Download Jee advance questions paper and more High school final essays Physics in PDF only on Docsity!
Q. Let f :[1, ) be a differentiable function such that (1) 1 3
f and 3 3 1 ( ) ( ) 3 , [1, )
x (^) x
f t dt^ ^ x f^ x^ ^ x ^ . Let^ e^ denote the base of the natural logarithm. Then the
value of f ( ) e is
(A)
e (^) (B) log 4 3
e ^ e (C)^42 3
e (^) (D)^2 3
e
Q.2 Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is
, then the probability that the experiment stops with head is
(A) 1
(B)
(C)
(D) 2
Q.
For any y^ ^ ^ , let cot ^1 ( ) y (0, )and tan 1 ( ) ,.
y ( )
^ Then the sum of all the solutions
of the equation
2 1 1 2 tan 6 cot^9 9 6 3
( y^ ) ( y )
y y
^
for 0 | y | 3,is equal to
(A) 2 3 3 (B) 3 2 3 (C) 4 3 6 (D) 6 4 3
SECTION 1 (Maximum Marks: 12)
This section contains FOUR (04) questions. Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct answer. For each question, choose the option corresponding to the correct answer. Answer to each question will be evaluated according to the following marking scheme: Full Marks : 3 If ONLY the correct option is chosen; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : െ1 In all other cases.
Mathematics
Q.4 Let the position vectors of the points P , Q R , and S be a^ ^ i ˆ 2 ˆ j 5 k ˆ, b^ ^ 3 i ˆ 6 ˆ j 3 k ˆ,
c^ ^ i j k and (^) d 2 i ˆ^ ˆ j k ˆ
, respectively. Then which of the following statements is true?
(A) The points P , Q R , and S are NOT coplanar
(B)
b d
is the position vector of a point which divides PR internally in the ratio 5 : 4
(C) 2 3
b d
is the position vector of a point which divides PR externally in the ratio 5 : 4 (D) The square of the magnitude of the vector b d
is 95
Q.
Let f : (0,1) be the function defined as
( ) [4 ] ,
f x x^ ^ x ^ x
where [ ] x denotes
the greatest integer less than or equal to x. Then which of the following statements is(are) true?
(A) The function f is discontinuous exactly at one point in (0,1)
(B) There is exactly one point in (0,1) at which the function f is continuous but NOT
differentiable
(C) The function f is NOT differentiable at more than three points in (0,1)
(D) The minimum value of the function f^ is
Q.
Let S be the set of all twice differentiable functions f from to such that
2 2 ( )^0
d f (^) x dx
for
all x ( 1,1). For f S , let X f be the number of points x ( 1,1) for which f ( ) x x .Then
which of the following statements is(are) true?
(A) There exists a function f S such that X f 0
(B) For every function f S , we have X f 2
(C) There exists a function f S such that X f 2
(D) There does NOT exist any function f^ in S such that X f 1
Q.8 For x , let tan^1 ( )^ , 2 2
x ( ^ ). Then the minimum value of the function
f : defined by
(^1) ( cos ) 2023 0
x tan x (^) et t f x dt t
(^)
is
Q.9 For x , let y x ( ) be a solution of the differential equation
( x^2^ 5)^ dy 2 x y 2 ( x x 2 5)^2 dx
such that y (2) 7.
Then the maximum value of the function y x ( ) is
Q.10 (^) Let X be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in X while 02244 and 44422 are not in X .Suppose that each element of X has an equal chance of being chosen. Let p be the conditional probability that an element chosen at random is a multiple
of 20 given that it is a multiple of 5. Then the value of 38 p^ is equal to
Q.11 (^) Let A 1 (^) , A 2 (^) , A 3 (^) , , A 8 be the vertices of a regular octagon that lie on a circle of radius 2. Let P be a point on the circle and let PAi denote the distance between the points P and Ai for i 1, 2, ,8. If P varies over the circle, then the maximum value of the product PA 1 (^) PA 2 (^) PA 8 , is
Q. Let
a b R c d a b c d
^ ^
. Then the number of invertible
matrices in R is
SECTION 3 (Maximum Marks: 24)
This section contains SIX (06) questions. The answer to each question is a NON-NEGATIVE INTEGER. For each question, enter the correct integer corresponding to the answer using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer. Answer to each question will be evaluated according to the following marking scheme: Full Marks : 4 If ONLY the correct integer is entered; Zero Marks : 0 In all other cases.
PARAGRAPH “I”
Consider an obtuse angled triangle ABC in which the difference between the largest and the smallest angle is 2
and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1. (There are two questions based on PARAGRAPH “I”, the question given below is one of them)
Q.14 (^) Let a be the area of the triangle ABC. Then the value of (^) 64 a (^) ^2 is
PARAGRAPH “I” Consider an obtuse angled triangle ABC in which the difference between the largest and the smallest angle is 2
and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie
on a circle of radius 1. (There are two questions based on PARAGRAPH “I”, the question given below is one of them)
Q.15 Then the inradius of the triangle ABC is
SECTION 4 (Maximum Marks: 12)
This section contains TWO (02) paragraphs. Based on each paragraph, there are TWO (02) questions. The answer to each question is a NUMERICAL VALUE. For each question, enter the correct numerical value of the answer using the mouse and the on- screen virtual numeric keypad in the place designated to enter the answer. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places. Answer to each question will be evaluated according to the following marking scheme: Full Marks : 3 If ONLY the correct numerical value is entered in the designated place; Zero Marks : 0 In all other cases.
PARAGRAPH “II” Consider the 6 6 square in the figure. Let A 1 (^) , A 2 (^) , , A 49 be the points of intersections (dots in the picture) in some order. We say that Ai and Aj are friends if they are adjacent along a row or along a column. Assume that each point Ai has an equal chance of being chosen.
(There are two questions based on PARAGRAPH “II”, the question given below is one of them)
Q.16 (^) Let pi be the probability that a randomly chosen point has i many friends, i 0,1, 2,3, 4. Let
X be a random variable such that for i 0,1, 2,3, 4, the probability P X ( i ) pi. Then the
value of 7 E ( X )is
PARAGRAPH “II” Consider the 6 6 square in the figure. Let A 1 (^) , A 2 (^) , , A 49 be the points of intersections (dots in the picture) in some order. We say that Ai and Aj are friends if they are adjacent along a row or along a column. Assume that each point Ai has an equal chance of being chosen.
(There are two questions based on PARAGRAPH “II”, the question given below is one of them)
Q.17 (^) Two distinct points are chosen randomly out of the points A 1 (^) , A 2 (^) , , A 49. Let p be the
probability that they are friends. Then the value of 7 p is
Q.1 An electric dipole is formed by two charges +𝑞 and −𝑞 located in 𝑥𝑦-plane at ( 0 , 2 ) mm and ( 0 , − 2 ) mm, respectively, as shown in the figure. The electric potential at point P ( 100 , 100 ) mm due to the dipole is 𝑉 0. The charges +𝑞 and −𝑞 are then moved to the points (− 1 , 2 ) mm and ( 1 , − 2 ) mm, respectively. What is the value of electric potential at P due to the new dipole?
(A) 𝑉 0 / 4 (B) 𝑉 0 / 2 (C) 𝑉 0 /√ 2 (D) 3 𝑉 0 / 4
Q.2 Young’s modulus of elasticity 𝑌 is expressed in terms of three derived quantities, namely, the gravitational constant 𝐺, Planck’s constant ℎ and the speed of light 𝑐, as 𝑌 = 𝑐𝛼ℎ𝛽𝐺𝛾. Which of the following is the correct option?
(A) 𝛼 = 7 , 𝛽 = − 1 , 𝛾 = − 2 (B) 𝛼 = − 7 , 𝛽 = − 1 , 𝛾 = − 2
(C) 𝛼 = 7 , 𝛽 = − 1 , 𝛾 = 2 (D) 𝛼 = − 7 , 𝛽 = 1 , 𝛾 = − 2
Q.3 A particle of mass 𝑚 is moving in the 𝑥𝑦-plane such that its velocity at a point (𝑥, 𝑦) is given as v⃗ = 𝛼(𝑦𝑥̂ + 2 𝑥𝑦̂ ), where 𝛼 is a non-zero constant. What is the force 𝐹 acting on the particle?
(A) 𝐹 = 2 𝑚𝛼^2 (𝑥𝑥̂ + 𝑦𝑦̂ ) (B) 𝐹 = 𝑚𝛼^2 (𝑦𝑥̂ + 2 𝑥𝑦̂ )
(C) 𝐹 = 2 𝑚𝛼^2 (𝑦𝑥̂ + 𝑥𝑦̂ ) (D) 𝐹 = 𝑚𝛼^2 (𝑥𝑥̂ + 2 𝑦𝑦̂ )
SECTION 1 (Maximum Marks: 12)
- This section contains FOUR (04) questions.
- Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct answer.
- For each question, choose the option corresponding to the correct answer.
- Answer to each question will be evaluated according to the following marking scheme: Full Marks : + 3 If ONLY the correct option is chosen; Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered); Negative Marks : − 1 In all other cases.
Physics
Q.4 An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is 𝑛. The internal energy of one mole of the gas is 𝑈𝑛 and the speed of sound in the gas is v𝑛. At a fixed temperature and pressure, which of the following is the correct option?
(A) v 3 < v 6 and 𝑈 3 > 𝑈 6 (B) v 5 > v 3 and 𝑈 3 > 𝑈 5 (C) v 5 > v 7 and 𝑈 5 < 𝑈 7 (D) v 6 < v 7 and 𝑈 6 < 𝑈 7
Q.5 A monochromatic light wave is incident normally on a glass slab of thickness 𝑑, as shown in the figure. The refractive index of the slab increases linearly from 𝑛 1 to 𝑛 2 over the height ℎ. Which of the following statement(s) is(are) true about the light wave emerging out of the slab?
(A) It will deflect up by an angle tan−^1 [(𝑛^2
(^2) −𝑛 12 )𝑑 2ℎ ]. (B) It will deflect up by an angle tan−^1 [(𝑛^2 − ℎ𝑛 1 )𝑑]. (C) It will not deflect. (D) The deflection angle depends only on (𝑛 2 − 𝑛 1 )^ and not on the individual values of 𝑛 1 and 𝑛 2.
Q.6 An annular disk of mass 𝑀, inner radius 𝑎 and outer radius 𝑏 is placed on a horizontal surface with coefficient of friction 𝜇, as shown in the figure. At some time, an impulse ℐ 0 𝑥̂ is applied at a height ℎ above the center of the disk. If ℎ = ℎ𝑚 then the disk rolls without slipping along the 𝑥-axis. Which of the following statement(s) is(are) correct?
(A) For 𝜇 ≠ 0 and 𝑎 → 0 , ℎ𝑚 = 𝑏/ 2. (B) For 𝜇 ≠ 0 and 𝑎 → 𝑏, ℎ𝑚 = 𝑏. (C) For ℎ = ℎ𝑚, the initial angular velocity does not depend on the inner radius 𝑎. (D) For 𝜇 = 0 and ℎ = 0 , the wheel always slides without rolling.
Q.7 The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by 𝐸⃗ = 30 ( 2 𝑥̂ + 𝑦̂ )sin [ 2 𝜋 ( 5 × 1014 𝑡 − 10
7 3 𝑧)]^ V^ m
− (^1). Which of the following option(s) is(are) correct?
[Given: The speed of light in vacuum, 𝑐 = 3 × 108 m s−^1 ]
(A) 𝐵𝑥 = − 2 × 10 −^7 sin [ 2 𝜋 ( 5 × 1014 𝑡 − 10
7 3 𝑧)]^ Wb^ m
(B) 𝐵𝑦 = 2 × 10 −^7 sin [ 2 𝜋 ( 5 × 1014 𝑡 − 10
7 3 𝑧)]^ Wb^ m
(C) The wave is polarized in the 𝑥𝑦-plane with polarization angle 30° with respect to the 𝑥-axis. (D) The refractive index of the medium is 2.
Q.10 (^) A string of length 1 m and mass 2 × 10 −^5 kg is under tension 𝑇. When the string vibrates, two successive harmonics are found to occur at frequencies 750 Hz and 1000 Hz. The value of tension 𝑇 is _____ Newton.
Q.11 An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius 0. 1 mm is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume 𝑉 0 to 100101 𝑉 0 with the movable piston. Considering air as an ideal gas, the height (ℎ) of the liquid column in the capillary above the liquid level in cm is_______.
[Given: Surface tension of the liquid is 0. 075 N m−^1 , atmospheric pressure is 105 N m−^2 , acceleration due to gravity (𝑔) is 10 m s−^2 , density of the liquid is 103 kg m−^3 and contact angle of capillary surface with the liquid is zero]
Q.12 (^) In a radioactive decay process, the activity is defined as 𝐴 = − 𝑑𝑁 𝑑𝑡 , where^ 𝑁(𝑡)^ is the number of radioactive nuclei at time 𝑡. Two radioactive sources, 𝑆 1 and 𝑆 2 have same activity at time 𝑡 = 0. At a later time, the activities of 𝑆 1 and 𝑆 2 are 𝐴 1 and 𝐴 2 , respectively. When 𝑆 1 and 𝑆 2 have just completed their 3rd^ and 7th^ half-lives, respectively, the ratio 𝐴 1 /𝐴 2 is __________.
Q.13 One mole of an ideal gas undergoes two different cyclic processes I and II, as shown in the 𝑃-𝑉 diagrams below. In cycle I, processes 𝑎, 𝑏, 𝑐 and 𝑑 are isobaric, isothermal, isobaric and isochoric, respectively. In cycle II, processes 𝑎′, 𝑏′, 𝑐′^ and 𝑑′^ are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle I is 𝑊𝐼 and that during cycle II is 𝑊𝐼𝐼. The ratio 𝑊𝐼/𝑊𝐼𝐼 is _______.
PARAGRAPH I
𝑆 1 and 𝑆 2 are two identical sound sources of frequency 656 Hz. The source 𝑆 1 is located at 𝑂 and 𝑆 2 moves anti-clockwise with a uniform speed 4 √ 2 m s−^1 on a circular path around 𝑂, as shown in the figure. There are three points 𝑃, 𝑄 and 𝑅 on this path such that 𝑃 and 𝑅 are diametrically opposite while 𝑄 is equidistant from them. A sound detector is placed at point 𝑃. The source 𝑆 1 can move along direction 𝑂𝑃.
[Given: The speed of sound in air is 324 m s−^1 ]
Q.15 (^) Consider both sources emitting sound. When 𝑆 2 is at 𝑅 and 𝑆 1 approaches the detector with a speed 4 m s−^1 , the beat frequency measured by the detector is _______Hz.
PARAGRAPH II
A cylindrical furnace has height (𝐻) and diameter (𝐷) both 1 m. It is maintained at temperature 360 K. The air gets heated inside the furnace at constant pressure 𝑃𝑎 and its temperature becomes 𝑇 = 360 𝐾. The hot air with density 𝜌 rises up a vertical chimney of diameter 𝑑 = 0. 1 m and height ℎ = 9 m above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density 𝜌𝑎 =
- 2 kg m−^3 , pressure 𝑃𝑎 and temperature 𝑇𝑎 = 300 K enters the furnace. Assume air as an ideal gas, neglect the variations in 𝜌 and 𝑇 inside the chimney and the furnace. Also ignore the viscous effects.
[Given: The acceleration due to gravity 𝑔 = 10 m s−^2 and 𝜋 = 3. 14 ]
Q.16 Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _______ gm s−^1.
