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JEE Main 2014 (09 Apr Online)

2014 · 90 questions · First 10 Solutions Free
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Q1 mcq
Three capacitances, each of 3 $\mu F$ , are provided. These cannot be combined to provide the resultant capacitance of :
a.
$2\mu F$
b.
$6\mu F$ Correct Answer
c.
$1.5\mu F$
d.
$1\mu F$

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Q2 mcq
A transverse wave is represented by : $y = \frac{10}{\pi} sin(\frac{2\pi}{T} t - \frac{2\pi}{\lambda}x)$ For what value of the wavelength the wave velocity is twice the maximum particle velocity?
a.
40 cm Correct Answer
b.
60 cm
c.
10 cm
d.
20 cm

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Q3 mcq
The magnitude of the average electric field normally present in the atmosphere just above the surface of the Earth is about $150\,N/C$ , directed inward towards the center of the Earth. This gives the total net surface charge carried by the Earth to be : [Given : $\in _{O}= 8.85 \times 10^{-12}\,\,C^{2}/N-m^{2},\,\,R_{E} = 6.37 \times 10^{6}m$ ]
a.
$-680\, kC$ Correct Answer
b.
$+\,670\,kC$
c.
$+\,680\, kC$
d.
$-\,670\, kC$

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Q4 mcq
A block A of mass 4 kg is placed on another block B of mass 5 kg, and the block B rests on a smooth horizontal table. If the minimum force that can be applied on A so that both the blocks move together is 12 N, the maximum force that can be applied on B for the blocks to move together will be :
a.
0 N Correct Answer
b.
25 N
c.
48 N
d.
27 N

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Q5 mcq
India's Mangalyan was sent to the Mars by launching it into a transfer orbit EOM around the sun. It leaves the earth at $E$ and meets Mars at $M$ . If the semi-major axis of Earth's orbit is $a_{e} = 1.5 \times 10^{11}m$ , that of Mar's orbit $a_{m} = 2.28 \times 10^{11}m$ , taking Kepler's laws, give the estimate of time for Mangalyan to reach Mars from Earth.
Question 5 image
a.
$220$ days.
b.
$500$ days.
c.
$260$ days. Correct Answer
d.
$320$ days.

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Q6 mcq
In materials like aluminium and copper, the correct order of magnitude of various elastic modulii is :
a.
Young's modulii $<$ shear modulii $<$ bulk modulii
b.
Bulk modulii $<$ shear modulii $<$ Young's modulii
c.
Shear modulii $<$ Young's modulii $<$ bulk modulii Correct Answer
d.
Bulk modulii $<$ Young's modulii $<$ shear modulii

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Q7 mcq
A cylinder of mass M and sphere of mass M are placed at points A and B of two inclines, respectively. (See c s figure). If they roll on the incline without slipping such that their accelerations are the same, then the ratio $\frac{sin\theta _{c}}{sin\theta _{s}}$ is :
Question 7 image
a.
$\frac{8}{7}$
b.
$\sqrt{\frac{8}{7}}$
c.
$\sqrt{\frac{15}{14}}$
d.
$\frac{15}{14}$ Correct Answer

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Q8 mcq
Water of volume 2 L in a closed container is heated with a coil of 1 kW. While water is heated, the container o o loses energy at a rate of 160 J/s. In how much time will the temperature of water rise from 27 C to 77 C ? (Specific heat of water is 4.2 kJ/kg and that of the container is negligible).
a.
8 min 20 s Correct Answer
b.
7 min
c.
14 min
d.
6 min 2 s

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Q9 mcq
A transmitting antenna at the top of a tower has a height $32\,m$ and the height of the receiving antenna is $50\,m$ . What is the maximum distance between them for satisfactory communication in line of sight (LOS) mode?
a.
$45. 5\, km$ Correct Answer
b.
$54. 5\, km$
c.
$55. 4\, km$
d.
$455\, km$

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Q10 mcq
Water is flowing at a speed of 1.5 m s-1 -2 2 through a horizontal tube of cross-sectional area 10 m and you are trying to stop the flow by your palm. Assuming that the water stops immediately after hitting the palm, the 3 -3 minimum force that you must exert should be (density of water = 10 kg m )
a.
33.7 N
b.
45 N
c.
15 N
d.
22.5 N Correct Answer

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Q11 mcq
The amplitude of a simple pendulum, oscillating in air with a small spherical bob, decreases from 10 cm to 8 cm in 40 seconds. Assuming that Stokes law is valid, and ratio of the coefficient of viscosity of air to that of carbon dioxide is 1.3, the time in which amplitude of this pendulum will reduce from 10 cm to 5 cm in carbondioxide will be close to (ln 5 = 1.601, ln 2 = 0.693).
a.
231 s
b.
142 s
c.
208 s
d.
161 s
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Q12 mcq
Identify the gate and match $A,B,Y$ in the bracket to check.
Question 12 image
a.
$OR\,(A= 1, \,B= 1, \,Y = 0)$
b.
$XOR\,(A= 0, \,B= 0, \,Y = 0)$
c.
$NOT\,(A= 1, \,B= 1, \,Y = 1)$
d.
$AND\,(A= 1, \,B= 1, \,Y = 1)$
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Q13 mcq
When the rms voltages $V_{L}, V_{C}$ and $V_{R}$ are measured respectively across the inductor $L,$ the capacitor $C$ and the resistor $R$ in a series $LCR$ circuit connected to an $AC$ source, it is found that the ratio $V_{L} :V_{C} :V_{R} = 1 : 2 : 3.$ If the rms voltage of the $AC$ source is $100 V,$ then $V_{R}$ is close to :
a.
$50 V$
b.
$70 V$
c.
$100 V$
d.
$90 V$
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Q14 mcq
The equation of state for a gas is given by $PV = nRT +\alpha V$ , where n is the number of moles and $\alpha$ is a positive constant. The initial temperature and pressure of one mole of the gas contained in a cylinder are T 0 and P respectively. The work done by the gas when its temperature doubles isobarically will be : 0
a.
$P_{0} T_{0} ln 2$
b.
$\frac{P_{0}T_{0}R}{P_{0}+\alpha}$
c.
$\frac{P_{0}T_{0}R}{P_{0}-\alpha}$
d.
$P_{0} T_{0}R$
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Q15 mcq
The mid points of two small magnetic dipoles of length $d$ in end-on positions, are separated by a distance $x$ $(x\gg d)$ . The magnitude of force between them is proportional to $x^{-n}$ where $n$ is :
a.
$3$
b.
$4$
c.
$2$
d.
$1$
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Q16 mcq
The focal lengths of objective lens and eye lens of a Galilean Telescope are respectively 30 cm and 3.0 cm. Telescope produces virtual, erect image of an object situated far away from it at least distance of distinct vision from the eye lens. In this condition the Magnifying Power of the Galilean Telescope should be :
a.
$+8. 8$
b.
$-11. 2$
c.
$+11. 2$
d.
$-8. 8$
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Q17 mcq
Using monochromatic light of wavelength $\lambda$ , an experimentalist sets up the Young's double slit experiment in three ways as shown. If she observes that $y=\beta '$ , the wavelength of light used is :
Question 17 image
a.
520 nm
b.
580 nm
c.
540 nm
d.
560 nm
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Q18 mcq
For which of the following particles will it be most difficult to experimentally verify the de-Broglie relationship?
a.
An electron
b.
An $\alpha -$ particle
c.
A dust particle
d.
A proton
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Q19 mcq
Match List I(Wavelength range of electromagnetic spectrum) with List II (Method of production of these waves) and select the correctoption from the options given below the lists. List I List II $700\, nm$ to (a) (i) Vibration of atoms and molecules. $1\, mm$ $1\, nm$ to Inner shell electrons in atoms moving from one (b) (ii) $400\, nm$ energy level to a lower level. (c) $< \,10^{-3}\, nm$ (iii) Radioactive decay of the nucleus. (d) $1\, mm$ to $0. 1\,m$ (iv) Magnetron valve.
a.
(a) - (iii), (b) - (iv), (c) - (i), (d) - (ii)
b.
(a) - (i), (b) - (ii), (c) - (iii), (d) - (iv)
c.
(a) - (iv), (b) - (iii), (c) - (ii), (d) - (i)
d.
(a) - (ii), (b) - (iii), (c) - (iv), (d) - (i)
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Q20 mcq
An n-p-n transistor has three leads A, B and C. Connecting B and C by moist fingers, A to the positive lead of an ammeter, and C to the negative lead of the ammeter, one finds large deflection. Then, A, B and C refer respectively to :
a.
Base, emitter and collector
b.
Base, collector and emitter
c.
Emitter, base and collector
d.
Collector, emitter and base
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Q21 mcq
A capillary tube is immersed vertically in water and the height of the water column is $x$ . When this arrangement is taken into a mine of depth d, the height of the water column is $y$ . If R is the radius of earth, the ratio $\frac{x}{y}$ is :
a.
$(\frac{R\,-\,d}{R\,+\,d})$
b.
$(\frac{R\,+\,d}{R\,-\,d})$
c.
$(1 - \frac{2d}{R} )$
d.
$(1 - \frac{d}{R})$
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Q22 mcq
An experiment is performed to obtain the value of acceleration due to gravity g by using a simple pendulum of length L. In this experiment time for 100 oscillations is measured by using a watch of 1 second least count and the value is 90.0 seconds. The length L is measured by using a meter scale of least count 1 mm and the value is 20.0 cm. The error in the determination of g would be :
a.
4.4%
b.
2.27%
c.
1.7%
d.
2.7%
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Q23 mcq
A d.c. main supply of e.m.f. $220\,V$ is connected across a storage battery of e.m.f. $200\,V$ through a resistance of $1\,\Omega$ . The battery terminals are connected to external resistance $R$ . The minimum value of $R$ , so that a current passes through the battery to charge it is:
a.
$9\,\Omega$
b.
$7\,\Omega$
c.
$11\,\Omega$
d.
$0$
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Q24 mcq
A diver looking up through the water sees the outside world contained in a circular horizon. The refractive index of water is $\frac{4}{3}$ , and the diver's eyes are $15\, cm$ below the surface of the water. Then the radius of the circle is :
a.
$\frac{15\times \sqrt{7}}{3}\, cm$
b.
$\frac{15\times 3}{\sqrt{7}}\, cm$
c.
$15 \times 3\sqrt{7}\, cm$
d.
$15 \times 3 \times \sqrt{5}\, cm$
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Q25 mcq
The magnetic field of earth at the equator is approximately $4 \times 10^{-5}T$ . The radius of earth is $6.4 \times 10^{6}m$ . Then the dipole moment of the earth will be nearly of the order of :
a.
1020 2 A m
b.
1016 2 A m
c.
1010 2 A m
d.
1023 2 A m
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Q26 mcq
The position of a projectile launched from the origin at t = 0 is given by $\vec{r}=(40\,ˆi+ 50\,ˆj)m$ at t = 2s. If the -2 projectile was launched at an angle $\theta$ from the horizontal, then $\theta$ is (take g = 10 ms ).
a.
$tan^{-1} \frac{3}{2}$
b.
$tan^{-1} \frac{2}{3}$
c.
$tan^{-1} \frac{7}{4}$
d.
$tan^{-1} \frac{4}{5}$
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Q27 mcq
A particle which is simultaneously subjected to two perpendicular simple harmonic motions represented by; $x=a_{1}cos\omega t$ and $y=a_{2}cos 2\omega t$ traces a curve given by :
Question 27 image
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Q28 mcq
Two bodies of masses $1\, kg$ and $4\, kg$ are connected to a vertical spring, as shown in the figure. The smaller mass executes simple harmonic motion of angular frequency $25\, rad \,s^{-1}$ , and amplitude $1. 6\, cm$ while the bigger mass remains stationary on the ground. The maximum force exerted by the system on the floor is (take $g\, = \,10\,m\,s^{-2}$ ).
Question 28 image
a.
$20\,N$
b.
$60\,N$
c.
$40\,N$
d.
$10\,N$
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Q29 mcq
Modern vacuum pumps can evacuate a vessel down to a pressure of $4.0 \times 10^{-15}$ atm. at room temperature -1 -1 5 $23 -1$ (300 K). Taking R = 8.3 JK mole , 1 atm = 10 Pa and $N_{Avogadro} = 6 \times 10 mole$ , the mean distance between molecules of gas in an evacuated vessel will be of the order of :
a.
$0. 2\,$ $\mu$ $m$
b.
$0. 2\, mm$
c.
$0. 2\, cm$
d.
$0. 2\, nm$
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Q30 mcq
If the binding energy of the electron in a hydrogen atom is $13. 6\, eV$ , the energy required to remove the electron from the first excited state of $Li^{++}$ is :
a.
$13. 6\, eV$
b.
$3. 4\, eV$
c.
$122. 4\, eV$
d.
$30. 6\, eV$
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Q31 mcq
Which of the following has unpaired electron(s)?
a.
$O^{-}_{2}$
b.
$N^{2+}_{2}$
c.
$O^{2-}_{2}$
d.
N 2
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Q32 mcq
Van der Waal's equation for a gas is stated as, $P = \frac{nRT}{V-nb} -a(\frac{n}{V})^{2}$ . This equation reduces to the perfect gas equation, $P = \frac{nRT}{V}$ When,
a.
temperature is sufficiently high and pressure is low.
b.
both temperature and pressure are very low.
c.
both temperature and pressure are very high.
d.
temperature is sufficiently low and pressure is high.
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Q33 mcq
In the hydroboration - oxidation reaction of propene with diborane, $H_{2}O_{2}$ and $NaOH,$ the organic compound formed is:
a.
$CH_{3}CH_{2}CH_{2}OH$
b.
$CH_{3}CH_{2}OH$
c.
$CH_{3}CHOHCH_{3}$
d.
$(CH_{3})_{3}COH$
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Q34 mcq
A compound of vanadium chloride has spin only magnetic moment of $1. 73$ BM. Its formula is
a.
VCl 2
b.
VCl 5
c.
VCl 4
d.
VCl 3
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Q35 mcq
The number and type of bonds in $C^{2-}_{2}$ ion in $CaC_{2}$ are:
a.
Two -bonds and two $\sigma$ $\pi$-bonds
b.
One -bond and two $\sigma$ $\pi$-bonds
c.
One -bond and one $\sigma$ $\pi$-bond
d.
Two -bonds and one -bond$\sigma$ $\pi$
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Q36 mcq
Structure of some important polymers are given. Which one represents Buna-S ?
Question 36 image
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Q37 mcq
The major product of the reaction is
Question 37 image
a.
​
b.
​
c.
​
d.
​
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Q38 mcq
The amount of oxygen in $3. 6$ moles of water is:
a.
$57. 6\,g$
b.
$115. 2\,g$
c.
$18. 4\,g$
d.
$28. 8\,g$
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Q39 mcq
At a certain temperature, only 50% HI is dissociated into H and I at equilibrium. The equilibrium constant is 2 2 :
a.
0.5
b.
4.0
c.
1.0
d.
0.25
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Q40 mcq
3+ An octahedral complex of Co is diamagnetic. The hybridisation involved in the formation of the complex is :
a.
dsp2
b.
d2 3 sp
c.
dsp3 d
d.
sp3 2 d
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Q41 mcq
Allyl phenyl ether can be prepared by heating :
a.
CH = CH - Br + C H - CH - ONa 2 6 5 2
b.
C H - CH = CH - Br + CH - ONa 6 5 3
c.
CH =​ CH - CH - Br + C H ONa C H Br + CH = CH - CH - ONa 2 2 6 5 6 5 2 2
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Q42 mcq
In which of the following pairs $A$ is more stable than $B?$
Question 42 image
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Q43 mcq
For the compounds CH Cl, CH Br, CH I and CH F, the correct order of increasing C-halogen bond length is : 3 3 3 3
a.
CH F < CH Br < CH Cl < CH I 3 3 3 3
b.
CH F < CH I < CH Br < CH Cl 3 3 3 3
c.
CH F < CH Cl < CH Br < CH I ​3 3 3 3
d.
CH Cl < CH Br < CH F < CH I 3 3 3 3
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Q44 mcq
The form of iron obtained from blast furnace is :
a.
Cast Iron
b.
Wrought Iron
c.
Steel
d.
Pig Iron
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Q45 mcq
$-^{In a nucleophilic substitution reaction :}R -Br + Cl^{-DMF}\rightarrow \rightarrow R -Cl + Br^{^{-}, which one of the following}$ undergoes complete inversion of configuration ?
a.
C H CHC H Br 6 5 6 5
b.
C H CCH C H Br 6 5 3 6 5
c.
C H CH Br 6 5 2
d.
C H CH CH Br 6 5 2 2
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Q46 mcq
Which one of the following class of compounds is obtained by polymerization of acetylene ?
a.
Poly-ene
b.
Poly-amide
c.
Poly-yne
d.
Poly-ester
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Q47 mcq
In a face centered cubic lattice atoms A are at the corner points and atoms B at the face centered points. If atom B is missing from one of the face centered points, the formula of the ionic compound is :
a.
A B 2 3
b.
AB 2
c.
A B 5 2
d.
A B 2 5
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Q48 mcq
The standard enthalpy of formation of $\, NH_{3}$ is $-\,46. 0\, kJ / mol$ . If bond enthalpy of $H_{2}$ is $-436\, kJ / mol$ and that of $N_{2}$ is $-712\, kJ / mol$ , the average bond enthalpy of $N -H$ bond in $NH_{3}$ is :
a.
$-\,964\, kJ / mol$
b.
$-\,1102\, kJ / mol$
c.
$+\,1056\, kJ / mol$
d.
$+\,352\, kJ / mol$
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Q49 mcq
Which is the major product formed when acetone is heated with iodine and potassium hydroxide?
a.
Acetophenone
b.
Iodoform
c.
Iodoacetone
d.
Acetic acid
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Q50 mcq
The temperature at which oxygen molecules have the same root mean square speed as helium atoms have at 300 K is : (Atomic masses : He = 4 u, O = 16 u)
a.
600 K
b.
2400 K
c.
1200 K
d.
300 K
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Q51 mcq
The standard electrode potentials $(E^{o}_{M^{+}/M})$ of four metals $A, B, C$ and $D$ are $-1.2 V,0.6 V$ , $0.85 V$ and $-0.76 V$ , respectively. The sequence of deposition of metals on applying potential is
a.
$B\, > \,D\, > \,C\, > \,A$
b.
$A\, > \,C\, > \,B\, > \,D$
c.
$C\, > \,B\, > \,D\, > \,A$
d.
$D\, > \,A\, > \,B\, > \,C$
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Q52 mcq
Which of the following is not formed when H S reacts with acidic K Cr O solution ? 2 2 2 7
a.
K SO 2 4
b.
Cr SO ) 2( 4 3
c.
S
d.
CrSO 4
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Q53 mcq
3- 3- The correct statement about the magnetic properties of [Fe(CN) ] and [FeF ] is : (Z = 26). 6 6
a.
Both are paramagnetic.
b.
Both are diamagnetic.
c.
[Fe(CN) 3- [Fe(CN) 3- ] is paramagnetic, ] is diamagnetic, 6 6 3- 3- [FeF ] is diamagnetic. [FeF ] is paramagnetic. 6 6
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Q54 mcq
Which one of the following reactions will not result in the formation of carbon-carbon bond ?
a.
Friedel Craft's acylation
b.
Reimer-Tieman reaction
c.
Cannizaro reaction
d.
Wurtz reaction
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Q55 mcq
In the following sets of reactants which two sets best exhibit the amphoteric character of Al O . xH O ? 2 3 2 $Set I Al_{2}O_{3}\cdot xH_{2}O_{(s)}+ OH^{-}_{(aq)}$ $Set II Al_{2}O_{3}x \cdot H_{2}O_{(s)}+ H_{2}O_{(l)}$ $Set III Al_{2}O_{3}x \cdot H_{2}O_{(s)}+ H^{+}_{(aq)}$ $Set IV Al_{2}O_{3}\cdot xH_{2}O_{(s)}+ NH_{3(aq)}$
a.
2 and 4
b.
1 and 2
c.
1 and 3
d.
3 and 4
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Q56 mcq
The gas evolved on heating $CaF_{2}$ and $SiO_{2}$ with concentrated $H_{2}SO_{4}$ , on hydrolysis gives a white gelatinous precipitate. The precipitate is:
a.
Silicic acid
b.
Calcium fluorosilicate
c.
Hydrofluorosilicic acid
d.
Silica gel
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Q57 mcq
A current of $10. 0\,A$ flows for $2. 00\,h$ through an electrolytic cell containing a molten salt of metal $X$ . This results in the decomposition of $0. 250\, mol$ of metal $X$ at the cathode. The oxidation state of $X$ in the molten salt is: $(F = 96, 500\,C)$
a.
$1+$
b.
$4+$
c.
$3+$
d.
$2+$
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Q58 mcq
The half-life period of a first-order reaction is $15\,minutes$ . The amount of substance left after one hour will be
a.
$\frac{1}{16}$ of the original amount
b.
$\frac{1}{8}$ of the original amount.
c.
$\frac{1}{32}$ of the original amount.
d.
$\frac{1}{4}$ of the original amount.
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Q59 mcq
Dissolving 120 g of a compound of (mol. wt. 60) in 1000 g of water gave a solution of density 1.12 g/mL. The molarity of the solution is :
a.
2.50 M
b.
1.00 M
c.
2.00 M
d.
4.00 M
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Q60 mcq
The energy of an electron in first Bohr's orbit of H atom is $-13.6 eV.$ The energy value of electron in the first excited state of $Li^{2+}$ is :
a.
$27.2 eV$
b.
$-30.6 eV$
c.
$30.6 eV$
d.
$-27.2 eV$
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Q61 mcq
$\int \frac{sin^{8}x-cos^{8}x}{(1-2 sin^{2}xcos^{2}x)}dx$ is equal to
a.
$-\frac{1}{2}sin 2x+c$
b.
$-sin^{2} x+c$
c.
$-\frac{1}{2}sinx+c$
d.
$\frac{1}{2}sin 2x+c$
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Q62 mcq
$\frac{1}{2}$ The integral $\int \frac{ln(1+2x)}{1+4x^{2}} dx$ equals $0$
a.
$\frac{\pi}{4} ln 2$
b.
$\frac{\pi}{16} ln 2$
c.
$\frac{\pi}{8} ln 2$
d.
$\frac{\pi}{32} ln 2$
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Q63 mcq
If $OB$ is the semi-minor axis of an ellipse, $F_{1}$ and $F_{2}$ are its focii and the angle between $F_{1}B$ and $F_{2}B$ is a right angle, then the square of the eccentricity of the ellipse is
a.
$\frac{1}{4}$
b.
$\frac{1}{\sqrt{2}}$
c.
$\frac{1}{2}$
d.
$\frac{1}{2\sqrt{2}}$
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Q64 mcq
The sum of the digits in the unit's place of all the $4$ - digit numbers formed by using the numbers $3, \,4, \,5$ and $\,6$ , without repetition is :
a.
$18$
b.
$36$
c.
$108$
d.
$432$
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Q65 mcq
If the differential equation representing the family of all circles touching $x$ -axis at the origin is $(x^{2} - y^{2})\frac{dy}{dx} =g(x)y$ , then $g(x)$ equals
a.
$\frac{1}{2}x^{2}$
b.
$2x$
c.
$\frac{1}{2}x$
d.
$2x^{2}$
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Q66 mcq
A line in the $3$ -dimensional space makes an angle $\theta (0 <\theta \leq \frac{\pi}{2})$ with both the $X$ and $Y-$ axes. Then, the set of all values of $\theta$ is in the interval :
a.
$(\frac{\pi}{3},\,\frac{\pi}{2} ]$
b.
$(0,\,\frac{\pi}{4}]$
c.
$[ \frac{\pi}{4} ,\,\frac{\pi}{2}]$
d.
$[\frac{\pi}{6},\,\frac{\pi}{3} ]$
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Q67 mcq
If $y=e^{nx}$ , then $\frac{d^{2}y}{dx^{2}}. \frac{d^{2}x}{dy^{2}}$ is equal to :
a.
$ne^{-nx}$
b.
$-ne^{-nx}$
c.
$ne^{nx}$
d.
$1$
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Q68 mcq
If $\vec{a} = 2, \, \vec{b} = 3$ and $2\vec{a}- \vec{b} = 5,$ then $2\vec{a}+ \vec{b}$ equals :
a.
$5$
b.
$7$
c.
$17$
d.
$1$
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Q69 mcq
If $B$ is a $3 \times 3$ matrix such that $B^{2} = 0$ , then $det. \,[(I+B)^{50} -50B]$ is equal to :
a.
$1$
b.
$2$
c.
$3$
d.
$50$
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Q70 mcq
Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than $200$ and less than $220.$ If the second term in it is $12$ , then its $4^{th}$ term is :
a.
$8$
b.
$24$
c.
$20$
d.
$16$
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Q71 mcq
If $f(x) =(\frac{3}{5})^{x}+(\frac{4}{5})^{x}-1, \,x\in R,$ then the equation $f(x) = 0$ has :
a.
No solution
b.
More than two solutions
c.
One solution
d.
Two solutions
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Q72 mcq
$|| || ||$ Equation of the plane which passes through the point of intersection of lines $\frac{x-1}{3} = \frac{y-2}{1} = \frac{z-3}{2}$ and $\frac{x-3}{1} = \frac{y-1}{2} = \frac{z-2}{3}$ and has the largest distance from the origin is:
a.
$4x+ 3y+ 5z= 50$
b.
$3x+ 4y+ 5z= 49$
c.
$5x+ 4y+ 3z= 57$
d.
$7x+ 2y+ 4z= 54$
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Q73 mcq
If $a, \,b, \,c$ are non - zero real numbers and if the system of equations $(a-1)x=y+z$ $(b-1)y=x+z$ $(c-1)z=x+y$ has a non - trivial solution, then $ab+bc+ca$ equals :
a.
$-1$
b.
$a+b+c$
c.
$abc$
d.
$1$
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Q74 mcq
If $A$ and $B$ are two events such that $P(A\cup B) =P(A\cap B)$ , then the incorrect statement amongst the following statements is :
a.
$P(A) +P(B) = 1$
b.
$P(A\cap B') = 0$
c.
$A&B$ are equally likely
d.
$P(A' \cap B) = 0$
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Q75 mcq
If equations $ax^{2} +bx+c= 0, \,(a, \,b, \,c\in R, \,a\neq 0)$ and $2x^{2} + 3x+ 4 = 0$ have a common root, then $a:b:c$ equals :
a.
$2 : 3 : 4$
b.
$4 : 3 : 2$
c.
$1 : 2 : 3$
d.
$3 : 2 : 1$
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Q76 mcq
The number of terms in the expansion of $(1 +x)^{101}(1 - x+x^{2})^{100}$ in powers of $x$ is
a.
$301$
b.
$302$
c.
$101$
d.
$202$
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Q77 mcq
If $cosec\,\theta = \frac{p+q}{\,\,p-q\,\,}(p \neq q,\,p \neq 0), \,$ then $cot(\frac{\pi}{4} + \frac{\theta}{2})$ is equals to:
a.
$pq$
b.
$\sqrt{\frac{p}{q}}$
c.
$\sqrt{\frac{q}{p}}$
d.
$\sqrt{pq}$
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Q78 mcq
Let $a$ and $b$ be any two numbers satisfying $\frac{1}{a^{2}} + \frac{1}{b^{2}} = \frac{1}{4}.$ Then, the foot of perpendicular from the origin on the variable line $\frac{x}{a} + \frac{y}{b} = 1$ lies on :
a.
A circle of radius $= 2$
b.
A hyperbola with each semi-axis $= \sqrt{2}$ .
c.
A hyperbola with each semi-axis $= 2$
d.
A circle of radius $= \sqrt{2}$
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Q79 mcq
If the point $(1, \,4)$ lies inside the circle $x^{2} +y^{2} -6x+ 10y+p= 0$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of $p$ is the interval
a.
$(25, \,39)$
b.
$(25, \,29)$
c.
$(0, \,25)$
d.
$(9, \,25)$
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Q80 mcq
If $f(x)$ is continuous and $f( \frac{9}{2})= \frac{2}{9},$ then $_{x}lim_{\rightarrow 0\rightarrow } f(\frac{1-cos 3x}{x^{2}})$ equals to
a.
$\frac{8}{9}$
b.
$0$
c.
$\frac{2}{9}$
d.
$\frac{9}{2}$
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Q81 mcq
Let $P$ be the relation defined on the set of all real numbers such that $P ={(a,b) : sec^{2} a-tan^{2} b= 1}$ . Then, $P$ is
a.
reflexive and symmetric but not transitive
b.
symmetric and transitive but not reflexive
c.
reflexive and transitive but not symmetric
d.
an equivalence relation
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Q82 mcq
Let $A={(x,y) :y^{2} \leq 4x, \,y-2x\geq -4}$ . The area of the region $A$ in square units is
a.
$10$
b.
$8$
c.
$9$
d.
$11$
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Q83 mcq
In a set of $2n$ distinct observations, each of the observation below the median of all the observations is increased by $5$ and each of the remaining observations is decreased by $3$ . Then, the mean of the new set of observations :
a.
Increases by $2$ .
b.
Increases by $1$ .
c.
Decreases by $2$ .
d.
Decreases by $1$ .
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Q84 mcq
Let $w(Im w\neq 0)$ be a complex number. Then, the set of all complex numbers $z$ satisfying the equation $w- wz=k(1 - z)$ , for some real number $k$ , is
a.
$\bar{{}z: \,z\neq 1}$
b.
${z: \,|z| = 1, \,z\neq 1}$
c.
${z: \,z=z}$
d.
$-{z: \,|z| = 1}$
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Q85 mcq
If the Rolle's theorem holds for the function $f(x) = 2x^{3} +ax^{2} +bx$ in the interval $[-1, 1]$ for the point $c= \frac{1}{2}$ , then the value of $2a+b$ is:
a.
$-1$
b.
$2$
c.
$1$
d.
$-2$
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Q86 mcq
Given three points $P,Q,R$ with $P(5, 3)$ and $R$ lies on the $x-$ axis. If the equation of $RQ$ is $x-2y= 2$ and $PQ$ is parallel to the $x-$ axis, then the centroid of $\Delta PQR$ lies on the line
a.
$x-2y+ 1 = 0$
b.
$2x+y-9 = 0$
c.
$2x-5y= 0$
d.
$5x-2y= 0$
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Q87 mcq
If $\frac{1}{\sqrt{\alpha}} , \frac{1}{\sqrt{\beta}}$ are the roots of the equation $ax^{2} +bx+ 1 = 0, \,(a\neq 0, \,a,b\in R)$ , then the equation $x(x+b^{3})+(a^{3} -3abx)= 0$ has roots:
a.
$\sqrt{\alpha \beta}$ and $\alpha \beta$
b.
$\alpha ^{-}\frac{3}{2}$ and $\beta ^{-}\frac{3}{2}$
c.
$\alpha \beta \frac{1}{2}$ and $\alpha \frac{1}{2}\beta$
d.
$\alpha \frac{3}{2}$ and $\beta \frac{3}{2}$
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Q88 mcq
The contrapositive of the statement "I go to school if it does not rain" is
a.
If it rains, I go to school.
b.
If it rains, I do not go to school.
c.
If I go to school, it rains.
d.
If I do not go to school, it rains.
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Q89 mcq
$||$ The number of values of in $\alpha$ $[0, \,2$ $\pi$ $]$ for which $2 sin^{3}$ $\alpha$ $-7 sin^{2}$ $\alpha$ $+ 7sin$ $\alpha$ $= 2$ , is :
a.
$3$
b.
$1$
c.
$6$
d.
$4$
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Q90 mcq
If the sum $\frac{3}{1^{2}} + \frac{5}{1^{2}+2^{2}} + \frac{7}{1^{2}+2^{2}+3^{2}}+ .... .+$ up to $20$ terms is equal to $\frac{k}{21} ,$ then $k$ is equal to
a.
$240$
b.
$120$
c.
$60$
d.
$180$
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