JEE Mains 2017 (Code-D)

Test Name: JEE Mains 2017 (Code-D)

Question 1:

It S is the set of distinct values of ‘b’ for which the following system of linear equations

x + y + z = 1

x + ay + z = 1

ax + by + z = 0

has no solution, then S is :

Option 1: an empty set
Option 2: an infinite set
Option 3: a finite set containing two or more elements
Option 4: a singleton
Correct Answer: 4

 

Solutions:The given system of equations can be written as

1111a1ab1. Determinant of the above matrix is as follows:

D=1111a1ab1=-a-12As the system of linear equations has no solution, therefore D = 0.

⇒-a-12=0⇒a=1For a = 1, the first two planes are co-incident to each other. So, x + y + z = 1 and ax + by + z = 0 It is given that S is the set of distinct values of ‘b’ for which the given system of linear equations has no solution. For no solution these two lines are parallel.

⇒1a=1b=11⇒a=1, b=1Therefore, set S has just one value i.e., it’s a singleton set. Hence the correct answer is option (D).

Question 2:

The following statement

(p → q) → [(~ p → q) → q ] is :

Option 1: a tautology
Option 2: equivalent to ~ p → q
Option 3: equivalent to p → ~ q
Option 4: a fallacy
Correct Answer: 1

 

Solutions:The truth table for the statement

p→q→~p→q→qis given below:

p ~p q p→q ~p→q ~p→q→q p→q→~p→q→q
T F T T T T T
T F F F T F T
F T T T T T T
F T F T F T T

So, the given statement is a tautology.

Hence, the correct answer is option A.

Question 3:

If 5(tan2x – cos2x) = 2cos2x + 9, then the value of cos4x is :

Option 1: -3   5
Option 2: 13
Option 3: 29
Option 4: -79
Correct Answer: 4

 

Solutions:

5tan2x-cos2x=2cos2x+9⇒5sin2xcos2x-cos2x=2cos2x+9      ∵tanx=sinxcosx⇒51-cos2x1+cos2x-1+cos2x2=2cos2x+9    ∵sin2x=1-cos2x2 and cos2x=1+cos2x2⇒521-cos2x-1+cos2x221+cos2x=2cos2x+9 ⇒51-cos22x-4cos2x=21+cos2x2cos2x+9⇒5-5cos22x-20cos2x=4cos22x+22cos2x+18⇒9cos22x+42cos2x+13=0 ⇒3cos2x+133cos2x+1=0⇒cos2x=-13 or cos2x=-133Now, cos4x=2cos22x-1=-79    ∵cos2x≠-133Hence, the correct answer is option D.

Question 4:

For three events A, B and C, P(Exactly one of A or B occurs) = P(Exactly one of B or C occurs) = P(Exactly one of C or A occurs) =

14and P (All the three events occur simultaneously) =

16. Then the probability that at least one of the events occurs, is :

Option 1: 732
Option 2: 716
Option 3: 764
Option 4: 316
Correct Answer: 2

 

Solutions:It is given that for three events A, B and C, P(Exactly one of A or B occurs) =

14

⇒PA+PB-2PA∩B=14            …..(1)

P(Exactly one of B or C occurs) =

14

⇒PB+PC-2PB∩C=14            …..(2) P(Exactly one of C or A occurs) =

14

⇒PA+PC-2PA∩C=14           …..(3) and P (All the three events occur simultaneously) =

116

⇒PA∩B∩C=16                        …..(4) Adding equations 1, 2 and 3, we get

⇒2PA+2PB+2PC-2PA∩B-2PB∩C-2PA∩C=34

⇒PA+PB+PC-PA∩B-PB∩C-PA∩C=38                                      …(5) The probability that at least one of the events occurs =

PA+PB+PC-PA∩B-PB∩C-PA∩C+PA∩B∩CSo, adding equation (5) and (6), we get

⇒PA+PB+PC-PA∩B-PB∩C-PA∩C+PA∩B∩C=38+116=6+116=716Hence the correct answer is option (B).

Question 5:

Let ω be a complex number such that 2ω + 1 = z where z =

–3. If

1111-ω2-1ω21ω2ω7= 3k, then k is equal to :

Option 1: –z
Option 2: z
Option 3: –1
Option 4: 1
Correct Answer: 1

 

Solutions:Given:

2ω+1=z where

z=-3 and

1111-ω2-1ω21ω2ω7=3kNow

z=-3=-1×3=i3So,

2ω+1=i3⇒2ω=i3-1⇒ω=i3-12          …(1)

We know

1+ω+ω2=0 and ω3=1                                …(2)

 

1111-ω2-1ω21ω2ω7=3k                                                     (given)

⇒1111ωω21ω2ω6×ω=3k⇒1111ωω21ω2(ω3)2×ω=3k⇒1111ωω21ω2ω=3k                           [∵ω3=1]

C1→C1+C2+C3⇒1+1+1111+ω+ω2ωω21+ω+ω2ω2ω=3k    ⇒3110ωω20ω2ω=3k                    from2On expanding we get,

3ω2-ω4=3k⇒k=ω2-ω4=ω2-ωOn putting the value of

ω we get,

-1-i32–1+i32=-i32-i32=-i3=-zHence, the correct answer is option (A).

Question 6:

Let k be an integer such that the triangle with vertices (k, –3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point :

Option 1: 2,-12
Option 2: 1,34
Option 3: 1,-34
Option 4: 2,12
Correct Answer: 4

 

Solutions:It is given that, the triangle with vertices (k, −3k), (5, k) and (−k, 2) has area 28 sq. units, for any integer k. By area formula, we have

12k-3k15k1-k21=±28⇒k-3k15k1-k21=±56⇒kk-2+3k5+k+110+k2=±56⇒k2-2k+15k+3k2+10+k2=±56 ⇒5k2+13k+66 =0         or       ⇒5k2+13k-46 =0The equation

5k2+13k+66 =0has no solution since

132-4×5×66=169-1320<0. Now, solve the equation

5k2+13k-46 =0. After solving we get the roots of the equation as

k=-13±169+92010=-13±3310⇒k=2 or -235As k is an integer therefore k = 2. Now, A(2, −6), B(5, 2), C(−2, 2). Equation of altitude dropped from vertex A is x = 2 ….. (i) Equation of altitude dropped from vertex C is 3x + 8y − 10 = 0  …..(ii) Consider the figure below:

Solving (i) and (ii), we get the orthocentre of the triangle as

2,12Hence the correct answer is option (D).

Question 7:

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is :

Option 1: 12.5
Option 2: 10
Option 3: 25
Option 4: 30
Correct Answer: 3

 

Solutions:Perimeter of the flower bed in the form of sector = Total length of the wire

⇒2r+l=20⇒2r+rθ=20⇒θ=20-2rr                   …1 Area of sector (A)=πr2θ360=12r2·20-2rr=r10-r⇒A=10r-r2For maximum or minimum area,

 

dAdr=0⇒d10r-r2dr=0⇒10-2r=0⇒r=5mPutting the value of r in (1) we get,

θ=20-255=20-105=2 radian

So, the maximum area comes out to be

A=12×r2×θ=12×52×2=25m2Hence the correct answer is option (C).

Question 8:

The area (in sq. units) of the region {(x, y) : x ≥ 0, x + y ≤ 3, x2 ≤ 4y and y ≤ 1 +

x} is :

Option 1: 5912
Option 2: 32
Option 3: 73
Option 4: 52
Correct Answer: 4

 

Solutions:Given that

x,y:x>0.x+y≤3,x2≤4y and y≤1+xConsider the following figure:

Now,

y≤1+x ⇒y-12≤xTherefore, the required area is

A=∫01[1+x-x24]dx+∫12[3-x-x24]dxA=x+23×32-x31201+3x-x22-x31212A=1+23-112+6-2-23-3+12+112A=2+12=52Hence the correct answer is option (D).

Question 9:

If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to the line,

x1=y4=z5is Q, then PQ is equal to :

Option 1: 35
Option 2: 242
Option 3: 42
Option 4: 65
Correct Answer: 2

 

Solutions:The equation of the line passing through P(1, −2, 3) and parallel to the line given line is

x-11=y+24=z-35      …..(1) Let R be the point of intersection of the given plane and line (1). Then the coordinates of R are given by (λ + 1, 4λ − 2, 5λ + 3). Now, R(λ + 1, 4λ − 2, 5λ + 3) lies on the plane 2x + 3− 4z + 22 = 0.

∴2λ+1+34λ-2-45λ+3+22=0⇒-6λ+6=0⇒λ=1So, coordinates of R = (1 + 1, 4 × 1 − 2, 5 × 1 + 3) = (2, 2, 8). Now,

PR=2-12+2+22+8-32=1+16+25=42So,

PQ=2PR=242Hence, the correct answer is option B.

Question 10:

If for

x∈0,14, the derivative of tan–1

6xx1-9x3is

x . g(x),then g(x) equals :

Option 1: 91+9×3
Option 2: 3xx1-9×3
Option 3: 3×1-9×3
Option 4: 31+9×3
Correct Answer: 1

 

Solutions:

Let y=tan-16xx1-9×3   for x∈0, 14     y=tan-12.3xx1-3xx2        ∵tan-12×1-x2=2tan-1x       =2tan-13xx

Now, dydx=2×11+3xx2×3×32×12        =91+9x3x∴ gx=91+9x3Hence, the correct answer is option A.

Question 11:

If (2 + sin x)

dydx+ (y + 1) cos x = 0 and y(0) = 1, then y

π2is equal to :

Option 1: 13
Option 2: -23
Option 3: -13
Option 4: 43
Correct Answer: 1

 

Solutions:It is given that,

2+sinxdydx+y+1cosx=0.

⇒dydx=-y+1cosx2+sinx

⇒dyy+1=-cosx2+sinxdxIntegrating both sides,

⇒∫dyy+1=∫-cosx2+sinxdx⇒logy+1=-log2+sinx+C⇒logy+1+log2+sinx=C⇒logy+12+sinx=C

⇒y+12+sinx=AFor x=0, y=1, ⇒A=4∴y+12+sinx=4For x=π2, ⇒y=13Hence the correct answer is option (A).

Question 12:

Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground such that AP = 2AB. If  ∠BPC = β, then tanβ is equal to

Option 1: 67
Option 2: 14
Option 3: 29
Option 4: 49
Correct Answer: 3

 

Solutions: Let

∠APC=α and ∠APB=θ.

tanθ=ABAP=x2x=12 tanα=ACAP=x22x=14Let

tanβ=y.

tanθ=tanα+β=tanα+tanβ1-tanαtanβ⇒12=14+y1-14y⇒12=1+4y4-y ⇒4-y=21+4y⇒4-y=2+8y⇒2=9y⇒29=yHence, the correct answer is option (3).

Question 13:

If A =

2-3-4   1,then adj (3A2 + 12A) is equal to

Option 1:    72-84-63   51
Option 2: 51638472
Option 3: 51846372
Option 4:    72-63-84   51
Correct Answer: 2

 

Solutions:It is given that,

A=2-3-41. To find 3A2 + 12A, first find out 3A2.

3A2=32-3-412-3-41=316-9-1213=48-27-3639Now,

12A=24-36-4812Therefore,

3A2 + 12A=48-27-3639+24-36-4812=72-63-8451.

So,

adj3A2+12A=51638472.Hence the correct answer is option (B).

Question 14:

For any three positive real numbers a, b and c, 9(25a2 + b2) + 25(c2 – 3ac) = 15b(3a + c), Then

Option 1: b , c and a are in G.P.
Option 2: b, c and a are in A.P.
Option 3: a, b and c are in A.P.
Option 4: a, b and c are in G.P.
Correct Answer: 2

 

Solutions:We have,

925a2+b2+25c2-3ac=15b3a+c⇒225a2+9b2+25c2-45ab-15bc-75ac=0⇒15a2+3b2+5c2-15a3b-3b5c-15a5c=0⇒1215a-3b2+3b-5c2+5c-15a2=0 ⇒15a=3b=5c⇒a=k15, b=k3, c=k5⇒a+b=2c∴b, c, a are in A.P.Hence, the correct answer is option B.

Question 15:

The distance of the point (1, 3, –7) from the plane passing through the point (1, –1, –1), having normal perpendicular to both the lines

x-11=y+2-2=z-43and

x-22=y+1-1=z+7-1,is

Option 1: 2074
Option 2: 1083
Option 3: 583
Option 4: 1074
Correct Answer: 2

 

Solutions:Consider the equation of lines,

x-11=y+2-2=z-43 and x-22=y+1-1=z+7-1. Normal vector is as follows:

i^j^k^1-232-1-1=5i^+7j^+3k^Since it is given that normal of the plane passing through the point (1, −1, −1) is perpendicular to both of the lines

x-11=y+2-2=z-43 and x-22=y+1-1=z+7-1. Therefore, equation of the plane is

5x-1+7y+1+3z+1=0⇒5x+7y+3z+5=0So, the distance of the point (1, 3, −7) from the above plane is as follows:

D=5+21-21+525+49+9=1083Hence the correct answer is option (B).

Question 16:

Let In =

∫tann x dx , (n > 1). If I4 + I6 = a tan5x + bx5 + C, where C is a constant of integration, then the ordered pair (a, b) is equal to

Option 1: -15, 1
Option 2: 15, 0
Option 3: 15,-1
Option 4: -15, 0
Correct Answer: 2

 

Solutions:

In=∫tannx dx=∫tan2x tann-2x dx=∫sec2x-1tann-2x dx=∫tann-2x sec2x dx-∫tann-2x dx=tann-1xn-1-In-2                               ∫fxnf’xdx=fxn+1n+1+C∴ In=tann-1xn-1-In-2          …..1Putting n = 6 in (1), we have

I6=tan5x5-I4⇒I6+I4=15tan5x+CComparing this expression with

I6+I4=atan5x+bx5+C, we have

a=15, b=0Thus, the ordered pair (a, b) is

15,0.

Hence, the correct answer is option B.

Question 17:

The eccentricity of an ellipse whose centre is at the origin is

12. If one of its directrices is x = –4, then the equation of the normal to it at

1,32is

Option 1: 2y – x = 2
Option 2: 4x – 2y = 1
Option 3: 4x + 2y = 7
Option 4: x + 2y = 4
Correct Answer: 2

 

Solutions: Given one of the directrices is = −4.

⇒-ae=-4⇒a=4e⇒a=4×12        Given e=12⇒a=2

Now e2=1-b2a2⇒b2=a21-e2=a21-122=3Equation of ellipse is

 

x2a2+y2b2=1⇒x24+y23=1Equation of normal at

1,32 is

a2xx1-b2yy1=a2-b2⇒4×1-3y32=4-3⇒4x-2y=1Hence, the correct answer is option B.

Question 18:

A hyperbola passes through the point

P 2 , 3and has foci at (±2, 0). Then the tangent to this hyperbola at P also passes through the point :

Option 1: 32, 23
Option 2: 22, 33
Option 3: 3, 2
Option 4: -2, -3
Correct Answer: 2

 

Solutions:Consider the following figure:

Here, the hyperbola passes through the point P(2, 3) and has foci at (±2, 0). Also, ae = 2. Now, the distance

PF=2-22+3=9-42=22-1Distance of the point P from F’ is

PF’=2+22+3=9+42=22+1

∴PF-PF’=2∴a=1 ⇒e=2∵e2=1+b2a2∴b2=3So, equation of hyperbola is

x2-y23=1 or 3×2-y2=3. Equation of the tangent at

2,3will be

32x-3y=3. Therefore, the tangent passes through the point

22,33. Hence, the correct answer is option B.

Question 19:

The function

f : R→-12,12defined as

fx=x1+x2,  is :

Option 1: Invertible
Option 2: injective but not surjective
Option 3: surjective but not injective
Option 4: neither injective nor surjective
Correct Answer: 3

 

Solutions:The function

f : R→-12,12is defined as

fx=x1+x2.

fx=x1+x2Diffentiating both sides w.r.t x, we have

f’x=1+x2×1-x×2×1+x22=1-x21+x22=-x-1x+11+x22Now, f ‘(x) = 0  for x = 1 and x = −1. f ‘(x) > 0 for −1 < x < 1 and f ‘(x) < 0 for x < −1 and x > 1 Also, f(0) = 0, f(1) =

12and f(−1) =

-12

limx→∞fx=limx→∞x1+x2=limx→∞11x+x→∞limx→-∞fx=limx→-∞x1+x2=limx→-∞11x+x→-∞So, the graph of the given function is It can be seen that the given function is many-one. So, the given function is not injective. Now, Range of f =

-12,12= Co-domain of f So, the given function is surjective. Thus, the given function is surjective but not injective.

Hence, the correct answer is option C.

Question 20:

limx→π2cotx-cosxπ-2x3equals

Option 1: 124
Option 2: 116
Option 3: 18
Option 4: 14
Correct Answer: 2

 

Solutions:Consider,

limx→π2cotx-cosxπ-2×3.

limx→π2cotx1-sinx-8x-π23=limx→π2tanπ2-x8π2-x1-cosπ2-xπ2-x2=18×1×12=116Hence the correct answer is option (B).

Question 21:

Let

a→=2i^+j^−2k^and

b→=i^+j^. Let

c→ be a vector such that

c→−a→=3,(a→×b→)×c→=3and the angle between

c→and

a→×b→be 30°. Then

a→·c→is equal to

Option 1: 258
Option 2: 2
Option 3: 5
Option 4: 18
Correct Answer: 2

 

Solutions:Consider

a⇀=2i^+j^-2k^, b⇀=i^+j^ and c⇀-a⇀=3, a⇀×b⇀×c⇀=3Now,

a⇀×b⇀=2i^-2j^+k^⇒ a⇀×b⇀=3We know that,

a⇀×b⇀×c⇀=a⇀×b⇀c⇀sin30°n^⇒ a⇀×b⇀×c⇀=3c⇀×12⇒3=3c⇀×12                       Given: a⇀×b⇀×c⇀=3⇒c⇀=2

Now, c⇀-a⇀=3Squaring , we get ⇒c⇀2+a⇀2-2c⇀.a⇀=9⇒4+9-2a⇀.c⇀=9⇒a⇀.c⇀=2Hence the correct answer is option (B).

Question 22:

The normal to the curve y(x – 2)(x – 3) = x + 6 at the point where the curve intersects the y-axis passes through the point :

Option 1: -12,-12
Option 2: 12, 12
Option 3: 12,-13
Option 4: 12, 13
Correct Answer: 2

 

Solutions:The given curve has the equation y(x – 2)(x – 3) = x + 6. Since the given curve intersects the y axis so, x coordinate will be 0. Therefore, y(0 – 2)(0 – 3) = 0 + 6

⇒y=1So, the point of intersection of the curve with the y axis is (0, 1). Now slope of the curve y(x – 2)(x – 3) = x + 6 can be calculated as

⇒y=x+6(x – 2)(x – 3)⇒dydx=x2-5x+6-x+62x-5×2-5x+62At (0, 1) the slope will be

 

dydx=6–3036⇒dydx=1Thus, slope of the curve is 1. So, the slope of the normal to this curve will be −1. So, the equation of the normal is

y-y1=mx-x1

⇒y-1=-1x-0⇒y-1=-x⇒x+y-1=0Thus,

12,12 satisfies the given equation of the normal. Hence, the correct answer is option B.

Question 23:

If two different numbers are taken from the set {0, 1, 2, 3,….., 10}; then the probability that their sum as well as absolute difference are both multiple of 4, is

Option 1: 655
Option 2: 1255
Option 3: 1445
Option 4: 755
Correct Answer: 1

 

Solutions:Let

A=0,1,2,…,10n(s) =

C211. Let B be the number of selecting two numbers whose sum as well as absolute difference both are multiple of 4. If a and b are two numbers selected from the given set then,

Sum=a+bAbsolute difference=a-bSo, B = {(0, 4), (0, 8), (2, 6), (2, 10), (4, 8), (6, 10)} n(B) = 6

∵ the order does not matters and which is clearly indicated in the sample space

i.e  11C2  Thus, P=nBnS=6C211=655. Hence, the correct answer is option A.

Question 24:

A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is

Option 1: 485
Option 2: 468
Option 3: 469
Option 4: 484
Correct Answer: 1

 

Solutions:According the question given :

Man X’ friends : 4L and 3M Wife Y’s friends : 3L and 4M

When a party is thrown there should be 3M and 3L on the whole. But 3 friends each of X and Y should appear.

Therefore, the no.of possibilities of throwing a party

X   3M    1L,2M       1M,2L      3L

Y   3L     2L,1M       2M,1L      3M

3C33C3 + 4C13C2 * 4C13C2 +  3C14C2 * 4C23C 4C34C3  = 485

Hence , the correct answer is option A.

Question 25:

The value of (21C110C1) + (21C210C2) + (21C310C3) + (21C410C4) +……..+ (21C1010C10) is

Option 1: 221 – 211
Option 2: 221 – 210
Option 3: 220 – 29
Option 4: 220 – 210
Correct Answer: 4

 

Solutions:S = (21C1 – 10C1) + (21C2 – 10C2) + (21C3 – 10C3)+(21C4 – 10C4)+(21C5 – 10C5)+(21C6 – 10C6)+(21C7 – 10C7)+(21C8 – 10C8)+(21C9 – 10C9)+(21C10 – 10C10)

S = ( 21C1 + 21C2 + 21C+ 21C4 + 21C21C21C21C21C21C10 ) – ( 10C10C10C+ 10C4  + 10C+10C+ 10C7  +10C10C+10C10 )

S= ( 21C1 + 21C2 + 21C+ 21C4 + 21C21C21C21C21C21C10 )

=

12( 21C1 + 21C2 +…..+ 21C20 )                                                                                                      

=

12( 21C0 + 21C1 + 21C2 +…..+ 21C20 + 21C21 – 2 )                                                                        {

∵ nC0+nC1+nC2 +…..+nCn =

2n}

=

12(

221-2 ) =

220 – 1S=  10C10C10C+ 10C4  + 10C+10C+ 10C7  +10C10C+10C10 )

= ( 10C+10C10C+…..+ 10C10 – 1 )

=

210 – 1

∴ S S= 

220 – 210Hence , the correct answer is option D

Question 26:

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is

Option 1: 125
Option 2: 6
Option 3: 4
Option 4: 625
Correct Answer: 1

 

Solutions:Total = 25 balls (15 green and 10 yellow) Let, p be the probability of drawing green ball and q be the probability of drawing yellow ball (with replacement)

p=1525=35

q=1025=25A random variable X which takes values 0,1,2….10 ( 10 balls to be drawn) is said to follow binomial distribution with parameters n and p and it is also satisfying the condition p+q = 1 and in that case Variance = npq. n = 10 ,

p=35, q=25Variance =

10×35×25=125Hence, the correct answer is option A.

Question 27:

Let a, b, c ∈ R. If f(x) = ax2 + bx + c is such that a + b + c = 3 and f(x + y) = f(x) + f(y) + xy,∀x, y ∈ R, then

∑10n=1fnis equal to

Option 1: 330
Option 2: 165
Option 3: 190
Option 4: 225
Correct Answer: 1

 

Solutions: Let a,b,c

∈R. If f(x) = ax2 + bx + c is such that a + b + c = 3 and f(x + y) = f(x) + f(y) + xy,

∀ x,y

∈ R, then

∑n=110f(n)  is equal to

f(x) = ax2 + bx + c f(x + y) = f(x) + f(y) + xy a(x + y)2 + b(x+ y) + c = ax2 + bx + c + ay2 + by + c + xy 2axy = c + xy

∀ x, y

∈R (2a – 1)xy – c = 0

∀ x, y

∈R

 

⇒c=0

∵ independent term is zero both sides

⇒a=12

∵ coefficient of quadratic term is zero both sides and thereby equating (2a – 1)=0

a + b + c = 3

12 + b + 0 = 3 b=

52 ∴f(x)=12×2+52x∑n=110f(n)=12∑n=110n2+52∑n=110n    ⇒12x10x11x216+52x10x112=330Hence , the correct answer is option A

Question 28:

The radius of a circle, having minimum area, which touches the curve y = 4 – x2 and the lines, y = |x| is

Option 1: 22+1
Option 2: 22-1
Option 3: 42-1
Option 4: 42+1
Correct Answer: 3

 

Solutions:The radius of a circle, having minimum area, which touches the curve y = 4 – x2 and the lines, y = |x| is

The equation of parabola  y = 4 – xon rearranging in standard form x= – (y – 4) and comparing it with x2 = – 4ay The parabola faces downwards due to negative  sign and since

a=14 and vertex will be at

y=4

let radius of circle be r, its center lies on y-axis as y-axis bisects the 2 rays of y = |x| due to symmetry center of the circle must be on y-axis let center be  (0, k) Length of perpendicular from (0, k) to y = x,

r=k2

∴Equation of circle

x2+(y-k)2=k22solving circle and parabola

4-y+y2-2ky+k22=0y2-(2k+1)y+k22+4=0Because circle just touches the parabola , so the quadratic equation formed by combining the quadratic equations of parabola and circle will give us the common solution (point where they meet)

∴D = 0

(2k+1)2=4k22+44k2+4k+1=2k2+16On solving we get

k=-4+1364Therefore radius =

k2radius

r =-4+13642 r≈1.354Option 1

2(2+1)≈4.828 Option 2

2(2-1)≈0.8284Option 3

4(2-1)≈1.656Option 4

4(2+1)≈9.656None of the option is exactly correct , however option C is the closest value to

r≈1.354Hence , the correct answer is option C.

Question 29:

If, for a positive integer n, the quadratic equation, x(x + 1) + (x + 1)(x + 2) +…..+ (x +

n–1)(x + n) = 10n has two consecutive integral solutions, then n is equal to

Option 1: 12
Option 2: 9
Option 3: 10
Option 4: 11
Correct Answer: 4

 

Solutions:Ans

Given equation,

x(x+1)+(x+1)(x+2)+(x+2)(x+3)+…+{x+(n-1)}{x+n}=10n

first term,                  x(x+1)=  x2 + x + 0Second term,  (x+1)(x+2) = x2 +3x + 2Third term,      (x+2)(x+3) = x2 +5x  + 6 Fourth term,   (x+3)(x+4)  = x2 +7x  + 12………               ……………nth term,    {x+(n-1)}{x+n}=x2 +(2n-1)x+n(n-1)Adding all the terms,

x(x+1)+(x+1)(x+2)+…+{x+(n-1)}{x+n}=nx2+(1+3+5+…to n terms)x+(0+2+6+12+…to n terms)                                                                             =

nx2+n2x+∑r=1nn(n-1)                                                                            =

nx2+n2x+n(n+1)(2n+1)6-n(n+1)2                                                                           =

nx2+n2x+n(n2-1)3Therefore, the given equation can be written as,

nx2+n2x+n(n2-1)3=10n

⇒x2+nx+n2-13-10=0⇒x2 +nx + n2-313=0Let the roots are

α and

β then

α+β=-n and α-β=1 (given)

(α+β)2=(α-β)2+4αβ⇒n2=12+4.n2-313

⇒3n2=3+4n2-124⇒n2=121⇒n=11.Hence, the correct answer is option D.

Question 30:

The integral

∫π43π4dx1+cosxis equal to

Option 1: −2
Option 2: 2
Option 3: 4
Option 4: −1
Correct Answer: 2

 

Solutions:The integral

∫π43π4dx1+cos x is equal to

I= ∫π43π4dx1+cos x  …(1)Using property

∫abf(x) dx=∫abf(a+b-x) dx

I= ∫π43π4dx1-cos x  …(2)Adding (1) and (2)

2I= ∫π43π42dx1-cos2x  ⇒I= ∫π43π4dxsin2x⇒I= ∫π43π4cosec2x dxI= -cot x  π43π4 = 2Hence, the correct answer is option B

Question 31:

A radioactive nucleus A with a half life T, decays into a nucleus B. At t = 0, there is no nucleus B. At sometime t, the ratio of the number of B to that of A is 0.3. Then, t is given by :

Option 1: t=Tlog1.3
Option 2: t=T2log2log1.3
Option 3: t=Tlog 1.3log 2
Option 4: t=T log 1.3
Correct Answer: 3

 

Solutions:Let NA be the total number of A present initially. Then after decaying into B, at time t, the number of A becomes equal to

NA = Noe-λtTherefore, number of B at time t is

NB = No(1 – e-λt )Now,

NBNA = No(1-e-λt)Noe-λtNBNA = eλt – 1Given, NBNA = 0.3Therefore, 0.3 = eλt – 11.3 = eλtλt = ln(1.3)   …(i)Now, it is given that half life of radioactive nucleus A is T which is equal toT =  ln(2)λ⇒λ =  ln(2)T   … (ii)Putting  (ii) in (i)t = ln(1.3)  ln(2)THence, the correct answer is option C.

Question 32:

The following observations were taken for determining surface tension T of water by capillary method: diameter of capillary, D = 1.25 × 10−2 m rise of water, h = 1.45 × 10−2 m.

Using g = 9.80 m/s2 and the simplified relation

T=rhg2×103N/m, the possible error in surface tension is closest to:

Option 1: 10%
Option 2: 0.15%
Option 3: 1.5%
Option 4: 2.4%
Correct Answer: 3

 

Solutions:Here, the information of least count of D and h measurement are not given. So, we will use maximum.

Permissible error in D and h = place value of last digit.

D = 1.25 × 10−2 m

So,

∆D = 0.01 × 10−2   mand, â€‹h = 1.45 × 10−2 m

So,

∆h = 0.01 × 10−2   m T =rgh2 × 103∆TT = ∆rr +∆hh = ∆DD +∆hh∆TT  = 0.01 × 10-21.25 × 10-2 +0.01 × 10-21.45 × 10-2∆TT  = (1125  +1145 ) × 100 %∆TT  = ( 0.008  + 0.0069) × 100 %∆TT  = 1.49 % ≈ 1.5 %Hence, the correct answer is option C.

 

Question 33:

An electron bean is acceleration by a potential difference V to hit a metallic target to produce X-rays. It produces continuous as well as characteristic X-rays. If λmin is the smallest possible wavelength of X-ray in the spectrum, the variation of log λmin with log V is correctly represented in:

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 2

 

Solutions:Energy of electron beam can be given as

eV = hcλ⇒ λmin = 12400eVTaking log on both sides⇒ log λmin =  log (12400) –  log (e)  –  log (V)log λmin =  C –  log (V)which is similar to line equationY = C – mx Hence, the correct answer is option B.

Question 34:

The moment of inertia of a uniform cylinder of length l and radius R about its perpendicular bisector is I. What is the ratio l/R such that the moment of inertia is minimum?

Option 1: 32
Option 2: 32
Option 3: 32
Option 4: 1
Correct Answer: 2

 

Solutions:The moment of inertia of a uniform cylinder of length l and radius R about its perpendicular bisector is given to be I which is equal to

I = Ml212 + MR24   …(i)where, M is the mass of the cylinder.

Now, we knowMass of the body  = volume of the body × density of the body⇒M =(π R2l) ρwhere, ρ = density of the bodyor, R2= Mπ lρ    …(ii)Putting (ii) in (i)I = Ml212 + M24π lρ   …(iii)Now, differentiating (iii) w.r.t ldIdl= M12 (2l) – M24π l2ρ   …(iv)For moment of inertia to be minimum, (iv) should be equal to 0.Hence, M12 (2l) – M24π l2ρ = 0l6 = M4π l2ρl3 = 3M2π ρ = 3(π R2l) ρ2π ρ    … using (ii)l2 = 3R22l2 R2 = 32l R = 32Hence, the correct answer is option B.

Question 35:

A slender uniform rod of mass M and length l is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. the angular acceleration of the rod when it makes an angle θ with the vertical is

Option 1: 2g3lcos θ
Option 2: 3g2lsin θ
Option 3: 2g3lsin θ
Option 4: 3g2lcos θ
Correct Answer: 2

 

Solutions: Taking torque about the pivot

τ = I αMgsinθl2 = Ml23α α = 3g2lsinθHence, the correct answer is option B.

Question 36:

Cp and Cv are specific heats at constant pressure and constant volume respectively. It is observed that Cp − Cv = a for hydrogen gas Cp − Cv = b for nitrogen gas The correct relation between a and b is:

Option 1: a = 28 b
Option 2: a=114b
Option 3: a = b
Option 4: a = 14 b
Correct Answer: 4

 

Solutions:We know that

C  = Mo S

For H2 as well as N2

Cp – Cv  = R where, Cp and Cv are molar specific heat capacities.

Mo Sp – Mo Sv  = R 

Sp – Sv  ​= R Mo

For H2 gas

Sp – Sv  ​= R 2 = aand for N2 gas

Sp – Sv  ​= R 28 = bSo,

a b = R2R28  = 14Therefore,

a = 14bHence, the correct answer is option D.

Question 37:

A copper ball of mass 100 gm is at a temperature T. It is dropped in a copper calorimeter of mass 100 gm, filled with 170 gm of water at room temperature. Subsequently, the temperature of the system is found to be 75° C. T is given by: (Given : room temperature = 30°C, specific heat of copper = 0.1 cal/gm° C)

Option 1: 825° C
Option 2: 800° C
Option 3: 885° C
Option 4: 1250° C
Correct Answer: 3

 

Solutions:Heat given by the copper ball = heat taken by the water + heat taken by the calorimeter system

Therefore, (100) (0.1) (T – 75) = (170) (1) (75 – 30) + (100) (0.1) (75 – 30)

10T – 750 = 8100

10T = 8850

T = 885°C

Hence, the correct answer is option C.

Question 38:

In amplitude modulation, sinusoidal carrier frequency used is denoted by ωc and the signal frequency is denoted by ωm. The bandwidth (∆ωm) of the signal is such that ∆ωm<< ωc. Which of the following frequency is not contained in the modulated wave?

Option 1: ωc − ωm
Option 2: ωm
Option 3: ωc
Option 4: ωm + ωc
Correct Answer: 2

 

Solutions:The amplitude modulated signal, cm(t) is given as

cm (t) = Ac sin ωc t + μAc2cos (ωc – ωm ) t – μAc2cos (ωc + ωm ) tSo, it can be inferred from above equation that frequency of ωis not contained in the modulated wave.

Hence, the correct answer is option B.

Question 39:

The temperature of an open room of volume 30 m3 increased from 17°C to 27°C due to the sunshine. The atmospheric pressure in the room remains 1 × 105 Pa. If ni and nf are the number of molecules in the room before and after heating, then nf − ni will be :

Option 1: −2.5 × 1025
Option 2: −1.61 × 1023
Option 3: 1.38 × 1023
Option 4: 2.5 × 1025
Correct Answer: 1

 

Solutions:Using ideal gas equation,

PV= NRT where n is the number of moles.

Before heating, the given temperature is Ti = 273 + 17 = 290 K

PoVo = NiR × 290   …(i)

After heating, the given temperature is Ti = 273 + 27 = 300 K

PoVo = NfR × 300   …(ii)

From equation (i) and (ii)

Nf – Ni = PoVoR×300 – PoVoR×290Therefore, difference in number of moles

Nf – Ni = – 10 PoVoR×290×300Hence,

 

nf – ni =   –  PoVo × 10R×290×300  × 6.023× 1023putting Po = 105 PA and Vo = 30 m3

Therefore, number of molecules nf – ni is

nf – ni = – 2.5 × 1025

Hence, the correct answer is option A.

Question 40:

In a Young’s double slit experiment, slits are separated by 0.5 mm, and the screen is placed 150 cm away. A beam of light consisting of two wavelengths, 650 nm and 520 nm, is used to obtain interference fringes on the screen. The least distance from the common central maximum to the point where the bright fringes due to both the wavelengths coincide is:

Option 1: 15.6 mm
Option 2: 1.56 mm
Option 3: 7.8 mm
Option 4: 9.75 mm
Correct Answer: 3

 

Solutions:Fringe width is given as

β1 = λDd where, d is the slits’ seperation and D is the separation between the screen and the slitβ1= 650 × 10-9 × 1.55 × 10-4      = 650 × 10-5 × 0.3      = 1.95 mmAlso,β2 = 520 × 10-9 × 1.55 × 10-4      = 520 × 10-5 × 0.3      = 1.56 mm4β1 = 5β2 = 7.8 mmHence, the correct answer is option C.

Question 41:

A particle A of mass m and initial velocity υ collides with a particle B of mass

m2which is at rest. The collision is head on, and elastic. The ratio of the de-Broglie wavelengths λA to λB after the collision is:

Option 1: λAλB=12
Option 2: λAλB=13
Option 3: λAλB=2
Option 4: λAλB=23
Correct Answer: 3

 

Solutions:        By conservation of linear momentum,

mv = mvA + m2vB   …(i)and by law of collisione = vB – vAv = 1   …(ii) From (i) and (ii)2(vB – vA) = 2vA + vB2vB – vB = 4vAvB= 4vAλAλB = mBvBmAvA =m2m × 4 = 2Hence, the correct answer is option C.

Question 42:

A magnetic needle of magnetic moment 6.7 × 10−2 Am2 and moment of inertia 7.5 × 10−6 kg m2 is performing simple harmonic oscillations in a magnetic field of 0.01 T. Time taken for 10 complete oscillations is:

Option 1: 8.76 S
Option 2: 6.65 S
Option 3: 8.89 S
Option 4: 6.98 S
Correct Answer: 2

 

Solutions:Given, magnetic moment, M =  6.7 × 10−2 A m2

Moment of inertia, I =  7.5 × 10−6 kg m2

and magnetic field, B = 0.01 T

Since, time period of oscillation is given as

T = 2πIMB    =  2π7.5 × 10-66.7 × 10 -2 × 10 -2    =  2π × 10-175 67To complete 10 oscillations, time taken, t , will bet = 10 Tt = 10 ×  2π × 10-175 67 = 6.65 secHence, the correct answer is option B.

Question 43:

An electric dipole has a fixed dipole moment

→p, which makes angle θ with respect to x-axis. When subjected to an electric field

E→1=Ei^, it experiences a torque

T→1=τk^. When subjected to another electric field

E→2=3E1j^it experiences a t orque

T→2=-T→1. The angle θ is:

Option 1: 90°
Option 2: 30°
Option 3: 45°
Option 4: 60°
Correct Answer: 4

 

Solutions:

We know torque is given as,

 

τ→ = p→ × E→

Since, both the torque in the given problem is related asT2→ =- T1→thus, T1→ + T2 →= 0τk^ + (- τk^) = 0(pxi^ + pyj^) × (Ei ^+ 3Ej^) = 0px × 3Ek ^+ py × E(-k ^)= 0Ek ^(3px –  py) = 0 pypx = 3Therefore, tanθ = 3θ = 60oHence, the correct answer is option D.

Question 44:

In a coil of resistance 100 Ω, a current is induced by changing the magnetic flux through it as shown in the figure. The magnitude of change in flux through the coil is:

Option 1: 275 Wb
Option 2: 200 Wb
Option 3: 225 Wb
Option 4: 250 Wb
Correct Answer: 4

 

Solutions:As we know emf induced (considering magnitude) in a coil due to change in magnetic flux is given as

e = dϕdt   …(i)

Therefore, current induced in the coil of resistance R is

I = eRI =1R dϕdt⇒dϕ = RIdtor, ϕ = R∫Idtor, ϕ = R × charge induced in the coil or, ϕ = R × area of given current-time graph   …(i)Now, area of given current-time graph = 12 × 10 × 0.5 = 2.5 coloumb   …(ii)Putting R = 100 Ω and (ii) in (i)ϕ = 100 × 2.5 = 250 WbHence, the correct answer is option D.

Question 45:

A time dependent force F = 6t acts on a particle of mass 1 kg. If the particle starts from rest, the work done by the force during the first 1 sec. will be:

Option 1: 18 J
Option 2: 4.5 J
Option 3: 22 J
Option 4: 9 J
Correct Answer: 2

 

Solutions: The acceleration, a, produced in the particle of mass 1 kg is

a=Fm=6t1Since, a =dvdt⇒dvdt=6tdv=6tdt∫0vdv=6∫0ttdtv=6t2201=3Therefore, the work done by the force during the first 1 sec. isW=∆KE=Kf-Ki⇒W=12(1)(3)2=4.5 JHence, the correct answer is option B.

Question 46:

Some energy levels of a molecule are shown in the figure. The ratio of the wavelengths r = λ12, is given by:

Option 1: r=13
Option 2: r=43
Option 3: r=23
Option 4: r=43
Correct Answer: 1

 

Solutions:Using,

∆E = hcλfor

λ1

(-E)-(-2E)=hcλ1      …(i)for λ2(-E)-(-43E)=hcλ2      …(ii)Equation (ii)/(i)43-12-1=λ1λ2λ1λ2=13Hence, the correct answer is option A.

Question 47:

In the above circuit the current in each resistance is:

Option 1: 0 A
Option 2: 1 A
Option 3: 0.25 A
Option 4: 0.5 A
Correct Answer: 1

 

Solutions: Take voltage at point A as 0. Then, voltage at other points can be written as shown in the figure. Hence, Potential Difference across each resistor is zero. so current is also zero. Hence, the correct answer is option A.

Question 48:

A body is thrown vertically upwards. Which one of the following graphs correctly represent the velocity vs time ?

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 4

 

Solutions: Here, acceleration on body is constant.

a=dvdt=-9.8Therefore, the slope of velocity time graph is also a negative constant.

Hence, the correct answer is option D.

Question 49:

A capacitance of 2μF is required in an electrical circuit across a potential difference of 1.0 kV. A large number of 1μF capacitors are available which can withstand a potential difference of not more than 300 V. The minimum number of capacitors required to achieve this is:

Option 1: 32
Option 2: 2
Option 3: 16
Option 4: 24
Correct Answer: 1

 

Solutions:To hold 1 kV potential difference, minimum four capacitors are required in series. Therefore, capacitance for a series =

14

μF So, for equivalent capacitance to be 2

μF, 8 parallel combinations are required.

Hence, the minimum number of capacitors required is 32 (

8 × 4 ).

Hence, the correct answer is option A.

Question 50:

In the given circuit diagram when the current reaches steady state in the circuit, the charge on the capacitor of capacitance C will be:

Option 1: CEr1r1+r
Option 2: CE
Option 3: CEr1r2+r
Option 4: CEr2r+r2
Correct Answer: 4

 

Solutions: At steady state, the capacitor will behave as an open circuit. So the current through r1 will be zero and hence the branch containing r1 can be ignored while considering the equivalent circuit.

Current in the circuit,

I=Er+r2Potential difference across r2 , V= Ir2

⇒V=Er2r+r2

Charge on capacitor, Q=CV

Q=CEr2r+r2Hence, the correct answer is option D.

Question 51:

In a common emitter amplifier circuit using an n-p-n transistor, the phase difference between the input and the output voltages will be:

Option 1: 180°
Option 2: 45°
Option 3: 90°
Option 4: 135°
Correct Answer: 1

 

Solutions:

∆Vcc=∆VCE +RL∆IC=0or ∆VCE=-RL∆ICThe change in VCE is the output voltage vo. vo=∆VCE=-β RL∆IBThe voltage gain of the amplifier is Av=v0vi=∆VCEr∆IB=-βacRLrThe negative voltage shows that the output voltage is opposite in phase with the input voltage.Hence, the correct answer is option A.

Question 52:

Which of the following statements is false ?

Option 1: Krichhoff’s second law represents energy conservation.
Option 2: Wheatstone bridge is the most sensitive when all the four resistance are of the same order of magnitude.
Option 3: In a balanced wheatstone bridge if the cell and the galvanometer are exchanged, the null point is disturbed.
Option 4: A rheostat can be used as a potential divider.
Correct Answer: 3

 

Solutions:In a balanced wheat stone bridge if the position of cell and galvanometer is exchanged the null point remains same.

Hence, the correct answer is option C.

Question 53:

A particle is executing simple harmonic motion with a time period T. At time = 0, it is at its position of equilibrium. The kinetic energy-time graph of the particle will look like:

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 1

 

Solutions:

x=Asin (ωt+φ)ϕ=0,πx=±Asin(ωt)KE=12mω2(A2-x2)KE=12KA2cos2 (ωt) Hence, the correct answer is option A.

Question 54:

An observer is moving with half the speed of light towards stationary microwave source emitting waves at frequency 10GHz. What is the frequency of the microwave measured by the observer? (speed of light = 3 × 108 ms−1)

Option 1: 15.3 GHz
Option 2: 10.1 GHz
Option 3: 12.1 GHz
Option 4: 17.3 GHz
Correct Answer: 4

 

Solutions:

ν’=ν1+vc1-vcν’=ν1+121-12=3νν’=10×1.73=17.3 GHzHence, the correct answer is option D.

Question 55:

A man grows into a giant such that his linear dimensions increase by a factor of 9. Assuming that his density remains same, the stress in the leg will change by factor of:

Option 1: 181
Option 2: 9
Option 3: 19
Option 4: 81
Correct Answer: 2

 

Solutions:Volume of man becomes (9)times. Since density is same, so weight of man becomes (9)times. Cross sectional area of leg becomes (9)times

Stress=WeightArea=9392=9 timesHence, the correct answer is option B.

Question 56:

When a current of 5 mA is passed through a galvanometer having a coil of resistance 15 Ω, it shows full scale defection. The value of the resistance to be put in series with the galvanometer to convert it into a voltmeter of range 0 − 10 V is:

Option 1: 4.005 ×103 Ω
Option 2: 1.985 × 103 Ω
Option 3: 2.045 × 103 Ω
Option 4: 2.535 × 103 Ω
Correct Answer: 2

 

Solutions:Full scale current, Ig =  5 mA Resistance of galvanometer, = 15 Ω

R = VIg-G= 105×10-3-15= 2000-15= 1.985 × 103 ΩHence, the correct answer is option B.

Question 57:

The variation of acceleration due to gravity g with distance d from centre of the earth is best represented by (R = Earth’s radius)

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 1

 

Solutions:

g=GMdR3                    d<Rg=GMR2                    d⩾R Hence, the correct answer is option A.

Question 58:

An external pressure P is applied on a cube at 0°C so that it is equally compressed from all sides. K is the bulk modulus of the material of the cube and α is its coefficient of linear expansion. Suppose we want to bring the cube to its original size by heating. The temperature should be raised by:

Option 1: 3PKα
Option 2: P3αK
Option 3: PαK
Option 4: 3αPK
Correct Answer: 2

 

Solutions:

∆VV=-γ∆θ=-3α∆θ∆p=-K∆VVp=K(3α∆θ)∆θ=p3αKHence, the correct answer is option B.

Question 59:

A diverging lens with magnitude of focal length 25 cm is placed at a distance of 15 cm from a converging lens of magnitude of focal length 20 cm. A beam of parallel light falls on the diverging lens. The final image formed is:

Option 1: real and at a distance of 6 cm from the convergent lens
Option 2: real and at a distance of 40 cm from convergent lens
Option 3: virtual and at a distance of 40 cm from convergent lens
Option 4: real and at distance of 40 cm from the divergent lens
Correct Answer: 2

 

Solutions:Image formed by first lens is I1 which is 25 cm left of diverging lens. For second lens u = 40 cm (i.e. at 2F), so final image will be 40 cm right of converging lens. Hence image will be real.

Hence, the correct answer is option B.

Question 60:

A body of mass m = 10−2 kg is moving in a medium and experiences a frictional force F = −kv2. Its initial speed is v0 = 10 ms−1. If after 10 s, its energy is

18mv02, the value of k will be:

Option 1: 10−1 kg m−1 s−1
Option 2: 10−3 kg m−1
Option 3: 10−3 kg s−1
Option 4: 10−4 kg m−1
Correct Answer: 4

 

Solutions:

F = -kv2mdvdt = -kv2∫vovv-2dv =∫0t-kmdtAlso, after 10 s,KE = 12mv2 = 18mvo2⇒ v= vo2Therefore, -1vvovo/2=-kmt2vo-1vo=-kmt⇒k=mvot=10-210×10=10-4 kg m-1Hence, the correct answer is option D.

Question 61:

1 gram of a carbonate (M2CO3) on treatment with excess HCl produces 0.01186 mole of CO2. The molar mass of M2CO3 in g mol–1 is :

Option 1: 84.3
Option 2: 118.6
Option 3: 11.86
Option 4: 1186
Correct Answer: 1

 

Solutions:Given: Mass of carbonate = 1 g Moles of carbon dioxide produced on reaction with hydrogen chloride = 0.01186 mol

To find: Molar mass of the carbonate = ?

Suppose, Molar mass of the metal carbonate (M2CO3) = M g/mol

The balanced chemical reaction between the carbonate and hydrogen chloride can be written as: M2CO3 + 2 HCl → 2 MCl2 + H2O + CO2

According to this reaction,

Moles of M2CO3 needed to produce 1 mole of CO2 = 1 mol Moles of M2CO3 needed to produce 0.01186 moles of CO2 = 0.01186 mol

Molar Mass of M2CO3 =

Mass of M2CO3Moles of M2CO3=1 g0.01186 mol = 84.3 g/molHence, the correct answer is option (1).

Question 62:

Given : C(graphite) + O2(g) → CO2(g) ;           ΔrH° = – 393.5 kJ mol–1 ; H2(g) +

12O2(g) → H2O(l) ;  ΔrH° = –285.8 kJ mol–1 ; CO2(g) + 2H2O(l) → CH4(g) + 2O2(g) ; ΔrH° = + 890.3 kJ mol–1 Based on the above thermochemical equations, the value of  ΔrH° at 298 K for the reaction C(graphite) + 2H2(g) → CH4(g) will be :

Option 1: +144.0 kJ mol–1
Option 2: –74.8 kJ mol–1
Option 3: –144.0 kJ mol–1
Option 4: +74.8 kJ mol–1
Correct Answer: 2

 

Solutions:Given: C(graphite) + O2(g) → CO2(g) ; ΔrHº = – 393.5 kJ mol–1                   …(I) H2(g) + 1/2 O2(g) → H2O(l) ; ΔrHº = – 285.8 kJ mol–1                      …(II) CO2(g) + 2 H2O(l) → CH4(g) + 2 O2(g) ; ΔrHº = + 890.3 kJ mol–1   …(III)

To Find: ΔrHº at 298 K for the reaction C(graphite) + 2 H2(g) → CH4(g)

The above reaction can be obtained as: ΔrHº = I + (2×II) + III ΔrHº = (− 393.5) + (− 285.8 × 2) + 890.3 ΔrHº = 890.3 − (393.5 + 571.6) ΔrHº = 890.3 − 965.1 ΔrHº = − 74.8 kJ / mol

Hence, the correct answer is option (2).

Question 63:

The freezing point of benzene decreases by 0.45°C when 0.2 g of acetic acid is added to 20 g of benzene. If acetic acid associates to form a dimer in benzene, percentage association of acetic acid in benzene will be : (Kf for benzene = 5.12 K kg mol–1)

Option 1: 80.4%
Option 2: 74.6%
Option 3: 94.6%
Option 4: 64.6%
Correct Answer: 3

 

Solutions:Given: Depression in freezing point (ΔTf) = 0.45 ºC Mass of acetic acid (CH3COOH) = 0.2 g Molar mass of acetic acid (CH3COOH)  = 60 g/mol Mass of benzene (C6H6) = 20 g Molal depression constant (Kf) = 5.12 K kg/mol Number of molecules of acetic acid that associate in benzene = 2

To find: Degree of association for acetic acid (α) = ?

The relation between degree of association and van’t Hoff factor is given as:

α = i-11n-1van’t Hoff factor can be obtained from the expression for depression in freezing point:

i = ΔTf × MCH3COOH × mC6H6Kf × mCH3COOH × 1000i = 0.45 × 60 × 205.12 × 0.2 × 1000 = 0.53Substituting this value of i in the expression for degree of association we get:

α = 0.53-112-1 = 0.94Percentage association of acetic acid in benzene = 94%

Hence, the correct answer is option (3).

Question 64:

The most abundant elements by mass in the body of a healthy human adult are : Oxygen (61.4%); Carbon (22.9%), Hydrogen (10.0%) ; and Nitrogen (2.6%). The weight which a 75 kg person would gain if all 1H atoms are replaced by 2H atoms is :

Option 1: 37.5 kg
Option 2: 7.5 kg
Option 3: 10 kg
Option 4: 15 kg
Correct Answer: 2

 

Solutions:75 kg person contain 10% hydrogen. Mass of hydrogen =

=10100×75 = 7.5 kgIf all 1H atom are replaced by 2H, the mass of hydrogen become twice. So, replacing 1H by 2H would replace 7.5 kg with 15 kg. Therefore, net gain is 7.5 kg.

Hence, the correct answer is option (2).

Question 65:

ΔU equal to :

Option 1: Isobaric work
Option 2: Adiabatic work
Option 3: Isothermal work
Option 4: Isochoric work
Correct Answer: 2

 

Solutions:Using first law of thermodynamics,

∆U = q + w

For adiabatic process, q = 0

i.e. ∆U = w

Therefore, ∆U is equal to adiabatic work.

Hence, the correct answer is option (2).

Question 66:

The formation of which of the following polymers involves hydrolysis reaction?

Option 1: Bakelite
Option 2: Nylon 6,6
Option 3: Terylene
Option 4: Nylon 6
Correct Answer: 4

 

Solutions:Nylon 6 is prepared by hydrolysis of caprolactum. The reaction can be shown as:

Hence, the correct answer is option (4).

Question 67:

Given

ECl2/Cl-°=1.36 V, ECr3+/Cr°=-0.74 VECr2O72-/Cr3+ °=1.33 V, EMnO4-/Mn2+°=1.51 VAmong the following, the strongest reducing agent is :

Option 1: Mn2+
Option 2: Cr3+
Option 3: Cl
Option 4: Cr
Correct Answer: 4

 

Solutions:The strongest reducing agent is determined by the least value of reduction potential. Therefore,

ECr3+/Cr°=-0.74 Vis the strongest reducing agent.

Hence, the correct answer is option (4).

Question 68:

The Tyndall effect is observed only when following conditions are satisfied :

(a) The diameter of the dispersed particles is much smaller than the wavelength of the light used.

(b) The diameter of the dispersed particles is not much smaller than the wavelength of the light used

(c) The refractive indices of the dispersed phase and dispersion medium are almost similar in magnitude.

(d) The refractive indices of the dispersed phase and dispersion medium differ greatly in magnitude.

Option 1: (b) and (d)
Option 2: (a) and (c)
Option 3: (b) and (c)
Option 4: (a) and (d)
Correct Answer: 1

 

Solutions:Tyndall effect is observed only when the diameter of the dispersed particles is not much smaller than the wavelength of the light used and the refractive indices of the dispersed phase and the dispersion medium differ greatly in magnitude.

Hence, the correct answer is option (1).

Question 69:

In the following reactions, ZnO is respectively acting as a/an :

(a) ZnO + Na2O → Na2ZnO2

(b) ZnO + CO2 → ZnCO3

Option 1: base and base
Option 2: acid and acid
Option 3: acid and base
Option 4: base and acid
Correct Answer: 3

 

Solutions:(a) ZnO + Na2O → Na2ZnO2

(b) ZnO + CO2 → ZnCO3

In equation (a), ZnO behaves like acid while in equation (b), ZnO behaves like base.

Hence, the correct answer is option (3).

Question 70:

Which of the following compounds will behave as a reducing sugar in an aqueous KOH solution?

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 3

 

Solutions:Reducing sugars are those in which there is presence of free anomeric –OH group.

Hence, the correct answer is option (3)

Question 71:

The major product obtained in the following reaction is :

Option 1: C6H5CH = CHC6H5
Option 2: (+) C6H5CH(OtBu)CH2C6H5
Option 3: (–) C6H5CH(OtBu)CH2C6H5
Option 4: (±) C6H5CH(OtBu)CH2C6H5
Correct Answer: 1

 

Solutions:

Hence, the correct answer is option (1).

Question 72:

Which of the following species is not paramagnetic?

Option 1: CO
Option 2: O2
Option 3: B2
Option 4: NO
Correct Answer: 1

 

Solutions:Molecular Species

Number of Electrons MOT Electronic Configuration Magnetic Nature
CO 6 + 8 = 14 σ1s2 σ*1s2 σ2s2 σ*2s2 π2px2 = π2py2 σ2pz2 Dimagnetic
O2 8 + 8 = 16 σ1s2 σ*1s2 σ2s2 σ*2s2 σ2pz2 π2px2 = π2py2 π*2px1 = π*2py1 Paramagnetic
B2 5 + 5 = 10 σ1s2 σ*1s2 σ2s2 σ*2s2 π2px1 = π2py1 Paramagnetic
NO 7 + 8 = 15 σ1s2 σ*1s2 σ2s2 σ*2s2 σ2pz2 π2px2 = π2py2 π*2px1 = π*2py0 Paramagnetic

Hence, the correct answer is option (1).

Question 73:

On treatment of 100 mL of 0.1 M solution of CoCl3.6H2O with excess AgNO3; 1.2 × 1022 ions are precipitated. The complex is:

Option 1: [Co(H2O)3Cl3].3H2O
Option 2: [Co(H2O)6]Cl3
Option 3: [Co(H2O)5Cl]Cl2.H2O
Option 4: [Co(H2O)4Cl2]Cl.2H2O
Correct Answer: 3

 

Solutions:

CoCl3.6H2O + AgNO3 → nAgCl↓1.2 × 10226.0 × 1023 = 2 × 10-2 mole = 0.020.01 × n = 0.02n = 2Therefore, the complex is [Co(H2O)5Cl]Cl2.H2O.

Hence, the correct answer is option (3).

Question 74:

pKa of a weak acid (HA) and pKb of a weak base (BOH) are 3.2 and 3.4, respectively. The pH of their salt (AB) solution is :

Option 1: 6.9
Option 2: 7.0
Option 3: 1.0
Option 4: 7.2
Correct Answer: 1

 

Solutions:Given: pH of weak acid, HA = 3.2 pH of weak base, BOH = 3.4

To find: pH of salt AB = ?

pH of the solution of salt AB formed by weak acid HA and weak acid BOH is given as :

pH=12pKw – pKb + pKapH = 1214 – 3.4 + 3.2pH = 6.9Hence, the correct answer is option (1).

Question 75:

The increasing order of the reactivity of the following halides for the SN1 reaction is :

Option 1: (II) < (I) < (III)
Option 2: (I) < (III) < (II)
Option 3: (II) < (III) < (I)
Option 4: (III) < (II) < (I)
Correct Answer: 1

 

Solutions:Order of reactivity for SN1 mechanism is in accordance with order of stability of carbocation involved:

Hence, the correct answer is option (1).

Question 76:

Both lithium and magnesium display several similar properties due to the diagonal relationship; however, the one which is incorrect, is :

Option 1: both form soluble bicarbonates
Option 2: both form nitrides
Option 3: nitrates of both Li and Mg yield NO2 and O2 on heating
Option 4: both form basic carbonates
Correct Answer: 4

 

Solutions:Magnesium form basic carbonate [MgCO3.Mg(OH)2], while lithium form normal carbonate (Li2CO3). So, the given statement in option 4, both form basic carbonate, is incorrect.

Hence, the correct answer is option (4).

Question 77:

The correct sequence of reagents for the following conversion will be :

Option 1: CH3MgBr, H+/CH3OH, [Ag(NH3)2]+OH
Option 2: CH3MgBr, [Ag(NH3)2]+OH, H+/CH3OH
Option 3: [Ag(NH3)2]+OH, CH3MgBr, H+/CH3OH
Option 4: [Ag(NH3)2]+OH, H+/CH3OH, CH3MgBr
Correct Answer: 4

 

Solutions:

Hence, the correct answer is option (4).

Question 78:

The products obtained when chlorine gas reacts with cold and dilute aqueous NaOH are :

Option 1: ClO2- and ClO3-
Option 2: Cl– and ClO–
Option 3: Cl– and ClO2-
Option 4: ClO– and ClO3-
Correct Answer: 2

 

Solutions:Complete chemical equation is:

2 NaOHCold and dilute + Cl2 → NaCl + NaOCl + H2OOr2 NaOH + Cl2 → Cl- + ClO- + Na+ + H2OHence, the correct answer is option (2).

Question 79:

Which of the following compounds will form significant amount of meta product during mono-nitration reaction ?

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 2

 

Solutions:Among the given compounds, aniline will give significant amount of meta product during mono-nitration reaction. Because aniline in acidic medium converts into anilinium ion and it is meta directing, therefore, significant amount of meta product is obtained.

Hence, the correct answer is option (2).

Question 80:

3-Methyl-pent-2-ene on reaction with HBr in presence of peroxide forms an addition product. The number of possible stereoisomers for the product is:

Option 1: Zero
Option 2: Two
Option 3: Four
Option 4: Six
Correct Answer: 3

 

Solutions: Total number of stereo centres present in the product of 2-bromo-3-methylpentane is two. Number of stereoisomers = 2n = 4 [n = Number of chiral carbon atom]

Hence, the correct answer is option (3).

Question 81:

Two reactions R1 and R2 have identical pre-exponential factors. Activation energy of R1 exceeds that of R2 by 10kJ mol–1. If k1 and k2 are rate constants for reactions R1 and R2 respectively at 300 K, then ln(k2/k1) is equal to : (R = 8.314 J mol–1 K–1)

Option 1: 12
Option 2: 6
Option 3: 4
Option 4: 8
Correct Answer: 3

 

Solutions:Given:

k1 = Ae-Ea1/RTk2 = Ae-Ea1 – 10/RTSince,lnk2k1 = 10RT = 108.314 × 10-3 × 300 =4.0093 ≈4Hence, the correct answer is option (3).

Question 82:

Which of the following molecules is least resonance stabilized?

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 3

 

Solutions:Given molecules in options 1, 2 and 4 are aromatic and stablized by according to Huckel’s rule. Molecule in option 3 is non-aromatic, hence it is least resonance stablised.

Hence, the correct answer is option (3).

Question 83:

The group having isoelectronic species is:

Option 1: O, F, Na, Mg+
Option 2: O2−, F, Na, Mg2+
Option 3: O, F, Na+, Mg2+
Option 4: O2−, F, Na+, Mg2+
Correct Answer: 4

 

Solutions:Isoelectronic species are those which have same number of electrons. Among the given group of species O2-, F, Na+ and Mg2+ have same number of electrons (all contain 10 electrons).

Hence, the correct answer is option (4).

Question 84:

The radius of the second Bohr orbit for hydrogen atom is :

(Planck’s Const. h = 6.6262 × 10–34 Js; mass of electron = 9.1091 × 10–31 kg; charge of electron e = 1.60210 × 10–19 C; permittivity of vacuum ∈0 = 8.854185 × 10–12 kg–1m–3A2)

Option 1: 4.76 Å
Option 2: 0.529 Å
Option 3: 2.12 Å
Option 4: 1.65 Å
Correct Answer: 3

 

Solutions:The radius expression for the second Bohr orbit for hydrogen atom is

rn = 52.9 n2Z pm = 0.529 n2Z A∘.

rn = 0.529 221 A∘    = 2.116 A∘ ≈ 2.12 A∘Hence, the correct answer is option (3).

Question 85:

The major product obtained in the following reaction is:

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 1

 

Solutions: DIBAL-H reduce both ester and carboxylic acid into aldehyde at low temperature.

Hence, the correct answer is option (1).

Question 86:

Which of the following reactions is an example of a redox reaction ?

Option 1: XeF2 + PF5 → [XeF]+

PF6-

Option 2: XeF6 + H2O → XeOF4 + 2HF
Option 3: XeF6 + 2H2O → XeO2F2 + 4HF
Option 4: XeF4 + O2F2 → XeF6 + O2
Correct Answer: 4

 

Solutions:

Xe+4F4 + O2+1F2 → Xe+6F6 + O02In the given reaction, oxidation of XeF4 and reduction of O2F2 occurs, hence, this is a kind of redox reaction.

Hence, the correct answer is option (4).

Question 87:

A metal crystallises in a face centred cubic structure. If the edge length of its unit cell is ‘a’, the closest approach between two atoms in metallic crystal will be :

Option 1: 22a
Option 2: 2a
Option 3: a2
Option 4: 2a
Correct Answer: 3

 

Solutions:In face centred cubic (fcc), atoms are in close contact along face diagonal of fcc unit cell. Hence,

4r = 2a2r = 2a2 = a2Hence, the correct answer is option (3).

Question 88:

Sodium salt of an organic acid ‘X’ produces effervescence with conc. H2SO4. ‘X’ reacts with the acidified aqueous CaCl2 solution to give a white precipitate which decolourises acidic solution of KMnO4. ‘X’ is :

Option 1: HCOONa
Option 2: CH3COONa
Option 3: Na2C2O4
Option 4: C6H5COONa
Correct Answer: 3

 

Solutions:Chemical reactions:

Sodium salt of organic acid (Na2C2O4) produces effervescence with conc. H2SO4.Na2C2O4 + H2SO4 → Na2SO4 + H2O + CO + CO2Na2C2O4 reacts with acidified aqueous CaCl2 to give white precipitate CaC2O4.Na2C2O4 + CaCl2 → CaC2O4 ↓ + 2 NaClWhite precipitate CaC2O4 decolourises acidic solution of KMnO4.8 H2SO4 + 5 CaC2O4 + 2 KMnO4 → 5 CaSO4 + K2SO4 + 2 MnSO4 + 8 H2O + 10 CO2Hence, the correct answer is option (3).

Question 89:

A water sample has ppm level concentration of following anions

F = 10 ;

SO42-= 100;

NO3-=50The anion/anions that make/makes the water sample unsuitable for drinking is/are :

Option 1: both

SO42-and

NO3-

Option 2: only F
Option 3: only

SO42-

Option 4: only

NO3-

Correct Answer: 2

 

Solutions:Internation standards for drinking water Fluoride ion (F-) concentration upto 1 ppm Nitrate ion (NO3) concentration upto 50 ppm Sulphate ion (SO42-) concentration upto 500 pmm From the given data, it can be conclude that only fluoride ion make the water sample unsuitable for drinking.

Hence, the correct answer is option (2).

Question 90:

Which of the following, upon treatment with tert-BuONa followed by addition of bromine water, fails to decolourize the colour of bromine

Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer: 4

 

Solutions:

Hence, the correct answer is option (4).

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