NTA JEE Main 12th April 2014 Online

NTA JEE Main 12th April 2014 Online - Question 1

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From the following combinations of physical constants (expressed through their usual symbols) the only combination, that would have the same value in different systems of units, is:

A

$$\frac{ch}{2\pi\varepsilon_0^2}$$

B

$$\frac{e^2}{2\pi\varepsilon_0 Gm_e^2}$$

C

$$\frac{\mu_0\varepsilon_0}{c^2} \cdot \frac{G}{he^2}$$

D

$$\frac{2\pi\sqrt{\mu_0\varepsilon_0}}{ce^2} \cdot \frac{h}{G}$$

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NTA JEE Main 12th April 2014 Online - Question 2

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A person climbs up a stalled escalator in 60 s. If standing on the same but escalator running with constant velocity he takes 40 s. How much time is taken by the person to walk up the moving escalator?

A

37 s

B

27 s

C

24 s

D

45 s

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NTA JEE Main 12th April 2014 Online - Question 3

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A bullet of mass 4 g is fired horizontally with a speed of 300 m/s into 0.8 kg block of wood at rest on a table. If the coefficient of friction between the block and the table is 0.3, how far will the block slide approximately?

A

0.19 m

B

0.379 m

C

0.569 m

D

0.758 m

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NTA JEE Main 12th April 2014 Online - Question 4

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A spring of unstretched length l has a mass m with one end fixed to a rigid support. Assuming spring to be made of a uniform wire, the kinetic energy possessed by it if its free end is pulled with uniform velocity v is:

A

$$\frac{1}{2}mv^2$$

B

$$mv^2$$

C

$$\frac{1}{3}mv^2$$

D

$$\frac{1}{6}mv^2$$

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NTA JEE Main 12th April 2014 Online - Question 5

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Three masses m, 2m and 3m are moving in the x-y plane with speeds 3u, 2u and u respectively as shown in the figure. The three masses collide at the same point at P and stick together. The velocity of the resulting mass will be:

A

$$\frac{u}{12}(\hat{i} + \sqrt{3}\hat{j})$$

B

$$\frac{u}{12}(\hat{i} - \sqrt{3}\hat{j})$$

C

$$\frac{u}{12}(-\hat{i} + \sqrt{3}\hat{j})$$

D

$$\frac{u}{12}(-\hat{i} - \sqrt{3}\hat{j})$$

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NTA JEE Main 12th April 2014 Online - Question 6

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A particle of mass 1 unit is moving in a circular path of radius a, with a constant velocity v as shown in the figure. The centre of circle is marked by 'C'. The angular momentum from the origin O can be written as:

A

$$va(1 + \cos 2\theta)$$

B

$$va(1 + \cos\theta)$$

C

$$va\cos 2\theta$$

D

$$va$$

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NTA JEE Main 12th April 2014 Online - Question 7

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Two hypothetical planets of masses m$$_1$$ and m$$_2$$ are at rest when they are infinite distance apart. Because of the gravitational force they move towards each other along the line joining their centres. What is their speed when their separation is 'd'? (Speed of m$$_1$$ is v$$_1$$ and that of m$$_2$$ is v$$_2$$)

A

$$v_1 = v_2$$

B

$$v_1 = m_2\sqrt{\frac{2G}{d(m_1+m_2)}}$$, $$v_2 = m_1\sqrt{\frac{2G}{d(m_1+m_2)}}$$

C

$$v_1 = m_1\sqrt{\frac{2G}{d(m_1+m_2)}}$$, $$v_2 = m_2\sqrt{\frac{2G}{d(m_1+m_2)}}$$

D

$$v_1 = m_2\sqrt{\frac{2G}{m_1}}$$, $$v_2 = m_2\sqrt{\frac{2G}{m_2}}$$

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NTA JEE Main 12th April 2014 Online - Question 8

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Steel ruptures when a shear of $$3.5 \times 10^8$$ N m$$^{-2}$$ is applied. The force needed to punch a 1 cm diameter hole in a steel sheet 0.3 cm thick is nearly:

A

$$1.4 \times 10^4$$ N

B

$$2.7 \times 10^4$$ N

C

$$3.3 \times 10^4$$ N

D

$$1.1 \times 10^4$$ N

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NTA JEE Main 12th April 2014 Online - Question 9

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A cylindrical vessel of cross-section A contains water to a height h. There is a hole in the bottom of radius 'a'. The time in which it will be emptied is:

A

$$\frac{2A}{\pi a^2}\sqrt{\frac{h}{g}}$$

B

$$\frac{\sqrt{2}A}{\pi a^2}\sqrt{\frac{h}{g}}$$

C

$$\frac{2\sqrt{2}A}{\pi a^2}\sqrt{\frac{h}{g}}$$

D

$$\frac{A}{\sqrt{2}\pi a^2}\sqrt{\frac{h}{g}}$$

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NTA JEE Main 12th April 2014 Online - Question 10

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Two soap bubbles coalesce to form a single bubble. If V is the subsequent change in volume of contained air and S change in total surface area, T is the surface tension and P atmospheric pressure, then which of the following relation is correct?

A

4PV + 3ST = 0

B

3PV + 4ST = 0

C

2PV + 3ST = 0

D

3PV + 2ST = 0

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NTA JEE Main 12th April 2014 Online - Question 11

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Hot water cools from 60°C to 50°C in the first 10 minutes and to 42°C in the next 10 minutes. The temperature of the surroundings is:

A

25°C

B

10°C

C

15°C

D

20°C

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NTA JEE Main 12th April 2014 Online - Question 12

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A Carnot engine absorbs 1000 J of heat energy from a reservoir at 127°C and rejects 600 J of heat energy during each cycle. The efficiency of engine and temperature of sink will be:

A

20% and $$-43$$°C

B

40% and $$-33$$°C

C

50% and $$-20$$°C

D

70% and $$-10$$°C

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NTA JEE Main 12th April 2014 Online - Question 13

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At room temperature a diatomic gas is found to have an r.m.s. speed of 1930 ms$$^{-1}$$. The gas is:

A

H$$_2$$

B

Cl$$_2$$

C

O$$_2$$

D

F$$_2$$

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NTA JEE Main 12th April 2014 Online - Question 14

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Which of the following expressions corresponds to simple harmonic motion along a straight line, where x is the displacement and a, b, c are positive constants?

A

$$a + bx - cx^2$$

B

$$bx^2$$

C

$$a - bx + cx^2$$

D

$$-bx$$

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NTA JEE Main 12th April 2014 Online - Question 15

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A source of sound A emitting waves of frequency 1800 Hz is falling towards ground with a terminal speed v. The observer B on the ground directly beneath the source receives waves of frequency 2150 Hz. The source A receives waves, reflected from ground of frequency nearly: (Speed of sound = 343 m/s)

A

2150 Hz

B

2500 Hz

C

1800 Hz

D

2400 Hz

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NTA JEE Main 12th April 2014 Online - Question 16

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A spherically symmetric charge distribution is characterised by the following charge density variations: $$\rho(r) = \rho_o\left(1 - \frac{r}{R}\right)$$ for $$r < R$$, $$\rho(r) = 0$$ for $$r \geq R$$. Where r is the distance from the centre of the charge distribution and $$\rho_o$$ is a constant. The electric field at an internal point ($$r < R$$) is:

A

$$\frac{\rho_o}{4\varepsilon_o}\left(\frac{r}{3} - \frac{r^2}{4R}\right)$$

B

$$\frac{\rho_o}{\varepsilon_o}\left(\frac{r}{3} - \frac{r^2}{4R}\right)$$

C

$$\frac{\rho_o}{3\varepsilon_o}\left(\frac{r}{3} - \frac{r^2}{4R}\right)$$

D

$$\frac{\rho_o}{12\varepsilon_o}\left(\frac{r}{3} - \frac{r^2}{4R}\right)$$

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NTA JEE Main 12th April 2014 Online - Question 17

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The space between the plates of a parallel plate capacitor is filled with a 'dielectric' whose 'dielectric constant' varies with distance as per the relation: $$K(x) = K_o + \lambda x$$ ($$\lambda$$ = a constant). The capacitance C, of the capacitor, would be related to its vacuum capacitance C$$_o$$ for the relation:

A

$$C = \frac{\lambda d}{\ln(1+K_o\lambda d)} C_o$$

B

$$C = \frac{\lambda}{d\ln(1+K_o\lambda d)} C_o$$

C

$$C = \frac{\lambda d}{\ln(1+\lambda d/K_o)} C_o$$

D

$$C = \frac{\lambda}{d\ln(1+K_o/\lambda d)} C_o$$

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NTA JEE Main 12th April 2014 Online - Question 18

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The circuit shown here has two batteries of 8.0 V and 16.0 V and three resistors 3$$\Omega$$, 9$$\Omega$$ and 9$$\Omega$$ and a capacitor of 5.0 $$\mu$$F. How much is the current I in the circuit in steady state?

A

1.6 A

B

0.67 A

C

2.5 A

D

0.25 A

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NTA JEE Main 12th April 2014 Online - Question 19

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In the experiment of calibration of voltmeter, a standard cell of e.m.f. 1.1 volt is balanced against 440 cm of potential wire. The potential difference across the ends of resistance is found to balance against 220 cm of the wire. The corresponding reading of voltmeter is 0.5 volt. The error in the reading of volmeter will be:

A

$$-0.15$$ volt

B

0.15 volt

C

0.5 volt

D

$$-0.05$$ volt

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NTA JEE Main 12th April 2014 Online - Question 20

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A positive charge 'q' of mass 'm' is moving along the +x axis. We wish to apply a uniform magnetic field B for time $$\Delta t$$ so that the charge reverses its direction crossing the y axis at a distance d. Then:

A

$$B = \frac{mv}{qd}$$ and $$\Delta t = \frac{\pi d}{v}$$

B

$$B = \frac{mv}{2qd}$$ and $$\Delta t = \frac{\pi d}{2v}$$

C

$$B = \frac{2mv}{qd}$$ and $$\Delta t = \frac{\pi d}{2v}$$

D

$$B = \frac{2mv}{qd}$$ and $$\Delta t = \frac{\pi d}{v}$$

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NTA JEE Main 12th April 2014 Online - Question 21

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Consider two thin identical conducting wires covered with very thin insulating material. One of the wires is bent into a loop and produces magnetic field B$$_1$$, at its centre when a current I passes through it. The ratio B$$_1$$ : B$$_2$$ is:

A

1:1

B

1 : 3

C

1 : 9

D

9 : 1

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NTA JEE Main 12th April 2014 Online - Question 22

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A sinusoidal voltage V(t) = 100 sin(500t) is applied across a pure inductance of L = 0.02H. The current through the coil is:

A

10 cos(500t)

B

$$-10$$ cos(500t)

C

10 sin(500t)

D

$$-10$$ sin(500t)

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NTA JEE Main 12th April 2014 Online - Question 23

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A lamp emits monochromatic green light uniformly in all directions. The lamp is 3% efficient in converting electrical power to electromagnetic waves and consumes 100 W of power. The amplitude of the electric field associated with the electromagnetic radiation at a distance of 5 m from the lamp will be nearly:

A

1.34 V/m

B

2.68 V/m

C

4.02 V/m

D

5.36 V/m

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NTA JEE Main 12th April 2014 Online - Question 24

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The refractive index of the material of a concave lens is $$\mu$$. It is immersed in a medium of refractive index $$\mu_1$$. A parallel beam of light is incident on the lens. The path of the emergent rays when $$\mu_1 > \mu$$ is:

A

B

C

D

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NTA JEE Main 12th April 2014 Online - Question 25

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Interference pattern is observed at 'P' due to superimposition of two rays coming out from a source 'S' as shown in the figure. The value of 'l' for which maxima is obtained at 'P' is: (R is perfect reflecting surface)

A

$$l = \frac{2n\lambda}{\sqrt{3}-1}$$

B

$$l = \frac{(2n-1)\lambda}{2(\sqrt{3}-1)}$$

C

$$l = \frac{(2n-1\lambda)\sqrt{3}}{4(2-\sqrt{3})}$$

D

$$l = \frac{(2n-1)\lambda}{\sqrt{3}-1}$$

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NTA JEE Main 12th April 2014 Online - Question 26

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In an experiment of single slit diffraction pattern, first minimum for red light coincides with first maximum of some other wavelength. If wavelength of red light is 6600 $$\text{Å}$$, then wavelength of first maximum will be:

A

3300 $$\text{Å}$$

B

4400 $$\text{Å}$$

C

5500 $$\text{Å}$$

D

6600 $$\text{Å}$$

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NTA JEE Main 12th April 2014 Online - Question 27

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A beam of light has two wavelengths of 4972 $$\text{Å}$$ and 6216 $$\text{Å}$$ with a total intensity of $$3.6 \times 10^{-3}$$ Wm$$^{-2}$$ equally distributed among the two wavelengths. The beam falls normally on an area of 1 cm$$^2$$ of a clean metallic surface of work function 2.3 eV. Assume that there is no loss of light by reflection and that each capable photon ejects one electron. The number of photoelectrons liberated in 2 s is approximately:

A

$$6 \times 10^{11}$$

B

$$9 \times 10^{11}$$

C

$$11 \times 10^{11}$$

D

$$15 \times 10^{11}$$

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NTA JEE Main 12th April 2014 Online - Question 28

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For LED's to emit light in visible region of electromagnetic light, it should have energy band gap in the range of:

A

0.1eV to 0.4eV

B

0.5eV to 0.8eV

C

0.9eV to 1.6eV

D

1.7eV to 3.0eV

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NTA JEE Main 12th April 2014 Online - Question 29

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A piece of bone of an animal from a ruin is found to have $$^{14}$$C activity of 12 disintegrations per minute per gm of its carbon content. The $$^{14}$$C activity of a living animal is 16 disintegrations per minute per gm. How long ago nearly did the animal die? (Given half life of $$^{14}$$C is $$t_{1/2}$$ = 5760 years)

A

1672 years

B

2391 years

C

3291 years

D

4453 years

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NTA JEE Main 12th April 2014 Online - Question 30

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For sky wave propagation, the radio waves must have a frequency range in between:

A

1MHz to 2MHz

B

5MHz to 25MHz

C

35MHz to 40MHz

D

45MHz to 50MHz

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NTA JEE Main 12th April 2014 Online - Question 31

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The amount of BaSO$$_4$$ formed upon mixing 100 mL of 20.8% BaCl$$_2$$ solution with 50 mL of 9.8% H$$_2$$SO$$_4$$ solution will be: (Ba = 137, Cl = 35.5, S = 32, H = 1 and O = 16)

A

23.3 g

B

11.65 g

C

30.6 g

D

33.2 g

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NTA JEE Main 12th April 2014 Online - Question 32

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If $$m$$ and $$e$$ are the mass and charge of the revolving electron in the orbit of radius $$r$$ for hydrogen atom, the total energy of the revolving electron will be:

A

$$\frac{1}{2}\frac{e^2}{r}$$

B

$$-\frac{e^2}{r}$$

C

$$\frac{me^2}{r}$$

D

$$-\frac{1}{2}\frac{e^2}{r}$$

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NTA JEE Main 12th April 2014 Online - Question 33

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The de-Broglie wavelength of a particle of mass 6.63 g moving with a velocity of 100 ms$$^{-1}$$ is:

A

$$10^{-33}$$ m

B

$$10^{-35}$$ m

C

$$10^{-31}$$ m

D

$$10^{-25}$$ m

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NTA JEE Main 12th April 2014 Online - Question 34

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Excited hydrogen atom emits light in the ultraviolet region at $$2.47 \times 10^{15}$$ Hz. With this frequency, the energy of a single photon is: (h = $$6.63 \times 10^{-34}$$ Js)

A

$$8.041 \times 10^{-40}$$ J

B

$$2.680 \times 10^{-19}$$ J

C

$$1.640 \times 10^{-18}$$ J

D

$$6.111 \times 10^{-17}$$ J

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NTA JEE Main 12th April 2014 Online - Question 35

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Similarity in chemical properties of the atoms of elements in a group of the Periodic table is most closely related to:

A

atomic numbers

B

atomic masses

C

number of principal energy levels

D

number of valence electrons

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NTA JEE Main 12th April 2014 Online - Question 36

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Which of the following arrangements represents the increasing order (smallest to largest) of ionic radii of the given species O$$^{2-}$$, S$$^{2-}$$, N$$^{3-}$$, P$$^{3-}$$?

A

O$$^{2-}$$ < N$$^{3-}$$ < S$$^{2-}$$ < P$$^{3-}$$

B

O$$^{2-}$$ < P$$^{3-}$$ < N$$^{3-}$$ < S$$^{2-}$$

C

N$$^3$$ < O$$^{2-}$$ < P$$^{3-}$$ < S$$^{2-}$$

D

N$$^{3-}$$ < S$$^{2-}$$ < O$$^{2-}$$ < P$$^{3-}$$

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NTA JEE Main 12th April 2014 Online - Question 37

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Which of the following molecules has two sigma ($$\sigma$$) and two pi ($$\pi$$) bonds?

A

C$$_2$$H$$_4$$

B

N$$_2$$ F$$_2$$

C

C$$_2$$H$$_2$$Cl$$_2$$

D

HCN

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NTA JEE Main 12th April 2014 Online - Question 38

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The (S°) of the following substances are: CH$$_4$$(g) 186.2 JK$$^{-1}$$ mol$$^{-1}$$, O$$_2$$(g) 205.2 JK$$^{-1}$$ mol$$^{-1}$$, CO$$_2$$(g) 213.6 JK$$^{-1}$$ mol$$^{-1}$$, H$$_2$$O(g) 69.9 JK$$^{-1}$$ mol$$^{-1}$$. The entropy change ($$\Delta S^\circ$$) for the reaction CH$$_4$$(g) + 2O$$_2$$(g) → CO$$_2$$(g) + 2H$$_2$$O(l) is:

A

$$-312.5$$ J K$$^{-1}$$ mol$$^{-1}$$

B

$$-242.8$$ JK$$^{-1}$$ mol$$^{-1}$$

C

$$-108.1$$ JK$$^{-1}$$ mol$$^{-1}$$

D

$$-37.6$$ J K$$^{-1}$$ mol$$^{-1}$$

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NTA JEE Main 12th April 2014 Online - Question 39

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The standard enthalpy of formation ($$\Delta_fH^\circ_{298}$$) for methane, CH$$_4$$ is $$-74.9$$ kJ mol$$^{-1}$$. In order to calculate the average energy given out in the formation of a C-H bond from this it is necessary to know which one of the following?

A

The dissociation energy of the hydrogen molecule, H$$_2$$.

B

The first four ionisation energies of carbon.

C

The dissociation energy of H$$_2$$ and enthalpy and sublimation of carbon (graphite).

D

The first four ionisation energies of carbon and electron affinity of hydrogen.

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NTA JEE Main 12th April 2014 Online - Question 40

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What happens when an inert gas is added to an equilibrium keeping volume unchanged?

A

More product will form

B

Less product will form

C

More reactant will form

D

Equilibrium will remain unchanged

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NTA JEE Main 12th April 2014 Online - Question 41

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The conjugate base of hydrazoic acid is:

A

N$$^{-3}$$

B

N$$_3^-$$

C

N$$_2^-$$

D

HN$$_3^-$$

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NTA JEE Main 12th April 2014 Online - Question 42

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Hydrogen peroxide acts both as an oxidising and as a reducing agent depending upon the nature of the reacting species. In which of the following cases H$$_2$$O$$_2$$ acts as a reducing agent in acid medium?

A

MnO$$_4^-$$

B

Cr$$_2$$O$$_7^{2-}$$

C

SO$$_3^{2-}$$

D

KI

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NTA JEE Main 12th April 2014 Online - Question 43

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Which one of the following acids does not exhibit optical isomerism?

A

Lactic acid

B

Tartaric acid

C

Maleic acid

D

$$\alpha$$-amino acids

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NTA JEE Main 12th April 2014 Online - Question 44

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 on mercuration-demercuration produces the major product:

A

B

C

D

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NTA JEE Main 12th April 2014 Online - Question 45

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In the presence of peroxide, HCl and HI do not give anti-Markownikoff's addition of alkenes because:

A

One of the steps is endothermic in HCl and HI

B

Both HCl and HI are strong acids

C

HCl is oxidizing and the HI is reducing

D

All the steps are exothermic is HCl and HI

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NTA JEE Main 12th April 2014 Online - Question 46

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Global warming is due to increase of:

A

methane and nitrous oxide in atmosphere

B

methane and CO$$_2$$ in atmosphere

C

methane and O$$_3$$ in atmosphere

D

methane and CO in atmosphere

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NTA JEE Main 12th April 2014 Online - Question 47

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In a monoclinic unit cell, the relation of sides and angles are respectively:

A

a = b $$\neq$$ c and $$\alpha = \beta = \gamma = 90$$°

B

a $$\neq$$ b $$\neq$$ c and $$\alpha = \beta = \gamma = 90$$°

C

a $$\neq$$ b $$\neq$$ c and $$\beta = \gamma = 90$$° $$\neq \alpha$$

D

a $$\neq$$ b $$\neq$$ c and $$\alpha \neq \beta \neq \gamma \neq 90$$°

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NTA JEE Main 12th April 2014 Online - Question 48

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How many electrons would be required to deposit 6.35 g of copper at the cathode during the electrolysis of an aqueous solution of copper sulphate? (Atomic mass of copper = 63.5u, N$$_A$$ = Avogadro's constant):

A

$$\frac{N_A}{20}$$

B

$$\frac{N_A}{10}$$

C

$$\frac{N_A}{5}$$

D

$$\frac{N_A}{2}$$

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NTA JEE Main 12th April 2014 Online - Question 49

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The rate coefficient (k) for a particular reactions is $$1.3 \times 10^{-4}$$ M$$^{-1}$$ s$$^{-1}$$ at 100°C, and $$1.3 \times 10^{-3}$$ M$$^{-1}$$ s$$^{-1}$$ at 150°C. What is the energy of activation (E$$_A$$) (in kJ) for this reaction? (R = molar gas constant = 8.314 JK$$^{-1}$$ mol$$^{-1}$$)

A

16

B

60

C

99

D

132

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NTA JEE Main 12th April 2014 Online - Question 50

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Which of the following xenon-oxo compounds may not be obtained by hydrolysis of xenon fluorides?

A

XeO$$_2$$F$$_2$$

B

XeOF$$_4$$

C

XeO$$_3$$

D

XeO$$_4$$

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NTA JEE Main 12th April 2014 Online - Question 51

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Which one of the following exhibits the large number of oxidation states?

A

Ti(22)

B

V(23)

C

Cr(24)

D

Mn(25)

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NTA JEE Main 12th April 2014 Online - Question 52

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Copper becomes green when exposed to moist air for a long period. This is due to:

A

the formation of a layer of cupric oxide on the surface of copper.

B

the formation of a layer of basic carbonate of copper on the surface of copper.

C

the formation of a layer of cupric hydroxide on the surface of copper.

D

the formation of basic copper sulphate layer on the surface of the metal.

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NTA JEE Main 12th April 2014 Online - Question 53

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Among the following species the one which causes the highest CFSE, $$\Delta_0$$ as a ligand is:

A

CN$$^-$$

B

NH$$_3$$

C

F$$^-$$

D

CO

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NTA JEE Main 12th April 2014 Online - Question 54

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Which one of the following complexes will most likely absorb visible light? (At. nos. Sc = 21, Ti = 22, V = 23, Zn = 30)

A

[Sc(H$$_2$$O)$$_6$$]$$^{3+}$$

B

[Ti(NH$$_3$$)$$_6$$]$$^{4+}$$

C

[V(NH$$_3$$)$$_6$$]$$^{3+}$$

D

[Zn(NH$$_3$$)$$_6$$]$$^{2+}$$

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NTA JEE Main 12th April 2014 Online - Question 55

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Conversion of benzene diazonium chloride to chlorobenzene is an example of which of the following reactions?

A

Claisen

B

Friedel-craft

C

Sandmeyer

D

Wurtz

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NTA JEE Main 12th April 2014 Online - Question 56

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The major product obtained in the photo catalysed bromination of 2-methylbutane is:

A

1-bromo-2-methylbutane

B

1-bromo-3-methylbutane

C

2-bromo-3-methylbutane

D

2-bromo-2-methylbutane

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NTA JEE Main 12th April 2014 Online - Question 57

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In the Victor-Meyer's test, the colour given by 1°, 2° and 3° alcohols are respectively:

A

Red, colourless, blue

B

Red, blue, colourless

C

Colourless, red, blue

D

Red, blue, violet

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NTA JEE Main 12th April 2014 Online - Question 58

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Phthalic acid reacts with resorcinol in the presence of concentrated H$$_2$$SO$$_4$$ to give:

A

Phenolphthalein

B

Alizarin

C

Coumarin

D

Fluorescein

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NTA JEE Main 12th April 2014 Online - Question 59

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Aminoglycosides are usually used as:

A

antibiotic

B

analgesic

C

hypnotic

D

antifertility

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NTA JEE Main 12th April 2014 Online - Question 60

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Which of the following will not show mutarotation?

A

Maltose

B

Lactose

C

Glucose

D

Sucrose

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NTA JEE Main 12th April 2014 Online - Question 61

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The sum of the roots of the equation, $$x^2 + |2x - 3| - 4 = 0$$, is:

A

2

B

$$-2$$

C

$$\sqrt{2}$$

D

$$-\sqrt{2}$$

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NTA JEE Main 12th April 2014 Online - Question 62

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Let $$z \neq -i$$ be any complex number such that $$\frac{z-1}{z+1}$$ is a purely imaginary number. Then $$z + \frac{1}{z}$$ is:

A

0

B

any non-zero real number other than 1

C

any non-zero real number.

D

a purely imaginary number.

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NTA JEE Main 12th April 2014 Online - Question 63

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8-digit numbers are formed using the digits 1, 1, 2, 2, 2, 3, 4, 4. The number of such numbers in which the odd digits do no occupy odd places, is:

A

160

B

120

C

60

D

48

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NTA JEE Main 12th April 2014 Online - Question 64

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Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of $$\frac{1}{a}$$ and $$\frac{1}{b}$$. If $$\frac{1}{M}$$ : G is 4 : 5, then a : b can be:

A

1 : 4

B

1 : 2

C

2 : 3

D

3 : 4

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NTA JEE Main 12th April 2014 Online - Question 65

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The least positive integer n such that $$1 - \frac{2}{3} - \frac{2}{3^2} - \ldots - \frac{2}{3^{n-1}} \lt \frac{1}{100}$$, is:

A

4

B

5

C

6

D

7

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NTA JEE Main 12th April 2014 Online - Question 66

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If $$1 + x^4 + x^5 = \sum_{i=0}^{5} a_i(1+x)^i$$, for all $$x$$ in R, then $$a_2$$ is:

A

$$-4$$

B

6

C

$$-8$$

D

10

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NTA JEE Main 12th April 2014 Online - Question 67

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If $$\left(2 + \frac{x}{3}\right)^{55}$$ is expanded in the ascending powers of x and the coefficients of powers of x in two consecutive terms of the expansion are equal, then these terms are:

A

7$$^{th}$$ and 8$$^{th}$$

B

8$$^{th}$$ and 9$$^{th}$$

C

28$$^{th}$$ and 29$$^{th}$$

D

27$$^{th}$$ and 28$$^{th}$$

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NTA JEE Main 12th April 2014 Online - Question 68

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If a line intercepted between the coordinate axes is trisected at a point A(4, 3), which is nearer to x-axis, then its equation is:

A

$$4x - 3y = 7$$

B

$$3x + 2y = 18$$

C

$$3x + 8y = 36$$

D

$$x + 3y = 13$$

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NTA JEE Main 12th April 2014 Online - Question 69

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If the three distinct lines $$x + 2ay + a = 0$$, $$x + 3by + b = 0$$ and $$x + 4ay + a = 0$$ are concurrent, then the point $$(a, b)$$ lies on a:

A

circle

B

hyperbola

C

straight line

D

parabola

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NTA JEE Main 12th April 2014 Online - Question 70

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For the two circles $$x^2 + y^2 = 16$$ and $$x^2 + y^2 - 2y = 0$$, there is/are:

A

one pair of common tangents

B

two pair of common tangents

C

three pair of common tangents

D

no common tangent

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NTA JEE Main 12th April 2014 Online - Question 71

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Two tangents are drawn from a point $$(-2, -1)$$ to the curve, $$y^2 = 4x$$. If $$\alpha$$ is the angle between them, then $$|\tan\alpha|$$ is equal to:

A

$$\frac{1}{3}$$

B

$$\frac{1}{\sqrt{3}}$$

C

$$\sqrt{3}$$

D

3

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NTA JEE Main 12th April 2014 Online - Question 72

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The minimum area of a triangle formed by any tangent to the ellipse $$\frac{x^2}{16} + \frac{y^2}{81} = 1$$ and the co-ordinate axes is:

A

12

B

18

C

26

D

36

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NTA JEE Main 12th April 2014 Online - Question 73

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Let p, q, r denote arbitrary statements. Then the logically equivalent of the statement $$p \Rightarrow (q \vee r)$$ is:

A

$$(p \vee q) \Rightarrow r$$

B

$$(p \Rightarrow q) \vee (p \Rightarrow r)$$

C

$$(p \Rightarrow \sim q) \wedge (p \Rightarrow r)$$

D

$$(p \Rightarrow q) \wedge (p \Rightarrow \sim r)$$

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NTA JEE Main 12th April 2014 Online - Question 74

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Let $$\bar{X}$$ and M.D. be the mean and the mean deviation about $$\bar{X}$$ of n observations $$x_i$$, i = 1, 2, n. If each of the observations is increased by 5, then the new mean and the mean deviation about the new mean, respectively, are:

A

$$\bar{X}$$, M.D.

B

$$\bar{X} + 5$$, M.D.

C

$$\bar{X}$$, M.D. +5

D

$$\bar{X} + 5$$, M.D. +5

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NTA JEE Main 12th April 2014 Online - Question 75

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A relation on the set A = {x : |x| < 3, x $$\in$$ Z}, where Z is the set of integers is defined by R = {(x, y) : y = |x|, x $$\neq$$ $$-1$$}. Then the number of elements in the power set of R is:

A

32

B

16

C

8

D

64

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NTA JEE Main 12th April 2014 Online - Question 76

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If $$A = \begin{bmatrix} 1 & 2 & x \\ 3 & -1 & 2 \end{bmatrix}$$ and $$B = \begin{bmatrix} y \\ x \\ 1 \end{bmatrix}$$ be such that AB = $$\begin{bmatrix} 6 \\ 8 \end{bmatrix}$$, then:

A

$$y = 2x$$

B

$$y = -2x$$

C

$$y = x$$

D

$$y = -x$$

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NTA JEE Main 12th April 2014 Online - Question 77

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If $$\begin{vmatrix} a^2 & b^2 & c^2\\ (a+\lambda)^2 & (b+\lambda)^2 & (c+\lambda)^2 \\ (a-\lambda)^2 & (b-\lambda)^2 & (c-\lambda)^2  \end{vmatrix}$$ = $$k\lambda \begin{vmatrix} a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1 \end{vmatrix}$$, $$\lambda \neq 0$$, then k is equal to:

A

4$$\lambda$$abc

B

$$-4\lambda$$abc

C

$$4\lambda^2$$

D

$$-4\lambda^2$$

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NTA JEE Main 12th April 2014 Online - Question 78

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If $$f(\theta) = \begin{vmatrix} 1 & \cos\theta & 1 \\ -\sin\theta & 1 & -\cos\theta \\ -1 & \sin\theta & 1 \end{vmatrix}$$ and A and B are respectively the maximum and the minimum values of $$f(\theta)$$, then (A, B) is equal to:

A

(3, $$-1$$)

B

$$(4, 2-\sqrt{2})$$

C

$$(2+\sqrt{2}, 2-\sqrt{2})$$

D

$$(2+\sqrt{2}, -1)$$

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NTA JEE Main 12th April 2014 Online - Question 79

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Statement I: The equation $$(\sin^{-1}x)^3 + (\cos^{-1}x)^3 - a\pi^3 = 0$$ has a solution for all $$a \geq \frac{1}{32}$$. 

Statement II: For any x $$\in$$ R, $$\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$$ and $$0 \leq \left(\sin^{-1}x - \frac{\pi}{4}\right)^2 \leq \frac{9\pi^2}{16}$$.

A

Both statements I and II are true.

B

Both statements I and II are false.

C

Statement I is true and statement II is false.

D

Statement I is false and statement II is true.

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NTA JEE Main 12th April 2014 Online - Question 80

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If $$f(x) = x^2 - x + 5$$, $$x > \frac{1}{2}$$, and g(x) is its inverse function, then g'(7) equals:

A

$$-\frac{1}{3}$$

B

$$\frac{1}{13}$$

C

$$\frac{1}{3}$$

D

$$-\frac{1}{13}$$

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NTA JEE Main 12th April 2014 Online - Question 81

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Let f, g : R → R be two functions defined by $$f(x) = \begin{cases} x\sin\left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$$ and $$g(x) = xf(x)$$. Statement I: f is a continuous function at x = 0. Statement II: g is a differentiable function at x = 0.

A

Both statement I and II are false.

B

Both statement I and II are true.

C

Statement I is true, statement II is false.

D

Statement I is false, statement II is true.

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NTA JEE Main 12th April 2014 Online - Question 82

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Let $$f$$ and $$g$$ be two differentiable functions on R such that $$f'(x) > 0$$ and $$g'(x) < 0$$ for all $$x \in R$$. Then for all x:

A

$$f(g(x)) > f(g(x-1))$$

B

$$f(g(x)) > f(g(x+1))$$

C

$$g(f(x)) > g(f(x-1))$$

D

$$g(f(x)) < g(f(x+1))$$

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NTA JEE Main 12th April 2014 Online - Question 83

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The integral $$\int \frac{\sin^2 x \cos^2 x}{(\sin^3 x + \cos^3 x)^2} dx$$ is equal to:

A

$$\frac{1}{(1+\cot^3 x)} + c$$

B

$$-\frac{1}{3(1+\tan^3 x)} + c$$

C

$$\frac{\sin^3 x}{(1+\cos^3 x)} + c$$

D

$$-\frac{\cos^3 x}{3(1+\sin^3 x)} + c$$

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NTA JEE Main 12th April 2014 Online - Question 84

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If [ ] denotes the greatest integer function, then the integral $$\int_0^{\pi} [\cos x] dx$$ is equal to:

A

$$\frac{\pi}{2}$$

B

0

C

$$-1$$

D

$$-\frac{\pi}{2}$$

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NTA JEE Main 12th April 2014 Online - Question 85

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If for a continuous function f(x), $$\int_{-\pi}^{t} (f(x) + x) dx = \pi^2 - t^2$$, for all $$t \geq -\pi$$, then $$f\left(-\frac{\pi}{3}\right)$$ is equal to:

A

$$\pi$$

B

$$\frac{\pi}{2}$$

C

$$\frac{\pi}{3}$$

D

$$\frac{\pi}{6}$$

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NTA JEE Main 12th April 2014 Online - Question 86

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The general solution of the differential equation, $$\sin 2x\left(\frac{dy}{dx} - \sqrt{\tan x}\right) - y = 0$$, is:

A

$$y\sqrt{\tan x} = x + c$$

B

$$y\sqrt{\cot x} = \tan x + c$$

C

$$y\sqrt{\tan x} = \cot x + c$$

D

$$y\sqrt{\cot x} = x + c$$

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NTA JEE Main 12th April 2014 Online - Question 87

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If $$\hat{x}$$, $$\hat{y}$$ and $$\hat{z}$$ are three unit vectors in threedimensional space, then the minimum value of $$|\hat{x} + \hat{y}|^2 + |\hat{y} + \hat{z}|^2 + |\hat{z} + \hat{x}|^2$$

A

$$\frac{3}{2}$$

B

3

C

$$3\sqrt{3}$$

D

6

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NTA JEE Main 12th April 2014 Online - Question 88

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A symmetrical form of the line of intersection of the planes $$x = ay + b$$ and $$z = cy + d$$ is:

A

$$\frac{x-b}{a} = \frac{y-1}{1} = \frac{z-d}{c}$$

B

$$\frac{x-b-a}{a} = \frac{y-1}{1} = \frac{z-d-c}{c}$$

C

$$\frac{x-a}{b} = \frac{y-0}{1} = \frac{z-c}{d}$$

D

$$\frac{x-b-a}{b} = \frac{y-1}{0} = \frac{z-d-c}{d}$$

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NTA JEE Main 12th April 2014 Online - Question 89

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If the distance between planes, $$4x - 2y - 4z + 1 = 0$$ and $$4x - 2y - 4z + d = 0$$ is 7, then d is:

A

41 or $$-42$$

B

42 or $$-43$$

C

$$-41$$ or 43

D

$$-42$$ or 44

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NTA JEE Main 12th April 2014 Online - Question 90

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A number x is chosen at random from the set {1, 2, 3, 4, ..., 100}. Define the event: A = the chosen number x satisfies $$\frac{(x-10)(x-50)}{(x-30)} \geq 0$$. Then P(A) is:

A

0.71

B

0.70

C

0.51

D

0.20

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