NTA JEE Main 2006

NTA JEE Main 2006 - Question 1

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Which of the following units denotes the dimensions $$ML^2/Q^2$$, where Q denotes the electric charge?

A

Weber (Wb)

B

$$Wb/m^2$$

C

Henry (H)

D

$$H/m^2$$

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NTA JEE Main 2006 - Question 2

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A particle located at $$x = 0$$ at time $$t = 0$$, starts moving along the positive $$x$$-direction with a velocity '$$v$$' that varies as $$v = \alpha\sqrt{x}$$. The displacement of the particle varies with time as

A

$$t^3$$

B

$$t^2$$

C

$$t$$

D

$$t^{1/2}$$

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NTA JEE Main 2006 - Question 3

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A body falling from rest under gravity passes a certain point $$P$$. It was at a distance of $$400\,m$$ from $$P$$, $$4\,s$$ prior to passing through $$P$$. If $$g = 10\,m/s^2$$, then the height above the point $$P$$ from where the body began to fall is

A

720 m

B

900 m

C

320 m

D

680 m

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NTA JEE Main 2006 - Question 4

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A mass of $$M\,kg$$ is suspended by a weightless string. The horizontal force that is required to displace it until the string makes an angle of $$45^\circ$$ with the initial vertical direction is

A

$$Mg(\sqrt{2} - 1)$$

B

$$Mg(\sqrt{2} + 1)$$

C

$$Mg\sqrt{2}$$

D

$$\dfrac{Mg}{\sqrt{2}}$$

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NTA JEE Main 2006 - Question 5

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A player caught a cricket ball of mass $$150\,g$$ moving at a rate of $$20\,m/s$$. If the catching process is completed in $$0.1\,s$$, the force of the blow exerted by the ball on the hand of the player is equal to

A

300 N

B

150 N

C

3 N

D

30 N

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NTA JEE Main 2006 - Question 6

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A particle of mass $$100\,g$$ is thrown vertically upwards with a speed of $$5\,m/s$$. The work done by the force of gravity during the time the particle goes up is

A

0.5 J

B

$$-0.5\,J$$

C

$$-1.25\,J$$

D

1.25 J

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NTA JEE Main 2006 - Question 7

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A ball of mass $$0.2\,kg$$ is thrown vertically upwards by applying a force by hand. If the hand moves $$0.2\,m$$ while applying the force and the ball goes upto $$2\,m$$ height further, find the magnitude of the force. Consider $$g = 10\,m/s^2$$

A

22 N

B

4 N

C

16 N

D

20 N

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NTA JEE Main 2006 - Question 8

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The potential energy of a $$1\,kg$$ particle free move along the x-axis is given by $$$V(x) = \left(\frac{x^4}{4} - \frac{x^2}{2}\right)J$$$ The total mechanical energy of the particle $$2\,J$$. Then, the maximum speed (in $$m/s$$) is

A

2

B

$$3/\sqrt{2}$$

C

$$\sqrt{2}$$

D

$$1/\sqrt{2}$$

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NTA JEE Main 2006 - Question 9

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A bomb of mass $$16\,kg$$ at rest explodes into two pieces of masses of $$4\,kg$$ and $$12\,kg$$. The velocity of the $$12\,kg$$ mass is $$4\,ms^{-1}$$. The kinetic energy of the other mass is

A

96 J

B

144 J

C

288 J

D

192 J

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NTA JEE Main 2006 - Question 10

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Consider a two particle system with particles having masses $$m_1$$ and $$m_2$$. If the first particle is pushed towards the centre of mass through a distance $$d$$, by what distance should the second particle be moved, so as to keep the centre of mass at the same position?

A

$$d$$

B

$$\dfrac{m_2}{m_1}d$$

C

$$\dfrac{m_1}{m_1+m_2}d$$

D

$$\dfrac{m_1}{m_2}d$$

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NTA JEE Main 2006 - Question 11

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A force of $$-F\hat{k}$$ acts on $$O$$, the origin of the coordinate system. The torque about the point $$(1, -1)$$ is

A

$$-F(\hat{i} - \hat{j})$$

B

$$F(\hat{i} - \hat{j})$$

C

$$-F(\hat{i} + \hat{j})$$

D

$$F(\hat{i} + \hat{j})$$

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NTA JEE Main 2006 - Question 12

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A thin circular ring of mass $$m$$ and radius $$R$$ is rotating about its axis with a constant angular velocity $$\omega$$. Two objects each of mass $$M$$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity $$\omega' =$$

A

$$\dfrac{\omega m}{(m+2M)}$$

B

$$\dfrac{\omega(m+2M)}{m}$$

C

$$\dfrac{\omega(m-2M)}{(m+2M)}$$

D

$$\dfrac{\omega m}{(m+M)}$$

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NTA JEE Main 2006 - Question 13

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Four point masses, each of value $$m$$, are placed at the corners of a square $$ABCD$$ of side $$\ell$$. The moment of inertia of system about an axis passing through $$A$$ and parallel to $$BD$$ is

A

$$m\ell^2$$

B

$$2m\ell^2$$

C

$$3m\ell^2$$

D

$$3\,m\ell^2$$

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NTA JEE Main 2006 - Question 14

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A wire elongates by $$\ell\,mm$$ when a load $$W$$ is hanged from it. If the wire goes over a pulley and two weights $$W$$ each are hung at the two ends, the elongation of the wire will be (in $$mm$$)

A

$$\ell/2$$

B

$$\ell$$

C

$$2\ell$$

D

zero

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NTA JEE Main 2006 - Question 15

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If the terminal speed of a sphere of gold (density $$= 19.5\,kg/m^3$$) is $$0.2\,m/s$$ in a viscous liquid (density $$= 1.5\,kg/m^3$$) of the same size in the same liquid.

A

0.2 m/s

B

0.4 m/s

C

0.133 m/s

D

0.1 m/s

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NTA JEE Main 2006 - Question 16

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Assuming the sun to be a spherical body of radius $$R$$ at a temperature of $$T\,K$$, evaluate the total radiant power, incident on Earth, at a distance $$r$$ from the Sun.

A

$$\dfrac{R^2\sigma T^4}{r^2}$$

B

$$\dfrac{4\pi r_0^2 R^2 \sigma T^4}{r^2}$$

C

$$\dfrac{\pi r_0^2 R^2 \sigma T^4}{r^2}$$

D

$$\dfrac{r_0^2 R^2 \sigma T^4}{4\pi r^2}$$

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NTA JEE Main 2006 - Question 17

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The work of $$146\,kJ$$ is performed in order to compress one kilo mole of gas adiabatically and in this process the temperature of the gas increases by $$7^\circ C$$. The gas is $$(R = 8.3\,J\,mol^{-1}\,K^{-1})$$

A

monoatomic

B

diatomic

C

triatomic

D

a mixture of monoatomic and diatomic

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NTA JEE Main 2006 - Question 18

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Two rigid boxes containing different ideal gases are placed on a table. Box A contains one mole of nitrogen at temperature $$T_0$$, while Box $$B$$ contains one mole of helium at temperature $$(7/3)T_0$$. The boxes are then put into thermal contact with each other and heat flows between them until the gases reach a common final temperature. (Ignore the heat capacity of boxes). Then, the final temperature of the gases, $$T_f$$, in terms of $$T_0$$ is

A

$$T_f = \dfrac{5}{2}T_0$$

B

$$T_f = \dfrac{3}{7}T_0$$

C

$$T_f = \dfrac{7}{3}T_0$$

D

$$T_f = \dfrac{3}{2}T_0$$

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NTA JEE Main 2006 - Question 19

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Starting from the origin, a body oscillates simple harmonically with a period of $$2\,s$$. After what time will its kinetic energy be 75% of the total energy?

A

$$\dfrac{1}{12}\,s$$

B

$$\dfrac{1}{6}\,s$$

C

$$\dfrac{1}{4}\,s$$

D

$$\dfrac{1}{3}\,s$$

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NTA JEE Main 2006 - Question 20

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The maximum velocity of a particle, executing simple harmonic motion with an amplitude $$7\,mm$$, is $$4.4\,m/s$$. The period of oscillation is

A

100 s

B

0.01 s

C

10 s

D

0.1 s

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NTA JEE Main 2006 - Question 21

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A coin is placed on a horizontal platform which undergoes vertical simple harmonic motion of angular frequency $$\omega$$. The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time

A

at the highest position of the platform

B

at the mean position of the platform

C

for an amplitude of $$\dfrac{g}{\omega^2}$$

D

for an amplitude of $$\dfrac{g^2}{\omega^2}$$

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NTA JEE Main 2006 - Question 22

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A whistle producing sound waves of frequencies $$9500\,Hz$$ and above is approaching a stationary person with speed $$v\,ms^{-1}$$. The velocity of sound in air is $$300\,ms^{-1}$$. If the person can hear frequencies upto a maximum of $$10{,}000\,Hz$$, the maximum value of $$v$$ upto which he can hear the whistle is

A

$$30\,ms^{-1}$$

B

$$15\sqrt{2}\,ms^{-1}$$

C

$$15/\sqrt{2}\,ms^{-1}$$

D

$$15\,ms^{-1}$$

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NTA JEE Main 2006 - Question 23

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A string is stretched between fixed points separated by $$75\,cm$$. It is observed to have resonant frequencies of $$420\,Hz$$ and $$315\,Hz$$. There are no other resonant frequencies between these two. Then, the lowest resonant frequency for this string is

A

10.5 Hz

B

105 Hz

C

1.05 Hz

D

1050 Hz

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NTA JEE Main 2006 - Question 24

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A electric dipole is placed at an angle of $$30^\circ$$ to a non-uniform electric field. The dipole will experience

A

a torque only

B

a translational force only in the direction of the field

C

a translational force only in a direction normal to the direction of the field

D

a torque as well as a translational force

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NTA JEE Main 2006 - Question 25

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Two insulating plates are both uniformly charged in such a way that the potential difference between them is $$V_2 - V_1 = 20\,V$$. (i.e. plate 2 is at a higher potential). The plates are separated by $$d = 0.1\,m$$ and can be treated as infinitely large. An electron is released from rest on the inner surface of plate 1. What is its speed when it hits plate 2? $$(e = 1.6\times 10^{-19}\,C, m_e = 9.11\times 10^{-31}\,kg)$$

A

$$32\times 10^{-19}\,m/s$$

B

$$2.65\times 10^{6}\,m/s$$

C

$$7.02\times 10^{12}\,m/s$$

D

$$1.87\times 10^{6}\,m/s$$

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NTA JEE Main 2006 - Question 26

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Two spherical conductors $$A$$ and $$B$$ of radii $$1\,mm$$ and $$2\,mm$$ are separated by a distance of $$5\,cm$$ and are uniformly charged. If the spheres are connected by a conducting wire then in equilibrium condition, the ratio of the magnitude of the electric fields at the surface of spheres $$A$$ and $$B$$ is

A

1 : 4

B

4 : 1

C

1 : 2

D

2 : 1

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NTA JEE Main 2006 - Question 27

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A material '$$B$$' has twice the specific resistance of '$$A$$'. A circular wire made of '$$B$$' has twice the diameter of a wire made of '$$A$$'. Then for the two wires to have the same resistance, the ratio $$\ell_A/\ell_B$$ of their respective lengths must be

A

2

B

1

C

$$\dfrac{1}{2}$$

D

$$\dfrac{1}{4}$$

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NTA JEE Main 2006 - Question 28

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The Kirchhoff's first law $$\left(\sum i = 0\right)$$ and second law $$\left(\sum iR = \sum E\right)$$, where the symbols have their usual meanings, are respectively based on

A

conservation of charge, conservation of energy

B

conservation of charge, conservation of momentum

C

conservation of energy, conservation of charge

D

conservation of momentum, conservation of charge

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NTA JEE Main 2006 - Question 29

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In a Wheatstone's bridge, there resistances $$P, Q$$ and $$R$$ connected in the three arms and the fourth arm is formed by two resistances $$S_1$$ and $$S_2$$ connected in parallel. The condition for bridge to be balanced will be

A

$$\dfrac{P}{Q} = \dfrac{R}{S_1 + S_2}$$

B

$$\dfrac{P}{Q} = \dfrac{2R}{S_1 + S_2}$$

C

$$\dfrac{P}{Q} = \dfrac{R(S_1 + S_2)}{S_1 S_2}$$

D

$$\dfrac{P}{Q} = \dfrac{R(S_1 + S_2)}{2 S_1 S_2}$$

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NTA JEE Main 2006 - Question 30

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Consider the circuit shown below. The current $$I$$ drawn from the 5 volt source will be

A

0.17 A

B

0.33 A

C

0.5 A

D

0.67 A

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NTA JEE Main 2006 - Question 31

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The resistance of a bulb filament is $$100\,\Omega$$ at a temperature of $$100^\circ C$$. If its temperature coefficient of resistance be $$0.005$$ per $$^\circ C$$, its resistance will become $$200\,\Omega$$ at a temperature of

A

$$200^\circ C$$

B

$$300^\circ C$$

C

$$400^\circ C$$

D

$$500^\circ C$$

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NTA JEE Main 2006 - Question 32

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A thermocouple is made from two metals, Antimony and Bismuth. If one junction of the couple is kept hot and the other is kept cold then, an electric current will

A

flow from Antimony to Bismuth at the cold junction

B

flow from Antimony to Bismuth at the hot junction

C

flow from Bismuth to Antimony at the cold junction

D

not flow through the thermocouple

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NTA JEE Main 2006 - Question 33

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An electric bulb is rated $$220$$ volt $$- 100$$ watt. The power consumed by it when operated on $$110$$ volt will be

A

50 watt

B

75 watt

C

40 watt

D

25 watt

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NTA JEE Main 2006 - Question 34

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Needles $$N_1, N_2$$ and $$N_3$$ are made of a ferromagnetic, a paramagnetic and a diamagnetic substance respectively. A magnet when brought close to them will

A

attract all three of them

B

attract $$N_1$$ and $$N_2$$ strongly but repel $$N_3$$

C

attract $$N_1$$ strongly, $$N_2$$ weakly and repel $$N_3$$ weakly

D

attract $$N_1$$ strongly, but repel $$N_2$$ and $$N_3$$ weakly

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NTA JEE Main 2006 - Question 35

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In a region, steady and uniform electric and magnetic fields are present. These two fields are parallel to each other. A charged particle is released from rest in this region. The path of the particle will be a

A

circle

B

helix

C

straight line

D

ellipse

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NTA JEE Main 2006 - Question 36

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A long solenoid has 200 turns per $$cm$$ and carries a current $$i$$. The magnetic field at its centre is $$6.28 \times 10^{-2}\,Wb/m^2$$. Another long solenoid has 100 turns per $$cm$$ and it carries a current $$i/3$$. The value of the magnetic field at its centre is

A

$$1.05 \times 10^{-4}\,Wb/m$$

B

$$1.05 \times 10^{-2}\,Wb/m^2$$

C

$$1.05 \times 10^{-5}\,Wb/m^2$$

D

$$1.05 \times 10^{-3}\,Wb/m^2$$

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NTA JEE Main 2006 - Question 37

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In an AC generator, a coil with $$N$$ turns, all of the same area $$A$$ and total resistance $$R$$, rotates with frequency $$\omega$$ in a magnetic field $$B$$. The maximum value of emf generated in the coil is

A

$$N.A.B.\omega$$

B

$$N.A.B.R.\omega$$

C

$$N.A.B$$

D

$$N.A.B.R$$

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NTA JEE Main 2006 - Question 38

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The flux linked with a coil at any instant '$$t$$' is given by $$\phi = 10t^2 - 50t + 250$$. The induced emf at $$t = 3\,s$$ is

A

190 V

B

$$-190$$ V

C

$$-10$$ V

D

10 V

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NTA JEE Main 2006 - Question 39

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In a series resonant LCR circuit, the voltage across $$R$$ is $$100$$ volts and $$R = 1\,k\Omega$$ with $$C = 2\,\mu F$$. The resonant frequency $$\omega$$ is $$200\,rad/s$$. At resonance the voltage across $$L$$ is

A

$$4 \times 10^{-3}\,V$$

B

$$2.5 \times 10^{-2}\,V$$

C

40 V

D

250 V

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NTA JEE Main 2006 - Question 40

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An inductor $$(L = 100\,mH)$$, a resistor $$(R = 100\,\Omega)$$ and a battery $$(E = 100\,V)$$ are initially connected in series as shown in the figure. After a long time the battery is disconnected after short circuiting the points $$A$$ and $$B$$.

What is the  current in the circuit  after $$1\,ms$$ ?

A

1 A

B

$$1/e\,A$$

C

$$e\,A$$

D

0.1 A

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NTA JEE Main 2006 - Question 41

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The rms value of the electric field of the light coming from the Sun is $$720\,N/C$$. The average total energy density of the electromagnetic wave is

A

$$3.3 \times 10^{-3}\,J/m^3$$

B

$$4.58 \times 10^{-6}\,J/m^3$$

C

$$6.37 \times 10^{-9}\,J/m^3$$

D

$$81.35 \times 10^{-12}\,J/m^3$$

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NTA JEE Main 2006 - Question 42

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The refractive index of glass is $$1.520$$ for red light and $$1.525$$ for blue light. Let $$D_1$$ and $$D_2$$ be the angles of minimum deviation for red and blue light respectively in a prism of this glass. Then

A

$$D_1 > D_2$$

B

$$D_1 < D_2$$

C

$$D_1 = D_2$$

D

$$D_1$$ can be less than or greater than depending upon the angle of prism

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NTA JEE Main 2006 - Question 43

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The time by a photoelectron to come out after the photon strikes is approximately

A

$$10^{-1}\,s$$

B

$$10^{-4}\,s$$

C

$$10^{-10}\,s$$

D

$$10^{-16}\,s$$

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NTA JEE Main 2006 - Question 44

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The threshold frequency for a metallic surface corresponds to an energy of $$6.2\,eV$$, and the stopping potential for a radiation incident on this surface $$5\,V$$. The incident radiation lies in

A

X-ray region

B

ultra-violet region

C

infra-red region

D

visible region

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NTA JEE Main 2006 - Question 45

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The anode voltage of a photocell is kept fixed. The wavelength $$\lambda$$ of the light falling on the cathode is gradually changed. The plate current $$I$$ of the photocell varies as follows :

A

B

C

D

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NTA JEE Main 2006 - Question 46

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An alpha nucleus of energy $$\dfrac{1}{2}mv^2$$ bombards a heavy nuclear target of charge $$Ze$$. Then the distance of closest approach for the alpha nucleus will be proportional to

A

$$\dfrac{1}{Ze}$$

B

$$v^2$$

C

$$\dfrac{1}{m}$$

D

$$\dfrac{1}{v^4}$$

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NTA JEE Main 2006 - Question 47

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The energy spectrum of $$\beta$$-particles [number $$N(E)$$ as a function of $$\beta$$-energy $$E$$] emitted from a radioactive source is

A

B

C

D

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NTA JEE Main 2006 - Question 48

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When $$_3Li^7$$ nuclei are bombarded by protons, and the resultant nuclei are $$_4Be^8$$, the emitted particles will be

A

neutrons

B

alpha particles

C

beta particles

D

gamma photons

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NTA JEE Main 2006 - Question 49

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A solid which is transparent to visible light and whose conductivity increases with temperature is formed by

A

Metallic binding

B

Ionic binding

C

Covalent binding

D

Van der Waals binding

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NTA JEE Main 2006 - Question 50

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The 'rad' is the correct unit used to report the measurement of

A

the rate of decay of radioactive source

B

the ability of a beam of gamma ray photons to produce ions in a target

C

the energy delivered by radiation to a target.

D

the biological effect of radiation

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NTA JEE Main 2006 - Question 51

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If the binding energy per nucleon in $$_3^7Li$$ and $$_2^4He$$ nuclei are $$5.60\,MeV$$ and $$7.06\,MeV$$ respectively, then in the reaction $$p + _3^7Li \to 2\,_2^4He$$ energy of proton must be

A

$$39.2\,MeV$$

B

$$28.24\,MeV$$

C

$$17.28\,MeV$$

D

$$1.46\,MeV$$

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NTA JEE Main 2006 - Question 52

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If the ratio of the concentration of electrons that of holes in a semiconductor is $$\dfrac{7}{5}$$ and the ratio of currents is $$\dfrac{7}{4}$$, then what is the ratio of their drift velocities?

A

$$\dfrac{4}{7}$$

B

$$\dfrac{5}{8}$$

C

$$\dfrac{4}{5}$$

D

$$\dfrac{5}{4}$$

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NTA JEE Main 2006 - Question 53

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In a common base mode of a transistor, the collector current is $$5.488\,mA$$ for an emitter current of $$5.60\,mA$$. The value of the base current amplification factor $$(\beta)$$ will be

A

48

B

49

C

50

D

51

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NTA JEE Main 2006 - Question 54

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If the lattice constant of this semiconductor is decreased, then which of the following is correct?

A

All $$E_c, E_g, E_v$$ decrease

B

All $$F_F$$ increase

C

$$E_c$$, and $$E_v$$ increase but $$E_g$$ decreases

D

$$E_c$$, and $$E_v$$ decrease $$E_g$$ increases

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NTA JEE Main 2006 - Question 55

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In the following, which one of the diodes is reverse biased?

A

B

C

D

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NTA JEE Main 2006 - Question 56

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The circuit has two oppositely connect ideal diodes in parallel. What is the current following in the circuit?

A

1.33 A

B

1.71 A

C

2.00 A

D

2.31 A

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NTA JEE Main 2006 - Question 57

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How many moles of magnesium phosphate, $$Mg_3(PO_4)_2$$ will contain $$0.25$$ mole of oxygen atoms?

A

0.02

B

$$3.125 \times 10^{-2}$$

C

$$1.25 \times 10^{-2}$$

D

$$2.5 \times 10^{-2}$$

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NTA JEE Main 2006 - Question 58

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Density of a $$2.05\,M$$ solution of acetic acid in water is $$1.02\,g/mL$$. The molality of the solution is

A

$$1.14\,mol\,kg^{-1}$$

B

$$2.28\,mol\,kg^{-1}$$

C

$$3.28\,mol\,kg^{-1}$$

D

$$0.44\,mol\,kg^{-1}$$

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NTA JEE Main 2006 - Question 59

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According to Bohr's theory, the angular momentum of an electron in $$5^{th}$$ orbit is

A

$$25\dfrac{h}{\pi}$$

B

$$1.0\dfrac{h}{\pi}$$

C

$$10\dfrac{h}{\pi}$$

D

$$2.5\dfrac{h}{\pi}$$

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NTA JEE Main 2006 - Question 60

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In the transformation of $$_{92}^{238}U$$ to $$_{92}^{234}U$$, if one emission is an $$\alpha$$-particle, what should be the other emission(s)?

A

Two $$\beta^-$$

B

Two $$\beta^-$$ and one $$\beta^+$$

C

One $$\beta^-$$ and one $$\gamma$$

D

One $$\beta^+$$ and one $$\beta^-$$

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NTA JEE Main 2006 - Question 61

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Uncertainty in the position of an electron (mass $$= 9.1 \times 10^{-31}\,kg$$) moving with a velocity $$300\,ms^{-1}$$, accurate upto $$0.001\%$$, will be

A

$$19.2 \times 10^{-2}\,m$$

B

$$5.76 \times 10^{-2}\,m$$

C

$$1.92 \times 10^{-2}\,m$$

D

$$3.84 \times 10^{-2}\,m$$ $$(h = 6.63 \times 10^{-34}\,Js)$$

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NTA JEE Main 2006 - Question 62

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Which one of the following sets of ions represents a collection of isoelectronic species?

A

$$K^+, Cl^-, Ca^{2+}, Sc^{3+}$$

B

$$Ba^{2+}, Sr^{2+}, K^+, S^{2-}$$

C

$$N^{3-}, O^{2-}, F^-, S^{2-}$$

D

$$Li^+, Na^+, Mg^{2+}, Ca^{2+}$$

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NTA JEE Main 2006 - Question 63

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The "spin-only" magnetic moment [in units of Bohr magneton, $$(\mu_B)$$] of $$Ni^{2+}$$ in aqueous solution would be (Atomic number of $$Ni = 28$$)

A

2.84

B

4.90

C

0

D

1.73

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NTA JEE Main 2006 - Question 64

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The increasing order of the first ionization enthalpies of the elements B, P, S and F (lowest first) is

A

F < S < P < B

B

P < S < B < F

C

B < P < S < F

D

B < S < P < F

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NTA JEE Main 2006 - Question 65

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The decreasing values of bond angles from $$NH_3$$ $$(106^\circ)$$ to $$SbH_3$$ $$(101^\circ)$$ down group-15 of the periodic table is due to

A

increasing bp-bp repulsion

B

increasing p-orbital character in $$sp^3$$

C

decreasing lp-bp repulsion

D

decreasing electronegativity

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NTA JEE Main 2006 - Question 66

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Following statements regarding the periodic trends of chemical reactivity of the alkali metals and the halogens are given. Which of these statements gives the correct picture?

A

The reactivity decreases in the alkali metals but increases in the halogens with increase in atomic number down the group

B

In both the alkali metals and the halogens the chemical reactivity decreases with increase in atomic number down the group

C

Chemical reactivity increases with increase in atomic number down the group in both the alkali metals and halogens

D

In alkali metals the reactivity increases but in the halogens it decreases with increase in atomic number down the group

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NTA JEE Main 2006 - Question 67

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Which of the following molecules/ions does not contain unpaired electrons?

A

$$O_2^{2-}$$

B

$$B_2$$

C

$$N_2^+$$

D

$$O_2$$

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NTA JEE Main 2006 - Question 68

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Among the following mixtures, dipole-dipole as the major interaction, is present in

A

benzene and ethanol

B

acetonitrile and acetone

C

KCl and water

D

benzene and carbon tetrachloride

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NTA JEE Main 2006 - Question 69

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A metal, $$M$$ forms chlorides in its $$+2$$ and $$+4$$ oxidation states. Which of the following statements about these chlorides is correct?

A

$$MCl_2$$ is more volatile than $$MCl_4$$

B

$$MCl_2$$ is more soluble in anhydrous ethanol than $$MCl_4$$

C

$$MCl_2$$ is more ionic than $$MCl_4$$

D

$$MCl_2$$ is more easily hydrolysed than $$MCl_4$$

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NTA JEE Main 2006 - Question 70

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In which of the following molecules/ions are all the bonds not equal?

A

$$SF_4$$

B

$$SiF_4$$

C

$$XeF_4$$

D

$$BF_4^-$$

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NTA JEE Main 2006 - Question 71

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Phosphorus pentachloride dissociates as follows, in a closed reaction vessel, $$PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g)$$ If total pressure at equilibrium of the reaction mixture is $$P$$ and degree of dissociation of $$PCl_5$$ is $$x$$, the partial pressure of $$PCl_3$$ will be

A

$$\left(\dfrac{x}{x+1}\right)P$$

B

$$\left(\dfrac{2x}{1-x}\right)P$$

C

$$\left(\dfrac{x}{x-1}\right)P$$

D

$$\left(\dfrac{x}{1-x}\right)P$$

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NTA JEE Main 2006 - Question 72

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The standard enthalpy of formation $$(\Delta_fH^\circ)$$ at $$298\,K$$ for methane, $$CH_4(g)$$, is $$-74.8\,kJ\,mol^{-1}$$. The additional information required to determine the average energy for $$C - H$$ bond formation would be

A

the dissociation energy of $$H_2$$ and enthalpy of sublimation of carbon

B

latent heat of vapourization of methane

C

the first four ionization energies of carbon and electron gain enthalpy of hydrogen

D

the dissociation energy of hydrogen molecule, $$H_2$$

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NTA JEE Main 2006 - Question 73

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An ideal gas is allowed to expand both reversibly and irreversibly in an isolated system. If $$T_i$$ is the initial temperature and $$T_f$$ is the final temperature, which of the following statements is correct?

A

$$(T_f)_{irrev} > (T_f)_{rev}$$

B

$$T_f > T_i$$ for reversible process but $$T_f = T_i$$ for irreversible process

C

$$(T_f)_{rev} = (T_f)_{irrev}$$

D

$$T_f = T_i$$ for both reversible and irreversible processes

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NTA JEE Main 2006 - Question 74

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The enthalpy changes for the following processes are listed below: $$Cl_2(g) = 2Cl(g),\ 242.3\,kJ\,mol^{-1}$$; $$I_2(g) = 2I(g),\ 151.0\,kJ\,mol^{-1}$$; $$ICl(g) = I(g) + Cl(g),\ 211.3\,kJ\,mol^{-1}$$; $$I_2(s) = I_2(g),\ 62.76\,kJ\,mol^{-1}$$. Given that the standard states for iodine and chlorine are $$I_2(s)$$ and $$Cl_2(g)$$, the standard enthalpy of formation for $$ICl(g)$$ is

A

$$-14.6\,kJ\,mol^{-1}$$

B

$$-16.8\,kJ\,mol^{-1}$$

C

$$+16.8\,kJ\,mol^{-1}$$

D

$$+244.8\,kJ\,mol^{-1}$$

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NTA JEE Main 2006 - Question 75

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$$(\Delta H - \Delta U)$$ for the formation of carbon monoxide (CO) from its elements at $$298\,K$$ is $$(R = 8.314\,J\,K^{-1}\,mol^{-1})$$

A

$$-1238.78\,J\,mol^{-1}$$

B

$$1238.78\,J\,mol^{-1}$$

C

$$-2477.57\,J\,mol^{-1}$$

D

$$2477.57\,J\,mol^{-1}$$

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NTA JEE Main 2006 - Question 76

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The equilibrium constant for the reaction $$$SO_3(g) \rightleftharpoons SO_2(g) + \frac{1}{2}O_2(g)$$$ is $$K_c = 4.9 \times 10^{-2}$$. The value of $$K_c$$ for the reaction $$$2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g)$$$ will be

A

416

B

$$2.40 \times 10^{-3}$$

C

$$9.8 \times 10^{-2}$$

D

$$4.9 \times 10^{-2}$$

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NTA JEE Main 2006 - Question 77

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Which of the following statements is true?

A

$$H_3PO_3$$ is a stronger acid than $$H_2SO_3$$

B

In aqueous medium HF is a stronger acid than HCl

C

$$HClO_4$$ is a weaker acid than $$HClO_3$$

D

$$HNO_3$$ is a stronger acid than $$HNO_2$$

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NTA JEE Main 2006 - Question 78

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The ionic mobility of alkali metal ions in aqueous solution is maximum for

A

$$K^+$$

B

$$Rb^+$$

C

$$Li^+$$

D

$$Na^+$$

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NTA JEE Main 2006 - Question 79

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The IUPAC name of the compound shown below is

A

2-bromo-6-chlorocyclohex-1-ene

B

6-bromo-2-chlorocyclohexene

C

3-bromo-1-chlorocyclohexene

D

1-bromo-3-chlorocyclohexene

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NTA JEE Main 2006 - Question 80

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The increasing order of the rate of HCN addition to compounds A - D is (A) HCHO (B) $$CH_3COCH_3$$ (C) $$PhCOCH_3$$ (D) PhCOPh

A

A < B < C < D

B

D < B < C < A

C

D < C < B < A

D

C < D < B < A

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NTA JEE Main 2006 - Question 81

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The increasing order of stability of the following free radicals is

A

$$(CH_3)_2\dot{C}H < (CH_3)_3\dot{C} < (C_6H_5)_2\dot{C}H < (C_6H_5)_3\dot{C}$$

B

$$(C_6H_5)_3\dot{C} < (C_6H_5)_2\dot{C}H < (CH_3)_3\dot{C} < (CH_3)_2\dot{C}H$$

C

$$(C_6H_5)_2\dot{C}H < (C_6H_5)_3\dot{C} < (CH_3)_3\dot{C} < (CH_3)_2\dot{C}H$$

D

$$(CH_3)_2\dot{C}H < (CH_3)_3\dot{C} < (C_6H_5)_3\dot{C} < (C_6H_5)_2\dot{C}H$$

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NTA JEE Main 2006 - Question 82

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Increasing order of stability among the three main conformations (i.e. Eclipse, Anti, Gauche) of 2-fluoroethanol is

A

Eclipse, Gauche, Anti

B

Gauche, Eclipse, Anti

C

Eclipse, Anti, Gauche

D

Anti, Gauche, Eclipse

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NTA JEE Main 2006 - Question 83

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The alkene formed as a major product in the above elimination reaction is

A

B

$$CH_2 = CH_2$$

C

D

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NTA JEE Main 2006 - Question 84

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Total volume of atoms present in a face-centre cubic unit cell of a metal is ($$r$$ is atomic radius)

A

$$\dfrac{20}{3}\pi r^3$$

B

$$\dfrac{24}{3}\pi r^3$$

C

$$\dfrac{12}{3}\pi r^3$$

D

$$\dfrac{16}{3}\pi r^3$$

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NTA JEE Main 2006 - Question 85

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$$18\,g$$ of glucose $$(C_6H_{12}O_6)$$ is added to $$178.2\,g$$ of water. The vapour pressure of water for this aqueous solution at $$100^\circ C$$ is

A

759.00 Torr

B

7.60 Torr

C

76.00 Torr

D

752.40 Torr

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NTA JEE Main 2006 - Question 86

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The molar conductivities $$\Lambda^\circ_{NaOAc}$$ and $$\Lambda^\circ_{HCl}$$ at infinite dilution in water at $$25^\circ C$$ are $$91.0$$ and $$426.2\,S\,cm^2/mol$$ respectively. To calculate $$\Lambda^\circ_{HOAc}$$, the additional value required is

A

$$\Lambda^\circ_{H_2O}$$

B

$$\Lambda^\circ_{KCl}$$

C

$$\Lambda^\circ_{NaOH}$$

D

$$\Lambda^\circ_{NaCl}$$

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NTA JEE Main 2006 - Question 87

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Given the data at $$25^\circ C$$, $$Ag + I^- \longrightarrow AgI + e^-;\ E^\circ = 0.152\,V$$; $$Ag \longrightarrow Ag^+ + e^-;\ E^\circ = -0.800\,V$$. What is the value of $$\log K_{sp}$$ for $$AgI$$? $$\left(2.303\dfrac{RT}{F} = 0.059\,V\right)$$

A

$$-8.12$$

B

$$+8.612$$

C

$$-37.83$$

D

$$-16.13$$

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NTA JEE Main 2006 - Question 88

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Resistance of a conductivity cell filled with a solution of an electrolyte of concentration $$0.1\,M$$ is $$100\,\Omega$$. The conductivity of this solution is $$1.29\,S\,m^{-1}$$. Resistance of the same cell when filled with $$0.2\,M$$ of the same solution is $$520\,\Omega$$. The molar conductivity of $$0.02\,M$$ solution of the electrolyte will be

A

$$124 \times 10^{-4}\,S\,m^2\,mol^{-1}$$

B

$$1240 \times 10^{-4}\,S\,m^2\,mol^{-1}$$

C

$$1.24 \times 10^{-4}\,S\,m^2\,mol^{-1}$$

D

$$12.4 \times 10^{-4}\,S\,m^2\,mol^{-1}$$

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NTA JEE Main 2006 - Question 89

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A reaction was found to be second order with respect to the concentration of carbon monoxide. If the concentration of carbon monoxide is doubled, with everything else kept the same, the rate of reaction will

A

remain unchanged

B

triple

C

increase by a factor of 4

D

double

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NTA JEE Main 2006 - Question 90

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Rate of a reaction can be expressed by Arrhenius equation as: $$$k = Ae^{-E/RT}$$$ In this equation, E represents

A

the energy above which all the colliding molecules will react

B

the energy below which colliding molecules will not react

C

the total energy of the reacting molecules at a temperature, $$T$$

D

the fraction of molecules with energy greater than the activation energy of the reaction

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NTA JEE Main 2006 - Question 91

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The following mechanism has been proposed for the reaction of NO with $$Br_2$$ to form NOBr : $$$NO(g) + Br_2(g) \rightleftharpoons NOBr_2(g)$$$ $$$NOBr_2(g) + NO(g) \longrightarrow 2NOBr(g)$$$ If the second step is the rate determining step, the order of the reaction with respect to $$NO(g)$$ is

A

1

B

0

C

3

D

2

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NTA JEE Main 2006 - Question 92

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In Langmuir's model of adsorption of a gas on a solid surface

A

the rate of dissociation of adsorbed molecules from the surface does not depend on the surface covered

B

the adsorption at a single site on the surface may involve multiple molecules at the same time

C

the mass of gas striking a given area of surface is proportional to the pressure of the gas

D

the mass of gas striking a given area of surface is independent of the pressure of the gas

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NTA JEE Main 2006 - Question 93

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Which of the following chemical reactions depicts the oxidizing behaviour of $$H_2SO_4$$?

A

$$2HI + H_2SO_4 \longrightarrow I_2 + SO_2 + 2H_2O$$

B

$$Ca(OH)_2 + H_2SO_4 \longrightarrow CaSO_4 + 2H_2O$$

C

$$NaCl + H_2SO_4 \longrightarrow NaHSO_4 + HCl$$

D

$$2PCl_5 + H_2SO_4 \longrightarrow 2POCl_3 + 2HCl + SO_2Cl_2$$

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NTA JEE Main 2006 - Question 94

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What products are expected from the disproportionation reaction of hypochlorous acid?

A

$$HClO_3$$ and $$Cl_2O$$

B

$$HClO_2$$ and $$HClO_4$$

C

$$HCl$$ and $$Cl_2O$$

D

$$HCl$$ and $$HClO_3$$

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NTA JEE Main 2006 - Question 95

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Lanthanoid contraction is caused due to

A

the appreciable shielding on outer electrons by 4f electrons from the nuclear charge

B

the appreciable shielding on outer electrons by 5d electrons from the nuclear charge

C

the same effective nuclear charge from Ce to Lu

D

the imperfect shielding on outer electrons by 4f electrons from the nuclear charge

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NTA JEE Main 2006 - Question 96

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The IUPAC name for the complex $$[Co(NO_2)(NH_3)_5]Cl_2$$ is

A

nitrito-N-pentaamminecobalt (III) chloride

B

nitrito-N-pentaamminecobalt (II) chloride

C

pentaammine nitrito-N-cobalt (II) chloride

D

pentaammine nitrito-N-cobalt (III) chloride

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NTA JEE Main 2006 - Question 97

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Nickel $$(Z = 28)$$ combines with a uninegative monodentate ligand $$X^-$$ to form a paramagnetic complex $$[NiX_4]^{2-}$$. The number of unpaired electron(s) in the nickel and geometry of this complex ion are, respectively

A

one, tetrahedral

B

two, tetrahedral

C

one, square planar

D

two, square planar

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NTA JEE Main 2006 - Question 98

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In $$Fe(CO)_5$$, the $$Fe - C$$ bond possesses

A

$$\pi$$-character only

B

both $$\sigma$$ and $$\pi$$ characters

C

ionic character

D

$$\sigma$$-character only

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NTA JEE Main 2006 - Question 99

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How many EDTA (ethylenediaminetetraacetic acid) molecules are required to make an octahedral complex with a $$Ca^{2+}$$ ion?

A

six

B

three

C

one

D

two

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NTA JEE Main 2006 - Question 100

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HBr reacts with $$CH_2 = CH - OCH_3$$ under anhydrous conditions at room temperature to give

A

$$CH_3CHO$$ and $$CH_3Br$$

B

$$BrCH_2CHO$$ and $$CH_3OH$$

C

$$BrCH_2 - CH_2 - OCH_3$$

D

$$H_3C - CHBr - OCH_3$$

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NTA JEE Main 2006 - Question 101

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$$$CH_3Br + Nu^- \longrightarrow CH_3 - Nu + Br^-$$$ The decreasing order of the rate of the above reaction with nucleophiles $$(Nu^-)$$ A to D is $$[Nu^- = (A)PhO^-, (B) AcO^-, (C) HO^-, (D) CH_3O^-]$$

A

D > C > A > B

B

D > C > B > A

C

A > B > C > D

D

B > D > C > A

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NTA JEE Main 2006 - Question 102

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Fluorobenzene $$(C_6H_5F)$$ can be synthesized in the laboratory

A

by heating phenol with HF and KF

B

from aniline by diazotisation followed by heating the diazonium salt with $$HBF_4$$

C

by direct fluorination of benzene with $$F_2$$ gas

D

by reacting bromobenzene with NaF solution

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NTA JEE Main 2006 - Question 103

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Question:

Consider the reaction of bromoethane $$\left(CH_3CH_2Br\right)$$ with two different cyanide reagents in an aqueous ethanolic solution:

Reaction 1:

$$CH_3CH_2Br + KCN \rightarrow \text{Product A}$$

Reaction 2:

$$CH_3CH_2Br + AgCN \rightarrow \text{Product B}$$

Identify the major products A and B, respectively.

A

Ethyl isocyanide and Propanenitrile (Ethyl cyanide)

B

Propanenitrile (Ethyl cyanide) and Ethyl isocyanide

C

Propanenitrile (Ethyl cyanide) in both reactions

D

Ethyl isocyanide in both reactions

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NTA JEE Main 2006 - Question 104

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Reaction of trans-2-phenyl-1-bromocyclopentane on reaction with alcoholic KOH produces

A

4-phenylcyclopentene

B

2-phenylcyclopentene

C

1-phenylcyclopentene

D

3-phenylcyclopentene

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NTA JEE Main 2006 - Question 105

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The structure of the compound that gives a tribromo derivative on treatment with bromine water is

A

B

C

D

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NTA JEE Main 2006 - Question 106

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Phenyl magnesium bromide reacts with methanol to give

A

a mixture of anisole and Mg(OH)Br

B

a mixture of benzene and Mg(OMe)Br

C

a mixture of toluene and Mg(OH)Br

D

a mixture of phenol and Mg(Me)Br

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NTA JEE Main 2006 - Question 107

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Among the following the one that gives positive iodoform test upon reaction with $$I_2$$ and NaOH is

A

$$CH_3CH_2CH(OH)CH_2CH_3$$

B

$$C_6H_5CH_2CH_2OH$$

C

D

$$PhCHOHCH_3$$

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NTA JEE Main 2006 - Question 108

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The electrophile involved in the above reaction is

A

B

dichlorocarbene $$(:CCl_2)$$

C

D

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NTA JEE Main 2006 - Question 109

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The correct order of increasing acid strength of the compounds is (a) $$CH_3CO_2H$$ (b) $$MeOCH_2CO_2H$$ (c) $$CF_3CO_2H$$ 

A

b < d < a < c

B

d < a < c < b

C

d < a < b < c

D

a < d < c < b

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NTA JEE Main 2006 - Question 110

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The term anomers of glucose refers to

A

isomers of glucose that differ in configurations at carbons one and four ($$C - 1$$ and $$C - 4$$)

B

a mixture of $$(D)$$-glucose and $$(L)$$-glucose

C

enantiomers of glucose

D

isomers of glucose that differ in configuration at carbon one ($$C - 1$$)

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NTA JEE Main 2006 - Question 111

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The pyrimidine bases present in DNA are

A

cytosine and adenine

B

cytosine and guanine

C

cytosine and thymine

D

cytosine and uracil

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NTA JEE Main 2006 - Question 112

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If the roots of the quadratic equation $$x^2 + px + q = 0$$ are $$\tan 30^\circ$$ and $$\tan 15^\circ$$, respectively then the value of $$2 + q - p$$ is

A

2

B

3

C

0

D

1

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NTA JEE Main 2006 - Question 113

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All the values of $$m$$ for which both roots of the equations $$x^2 - 2mx + m^2 - 1 = 0$$ are greater than $$-2$$ but less than $$4$$, lie in the interval

A

$$-2 < m < 0$$

B

$$m > 3$$

C

$$-1 < m < 3$$

D

$$1 < m < 4$$

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NTA JEE Main 2006 - Question 114

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If $$z^2 + z + 1 = 0$$, where $$z$$ is a complex number, then the value of $$$\left(z + \frac{1}{z}\right)^2 + \left(z^2 + \frac{1}{z^2}\right)^2 + \left(z^3 + \frac{1}{z^3}\right)^2 + \cdots + \left(z^6 + \frac{1}{z^6}\right)^2$$$ is

A

18

B

54

C

6

D

12

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NTA JEE Main 2006 - Question 115

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At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote is

A

5040

B

6210

C

385

D

1110

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NTA JEE Main 2006 - Question 116

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The value of $$\sum_{k=1}^{10}\left(\sin\dfrac{2k\pi}{11} + i\cos\dfrac{2k\pi}{11}\right)$$ is

A

i

B

1

C

-1

D

-i

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NTA JEE Main 2006 - Question 117

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Let $$a_1, a_2, a_3, \ldots$$ be terms of an A.P. If $$\dfrac{a_1 + a_2 + \cdots + a_p}{a_1 + a_2 + \cdots + a_q} = \dfrac{p^2}{q^2}, p \ne q$$, then $$\dfrac{a_6}{a_{21}}$$ equals

A

$$\dfrac{41}{11}$$

B

$$\dfrac{7}{2}$$

C

$$\dfrac{2}{7}$$

D

$$\dfrac{11}{41}$$

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NTA JEE Main 2006 - Question 118

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If $$a_1, a_2, \ldots, a_n$$ are in H.P., then the expression $$a_1a_2 + a_2a_3 + \ldots + a_{n-1}a_n$$ is equal to

A

$$n(a_1 - a_n)$$

B

$$(n - 1)(a_1 - a_n)$$

C

$$na_1a_n$$

D

$$(n - 1)a_1a_n$$

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NTA JEE Main 2006 - Question 119

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If the expansion in powers of $$x$$ of the function $$\dfrac{1}{(1 - ax)(1 - bx)}$$ is $$a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots$$, then $$a_n$$ is

A

$$\dfrac{b^n - a^n}{b - a}$$

B

$$\dfrac{a^n - b^n}{b - a}$$

C

$$\dfrac{a^{n+1} - b^{n+1}}{b - a}$$

D

$$\dfrac{b^{n+1} - a^{n+1}}{b - a}$$

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NTA JEE Main 2006 - Question 120

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For natural numbers $$m, n$$ if $$(1 - y)^m(1 + y)^n = 1 + a_1y + a_2y^2 + \ldots$$, and $$a_1 = a_2 = 10$$ then $$(m, n)$$ is

A

(20, 45)

B

(35, 20)

C

(45, 35)

D

(35, 45)

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NTA JEE Main 2006 - Question 121

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The number of values of $$x$$ in the interval $$[0, 3\pi]$$ satisfying the equation $$2\sin^2 x + 5\sin x - 3 = 0$$ is

A

4

B

6

C

1

D

2

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NTA JEE Main 2006 - Question 122

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If $$0 < x < \pi$$ and $$\cos x + \sin x = \dfrac{1}{2}$$, then $$\tan x$$ is

A

$$\dfrac{(1 - \sqrt{7})}{4}$$

B

$$\dfrac{(4 - \sqrt{7})}{3}$$

C

$$-\dfrac{(4 + \sqrt{7})}{3}$$

D

$$\dfrac{(1 + \sqrt{7})}{4}$$

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NTA JEE Main 2006 - Question 123

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A straight line through the point $$A(3, 4)$$ is such that its intercept between the axes is bisected at $$A$$. Its equation is

A

$$x + y = 7$$

B

$$3x - 4y + 7 = 0$$

C

$$4x + 3y = 24$$

D

$$3x + 4y = 25$$

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NTA JEE Main 2006 - Question 124

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The two lines $$x = ay + b, z = cy + d$$; and $$x = a'y + b', z = c'y + d'$$ are perpendicular to each other if

A

$$aa' + cc' = -1$$

B

$$aa' + cc' = 1$$

C

$$\dfrac{a}{a'} + \dfrac{c}{c'} = -1$$

D

$$\dfrac{a}{a'} + \dfrac{c}{c'} = 1$$

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NTA JEE Main 2006 - Question 125

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If $$(a, a^2)$$ falls inside the angle made by the lines $$y = \dfrac{x}{2}, x > 0$$ and $$y = 3x, x > 0$$, then $$a$$ belongs to

A

$$\left(0, \dfrac{1}{2}\right)$$

B

$$(3, \infty)$$

C

$$\left(\dfrac{1}{2}, 3\right)$$

D

$$\left(-3, -\dfrac{1}{2}\right)$$

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NTA JEE Main 2006 - Question 126

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If the lines $$3x - 4y - 7 = 0$$ and $$2x - 3y - 5 = 0$$ are two diameters of a circle of area $$49\pi$$ square units, the equation of the circle is

A

$$x^2 + y^2 + 2x - 2y - 47 = 0$$

B

$$x^2 + y^2 + 2x - 2y - 62 = 0$$

C

$$x^2 + y^2 - 2x + 2y - 62 = 0$$

D

$$x^2 + y^2 - 2x + 2y - 47 = 0$$

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NTA JEE Main 2006 - Question 127

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Let $$C$$ be the circle with centre $$(0, 0)$$ and radius $$3$$ units. The equation of the locus of the mid points of the chords of the circle $$C$$ that subtend an angle of $$\dfrac{2\pi}{3}$$ at its centre is

A

$$x^2 + y^2 = \dfrac{3}{2}$$

B

$$x^2 + y^2 = 1$$

C

$$x^2 + y^2 = \dfrac{27}{4}$$

D

$$x^2 + y^2 = \dfrac{9}{4}$$

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NTA JEE Main 2006 - Question 128

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The locus of the vertices of the family of parabolas $$y = \dfrac{a^3 x^2}{3} + \dfrac{a^2 x}{2} - 2a$$ is

A

$$xy = \dfrac{105}{64}$$

B

$$xy = \dfrac{3}{4}$$

C

$$xy = \dfrac{35}{16}$$

D

$$xy = \dfrac{64}{105}$$

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NTA JEE Main 2006 - Question 129

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Angle between the tangents to the curve $$y = x^2 - 5x + 6$$ at the points $$(2, 0)$$ and $$(3, 0)$$ is

A

$$\dfrac{\pi}{2}$$

B

$$\dfrac{\pi}{3}$$

C

$$\dfrac{\pi}{6}$$

D

$$\dfrac{\pi}{4}$$

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NTA JEE Main 2006 - Question 130

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In an ellipse, the distance between its foci is $$6$$ and minor axis is $$8$$. Then its eccentricity is

A

$$\dfrac{3}{5}$$

B

$$\dfrac{1}{2}$$

C

$$\dfrac{4}{5}$$

D

$$\dfrac{1}{\sqrt{5}}$$

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NTA JEE Main 2006 - Question 131

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Suppose a population $$A$$ has $$100$$ observations $$101, 102, \ldots, 200$$, and another population $$B$$ has $$100$$ observations $$151, 152, \ldots, 250$$. If $$V_A$$ and $$V_B$$ represent the variances of the two populations, respectively, then $$\dfrac{V_A}{V_B}$$ is

A

1

B

$$\dfrac{9}{4}$$

C

$$\dfrac{4}{9}$$

D

$$\dfrac{2}{3}$$

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NTA JEE Main 2006 - Question 132

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A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $$x$$. The maximum area enclosed by the park is

A

$$\dfrac{3}{2}x^2$$

B

$$\sqrt{\dfrac{x^3}{8}}$$

C

$$\dfrac{1}{2}x^2$$

D

$$\pi x^2$$

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NTA JEE Main 2006 - Question 133

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Let $$W$$ denote the words in the English dictionary. Define the relation $$R$$ by : $$R = \{(x, y) \in W \times W \mid$$ the words $$x$$ and $$y$$ have at least one letter in common $$\}$$. Then $$R$$ is

A

not reflexive, symmetric and transitive

B

reflexive, symmetric and not transitive

C

reflexive, symmetric and transitive

D

reflexive, not symmetric and transitive

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NTA JEE Main 2006 - Question 134

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If $$A$$ and $$B$$ are square matrices of size $$n \times n$$ such that $$A^2 - B^2 = (A - B)(A + B)$$, then which of the following will be always true?

A

$$A = B$$

B

$$AB = BA$$

C

either of $$A$$ or $$B$$ is a zero matrix

D

either of $$A$$ or $$B$$ is an identity matrix

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NTA JEE Main 2006 - Question 135

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Let $$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$ and $$B = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}, a, b \in N$$. Then

A

there cannot exist any $$B$$ such that $$AB = BA$$

B

there exist more than one but finite number of $$B$$'s such that $$AB = BA$$

C

there exists exactly one $$B$$ such that $$AB = BA$$

D

there exist infinitely many $$B$$'s such that $$AB = BA$$

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NTA JEE Main 2006 - Question 136

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The set of points where $$f(x) = \dfrac{x}{1 + |x|}$$ is differentiable is

A

$$(-\infty, 0) \cup (0, \infty)$$

B

$$(-\infty, -1) \cup (-1, \infty)$$

C

$$(-\infty, \infty)$$

D

$$(0, \infty)$$

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NTA JEE Main 2006 - Question 137

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If $$x^m \cdot y^n = (x + y)^{m+n}$$, then $$\dfrac{dy}{dx}$$ is

A

$$\dfrac{y}{x}$$

B

$$\dfrac{x + y}{xy}$$

C

$$xy$$

D

$$\dfrac{x}{y}$$

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NTA JEE Main 2006 - Question 138

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If $$x$$ is real, the maximum value of $$\dfrac{3x^2 + 9x + 17}{3x^2 + 9x + 7}$$ is

A

$$1/4$$

B

41

C

1

D

$$17/7$$

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NTA JEE Main 2006 - Question 139

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The function $$f(x) = \dfrac{x}{2} + \dfrac{2}{x}$$ has a local minimum at

A

$$x = 2$$

B

$$x = -2$$

C

$$x = 0$$

D

$$x = 1$$

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NTA JEE Main 2006 - Question 140

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The value of the integral, $$\displaystyle\int_3^6 \dfrac{\sqrt{x}}{\sqrt{9 - x} + \sqrt{x}}\,dx$$ is

A

$$1/2$$

B

$$3/2$$

C

2

D

1

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NTA JEE Main 2006 - Question 141

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$$\displaystyle\int_0^\pi xf(\sin x)\,dx$$ is equal to

A

$$\pi\displaystyle\int_0^\pi f(\cos x)\,dx$$

B

$$\pi\displaystyle\int_0^\pi f(\sin x)\,dx$$

C

$$\dfrac{\pi}{2}\displaystyle\int_0^{\pi/2} f(\sin x)\,dx$$

D

$$\pi\displaystyle\int_0^{\pi/2} f(\cos x)\,dx$$

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NTA JEE Main 2006 - Question 142

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$$\displaystyle\int_{-3\pi/2}^{-\pi/2}\left[(x + \pi)^3 + \cos^2(x + 3\pi)\right]dx$$ is equal to

A

$$\dfrac{\pi^4}{32}$$

B

$$\dfrac{\pi^4}{32} + \dfrac{\pi}{2}$$

C

$$\dfrac{\pi}{2}$$

D

$$\dfrac{\pi}{4} - 1$$

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NTA JEE Main 2006 - Question 143

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The value of $$\displaystyle\int_1^a [x]f'(x)\,dx, a > 1$$, where $$[x]$$ denotes the greatest integer not exceeding $$x$$ is

A

$$af(a) - \{f(1) + f(2) + \ldots + f([a])\}$$

B

$$[a]f(a) - \{f(1) + f(2) + \ldots + f([a])\}$$

C

$$[a]f([a]) - \{f(1) + f(2) + \ldots + f(a)\}$$

D

$$af([a]) - \{f(1) + f(2) + \ldots + f(a)\}$$

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NTA JEE Main 2006 - Question 144

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The differential equation whose solution is $$Ax^2 + By^2 = 1$$, where $$A$$ and $$B$$ are arbitrary constants is of

A

second order and second degree

B

first order and second degree

C

first order and first degree

D

second order and first degree

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NTA JEE Main 2006 - Question 145

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$$ABC$$ is a triangle, right angled at $$A$$. The resultant of the forces acting along $$\overrightarrow{AB}, \overrightarrow{AC}$$ with magnitudes $$\dfrac{1}{AB}$$ and $$\dfrac{1}{AC}$$ respectively is the force along $$\overrightarrow{AD}$$, where $$D$$ is the foot of the perpendicular from $$A$$ onto $$BC$$. The magnitude of the resultant is

A

$$\dfrac{AB^2 + AC^2}{(AB)^2(AC)^2}$$

B

$$\dfrac{(AB)(AC)}{AB + AC}$$

C

$$\dfrac{1}{AB} + \dfrac{1}{AC}$$

D

$$\dfrac{1}{AD}$$

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NTA JEE Main 2006 - Question 146

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If $$(\overline{a} \times \overline{b}) \times \overline{c} = \overline{a} \times (\overline{b} \times \overline{c})$$, where $$\overline{a}, \overline{b}$$ and $$\overline{c}$$ are any three vectors such that $$\overline{a} \cdot \overline{b} \ne 0, \overline{b} \cdot \overline{c} \ne 0$$, then $$\overline{a}$$ and $$\overline{c}$$ are

A

inclined at an angle of $$\pi/3$$ between them

B

inclined at an angle of $$\pi/6$$ between them

C

perpendicular

D

parallel

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NTA JEE Main 2006 - Question 147

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A particle has two velocities of equal magnitude inclined to each other at an angle $$\theta$$. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then $$\theta$$ is

A

$$90^\circ$$

B

$$120^\circ$$

C

$$45^\circ$$

D

$$60^\circ$$

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NTA JEE Main 2006 - Question 148

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The values of $$a$$, for which the points $$A, B, C$$ with position vectors $$2\hat{i} - \hat{j} + \hat{k}, \hat{i} - 3\hat{j} - 5\hat{k}$$ and $$a\hat{i} - 3\hat{j} + \hat{k}$$ respectively are the vertices of a right-angled triangle with $$C = \dfrac{\pi}{2}$$ are

A

2 and 1

B

$$-2$$ and $$-1$$

C

$$-2$$ and 1

D

2 and $$-1$$

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NTA JEE Main 2006 - Question 149

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The image of the point $$(-1, 3, 4)$$ in the plane $$x - 2y = 0$$ is

A

$$\left(-\dfrac{17}{3}, -\dfrac{19}{3}, 4\right)$$

B

$$(15, 11, 4)$$

C

$$\left(-\dfrac{17}{3}, -\dfrac{19}{3}, 1\right)$$

D

none of these

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NTA JEE Main 2006 - Question 150

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At a telephone enquiry system the number of phone cells regarding relevant enquiry follow Poisson distribution with an average of $$5$$ phone calls during $$10$$-minute time intervals. The probability that there is at the most one phone call during a $$10$$-minute time period is

A

$$\dfrac{6}{5^e}$$

B

$$\dfrac{5}{6}$$

C

$$\dfrac{6}{55}$$

D

$$\dfrac{6}{e^5}$$

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