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Question 1. Let A=\begin{bmatrix}2&1&2\\6&2&11\\3&3&2\end{bmatrix} and P=\begin{bmatrix}1&2&0\\5&0&2\\7&1&5\end{bmatrix}.
The sum of the prime factors of \left|P^{-1}AP-2I\right| is equal to
(1) 26
(2) 27
(3) 66
(4) 23
Answer: (1)
Question 2. Number of ways of arranging 8 identical books into 4 identical shelves, where any number of shelves may remain empty, is equal to
(1) 18
(2) 16
(3) 12
(4) 15
Answer: (4)
Question 3. Let P(3,2,3), Q(4,6,2) and R(7,3,2) be the vertices of triangle \triangle PQR.
Then the angle \angle QPR is
(1) \frac{\pi}{6}
(2) \cos^{-1}\left(\frac{7}{18}\right)
(3) \cos^{-1}\left(\frac{1}{18}\right)
(4) \frac{\pi}{3}
Answer: (4)
Question 4. If the mean and variance of five observations are \frac{24}{5} and \frac{194}{25} respectively, and the mean of the first four observations is \frac{7}{2}, then the variance of the first four observations is equal to
(1) \frac{4}{5}
(2) \frac{77}{12}
(3) \frac{5}{4}
(4) \frac{105}{4}
Answer: (3)
Question 5. The function f(x)=2x+3x^{\frac{2}{3}}, where x\in\mathbb{R}, has
(2) exactly one point of local maxima and no point of local minima
(1) exactly one point of local minima and no point of local maxima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima
Answer: (3)
Question 6. Let r and \theta respectively be the modulus and amplitude of the complex number z=2-i\left(2\tan\frac{5\pi}{8}\right).
Then is equal to
(1) \left(2\sec\frac{3\pi}{8},\frac{3\pi}{8}\right)
(2) \left(2\sec\frac{3\pi}{8},\frac{5\pi}{8}\right)
(3) \left(2\sec\frac{5\pi}{8},\frac{3\pi}{8}\right)
(4) \left(2\sec\frac{11\pi}{8},\frac{11\pi}{8}\right)
Answer: (1)
Question 7. The sum of the solutions x\in\mathbb{R} of the equation
\frac{3\cos2x+\cos^{3}2x}{\cos^{6}x-\sin^{6}x}=x^{3}-x^{2}+6 is
(1) 0
(2) 1
(3) -1
(4) 3
Answer: (3)
Question 8. Let \overrightarrow{OA}=\vec a, \overrightarrow{OB}=12\vec a+4\vec b and \overrightarrow{OC}=\vec b, where O is the origin.
If S is the parallelogram with adjacent sides OA and OC, then Area of the quadrilateral OABC Area of S is equal to ____.
(1) 6
(2) 10
(3) 7
(4) 8
Answer: (4)
Question 9. If \log_e a, \log_e b, \log_e c are in an A.P. and
\log_e\left(\frac{a}{2b}\right), \log_e\left(\frac{2b}{3c}\right),
\log_e\left(\frac{3c}{a}\right) are also in an A.P., then a:b:c is equal to
(1) 9:6:4
(2) 16:4:1
(3) 25:10:4
(4) 6:3:2
Answer: (1)
Question 10. If
\int\frac{\sin^\frac32x+\cos^\frac32x}{\sqrt{\sin^3x\cos^3x\sin(x-\theta)}}\,dx=A\sqrt{\cos\theta\tan x-\sin\theta}+B\sqrt{\cos\theta-\sin\theta\cot x}+C,
where C is the constant of integration, then AB is equal to
(1) 4\operatorname{cosec}(2\theta)
(2) 4\sec\theta
(3) 2\sec\theta
(4) 8\operatorname{cosec}(2\theta)
Answer: (4)
Question 11. The distance of the point from the line 2x-3y+28=0,
measured parallel to the line \sqrt{3}x-y+1=0, is equal to
(1) 4\sqrt{2}
(2) 6\sqrt{3}
(3) 3+4\sqrt{2}
(4) 4+6\sqrt{3}
Answer: (4)
Question 12. If \sin\left(\frac{y}{x}\right)=\log_e|x|+\frac{\alpha}{2}
is the solution of the differential equation
x\cos\left(\frac{y}{x}\right)\frac{dy}{dx}=y\cos\left(\frac{y}{x}\right)+xand y(1)=\frac{\pi}{3}, then \alpha^2 is equal to
(1) 3
(2) 12
(3) 4
(4) 9
Answer: (1)
Question 13. If each term of a geometric progression a_1,a_2,a_3,\ldots with
a_1=\frac{1}{8} and a_2\ne a_1, is the arithmetic mean of the next two terms,
and S_n=a_1+a_2+\cdots+a_n, then S_{20}-S_{18} is equal to
(1) 2^{15}
(2) -2^{18}
(3) 2^{18}
(4) -2^{15}
Answer: (4)
Question 14. Let A be the point of intersection of the lines 3x+2y=14 and 5x-y=6, and B be the point of intersection of the lines 4x+3y=8 and 6x+y=5.
The distance of the point P(5,-2) from the line AB is
(1) \frac{13}{2}
(2) 8
(3) \frac{5}{2}
(4) 6
Answer: (4)
Question 15. Let x=\frac{m}{n}, where m and n are co-prime natural numbers,
be a solution of the equation \cos\left(2\sin^{-1}x\right)=\frac{1}{9}.
Let \alpha,\beta with \alpha>\beta be the roots of the equationmx^{2}-nx-m+n=0.
Then the point lies on the line
(1) 3x+2y=2
(2) 5x-8y=-9
(3) 3x-2y=-2
(4) 5x+8y=9
Answer: (4)
Question 16. The function f(x)=\frac{x}{x^{2}-6x-16}, where
x\in\mathbb{R}-\left(-2,8\right),
(1) decreases in (-2, 8) and increases in \left(-\infty,\;-2\right)\cup\left(8,\;\infty\right)
(2) decreases in \left(-\infty,\;-2\right)\cup\left(-2,\;8\right)\cup\left(8,\;\infty\right)
(3) decreases in \left(-\infty,\;-2\right) and increases in \left(8,\;\infty\right)
(4) increases in \left(-\infty,-2\right)\cup(-2,8)\cup(8,\infty)
Answer: (2)
Question 17. Let y=\log_{e}\left(\frac{1-x^{2}}{1+x^{2}}\right), where -1<x<1.
At x=\frac{1}{2}, find the value of 225\left(y'-y''\right).
(1) 732
(2) 746
(3) 742
(4) 736
Answer: (4)
Question 18. If R is the smallest equivalence relation on the set {1,2,3,4} such that
{(1,2),(1,3)}\subset R, then the number of elements in R is ____.
(1) 10
(2) 12
(3) 8
(4) 15
Answer: (1)
Question 19. An integer is chosen at random from the integers 1, 2,3,\ldots,50.
The probability that the chosen integer is a multiple of at least one of 4, 6, and 7 is:
(1) \frac{8}{25}
(2) \frac{21}{50}
(3) \frac{9}{50}
(4) \frac{14}{25}
Answer: (2)
Question 20. Let a unit vector \hat{u}=x\hat{i}+y\hat{j}+z\hat{k} make angles
\frac{\pi}{2}, \frac{\pi}{3}, and \frac{2\pi}{3}
with the vectors \frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{k},
\frac{1}{\sqrt{2}}\hat{j}+\frac{1}{\sqrt{2}}\hat{k}, and
\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j} respectively.
If \vec{v}=\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\frac{1}{\sqrt{2}}\hat{k},
then the value of |\hat{u}-\vec{v}|^{2} is:
(1) \frac{11}{2}
(2) \frac{5}{2}
(3) 9
(4) 7
Answer: (2)
Question 21. Let \alpha,\beta be the roots of the equation
x^{2}-\sqrt{6}\,x+3=0 such that \operatorname{Im}(\alpha)>\operatorname{Im}(\beta).
Let a,b be integers not divisible by 3, and let n be a natural number such that
\frac{\alpha^{99}}{\beta}+\alpha^{98}=3^{n}(a+ib), where i=\sqrt{-1}.
Then the value of n+a+b is ____ .
Answer: 49
Question 22. Let for any three distinct consecutive terms a,b,c of an A.P.,
the lines ax+by+c=0 be concurrent at a point P. Let Q(\alpha,\beta)
be a point such that the system of equations x+y+z=6, 2x+5y+\alpha z=\beta,
x+2y+3z=4 has infinitely many solutions. Then the value of ^{2} is _____.
Answer: 113
Question 23. Let P(\alpha,\beta) be a point on the parabola y^{2}=4x.
If P also lies on the chord of the parabola x^{2}=8y whose midpoint is
\left(1,\frac{5}{4}\right), then the value of (\beta-8) is ____ .
Answer: 192
Question 24. If
\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\sqrt{1-\sin 2x}\,dx=\alpha+\beta\sqrt{2}+\gamma\sqrt{3},
where \alpha,\beta,\gamma are rational numbers,
then the value of 3\alpha+4\beta-\gamma is ____.
Answer: 6
Question 25. Remainder when 64^{32^{32}} is divided by 9 is equal to ____.
Answer: 1
Question 26. Two sources of light emit with a power of 200 W each. The ratio of the number of photons of visible light emitted by the two sources, having wavelengths 300 nm and 500 nm respectively, is asked.
(1) 1:5
(2) 1:3
(3) 5:3
(4) 3:5
Answer: (4)
Question 27. The truth table for the given circuit is shown below.
Answer: (2)
Question 28. A physical quantity Q depends on the quantities a, b and c according to the relation
Q=\frac{a^4b^3}{c^2}. The percentage errors in a, b and c are 3 %, 4 % and 5 % respectively.
The percentage error in Q is asked.
(1) 66%
(2) 43%
(3) 34%
(4) 14%
Answer: (3)
Question 29. In an a.c. circuit, the voltage and current are given by V=100\sin(100t) V
I=100\sin\left(100t+\frac{\pi}{3}\right) mA respectively. The average power dissipated in one cycle is asked.
(1) 5 W
(2) 10 W
(3) 2.5 W
(4) 25 W
Answer: (3)
Question 30. The temperature of a gas having 2.0\times10^{25} molecules per cubic meter at 1.38 atm is asked.
Given, k=1.38\times10^{-23} JK⁻¹
(1) 500 K
(2) 200 K
(3) 100 K
(4) 300 K
Answer: (1)
Question 31. A stone of mass 900 g is tied to a string and moved in a vertical circle of radius 1 m.
It makes 10 rpm. The tension in the string at the lowest point is asked.
Given, \pi^{2}=9.8 and g=9.8 m/s²
(1) 97 N
(2) 9.8 N
(3) 8.82 N
(4) 17.8 N
Answer: (2)
Question 32. The bob of a pendulum is released from a horizontal position. The length of the pendulum is 10 m. It dissipates 10\% of its initial energy due to air resistance. The speed of the bob at the lowest point is asked.
Given, g=10 m/s²
(1) 6\sqrt{5} m/s
(2) 5\sqrt{6} m/s
(3) 5\sqrt{5} m/s
(4) 2\sqrt{5} m/s
Answer: (1)
Question 33. If the distance between an object and its two times magnified virtual image formed by a curved mirror is 15 cm, the focal length of the mirror is asked.
(1) 15 cm
(2) -12 cm
(3) -10 cm
(4) \frac{10}{3} cm
Answer: (3)
Question 34. Two particles X and Y have equal charges. They are accelerated through the same potential difference. They then enter normally into a region of uniform magnetic field. They describe circular paths of radii R_{1} and R_{2} respectively. The ratio of the masses of X and Y is asked.
(1) \left(\frac{R_{2}}{R_{1}}\right)^{2}
(2) \left(\frac{R_{1}}{R_{2}}\right)^{2}
(3) \frac{R_{1}}{R_{2}}
(4) \frac{R_{2}}{R_{1}}
Answer: (2)
Question 35. In Young’s double slit experiment, light from two identical sources are superimposing on a screen. The path difference between the two lights reaching at a point on the screen is \frac{7 \lambda}{4}. The ratio of intensity of fringe at this point with respect to the maximum intensity of the fringe is :
(1) 1 / 2
(2) 3 / 4
(3) 1 / 3
(4) 1 / 4
Answer: (1)
Question 36. A small liquid drop of radius R is divided into 27 identical liquid drops. The surface tension of the liquid is T. The work done in the process is asked.
(1) 8\pi R^{2}T
(2) 3\pi R^{2}T
(3) \frac{1}{8}\pi R^{2}T
(4) 4\pi R^{2}T
Answer: (1)
Question 37. A bob of mass m is suspended by a light string of length L. It is given a minimum horizontal velocity at the lowest point A. It just completes a half circle and reaches the topmost point B.
The ratio \frac{(\text{K.E.}){A}}{(\text{K.E.}){B}} is asked.
(1) 3:2
(2) 5:1
(3) 2:5
(4) 1:5
Answer: (2)
Question 38. A wire of length L and radius r is clamped at one end. A force F is applied at the other end.
The length of the wire increases by \ell. Now, the radius of the wire and the applied force are both reduced to half. The original length L is kept the same. The new increase in length is asked.
(1) 3 times
(2) \frac{3}{2} times
(3) 4 times
(4) 2 times
Answer: (4)
Question 39. A planet takes 200 days to complete one revolution around the Sun. The distance of the planet from the Sun is reduced to one fourth of its original distance. The new time period is asked.
(1) 25 days
(2) 50 days
(3) 100 days
(4) 20 days
Answer: (1)
Question 40. A plane electromagnetic wave of frequency 35 MHz travels in free space along the X-direction.
At a particular point in space and time, \vec{E}=9.6\hat{j} V/m. The magnetic field at this point is asked.
(1) 3.2\times10^{-8}\hat{k} T
(2) 3.2\times10^{-8}\hat{i} T
(3) 9.6\hat{j} T
(4) 9.6\times10^{-8}\hat{k} T
Answer: (1)
Question 41. In the given circuit, the current in resistance R_{3} is asked.
(1) 1 A
(2) 1.5 A
(3) 2 A
(4) 2.5 A
Answer: (1)
Question 42. A particle is moving in a straight line. The variation of position x with time t is given by x = t^{3} - 6t^{2} + 20t + 15 m. The velocity of the particle when its acceleration becomes zero is:
(1) 4 m/s
(2) 8 m/s
(3) 10 m/s
(4) 6 m/s
Answer: (2)
Question 43. N moles of a polyatomic gas with degrees of freedom f = 6 are mixed with two moles of a monoatomic gas. The mixture behaves like a diatomic gas. The value of N is:
(1) 6
(2) 3
(3) 4
(4) 2
Answer: (3)
Question 44. Given below are two statements:
Statement I:
Most of the mass of the atom and all its positive charge are concentrated in a tiny nucleus, and electrons revolve around it. This is Rutherford’s atomic model.
Statement II:
An atom is a spherical cloud of positive charge with electrons embedded in it.
This is a special case of Rutherford’s atomic model.
In the light of the above statements, choose the most appropriate option:
(1) Both statement I and statement II are false
(2) Statement I is false but statement II is true
(3) Statement I is true but statement II is false
(4) Both statement I and statement II are true
Answer: (3)
Question 45. An electric field is given by \vec{E} = 6\hat{i} + 5\hat{j} + 3\hat{k} N/C. The electric flux through a surface of area 30, lying in the YZ-plane, is required. The answer is in SI units.
(1) 90
(2) 150
(3) 180
(4) 60
Answer: (3)
Question 46. Two metallic wires P and Q have the same volume and are made of the same material. The ratio of their cross-sectional areas is 4:1. A force F_{1} applied to wire P produces an extension \Delta l. The force required to produce the same extension in wire Q is F_{2}. The value of \frac{F_{1}}{F_{2}} is ____.
Answer: (4)
Question 47. A horizontal straight wire 5 m long extending from east to west falling freely at right angle to horizontal component of earth’s magnetic field 0.60 \times 10^{-4} \mathrm{Wbm}^{-2}. The instantaneous value of emf induced in the wire when its velocity is 10 \mathrm{~ms}^{-1} is _____ \times 10^{-3} \mathrm{~V}.
Answer: (3)
Question 48. A hydrogen atom is bombarded with electrons accelerated through a potential difference V, causing excitation of hydrogen atoms. The experiment is performed at temperature T = 0 K. The minimum potential difference required to observe any Balmer series line in the emission spectrum is \frac{\alpha}{10} V. The value of \alpha is ____.
(Answer in integer form)
Answer: (121)
Question 49. A charge of 4.0\mu C is moving with a velocity of 4.0 \times 10^{6} m/s along the positive y-axis.
It is placed in a magnetic field \vec{B} = 2\hat{k} T.
The magnetic force acting on the charge is x\hat{i} N. The value of x is ___.
Answer: (32)
Question 50. A body of mass 5 kg moves with a uniform speed 3\sqrt{2} m/s in the X–Y plane. It moves along the line y = x + 4. The angular momentum of the particle about the origin is ______ \mathrm{kg}\,\mathrm{m}^{2}\,\mathrm{s}^{-1}.
Answer: (60)
Question 51. The ascending order of acidity of the following H atoms is given below.
(1) C < D < B < A
(2) A < B < C < D
(3) A < B < D < C
(4) D < C < B < A
Answer: (1)
Question 52. Match List I with List II.
| List I (Bio Polymer) | List II (Monomer) |
|---|---|
| A. Starch | I. Nucleotide |
| B. Cellulose | II. \alpha-glucose |
| C. Nucleic acid | III. \beta-glucose |
| D. Protein | IV. \alpha-amino acid |
Choose the correct answer from the options given below.
(1) A-II, B-I, C-III, D-IV
(2) A-IV, B-II, C-I, D-III
(3) A-I, B-III, C-IV, D-II
(4) A-II, B-III, C-I, D-IV
Answer: (4)
Question 53. Match List I with List II.
| List I (Compound) | List II (\mathrm{p}K_{\mathrm{a}} value) |
|---|---|
| A. Ethanol | I. 10.0 |
| B. Phenol | II. 15.9 |
| C. m-Nitrophenol | III. 7.1 |
| D. p-Nitrophenol | IV. 8.3 |
Choose the correct answer from the options given below.
(1) A-I, B-II, C-III, D-IV
(2) A-IV, B-I, C-II, D-III
(3) A-III, B-IV, C-I, D-II
(4) A-II, B-I, C-IV, D-III
Answer: (4)
Question 54. Which of the following reactions is correct?
Answer: (2)
Question 55. According to IUPAC system, the compound 
is named as:
(1) Cyclohex-1-en-2-ol
(2) 1-Hydroxyhex-2-ene
(3) Cyclohex-1-en-3-ol
(4) Cyclohex-2-en-1-ol
Answer: (4)
Question 56. The correct IUPAC name of \mathrm{K}{2}\mathrm{MnO}{4} is:
(1) Potassium tetraoxopermanganate (VI)
(2) Potassium tetraoxidomanganate (VI)
(3) Dipotassium tetraoxidomanganate (VII)
(4) Potassium tetraoxidomanganese (VI)
Answer: (2)
Question 57. A reagent which gives brilliant red precipitate with Nickel ions in basic medium is:
(1) Sodium nitroprusside
(2) Neutral \mathrm{FeCl}_{3}
(3) Meta-dinitrobenzene
(4) Dimethyl glyoxime
Answer: (4)
Question 58. Phenol treated with chloroform in the presence of sodium hydroxide, followed by hydrolysis in the presence of an acid, results in:
(1) Salicylic acid
(2) Benzene-1,2-diol
(3) Benzene-1,3-diol
(4) 2-Hydroxybenzaldehyde
Answer: (4)
Question 59. Match List I with List II.
| List I (Spectral Series for Hydrogen) | List II (Spectral Region / Higher Energy State) |
|---|---|
| A. Lyman | I. Infrared region |
| B. Balmer | II. UV region |
| C. Paschen | III. Infrared region |
| D. Pfund | IV. Visible region |
Choose the correct answer from the options given below.
(1) A-II, B-III, C-I, D-IV
(2) A-I, B-III, C-II, D-IV
(3) A-II, B-IV, C-III, D-I
(4) A-I, B-II, C-III, D-IV
Answer: (3)
Question 60. On passing a gas X through Nessler’s reagent, a brown precipitate is obtained.
The gas X is:
(1) \mathrm{H}{2}\mathrm{S}
(2) \mathrm{CO}{2}
(3) \mathrm{NH}{3}
(4) \mathrm{Cl}{2}
Answer: (3)
Question 61. The product A formed in the following reaction is:
Answer: (3)
Question 62. Identify the reagents used for the following conversion:
Answer: (4)
Question 63. Which of the following acts as a strong reducing agent?
(Atomic number: \mathrm{Ce}=58, \mathrm{Eu}=63, \mathrm{Gd}=64, \mathrm{Lu}=71)
(1) \mathrm{Lu}^{3+}
(2) \mathrm{Gd}^{3+}
(3) \mathrm{Eu}^{2+}
(4) \mathrm{Ce}^{4+}
Answer: (3)
Question 64. Chromatographic technique/s based on the principle of differential adsorption is/are:
A. Column chromatography
B. Thin layer chromatography
C. Paper chromatography
Choose the most appropriate answer from the options given below.
(1) B only
(2) A only
(3) A & B only
(4) C only
Answer: (3)
Question 65. Which of the following statements are correct about \mathrm{Zn}, \mathrm{Cd} and Hg?
A. They exhibit high enthalpy of atomization as the d-subshell is full.
B. Zn and Cd do not show variable oxidation state while Hg shows +I and +II.
C. Compounds of \mathrm{Zn}, \mathrm{Cd} and Hg are paramagnetic in nature.
D. \mathrm{Zn}, \mathrm{Cd} and Hg are called soft metals.
Choose the most appropriate answer from the options given below.
(1) B, D only
(2) B, C only
(3) A, D only
(4) C, D only
Answer: (1)
Question 66. The element having the highest first ionization enthalpy is:
(1) Si
(2) Al
(3) N
(4) C
Answer: (3)
Question 67. Alkyl halide is converted into alkyl isocyanide by reaction with:
(1) NaCN
(2) \mathrm{NH}_{4}\mathrm{CN}
(3) KCN
(4) AgCN
Answer: (4)
Question 68. Which one of the following will show geometrical isomerism?
Answer: (3)
Question 69. Given below are two statements:
Statement I:
Fluorine has the most negative electron gain enthalpy in its group.
Statement II:
Oxygen has the least negative electron gain enthalpy in its group.
In the light of the above statements, choose the most appropriate answer from the options given below.
(1) Both Statement I and Statement II are true
(2) Statement I is true but Statement II is false
(3) Both Statement I and Statement II are false
(4) Statement I is false but Statement II is true
Answer: (4)
Question 70. Anomalous behaviour of oxygen is due to its:
(1) Large size and high electronegativity
(2) Small size and low electronegativity
(3) Small size and high electronegativity
(4) Large size and low electronegativity
Answer: (3)
Question 71. The total number of antibonding molecular orbitals formed from 2s and 2p atomic orbitals in a diatomic molecule is ______ .
Answer: 4
Question 72. The oxidation number of iron in the compound formed during the brown ring test for \mathrm{NO}_{3}^{-} ion is ____.
Answer: 1
Question 73. The following concentrations were observed at 500 K for the formation of \mathrm{NH}{3} from \mathrm{N}{2} and \mathrm{H}_{2}.
At equilibrium:
[\mathrm{N}{2}]=2\times10^{-2}\,\mathrm{M} [\mathrm{H}{2}]=3\times10^{-2}\,\mathrm{M} [\mathrm{NH}_{3}]=1.5\times10^{-2}\,\mathrm{M}The equilibrium constant for the reaction is _____.
Answer: 417
Question 74. Molality of 0.8\,\mathrm{M}\,\mathrm{H}{2}\mathrm{SO}{4} (density 1.06\,\mathrm{g}\,\mathrm{cm}^{-3}) is_____. \times10^{-3}\,\mathrm{m}.
Answer: 815
Question 75. If 50 mL of 0.5 M oxalic acid is required to neutralise 25 mL of NaOH solution, the amount of NaOH in 50 mL of the given NaOH solution is _____ g.
Answer: 4