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JEE Main 2017 Question Paper with Solutions (2nd April)


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JEE Main 2017 Question Paper with Solutions (2nd April) - Question 1

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 1

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 2

The temperature of an open room of volume 30 m3 increases 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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 2

n1 = initial number of moles;

n2 = final number of moles

Change of number of molecules
nf – ni = (n2 – n1) × 6.023 × 1023
≈ – 2.5 × 1025

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 3

Which of the following statements is false?

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 3

In a balanced Wheatstone bridge, the null point remains unchanged even if cell and galvanometer are interchanged.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 4

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  the possible error in surface tension is closest to

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 4




= 0.8 + 0.689
= 1.489
≈ 1.5%

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 5

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 5

Modulated wave has frequency range.
ω± ωm
∴ Since ωc >> ωm
∴ ωm is excluded.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 6

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 6


For converging lens
u = –40 cm which is equal to 2f
∴ Image will be real and at a distance of 40 cm from convergent lens.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 7

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?

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 7





JEE Main 2017 Question Paper with Solutions (2nd April) - Question 8

An electron beam is accelerated 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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 8

In X-ray tube


Slope is negative
Intercept on y-axis is positive

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 9

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 9




JEE Main 2017 Question Paper with Solutions (2nd April) - Question 10

An electric dipole has a fixed dipole moment  which makes angle θ with respect to x-axis. When subjected to an electric field it experiences a torque  When subjected to another electric field  it experiences a torque  The angle θ is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 10






From (i) and (ii)

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 11

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 11

In common emitter configuration for n-p-n transistor, phase difference between output and input voltage is 180°.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 12

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 :

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 12

Let molar heat capacity at constant pressure = Xand molar heat capacity at constant volume = Xv
Xp – Xv =R
MCp  – MCv = R
Cp  – Cv = R/M
For hydrogen; a = R/2
For N2; b = R/28
a/b = 14
a = 14b

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 13

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)

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 13

100 × 0.1 × (t – 75) = 100 × 0.1 × 45 + 170 × 1 × 45
10t – 750 = 450 + 7650
10t = 1200 + 7650
10t = 8850
t = 885°C

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 14

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 1/8 mv02 the value of k will be

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 14







JEE Main 2017 Question Paper with Solutions (2nd April) - Question 15

When a current of 5 mA is passed through a galvanometer having a coil of resistance 15 Ω, it shows full scale deflection. 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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 15

ig = 5 × 10–3 A
G = 15 Ω
Let series resistance be R.
V = ig (R + G)
10 = 5 × 10–3 (R + 15)
R = 2000 – 15 = 1985 = 1.985 × 103
 

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 16

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 16

Torque at angle θ



JEE Main 2017 Question Paper with Solutions (2nd April) - Question 17

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 17

From energy level diagram


JEE Main 2017 Question Paper with Solutions (2nd April) - Question 18

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 a factor of

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 18


∵ Density remains same
So, mass ∝ Volume

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 19

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 19



Magnitude of change in flux = R × area under current vs time graph

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 20

In a Young's double slit experiment, slit s 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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 20


JEE Main 2017 Question Paper with Solutions (2nd April) - Question 21

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 21



For 10 oscillations,
t = 10T = 2π × 1.06
= 6.6568 ≈ 6.65 s

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 22

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 22


Variation of g inside earth surface


JEE Main 2017 Question Paper with Solutions (2nd April) - Question 23


In the above circuit the current in each resistance is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 23

The potential difference in each loop is zero.
∴ No current will flow. 

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 24

with a particle B of mass m/2 which is at rest. Thecollision is head on, and elastic. The ratio of the de-Broglie wavelengths λA to λB after the collision is 

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 24




JEE Main 2017 Question Paper with Solutions (2nd April) - Question 25

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 a 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 :

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 25




JEE Main 2017 Question Paper with Solutions (2nd April) - Question 26

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 second will be

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 26


JEE Main 2017 Question Paper with Solutions (2nd April) - Question 27

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 27

For relativistic motion
; v = relative speed of approach

 

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 28

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 28

In steady state, flow of current through capacitor will be zero.



JEE Main 2017 Question Paper with Solutions (2nd April) - Question 29

A capacitance of 2 μF is required in an electrical circuit across a potential difference of 1.0 kV. A large number of 1 μF 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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 29

Following arrangement will do the needful :
8 capacitors of 1μF in parallel with four such branches in series.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 30

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 30

Acceleration is constant and negative

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 31

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 31

(a) Nitration reactions take place in presence of concentrated HNO3+ concentrated H2SO4.
(b) Aniline acts as a base. In presence of H2SO4, its protonation takes place and anilinium ion is formed
(c)Anilinium ion is a strongly deactivating group and meta directing in nature so it gives meta nitration product in a significant amount.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 32

ΔU is equal to

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 32

For adiabatic process, q = 0
∴ As per 1st law of thermodynamics,
ΔU = W

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 33

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 33

Rate of SN1 reaction ∝ stability of carbocation


So, II < I < III
Increase stability of carbocation and hence increase reactivity of halides.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 34

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
ε0 = 8.854185 × 10–12 kg–1 m–3 A2)

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 34


 = 2.12 Å

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 35

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 35



= 6.9

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 36

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 36

Caprolactam is hydrolysed to produce caproic acid which undergoes condensation to produce Nylon-6.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 37

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 37

Mass of hydrogen = 10/100 x 75 = 7.5 kg
Replacing 1H by 2H would replace 7.5 kg with 15 kg
∴ Net gain = 7.5 kg

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 38

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 38


The above product does not have any C = C or C ≡ C bond, so, it will not give Br2-water test.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 39

In the following reactions, ZnO is respectively acting as a/an
(a) ZnO + Na2O → Na2ZnO2
(b) ZnO + CO2 → ZnCO3

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 39

In (a), ZnO acts as acidic oxide as Na2
O is basic oxide.
In (b), ZnO acts as basic oxide as CO2
 is acidic oxide.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 40

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 40

Mg forms basic carbonate
3MgCO3 · Mg(OH)2 ·3H2O but no such basic carbonate is formed by Li.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 41

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 41


Since product (X) contains two chiral centres and it is unsymmetrical.
So, its total stereoisomers = 22 = 4.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 42

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 42

In FCC, one of the face is like

By ΔABC,
2a2 = 16r2

Distance of closest approach = 2r = a/√2

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 43

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 43

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 44

The correct sequence of reagents for the following conversion will be

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 44


JEE Main 2017 Question Paper with Solutions (2nd April) - Question 45

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 particle 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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 45

For Tyndall effect refractive index of dispersion phase and dispersion medium must differ significantly. Secondly, size of dispersed phase should not differ much from wavelength used.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 46

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 46

Sugars in which there is free anomeric –OH group are reducing sugars

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 47

Given
C(graphite) + O2(g) →CO2(g);
ΔrHº = –393.5 kJ mol–1
H2(g) + 1/2 O2 (g) → H2O(l);
ΔrH0 = -285.8 kJ mol-1
CO2(g) + 2H2O(l) → CH4(g) + 2O2(g);
ΔrH0 = +890.3 kJ mol-1
Based on the above thermochemical equations, the value of ΔrHº at 298 K for the reaction
C(graphite) + 2H2(g) → CH4(g) wil1 be

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 47



By applying the operation
(i) + 2 × (ii) + (iii), we get

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 48

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 48

Xe is oxidised from +4(in XeF4) to +6(in XeF6)
Oxygen is reduced from +1 (in O2F2) to zero (in O2)

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 49

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 49

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 50

The major product obtained in the following reaction is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 50

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 51

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 51


JEE Main 2017 Question Paper with Solutions (2nd April) - Question 52

Which of the following species is not paramagnetic?

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 52

CO has 14 electrons (even) ∴ it is diamagnetic
NO has 15e-(odd) ∴ it is paramagnetic and has 1 unpaired electron in π*2p molecular orbital.
B2 has 10e- (even) but still paramagnetic and has two unpaired electrons in π2px and π2py (s-p mixing).
O2 has 16 e- (even) but still paramagnetic and has two unpaired electrons in π*2px and π*2py molecular

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 53

The freezing point of benzene decreases by 0.45ºC 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)

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 53


⇒ i = 0.527




α = 0.946
∴ % association = 94.6%

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 54

Which of the following molecules is least resonance stabilized?

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 54

However, all molecules given in options are stabilised by resonance but compound given in option (4) is least resonance stabilised (other three are aromatic)

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 55

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 55

Millimoles of AgNO3  = 
Millimoles of CoCl3·6H2O = 0.1 × 100 = 10
∴ Each mole of CoCl3·6H2O gives two chloride ions.
∴ [Co(H2O)5Cl]Cl2-H2O

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 56

The major product obtained in the following reaction is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 56

DIBAL — H reduces esters and carboxylic acids into aldehydes

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 57

A water sample has ppm level concentration of following anions
F- = 10; SO42- = 100; NO3- = 50
The anion/anions that make/makes the water sample unsuitable for drinking is/are

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 57

Permissible limit of F in drinking water is upto 1 ppm. Excess concentration of F
> 10 ppm causes decay of bones. 

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 58

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 58

M2CO3 + 2HCl → 2MCl + H2O + CO2

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 59

Given


Among the following, the strongest reducing agent is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 59


Positive E° is for Cr, hence it is strongest reducing agent.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 60

The group having isoelectronic species is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 60

Mg2+, Na+, O2– and Fall have 10 electrons each.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 61

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 61



(k2 - 7k+ 10) + 4k2 + 20k = ± 56
5k2 +13k+ 10 = ± 56

= 2, –4.6


Equation of AD,
x = 2 ...(i)
Also equation of BE,

2y -4=x - 5
x - 2y-1 = 0 ...(ii)
Solving (i) & (ii), 2y = 1
y = 1/2
Orthocentre is 

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 62

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 62

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 63

The function  defined as is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 63


f '(x) changes sign in different intervals.
∴ Not injective.

yx2 - x+ y = 0
For y ≠ 0

For, y = 0  ⇒ x = 0
∴ Part of range

∴ Surjective but not injective.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 64

The following statement (p → q) → [(~ p → q) → q] is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 64

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 65

If 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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 65


⇒ –(1 – a)2 = 0
⇒ a = 1
For a = 1
Eq. (1) & (2) are identical i.e.,x + y + z = 1
To have no solution with x + by + z = 0
b = 1

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 66

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 66


Area of shaded region

= 5/2 sq. unit

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 67

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 67

9(25a2 + b2) + 25 (c2 - 3ac) = 15b (3a + c)
⇒ (15a)2 + (3b)2 + (5c)2 - 45ab - 15bc - 75ac = 0
⇒ (15a - 3b)2 + (3b - 5c)2 + (15a - 5c)2 = 0
It is possible when
15a - 3b= 0 and 3b -5c= 0 and 15a - 5c= 0
15a = 3b= 5c

∴ b, c, a are in A.P.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 68

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 68

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 69

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 69

y (x -2)( x- 3) = x + 6
At y-axis, x = 0, y = 1
Now, on differentiation.

Now slope of normal = –1
Equation of normal  y – 1 = –1(x – 0)
y + x – 1 = 0 ... (i)
Line (i) passes through (1/2,1/2)

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 70

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 70


a2 +b2= 4


⇒ b2 = 3
∴ a2= 1

∴ Tangent at 
Clearly it passes through (2√2, 3√3 )

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 71

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

then is equal to

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 71

As, f (x + y)= f (x) + f (y) + xy
Given, f (1) = 3
Putting, x =y= 1 ⇒ f (2) = 2f (1)+ 1 = 7
Similarly, x = 1,y= 2 ⇒ f (3) =f (1)+ f (2) + 2 = 12

= 3 + 7 + 12 + 18 + ... = S (let)
Now, S = 3+ 7 + 12 + 18 + ... +tn
Again, S = 3+7 + 12 + ... +tn -1 + tn
We get, tn = 3+ 4 + 5 + ... terms




JEE Main 2017 Question Paper with Solutions (2nd April) - Question 72

Let  and be a vector such that  and the angle between be 30°. Then is equal to

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 72




JEE Main 2017 Question Paper with Solutions (2nd April) - Question 73

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 tan β is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 73





JEE Main 2017 Question Paper with Solutions (2nd April) - Question 74

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 74


2r + θr =20 ... (i)



A to be maximum


Hence for r = 5, A is maximum
Now, 10 + θ·5 = 20 ⇒ θ = 2 (radian)

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 75

The integral  is equal to

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 75






JEE Main 2017 Question Paper with Solutions (2nd April) - Question 76

If ( 2 + sinx) dy/dx + (y + 1)cos x = 0 and y(0) = 1, then y(π/2) is equal to 

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 76




ln| y + 1 |+ ln (2 + sinx ) = lnC
(y + 1) (2+ sinx ) = C
Put x = 0, y = 1
(1 + 1)× 2 = C ⇒ C = 4
Now, ( y +1) (2+ sinx ) = 4

( y +1) (2+ 1) = 4

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 77

Let In = tann xdx,(n > 1). If I4 +I6= a tan5 x + bx5 + C, where C is a constant of integration, then the ordered pair (a, b) is equal to

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 77



Let tanx = t
sec2x dx = dt

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 78

Let ω be a complex number such that 2ω + 1 = z where z =√-3 . If
 then k is equal to

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 78

2ω + 1 = z , z = √3i

C1 → C1 + C2 + C3

= 3 (ω2 - ω4)

= -3√3i
= –3z
∴ k = –z

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 79

The value of
(21C1 - 10C1) + (21C2 - 10C2) + (21C3 - 10C3) + (21C4 - 10C4) +... + (21C10 - 10C10) is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 79

21C1 + 21C2 +... + 21C10 = 1/2 {21C0 + 21C1 +... + 21C21}-1
= 220 – 1
(10C1 + 10C2 +... + 10C10) = 210 -1
∴ Required sum = (220 – 1) – (210 – 1)
= 220 – 210

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 80

equals

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 80





= 1/16.

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 81

If 5 (tan2x – cos2x) = 2cos 2x + 9, then the value of cos 4x is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 81

5 tan2x = 9 cos2x + 7
5 sec2x – 5 = 9 cos2x + 7
Let cos2x = t



 cos4x = 2 cos2 2x – 1

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 82

If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to the line, x/1 = y/4 = z/5 is Q, then PQ is equal to

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 82

Equation of PQ,

Let M be (λ+1, 4λ- 2, 5λ+ 3)

As it lies on 2x + 3y – 4z + 22 = 0
λ = 1
For Q, λ = 2
Distance PQ 

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 83

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  and  is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 83

Let the plane be
a(x - 1)+ b( y + 1) + c(z +1) = 0
It is perpendicular to the given lines
a – 2b + 3c = 0
Solving, a : b : c = 5 : 7 : 3
∴ The plane is 5x + 7y + 3z + 5 = 0
Distance of (1, 3, –7) from this plane = 10/√83

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 84

If for x∈  the derivative of tan -1  is √x × g(x) , then g(x) equals

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 84



JEE Main 2017 Question Paper with Solutions (2nd April) - Question 85

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 85


x2 =-(y -4)
Let a point on the parabola 
Equation of normal at P is


It passes through centre of circle, say (0, k)


(Length of perpendicular from (0, k) to y = x)

Equation of circle is

It passes through point P

t4 +t2 (8k- 28) + 8k2 - 128k + 256 = 0
For t = ⇒  k2 - 16k+ 32 = 0
k = 8±4√2

(14 - 4k )2+ (14 - 4k ) (8k - 28) + 8k2 -128k + 256 = 0
2k2 +4k-15 = 0


From (iii) & (iv),

But from options, r = 4(√2-1)

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 86

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 86

n = 10
p(Probability of drawing a green ball) = 15/25

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 87

The eccentricity of an ellipse whose centre is at the origin is 1/2. If one of its directrices is x = – 4, then
the equation of the normal to it at  is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 87


x = –4


-a= -4 x e
a = 2
Now,  b2 =a2 (1- e2 ) = 3
Equation to ellipse

Equation of normal is

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 88

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

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 88

Total number of ways = 11C2 = 55
Favourable ways are
(0, 4), (0, 8), (4, 8), (2, 6), (2, 10), (6, 10)
Probability = 6/55

JEE Main 2017 Question Paper with Solutions (2nd April) - Question 89

 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) = 1/4 and P(All the three events occur simultaneously) = 1/16 Then the probability that at least one of the events occurs, is

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 89






JEE Main 2017 Question Paper with Solutions (2nd April) - Question 90

,then adj (3A2 + 12A) is equal to

Detailed Solution for JEE Main 2017 Question Paper with Solutions (2nd April) - Question 90



= (2 – 2λ- λ + λ2 ) - 12
f (λ)= λ2 - 3λ - 10
∵ A satisfies f (λ )
∴ A2 – 3A –10I = 0
A2 – 3A = 10I
3A2 – 9A = 30I
3A2 + 12A = 30I + 21A


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