. Given m1/m2=2 and R1/R2=1/4, match the ratios in List-I to the numbers in List-II.

LIST - I LIST - II
P. v1/v2 1. 1/8
Q. L1/L2 2. 1
R. K1/K2 3. 2
S. T1/T2 4. 8
A planet of mass M, has two natural satellites with masses m1 and m2. The radii of their circular orbits are R1 and R2 respectively, Ignore the gravitational force between the satellites. Define v1,L1,K1 and T1 to be , respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite 1; and v2,L2,K2, and T2 to be the corresponding quantities of satellite 2. Given m1/m2=2 and R1/R2=1/4, match the ratios in List-I to the numbers in List-II.

LIST - I LIST - II
P. v1/v2 1. 1/8
Q. L1/L2 2. 1
R. K1/K2 3. 2
S. T1/T2 4. 8
Q.17 Correct
Q.17 In-correct
Q.17 Unattempt
One mole of a monatomic ideal gas undergoes four thermodynamic processes as shown schematically in the PV-diagram below. Among these four processes, one is isobaric, one is isochoric, one is isothermal and one is adiabatic. Match the processes mentioned in List-I with the corresponding statements in List-II.



LIST - I LIST - II
P. In process I 1. Work done by the gas is zero
Q. In process II 2. Temperature of the gas remains
unchanged
R. In process III 3. No heat is exchanged between
the gas and its surroundings
S. In process IV 4. Work done by the gas is 6P0V0
One mole of a monatomic ideal gas undergoes four thermodynamic processes as shown schematically in the PV-diagram below. Among these four processes, one is isobaric, one is isochoric, one is isothermal and one is adiabatic. Match the processes mentioned in List-I with the corresponding statements in List-II.



LIST - I LIST - II
P. In process I 1. Work done by the gas is zero
Q. In process II 2. Temperature of the gas remains
unchanged
R. In process III 3. No heat is exchanged between
the gas and its surroundings
S. In process IV 4. Work done by the gas is 6P0V0
Q.18 Correct
Q.18 In-correct
Q.18 Unattempt
In the List-I below, four different paths of a particle are given as functions of time. In these functions, α and β are positive constants of appropriate dimensions and α≠β In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned p→ is the linear momentum, L→ is the angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path.

LIST - I LIST - II
P. r→(t)=α ti^+βtj^ 1. p→
Q. r→(t)=αcosωti^+βsin⁡ωtj^ 2. L→
R. r→(t)=α(cos⁡ωti^+sin⁡ωtj^) 3. K
S. r→(t)=αti^+β2t2j^ 4. U
 
In the List-I below, four different paths of a particle are given as functions of time. In these functions, α and β are positive constants of appropriate dimensions and α≠β In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned p→ is the linear momentum, L→ is the angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path.

LIST - I LIST - II
P. r→(t)=α ti^+βtj^ 1. p→
Q. r→(t)=αcosωti^+βsin⁡ωtj^ 2. L→
R. r→(t)=α(cos⁡ωti^+sin⁡ωtj^) 3. K
S. r→(t)=αti^+β2t2j^ 4. U
 
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