Measures of Central Tendency: Mean, Median and Mode
ย
Full jee mains 2015 online paper 10th april final
1. JEE Mains 2015 10th April (online)
Physics
Single Correct Answer Type:
1. In an ideal at temperature T, the average force that a molecule applies on the walls of a closed
container depends on ๐ ๐๐ ๐ ๐
. A good estimate for q is:
(A) 2 (B)
1
2
(C) 1 (D)
1
4
Answer: (C)
Solution:
Average linear for collision to occur
๐ก =
2๐
๐ข
Change in momentum in 1 collision
ฮ๐ = 2 ๐๐ข
โด average force in collision
=
ฮ๐
๐ก
๐ข = root mean square speed
=
2 ๐๐ข
2๐
ร ๐ข
โ ๐ โ ๐ข2
โด ๐ข2
โ ๐
โ ๐ ร ๐
โ ๐ = 1
2. In an unbiased n โ p junction electrons diffuse from n-region to p-region because:
(A) Electrons travel across the junction due to potential difference
(B) Only electrons move from n to p region and not the vice โ versa
(C) Electron concentration in n โ region is more as compared to that in p โ region
(D) Holes in p โ region attract them
Answer: (C)
Solution:
In a ๐ โ ๐ junction diffusion occurs due to spontaneous movement of majority charge carrier from
the region of high concentration to low concentration so option 3 in correct.
3. A 10V battery with internal resistance 1ฮฉ ๐๐๐ ๐ 15๐ battery with internal resistance 0.6ฮฉ are
connected in parallel to a voltmeter (see figure). The reading in the voltmeter will be close to:
2. (A) 11.9 ๐ (B) 13.1 ๐ (C) 12.5 ๐ (D) 24.5 ๐
Answer: (B)
Solution:
The equivalent ems of the battery combination in given as
Equation =
๐ธ1
๐1
+
๐ธ1
๐2
1
๐1
+
1
๐2
=
10
1
+
15
0.6
1
1
+
1
0.6
=
10+
150
6
1+
10
6
=
105
8
= 13.1 ๐ฃ๐๐๐ก
โด The reading measured by voltmeter = 13.1 ๐ฃ๐๐๐ก
4. A proton (mass m) accelerate by a potential difference V flies through a uniform transverse
magnetic field B. The field occupies a region of space by width โฒ๐โฒ
. ๐ผ๐ โฒ๐ผโฒ be the angle of
deviation of proton from initial direction of motion (see figure), the value of sin ๐ผ will be:
(A)
๐ต
2
โ
๐๐
๐๐
(B) ๐ต๐โ
๐
2๐๐
(C)
๐ต
๐
โ
๐
2๐๐
(D) ๐ ๐ โ
๐ต๐
2๐
Answer: (B)
3. Solution:
Due to potential difference V speed acquired by proton in ๐ฃ0
โ ๐ = ๐ ฮ ๐ = ฮ๐
โ ๐๐ฃ =
1
2
๐ ๐ฃ0
2
โ ๐ฃ0 = โ
2๐๐ฃ
๐
Radius of circular path acquired is ๐ =
๐๐ฃ0
๐๐ต
โ ๐ =
๐
๐๐ต
โ
2๐๐ฃ
๐
= โ
2๐ฃ๐
๐
ร
1
๐ต
In โ๐ถ๐๐ท,sin ๐ผ =
๐
๐
= ๐โ
๐
2 ๐ฃ๐
๐ต = ๐ต๐โ
๐
2 ๐๐ฃ
5. de โ Broglie wavelength of an electron accelerated by a voltage of 50 V is close to
(|๐| = 1.6 ร 10โ19
๐ถ, ๐ ๐ = 9.1 ร 10โ31
๐๐, โ = 6.6 ร 10โ34
๐ฝ๐ ):
(A) 0.5 โซ (B) 1.2 โซ (C) 1.7 โซ (D) 2.4 โซ
Answer: (B)
Solution:
De broglie wavelength ๐ in given by
๐ =
โ
๐
=
โ
โ2 ๐๐
โด ๐๐๐๐๐ก๐๐ ๐๐๐๐๐๐ฆ = ๐ = ๐ ฮ๐ฃ
โ ๐ =
โ
โ2๐๐โ๐ฃ
=
6.6 ร10โ34
โ2 ร9.1 ร 10โ3 ร 1.6 ร10โ19 ร 50
=
6.6 ร10โ34
โ3.2 ร9.1 ร 10โ31โ19 + 2
=
6.6 ร10โ34
โ3.2 ร9.1 ร 10โ48
=
6.6 ร10โ34
โ5.396 ร 10โ24
= 1.22 ร 10โ10
= 1.2 ๐ดยฐ
6. Suppose the drift velocity ๐ฃ ๐ in a material varied with the applied electric field E as ๐ฃ ๐ โ โ๐ธ.
Then ๐ โ ๐ผ graph for a wire made of such a material is best given by:
4. (A)
(B)
(C)
(D)
Answer: (C)
Solution:
โด ๐ฃ ๐ = ๐โ๐ธ and ๐ผ = ๐ ๐ ๐ด ๐ฃ ๐
โ ๐ผ = ๐ ๐๐ด ๐โ๐ธ
โด ๐ธ =
๐ฃ
๐
โ ๐ผ = ๐๐๐ด๐ โ
๐ฃ
๐
โ ๐ผ โ โ ๐ฃ โ ๐ฃ โ ๐ผ2
So
7. A parallel beam of electrons travelling in x โ direction falls on a slit of width d (see figure). If
after passing the slit, an electron acquires momentum ๐๐ฆ in the y โ direction then for a majority
of electrons passing through the slit (h is Planckโs constant):
5. (A) |๐๐ฆ|๐ < โ (B) |๐๐ฆ|๐ > โ (C) |๐๐ฆ|๐ โ โ (D) |๐๐ฆ|๐ > > โ
Answer: (D)
Solution:
The electron beam will be diffractive at an angle ฮธ
For central maxima
๐ sin ๐ = ๐
๐ sin ๐ =
๐
๐
Also ๐ sin ๐ = ๐ ๐ฆ
โ ๐ ๐ ๐ฆ = โ
โด For majority of ๐ ๐
โฒ๐ passing through the shit lyeing in the central maxima ๐ ๐ ๐ฆ โ โ
8. A block of mass ๐ = 10 ๐๐ rests on a horizontal table. The coefficient of friction between the
block and the table is 0.05. When hit by a bullet of mass 50 g moving with speed v, that gets
embedded in it, the block moves and comes to stop after moving a distance of 2 m on the table.
If a freely falling object were to acquire speed
๐ฃ
10
after being dropped from height H, then
neglecting energy losses and taking ๐ = 10 ๐๐ โ2
, the value of H is close to:
(A) 0.2 km (B) 0.5 km (C) 0.3 km (D) 0.4 km
Answer: ()
Solution:
9. When current in a coil changes from 5 A to 2 A in 0.1 s, an average voltage of 50 V is
produced. The self โ inductance of the coil is:
(A) 1.67 H (B) 6 H (C) 3 H (D) 0.67 H
Answer: (A)
Solution:
6. Area of coil
๐ = ๐ฟ๐ผ โ
โ๐
โ๐ก
= ๐ฟ
โ๐ผ
โ๐ก
โด (๐๐๐๐) ๐๐ฃ๐๐๐๐๐ = |
โ๐
โ๐ก
| = ๐ฟ |
โ๐ผ
โ๐ก
|
โ 50 = ๐ฟ ร
5โ2
0.1
โ
5
3
= ๐ฟ
โ ๐ฟ = 1.674
10. ๐ฅ ๐๐๐ ๐ฆ displacements of a particle are given as ๐ฅ(๐ก) = ๐ sin ๐๐ก ๐๐๐ ๐ฆ(๐ก) = ๐ sin 2๐๐ก. Its
trajectory will look like:
(A)
(B)
(C)
(D)
Answer: (C)
Solution:
โต ๐ฅ = ๐ด sin ๐๐ก โ ๐ ๐๐ ๐๐ก =
๐ฅ
๐ด
Also, ๐๐๐ ๐๐ก = โ1 โ sin2 ๐๐ก = โ1 โ
๐ฅ2
๐ด2
7. โ cos ๐๐ก =
โ๐ด2โ๐ฅ2
๐ด
As, ๐ฆ = 2๐ด sin ๐๐ก cos ๐๐ก
โ ๐ฆ = 2 ๐ด
๐ฅ
๐ด
โ๐ด2 โ ๐ฅ2
๐ด
โ ๐ฆ =
2
๐ด
๐ฅ โ ๐ด2 โ ๐ฅ2
โ ๐ฆ = 0 ๐๐ก ๐ฅ = 0 ๐๐๐ ๐ฅ = ยฑ ๐ด
Which in possible only in option (3)
11. Consider a thin uniform square sheet made of a rigid material. If its side is โaโ, mass m and
moment of inertia I about one of its diagonals, then:
(A) ๐ผ =
๐๐2
24
(B)
๐๐2
24
< ๐ผ <
๐๐2
12
(C) ๐ผ >
๐๐2
12
(D) ๐ผ =
๐๐2
12
Answer: (D)
Solution:
In a uniform square plate due to symmetry moment of Inertia about all the axis passing through
centre and lying in the blank of the plate is same.
โด ๐ผ ๐๐๐๐๐๐๐๐ = ๐ผ ๐๐๐๐๐๐๐๐ ๐ก๐ ๐ ๐๐๐
=
๐๐2
12
12. Diameter of a steel ball is measured using a Vernier calipers which has divisions of 0.1 cm on
its main scale (MS) and 10 divisions of its vernier scale (VS) match 9 divisions on the main
scale. Three such measurements for a ball are given as:
S.No. MS (cm) VS divisions
1. 0.5 8
2. 0.5 4
3. 0.5 6
If the zero error is โ 0.03 cm, then mean corrected diameter is:
(A) 0.53 cm
8. (B) 0.56 cm
(C) 0.59 cm
(D) 0.52 cm
Answer: (C)
Solution:
L.C. of Vernier calipers
=
1 ๐๐๐๐ ๐ ๐๐๐๐ ๐๐๐๐๐๐ก๐๐
๐๐๐ก๐๐ ๐๐๐ฃ๐๐ ๐๐๐ vernier ๐ ๐๐๐๐
=
0.1
10
= 0.01 ๐๐
Required of Vernier calipers
= ๐. ๐. ๐ . +(๐ฟ. ๐ถ) ร ๐ฃ๐ ๐๐๐ฃ๐๐ ๐๐๐๐ .
โด Measured diameter are respecting
0.52 ๐๐ 0.54 ๐๐, 0.56 ๐๐
โด ๐๐ฃ๐๐๐๐๐ ๐๐๐๐๐๐ก๐๐ =
0.58 + 0.54 + 0.56
3
=
1.68
3
= 0.56
โด ๐๐๐๐๐๐๐ก๐๐ ๐๐๐๐๐๐ก๐๐ = 0.56 โ (โ0.03)
= 0.56 + 0.03 = 0.59 ๐๐
13. A very long (length L) cylindrical galaxy is made of uniformly distributed mass and has radius
R (R < < L). A star outside the galaxy is orbiting the galaxy in a plane perpendicular to the
galaxy and passing through its centre. If the time period of star is T and its distance from the
galaxyโs axis is r, then:
(A) ๐ โ โ ๐
(B) ๐ โ ๐
(C) ๐ โ ๐2
(D) ๐2
โ ๐3
Answer: (B)
Solution:
Due to a long solid cylinder gravitational field strong can be given as:
๐โฒ =
2 ๐บ ๐
๐ฅ
Where
๐ = ๐๐๐๐๐๐ ๐๐๐ ๐ ๐๐๐๐ ๐๐ก๐ฆ ๐๐ ๐๐๐๐๐ฅ๐ฆ.
๐น๐๐ ๐กโ๐ ๐๐๐๐๐ก๐๐ ๐๐๐ก๐๐๐ ๐๐๐๐ข๐๐ ๐กโ๐ ๐๐๐๐๐ฅ๐ฆ.
9. ๐๐ = ๐๐๐๐๐ก๐๐๐๐๐ก๐๐
โ ๐๐ = ๐ ๐2
๐ฅ
โ
2๐บ๐
๐ฅ
= ๐2
๐ฅ
โ ๐2
โ
1
๐ฅ2
โ ๐ โ
1
๐ฅ
โ
2๐
๐
โ
1
๐ฅ
โ ๐ โ ๐ฅ
So option 2 is correct
14. An electromagnetic wave travelling in the x โ direction has frequency of 2 ร 1014
๐ป๐ง and
electric field amplitude of 27 ๐๐โ1
. From the options given below, which one describes the
magnetic field for this wave?
(A) ๐ตโ (๐ฅ, ๐ก) = (9 ร 10โ8
๐)๐ฬ sin[1.5 ร 10โ6
๐ฅ โ 2 ร 1014
๐ก]
(B) ๐ตโ (๐ฅ, ๐ก) = (9 ร 10โ8
๐)๐ฬ sin[2๐(1.5 ร 10โ8
๐ฅ โ 2 ร 1014
๐ก)]
(C) ๐ตโ (๐ฅ, ๐ก) = (3 ร 10โ8
๐)๐ฬ sin[2๐(1.5 ร 10โ8
๐ฅ โ 2 ร 1014
๐ก)]
(D) ๐ตโ (๐ฅ, ๐ก) = (9 ร 10โ8
๐)๐ฬ sin[2๐ (1.5 ร 10โ6
๐ฅ โ 2 ร 1014
๐ก)]
Answer: (D)
Solution:
๐โ๐๐ ๐ธ = ๐ธ0 ๐ ๐๐ ๐ถ ๐๐ฅ โ ๐๐ก
๐โ๐๐ ๐ต = ๐ต0 ๐ ๐๐ ๐ถ ๐๐ฅ โ ๐๐ก
Of light in travelling along ๐ฬ then ๐ตโ in either along ๐ or๐โ .
โด ๐๐๐๐๐ ๐๐ ๐๐๐โ๐ก ๐ถ =
๐ธ0
๐ต0
โ ๐ต0 =
๐ธ0
๐ถ
โ ๐ต0 =
27
3ร108 = 9 ร 10โ8
๐
also, ๐ = 2๐ f = 2ฯ ร 2 ร 1014
= 4 ๐ ร 1014
Looking into the option the correct
Answer is ๐ตโ = 9 ร 10โ8
sin2๐ (1.5 ร 10โ6
๐ฅ โ 2 ร 1014
๐ก) ๐ฬ
15. A telescope has an objective lens of focal length 150 cm and an eyepiece of focal length 5 cm. If
a 50 m tall tower at a distance of 1 km is observed through this telescope in normal setting, the
angle formed by the image of the tower is ๐, then ๐ is close to:
10. (A) 30ยฐ
(B) 15ยฐ
(C) 1ยฐ
(D) 60ยฐ
Answer: (D)
Solution:
16. A block of mass ๐ = 0.1 ๐๐ is connected to a spring of unknown spring constant k. It is
compressed to a distance x from its equilibrium position and released from rest. After
approaching half the distance (
๐ฅ
2
) from equilibrium position, it hits another block and comes
to rest momentarily, while the other block moves with a velocity 3 ๐๐ โ1
. The total initial
energy of the spring is:
(A) 0.6 ๐ฝ
(B) 0.8 ๐ฝ
(C) 1.5 ๐ฝ
(D) 0.3 ๐ฝ
Answer: (A)
Solution: By energy conservation between compression positions ๐ฅ and
๐ฅ
2
1
2
๐๐ฅ2
=
1
2
๐ (
๐ฅ
2
)
2
+
1
2
๐๐ฃ2
1
2
๐๐ฅ2
โ
1
2
๐
๐ฅ2
4
=
1
2
๐๐ฃ2
1
2
๐๐ฅ2
(
3
4
) =
1
2
๐๐ฃ2
๐ฃ = โ
3๐๐ฅ2
4๐
= โ
3๐
๐
๐ฅ
2
On collision with a block at rest
โต Velocities are exchanged โ elastic collision between identical masses.
โด ๐ฃ = 3 = โ
3๐
๐
๐ฅ
2
โ 6 = โ
3๐
๐
๐ฅ
โ ๐ฅ = 6โ
๐
3๐
โด The initial energy of the spring is
11. ๐ =
1
2
๐ ๐ฅ2
=
1
2
๐ ร 36
๐
3๐
= 6๐
๐ = 6 ร 0.1 = 0.6 ๐ฝ
17. Shown in the figure are two point charges + Q and โ Q inside the cavity of a spherical shell. The
charges are kept near the surface of the cavity on opposite sides of the centre of the shell. If ๐1is
the surface charge on the inner surface and ๐1net charge on it and ๐2 the surface charge on the
other surface and ๐2 net charge on it then:
(A) ๐1 = 0, ๐1 = 0, ๐2 = 0, ๐2 = 0
(B) ๐1 โ 0, ๐1 = 0, ๐2 โ 0, ๐2 = 0
(C) ๐1 โ 0, ๐1 โ 0, ๐2 โ 0, ๐2 โ 0
(D) ๐1 โ 0, ๐1 = 0, ๐2 = 0, ๐2 = 0
Answer: (D)
Solution: By the property of electrostatic shielding in the conductors ๐ = 0 in the conductor.
So electric flux = 0 through a dotted Gaussian surface as shown
The net enclosed charge through Gaussian surface = 0
โ Net charge ๐1 on the inner surface = 0 but the equal and opposite induced charge on the surface
will be distributed non uniformly on the inner surface
So, ๐1 โ 0
โต ๐1 = 0 on the inner surface
So, net charge ๐2 = 0 on the outer surface as conductor is neutral but โต outer surface is free from
any electric field so no charge density exists on the outer surface. So, ๐2 = 0.
18. You are asked to design a shaving mirror assuming that a person keeps it 10 cm from his face
and views the magnified image of the face at the closest comfortable distance of 25 cm. The
radius of curvature of the mirror would then be:
(A) 24 ๐๐
(B) 30 ๐๐
(C) 60 ๐๐
12. (D) โ24 ๐๐
Answer: (C)
Solution:
If AB is the position of face of man then A โBโ is the position of image of face.
As image is formed at 25cm form the object.
โด From concave mirror image is 15cm behind the mirror.
So, ๐ข = โ10 ๐๐, ๐ฃ = +15 ๐๐
โ
1
๐
=
1
๐ข
+
1
๐ฃ
โ
1
๐
=
1
โ10
+
1
15
=
โ3 + 2
30
โ ๐ = โ300 ๐๐
So, radius of curvature = 60 ๐๐
19. A thin disc of radius ๐ = 2๐ has a concentric hole of radius โaโ in it (see figure). It carries
uniform surface charge โฒ๐โฒ on it. If the electric field on its axis at height โฒโโฒ
(โ < < ๐) from its
centre is given as โChโ then value of โCโ is:
(A)
๐
4 ๐ผ๐0
(B)
๐
๐ผ๐0
(C)
๐
๐๐ผ๐0
(D)
๐
2๐ผ๐0
Answer: (A)
Solution: โต at the axial point of a uniformly charged disc electric field is given by
๐ธ =
๐
2๐0
(1 โ ๐๐๐ ๐)
13. By superposition principle when inner disc is removed then electric field due to remaining disc is
๐ธ =
๐
2๐0
[(1 โ ๐๐๐ ๐2) โ (1 โ ๐๐๐ ๐1)]
=
๐
2๐0
[๐๐๐ ๐1 โ ๐๐๐ ๐2]
=
๐
2๐0
[
โ
โโ2 + ๐2
โ
โ
โโ2 + ๐2
]
=
๐
2๐0
[
โ
๐โ1 +
โ2
๐2
โ
โ
โ1 +
โ2
๐2 ]
โต โ โช ๐ and b
โด ๐ธ =
๐
2๐0
[
โ
๐
โ
โ
๐
]
=
๐
2๐0
[
โ
๐
โ
โ
2๐
] =
๐โ
4๐0 ๐
โ ๐ถ =
๐
4๐๐0
20. An ideal gas goes through a reversible cycle ๐ โ ๐ โ ๐ โ ๐ has the V โ T diagram shown below.
Process ๐ โ ๐ ๐๐๐ ๐ โ ๐ are adiabatic.
The corresponding P โ V diagram for the process is (all figures are schematic and not drawn to
scale) :
(A)
14. (B)
(C)
(D)
Answer: (A)
Solution: Is an adiabatic process
๐๐ ๐พโ1
= ๐๐๐๐ ๐ก โ ๐๐
1
๐พโ1 = ๐๐๐๐ ๐ก
โ as T increase V decreases at non-uniform rate
In process ๐ โ ๐ P = constant as ๐ โ ๐
In process ๐ โ ๐ ๐โฒ
= constant s ๐ โ ๐
But since slope of V โ T graph โ
1
๐
since slope of ab < slope of cd
โ ๐๐๐ > ๐๐๐
Also in adiabatic process ๐ โ ๐ as T is increasing V in decreasing
โ P is increasing, so P โ V diagram is as below
15. 21. A uniform solid cylindrical roller of mass โmโ is being pulled on a horizontal surface with force F
parallel to the surface and applied at its centre. If the acceleration of the cylinder is โaโ and it is
rolling without slipping then the value of โFโ is:
(A)
3
2
๐๐
(B) 2 ๐๐
(C)
5
3
๐๐
(D) ๐๐
Answer: (A)
Solution:
From free body diagram of cylinder
๐น โ ๐๐ = ๐๐ โฆ..(1)
โต โ ๐๐๐ฅ๐ก = ๐๐ ๐๐
๐๐๐ ๐ โ ๐ ๐๐ฅ๐ก = ๐ผ๐๐ โ
โน ๐๐ ๐ = ๐ผ๐๐ โ
โน ๐๐ ๐ =
1
2
๐๐ 2
โ โฆ.. (2)
For rolling without slipping
๐ = ๐ โ โฆโฆ (3)
โน โ=
๐
๐
โด ๐๐ ๐ =
1
2
๐๐ 2 ๐
๐
โน ๐๐ =
1
2
๐๐
Put in (1)
๐ โ
1
2
๐๐ = ๐๐
16. โน ๐ =
3
2
๐๐
22. A 25 cm long solenoid has radius 2 cm and 500 total number of turns. It carries a current of 15
A. If it is equivalent to a magnet of the same size and magnetization
๐โโ (๐๐๐๐๐๐ก๐๐ ๐๐๐๐๐๐ก ๐๐๐๐ข๐๐โ ), ๐กโ๐๐ |๐โโ | is:
(A) 3๐ ๐ด๐โ1
(B) 30000 ๐ด๐โ1
(C) 30000๐ ๐ด๐โ1
(D) 300 ๐ด๐โ1
Answer: (B)
Solution:
๐๐๐๐ข๐๐ = ๐ด๐
๐๐๐๐๐๐ก๐๐ง๐๐ก๐๐๐ ๐โโ =
๐๐๐๐๐๐ก๐๐ ๐๐๐๐๐๐ก
๐๐๐๐ข๐๐
=
(๐๐.๐๐ ๐ก๐ข๐๐๐ )ร(๐ถ๐ข๐๐๐๐๐ก)ร๐ด๐๐๐
๐๐๐๐ข๐๐
=
๐ ๐ผ ๐ด
๐ด โ
=
๐๐ผ
โ
=
500ร15ร100
25
= 60 ร 500
= 30 ร 103
= 30000 ๐ด๐โ1
23. In the circuits (a) and (b) switches ๐1 ๐๐๐ ๐2 are closed at t = 0 and are kept closed for a long
time. The variation of currents in the two circuits for ๐ก โฅ 0 are roughly shown by (figures are
schematic and not drawn to scale):
(A)
18. For L โ R circuit
24. If two glass plates have water between them and are separated by very small distance (see
figure), it is very difficult to pull them apart. It is because the water in between forms
cylindrical surface on the side that gives rise to lower pressure in the water in comparison to
atmosphere. If the radius of the cylindrical surface is R and surface tension of water is T then
the pressure in water between the plates is lower by:
(A)
2๐
๐
(B)
๐
4๐
(C)
4๐
๐
(D)
๐
2๐
Answer: (A)
Solution:
๐ = 2๐ ๐๐๐ ๐
โด ๐๐๐๐ ๐ ๐ข๐๐ ๐๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐ ๐ ๐ ๐๐๐ข๐๐๐ ๐๐ข๐๐ฃ๐๐ก๐ข๐๐ ๐๐๐๐
โ๐ = 2๐ (
1
๐ 1
+
1
๐ 2
)
โต ๐ 1 = ๐ ๐๐๐ ๐ 2 = โ
โ๐ = 2๐ (
1
๐
+
1
โ
)
19. โ๐ =
2๐ผ
๐
โด Pressure is more in the concave side hence pressure in water between the plates is lower by
2๐
๐
25. A simple harmonic oscillator of angular frequency 2 rad ๐ โ1
is acted upon by an external force
๐น = sin ๐ก ๐. If the oscillator is at rest in its equilibrium position at ๐ก = ๐, its position at later
times is proportional to:
(A) sin ๐ก +
1
2
cos 2๐ก
(B) ๐๐๐ ๐ก โ
1
2
sin2๐ก
(C) sin ๐ก โ
1
2
sin2๐ก
(D) sin ๐ก +
1
2
sin2๐ก
Answer: (C)
Solution:
It is given that oscillator at rest at t = 0 i.e. at t = 0, v = 0
So, in option we can check by putting ๐ฃ =
๐๐ฅ
๐๐ก
= 0
(1) ๐ผ๐ ๐ฅ โ sin ๐ก +
1
2
cos2๐ก
โน ๐ฃ โ cos ๐ก +
1
2
ร 2 (โ sin 2๐ก)
โน ๐๐ก ๐ก = 0, ๐ฃ โ 1 โ 0 โ 0
(2) ๐ผ๐ ๐ฅ โ cos ๐ก โ
1
2
sin ๐ก
โน ๐ฃ โ โ sin ๐ก โ
1
2
cos ๐ก
โน ๐๐ก ๐ก = 0, ๐ฃ โ โ
1
2
โ 0
(3) ๐ผ๐ ๐ฅ โ sin ๐ก โ
1
2
๐ ๐๐๐ 2๐ก
๐กโ๐๐ ๐ โ cos ๐ก โ
1
2
ร 2 cos 2๐ก
โน ๐๐ก ๐ก = 0, ๐ฃ โ 1 โ 1 = 0
(4) ๐ผ๐ ๐ฅ โ sin ๐ก +
1
2
sin2๐ก
โน ๐ฃ โ cos ๐ก +
1
2
ร 2 cos2๐ก
โน ๐๐ก ๐ก = 0, ๐ฃ โ 1 + 1
โน ๐ฃ โ 2 โ 0
โด ๐๐ ๐๐๐ก๐๐๐ (3) ๐ฃ = 0 ๐๐ก ๐ก = 0
26. If a body moving in a circular path maintains constant speed of 10 ๐๐ โ1
, then which of the
following correctly describes relation between acceleration and radius?
(A)
20. (B)
(C)
(D)
Answer: (D)
Solution:
V = constant
โน No tangential acceleration
โน Only centripetal acceleration
๐ =
๐ฃ2
๐
โน ๐๐ = ๐๐๐๐ ๐ก๐๐๐ก
21. โน ๐ โ
1
๐
27. If it takes 5 minutes to fill a 15 litre bucket from a water tap of diameter
2
โ ๐
๐๐ then the
Reynolds number for the flow is (density of water =103 ๐๐ ๐3โ
๐๐๐ ๐ฃ๐๐ ๐๐๐ ๐๐ก๐ฆ ๐๐ ๐ค๐๐ก๐๐ =
10โ3
๐๐. ๐ ) close to:
(A) 5500 (B) 550 (C) 1100 (D) 11,000
Answer: (A)
Solution:
Reynolds number
๐ =
๐๐๐ท
๐
๐ท = Diameter of litre
Also rate of flow =
๐๐๐๐ข๐๐
๐ก๐๐๐
= ๐ด ๐
๐
๐ก
=
๐ ๐ท2
4
ร ๐ โ ๐ =
4๐
๐๐ท2 ๐ก
โด ๐ =
๐ ๐ท
๐
ร
4 ๐
๐ ๐ท2 ๐ก
=
4 ๐ ๐
๐ ๐ ๐ท ๐ก
=
4 ร 103
ร 15 ร 10โ3
๐ ร 10โ3 ร 2 ร 5 ร 60
โ ๐ ร 102
=
10000
โ ๐
โ 5500
28. If one were to apply Bohr model to a particle of mass โmโ and charge โqโ moving in a plane
under the influence of a magnetic field โBโ, the energy of the charged particle in the ๐ ๐กโ
level
will be:
(A) ๐ (
โ๐๐ต
๐๐
) (B) ๐ (
โ๐๐ต
4๐๐
) (C) ๐ (
โ๐๐ต
2๐๐
) (D) ๐ (
โ๐๐ต
8๐๐
)
Answer: (B)
Solution:
22. For a charge q moving in a +r uniform magnetic field B
๐๐ =
๐๐ฃ2
๐
๐๐๐ต =
๐๐ฃ2
๐
โ ๐๐ฃ2
= ๐๐๐ต๐
โ
1
2
๐๐ฃ2
=
๐๐๐ต๐
2
โ ๐ธ๐๐๐๐๐ฆ =
๐๐๐ต๐
2
(1)
By Bohrโs quantisation condition
Angular momentum ๐ฟ = ๐
โ
2๐
โ ๐๐ฃ๐ =
๐โ
2๐
โ ๐ฃ๐ =
๐โ
2๐ ๐
(2)
Put (2) in (2)
โ ๐ธ๐๐๐๐๐ฆ =
๐๐ต
2
(
โ
2 ๐ ๐
)
=
๐๐ต ๐โ
4 ๐ ๐
29. If the capacitance of a nanocapacitor is measured in terms of a unit โuโ made by combining the
electronic charge โeโ, Bohr radius โฒ๐0
โฒ
, Planckโs constant โhโ and speed of light โcโ then:
(A) ๐ข =
๐2 ๐0
โ๐
(B) ๐ข =
โ๐
๐2 ๐0
(C) ๐ข =
๐2 ๐
โ๐0
(D) ๐ข =
๐2โ
๐๐0
Answer: (A)
Solution:
โต ๐ถ๐๐๐๐๐๐ก๐๐๐๐ ๐ถ =
๐
โ๐ฃ
๐ด๐๐ ๐ [
โ๐
๐
] = [
โ๐
๐0
] = [๐ธ๐๐๐๐๐ฆ]
โด [๐ถ] =
[๐]
[โ๐ฃ]
=
[๐] [๐]
[โ๐ฃ] [๐]
โต ๐ = ๐โ๐ฃ โ [๐] [โ๐ฃ] = [๐ธ๐๐๐๐๐ฆ]
โด [๐ถ] =
[๐2]
[๐ธ๐๐๐๐๐ฆ]
=
[๐2] [๐0]
[โ๐]
โด [๐ถ๐๐๐๐๐๐ก๐๐๐๐ ] =
[๐2] [๐0]
[โ๐]
โ ๐ข =
๐2 ๐0
โ๐
23. 30. A bat moving at 10 ๐๐ โ1
towards a wall sends a sound signal of 8000 Hz towards it. On
reflection it hears a sound of frequency๐. The value of ๐ in Hz is close to
(๐ ๐๐๐๐ ๐๐ ๐ ๐๐ข๐๐ = 320 ๐๐ โ1)
(A) 8258
(B) 8424
(C) 8000
(D) 8516
Answer: (D)
Solution:
We can assume that reflected wave is due to image of B coming with same speed in opposite
direction
Observer
๐ =
๐ฃ + 10
๐ฃ โ 10
ร ๐0
=
320 + 10
320 โ 10
ร 8000
=
330
310
ร 8000
=
33
31
ร 8000
= 8516 ๐ป๐ง
24. JEE Mains 2015 10th April (online)
Chemistry
Single correct answer type
1. 1.4 g of an organic compound was digested according to Kjeldahlโs method and the ammonia
evolved was absorbed in 60 mL of M/10 ๐ป2 ๐๐4 solution. The excess sulphuric acid required 20
mL of M/10 NaOH solution for neutralization. The percentage of nitrogen in the compound is:
(A) 24 (B)3 (C)5 (D)10
Solution: (D) 60 ร
1
10
= 6 ๐๐ ๐ป2 ๐๐4 used
Excess ๐ป2 ๐๐4 โก 20 ร
1
10
ร
1
2
= 1 ๐๐ ๐ป2 ๐๐4
๐ป2 ๐๐4 used = 6 โ 1 = 5 ๐๐
2๐๐ป3 + ๐ป2 ๐๐4 โถ (๐๐ป4)2 ๐๐4
mM of ๐๐ป3 = 10 ๐๐
Mass of ๐ = 10 ร 10โ3
ร 14 (
๐
๐๐๐๐
) = 0.140๐
% ๐2 =
0.140
1.4
ร 100 = 10%
2. The optically inactive compound from the following is:
(A) 2-chloropropanal
(B) 2-chloro-2-methylbutane
(C) 2-chlorobutane
(D) 2-chloropentane
Solution: (B)
(Optically active)
(Optically inactive because of 2 โ ๐ถ๐ป3 groups present on same C atom)
25. (Optically active)
3. The least number of oxyacids are formed by:
(A) Chlorine
(B) Fluorine
(C) Sulphur
(D) Nitrogen
Solution: (B) Fluorine does not form oxyacids as it is more electronegative than oxygen.
4. Gaseous ๐2 ๐4 dissociates into gaseous ๐๐2according to the reaction๐2 ๐4(๐) โ 2๐๐2(๐)
At 300 K and 1 atm pressure, the degree of dissociation of ๐2 ๐4 is 0.2. If one mole of ๐2 ๐4 gas is
contained in a vessel, then the density of the equilibrium mixture is:
(A) 3.11 g/L
(B) 1.56 g/L
(C) 4.56 g/L
(D) 6.22 g/L
Solution: (A)
๐2 ๐4 โ 2๐๐2
(1 โ ๐ผ) 2๐ผ
Total moles at equilibrium = 1 โ ๐ผ + 2๐ผ = 1 + ๐ผ = 1.2
M avg for equilibrium mixture =
92
๐
๐๐๐๐
(๐2 ๐4)
1.2
๐ ๐๐ฃ๐๐๐๐๐ =
๐๐ ๐๐ฃ๐
๐ ๐
=
1 ร 76.67
0.082 ร 300
=
76.67
24.6
= 3.11 ๐๐ฟโ1
5. Arrange the following amines in the order of increasing basicity.
(A)
26. (B)
(C)
(D)
Solution: (C)
Most basic due to +I effect of methyl group. Methoxy group provides electron density at -
๐๐ป2
-๐๐2 group with draws electron density from N of -๐๐ป2
6.
27. A is;
(A)
(B)
(C)
(D)
Solution: (A)
7. A solution at 20 ๐
๐ถ is composed of 1.5 mol of benzene and 3.5 mol of toluene. If the vapour
pressure of pure benzene and pure toluene at this temperature are 74.7 torr and 22.3 torr,
respectively, then the total vapour pressure of the solution and the benzene mole fraction in
equilibrium with it will be, respectively:
(A) 30.5 torr and 0.389
(B) 35.0 torr and 0.480
(C) 38.0 torr and 0.589
(D) 35.8 torr and 0.280
Solution: (C) ๐ ๐ต๐๐๐ง๐๐๐ =
1.5
5
= 0.3
๐ ๐๐๐๐ข๐๐๐ =
3.5
5
= 0.7
๐๐ก๐๐ก๐๐ = 0.3 ร 74.7 + 0.7 ร 22.3
28. = 22.41 + 15.61 = 38.02
โ 38 ๐๐๐๐
By Daltonโs law to vapour phase
๐ ๐ต๐๐๐ง๐๐๐
โฒ (๐ฃ๐๐ ๐โ๐๐ ๐) =
0.3 ร 74.7
38
=
22.41
38
= 0.589
8. Which molecule/ion among the following cannot act as a ligand in complex compounds?
(A) ๐ถ๐โ
(B) ๐ถ๐ป4
(C) ๐ถ๐
(D) ๐ต๐โ
Solution: (B) ๐ถ๐ป4 does not have either a lone pair or ๐-electron pair it cannot act as ligand.
9. A compound A with molecular formula ๐ถ10 ๐ป13 ๐ถ๐ gives a white precipitate on adding silver
nitrate solution. A on reacting with alcoholic KOH gives compound B as the main product. B on
ozonolysis gives C and D. C gives Cannizaro reaction but not aldol condensation. D gives aldol
condensation but not Cannizaro reaction. A is:
(A)
(B)
(C)
(D)
Solution: (B) Chlorine attached to tertiary carbon will give a white precipitate on adding ๐ด๐๐๐3
(Saytzeff Rule)
29. 10.
is used as:
(A) Antacid
(B) Insecticide
(C) Antihistamine
(D) Analgesic
Solution: (D) Acetyl salicylic acid is analgesic.
11. An aqueous solution of a salt X turns blood red on treatment with ๐๐ถ๐โ
and blue on
treatment with ๐พ4[๐น๐(๐ถ๐)6], X also gives a positive chromyl chloride test. The salt X is:
(A) ๐น๐๐ถ๐3
(B) ๐น๐(๐๐3)3
(C) ๐ถ๐ข๐ถ๐2
(D) ๐ถ๐ข(๐๐3)2
Solution: (A)
๐น๐๐ถ๐ฟ3 + 3 ๐๐ถ๐๐๐
โ
โ ๐น๐(๐๐ถ๐)3 + 3 ๐ถ๐โ
(๐ต๐๐๐๐ ๐๐๐)
4 ๐น๐๐ถ๐3 + 3๐พ4[๐น๐(๐ถ๐)6] โถ 12 ๐พ๐ถ๐ + ๐น๐4[๐น๐(๐ถ๐)6]3
๐๐๐ข๐ ๐ ๐๐๐ ๐๐๐ข๐
2๐น๐๐ถ๐3 + 3๐ป2 ๐๐4 โถ ๐น๐2(๐๐4)3 + 6๐ป๐ถ๐
๐พ2 ๐ถ๐2 ๐7 + 2๐ป2 ๐๐4 โถ 2๐พ๐ป๐๐4 + 2๐ถ๐๐3 + ๐ป2 ๐
๐ถ๐๐3 + 2๐ป๐ถ๐ โถ ๐ถ๐๐2 ๐ถ๐2 + ๐ป2 ๐
(๐ถโ๐๐๐๐ฆ๐๐โ๐๐๐๐๐๐)
๐ถ๐๐2 ๐ถ๐2 + 4 ๐ ๐๐๐ป โถ ๐๐2 ๐ถ๐๐4 + 2๐๐๐ถ๐ + 2๐ป2 ๐
(๐ฆ๐๐๐๐๐ค)
30. ๐๐2 ๐ถ๐๐4 + ๐๐(๐ถ๐ป3 ๐ถ๐๐)2 โถ ๐๐๐ถ๐๐4 + 2๐ถ๐ป3 ๐ถ๐๐๐๐
(๐ฆ๐๐๐๐๐ค ๐๐๐ก)
12. The correct statement on the isomerism associated with the following complex ions,
(A) [๐๐(๐ป2 ๐)5 ๐๐ป3]2+
(B) [๐๐(๐ป2 ๐)4(๐๐ป3)2]2+
and
(C) [๐๐(๐ป2 ๐)3(๐๐ป3)3]2+
is
(D) (A) and (B) show only geometrical isomerism
Solution: (D) [๐๐ (๐ป2 ๐)4(๐๐ป3)2]2+
Show c is & trans geometrical isomerism [๐๐ (๐ป2 ๐)3(๐๐ป3)3]2+
Show facial & meridional geometrical isomerism.
13. In the presence of a small amount of phosphorous, aliphatic carboxylic acids react with ๐ผ-
hydrogen has been replaced by halogen. This reaction is known as:
(A) Etard reaction
(B) Wolff-Kischner reaction
(C) Rosenmund reaction
(D) Hell-volhard-zelinsky reaction
Solution: (D) This reaction is known as HVZ reaction.
14. The reaction 2N2O5(g) โ 4NO2(g) + O2(g) follows first order kinetics. The pressure of a
vessel containing only N2O5 was found to increase from 50 mm Hg to 87.5 mm Hg in 30 min.
The pressure exerted by the gases after 60 min. Will be (Assume temperature remains
constant) :
(A) 106.25 mm Hg
(B) 125 mm Hg
(C) 116.25 mm Hg
(D) 150 mm Hg
Solution: (A)
2๐2 ๐5(๐) โถ 4 ๐๐2(๐)
(๐0 โ ๐ฅ) 2๐ฅ
+ ๐2(๐)
๐ฅ
2
โ ๐๐๐๐ ๐ ๐ข๐๐ = ๐0 โ ๐ฅ + 2๐ฅ +
๐ฅ
2
= ๐0 +
3๐ฅ
2
= ๐๐ก๐๐ก๐๐
87.5 = 50 +
3๐ฅ
2
3๐ฅ
2
= 37.5
โด ๐ฅ = 37.5 ร
2
3
= 25
31. For first order kinetics
๐๐ก = ln
๐0
๐0 โ ๐ฅ
= ๐๐
50
25
= ln 2
๐ =
1
๐ก
ln 2 =
1
30
ln 2
After 60 min
๐ =
1
๐กโฒ
ln
๐0
๐0 โ ๐ฅโฒ
โ
1
30
ln 2 =
1
60
ln
๐0
๐0 โ ๐ฅโฒ
2 ln 2 = ln
๐0
๐0 โ ๐ฅโฒ
โ ln 4
๐0
๐0 โ ๐ฅโฒ
= 4 โ ๐0
= 4 ๐0
โ 4๐ฅโฒ
๐ฅโฒ
=
4๐0 โ ๐0
4
=
3๐0
4
=
3 ร 50
4
= 37.5
ฮฃ60 ๐๐๐ ๐๐๐ก๐๐ ๐๐๐๐ ๐ ๐ข๐๐ = ๐0 +
3๐ฅโฒ
2
= 50 + 3 ร
37.5
2
= 50 + 56.25 = 106.25 ๐๐
15. If the principal quantum number n = 6, the correct sequence of filling of electrons will be:
(A) ns โ (n โ 1) d โ (n โ 2) f โ np
(B) ns โ np โ (n โ 1)d โ (n โ 2)f
(C) ns โ (n โ 2)f โ np โ (n โ 1)d
(D) ns โ (n โ 2)f โ (n โ 1)d โ np
Solution: (D) As per (n + โ) rule when n = 6
ns subshell โ 6 + 0 = 6
(n โ 1) d subshell โ 5 + 2 = 7
(n โ 2) f subshell โ 4 + 3 = 7
np subshell โ 6 + 1 = 7
When n + โ values are same, the one have lowest n value filled first.
ns , (n โ 2)f, (n โ 1)d, np
(n + โ) values โ 7 , 7 , 7
n value โ 4 , 5 , 6
16. The cation that will not be precipitated by H2S in the presence of dil HCl is:
(A) Co2+
(B) As3+
(C) Pb2+
(D) Cu2+
Solution: (A) Co2+
precipitated by H2S in presence of NH4OH in group IV as CoS (Black ppt.)
Other are precipitated as sulphide in presence of dil HCl in group II.
32. 17. The geometry of XeOF4 by VSEPR theory is:
(A) Trigonal bipyramidal
(B) Square pyramidal
(C) Pentagonal planar
(D) Octahedral
Solution: (B) H =
1
2
(V + M โ C + A)
=
1
2
(8 + 4) = 6
sp3
d2
Hybridization
4 B.P + 1 B.P (Double bonded) + 1 L.P,
Square pyramidal
Oxygen atom doubly bonded to Xe lone pair of electrons on apical position.
18. The correct order of thermal stability of hydroxides is:
(A) Mg(OH)2 < Sr(OH)2 < Ca(OH)2 < Ba(OH)2
(B) Mg(OH)2 < Ca(OH)2 < Sr(OH)2 < Ba(OH)2
(C) Ba(OH)2 < Sr(OH)2 < Ca(OH)2 < Mg(OH)2
(D) Ba(OH)2 < Ca(OH)2 < Sr(OH)2 < Mg(OH)2
Solution: (B) Thermal stabilities of hydroxides of group II A elements increase from
Be(OH)2 to Ba(OH)2 because going down the group the cation size increases & covalent
character decreases & ionic character increases i.e. Mg(OH)2 < Ca(OH)2 < Sr(OH)2 <
Ba(OH)2
19. Photochemical smog consists of excessive amount of X, in addition to aldehydes, ketones,
peroxy acetyl nitrile (PAN), and so forth. X is:
(A) CH4
(B) CO2
(C) O3
(D) CO
Solution: (C) Photochemical smog is the chemical reaction of sunlight, nitrogen oxides and VOCs in
the atmosphere.
33. NO2
hv
โ NO + O
O + O2 โ O3
So, it consists of excessive amount of ozone molecules as atomic oxygen reacts with one of the
abundant oxygen molecules producing ozone.
20. A sample of a hydrate of barium chloride weighing 61 g was heated until all the water of
hydration is removed. The dried sample weighed 52 g. The formula of the hydrated salt is:
(atomic mass, Ba = 137 amu, Cl = 35.5 amu)
(A) BaCl2 โ H2O
(B) BaCl2 โ 3H2O
(C) BaCl2 โ 4H2O
(D) BaCl2 โ 2H2O
Solution: (D) BaCl2 โ xH2O โ BaCl2 + x H2O
(137 + 2 ร 35.5 + 18x)
= (208 + 18x) g/mole
208 + 18 x
208
=
61
52
10816 + 936 x = 12688
936 x = 1872
x = 2
Formula is BaCl2 โ 2H2O
21. The following statements relate to the adsorption of gases on a solid surface. Identify the
incorrect statement among them:
(A) Entropy of adsorption is negative
(B) Enthalpy of adsorption is negative
(C) On adsorption decrease in surface energy appears as heat
(D) On adsorption, the residual forces on the surface are increased
Solution: (D) Adsorption is spontaneous process โG is โve
During adsorption randomness of adsorbate molecules reduced โS is โve
โG = โH โ TโS
โH = โG + TโS
โH is highly โve and residual forces on surface are satisfied.
22. In the isolation of metals, calcination process usually results in:
(A) Metal oxide
(B) Metal carbonate
(C) Metal sulphide
(D) Metal hydroxide
34. Solution: (A) Calcination used for decomposition of metal carbonates
M CO3
โ
โ MO + CO2 โ
23. A variable, opposite external potential (Eext) is applied to the cell Zn | Zn2+ (1M) โฅ
Cu2+ (1 M)| Cu , of potential 1.1 V. When Eext < 1.1 V and Eext > 1.1 V, respectively electrons
flow from:
(A) Anode to cathode in both cases
(B) Anode to cathode and cathode to anode
(C) Cathode to anode and anode to cathode
(D) Cathode to anode in both cases
Solution: (B) For the Daniel cell
Ecell = 0.34 โ (โ0.76) = 1.10 V
When Eext < 1.10 V electron flow from anode to cathode in external circuit
When Eext > 1.10 V electrons flow from cathode to anode in external circuit (Reverse
Reaction)
24. Complete hydrolysis of starch gives:
(A) Galactose and fructose in equimolar amounts
(B) Glucose and galactose in equimolar amouunts
(C) Glucose and fructose in equimolar amounts
(D) Glucose only
Solution: (D) On complete hydrolysis of starch, glucose is formed. Amylase is an enzyme that
catalyses the hydrolysis of starch into sugars.
25. Match the polymers in column-A with their main uses in column-B and choose the correct
answer:
Column - A Column - B
A. Polystyrene i. Paints and lacquers
B. Glyptal ii. Rain coats
C. Polyvinyl chloride
chloride
iii. Manufacture of toys
D. Bakelite iv. Computer discs
(A) A โ iii , B โ i , C โ ii , D โ iv
(B) A โ ii , B โ i , C โ iii , D โ iv
(C) A โ ii , B โ iv , C โ iii , D โ i
(D) A โ iii , B โ iv , C โ ii , D โ i
Solution: (A) A โ iii , B โ i , C โ ii , D โ iv
35. 26. Permanent hardness in water cannot be cured by:
(A) Treatment with washing soda
(B) Ion exchange method
(C) Calgonโs methos
(D) Boiling
Solution: (D) Permanent hardness due to SO4
2โ
, Clโ
of Ca2+
and Mg2+
cannot be removed by boiling.
27. In the long form of periodic table, the valence shell electronic configuration of 5s2
5p4
corresponds to the element present in:
(A) Group 16 and period 5
(B) Group 17 and period 5
(C) Group 16 and period 6
(D) Group 17 and period 6
Solution: (A) 5s2
, 5p4
configuration is actually 36[Kr]5s2
, 4d10
, 5p4
i.e. 5th period and group 16 and
element Tellurium.
28. The heat of atomization of methane and ethane are 360 kJ/mol and 620 kJ/mol, respectively.
The longest wavelength of light capable of breaking the C โ C bond is (Avogadro number =
6.023 ร 1023
, h = 6.62 ร 10โ34
J s):
(A) 2.48 ร 104
nm
(B) 1.49 ร 104
nm
(C) 2.48 ร 103
nm
(D) 1.49 ร 103
nm
Solution: (D) 4 B.E (C โ H) bond = 360 kJ
B.E (C โ H) bond = 90 kJ/mole
In C2H6 โ B. E(CโC) + 6B. E(CโH) = 620 kJ
B. E(CโC) bond = 620 โ 6 ร 90 = 80 kJ moleโ
B. E(CโC) bond =
80
96.48
= 0.83 eV bondโ
ฮป(Photon in โซ) for rupture of
C โ C bond =
12408
0.83
= 14950โซ
= 1495 nm
โ 1.49 ร 103
nm
29. Which of the following is not an assumption of the kinetic theory of gases?
36. (A) Collisions of gas particles are perfectly elastic.
(B) A gas consists of many identical particles which are in continual motion.
(C) At high pressure, gas particles are difficult to compress.
(D) Gas particles have negligible volume.
Solution: (C) At high pressures gas particles difficult to compress rather they are not compressible at
all.
30. After understanding the assertion and reason, choose the correct option.
Assertion: In the bonding molecular orbital (MO) of H2 , electron density is increases between
the nuclei.
Reason: The bonding MO is ฯA + ฯB , which shows destructive interference of the combining
electron waves.
(A) Assertion and Reason are correct, but Reason is not the correct explanation for the Assertion.
(B) Assertion and Reason are correct and Reason is the correct explanation for the Assertion.
(C) Assertion is incorrect, Reason is correct.
(D) Assertion is correct, Reason is incorrect.
Solution: (D) Electron density between nuclei increased during formation of BMO in H2.
BMO is ฯA + ฯB (Linear combination of Atomic orbitals) provides constructive interference.
37. JEE Mains 2015 10th April (online)
Mathematics
1. If the coefficient of the three successive terms in the binomial expansion of (1 + ๐ฅ) ๐
are in the
ratio 1 : 7 : 42, then the first of these terms in the expansion is :
1. 9 ๐กโ
2. 6 ๐กโ
3. 8 ๐กโ
4. 7 ๐กโ
Answer: (4)
Solution: Let ๐
๐ถ๐ be the first term, then
๐ ๐ถ ๐
๐ ๐ถ ๐+1
=
1
7
โ
๐ + 1
๐ โ ๐
=
1
7
โ 7๐ + 7 = ๐ โ ๐
๐ โ 8๐ = 7 โฆ..(i)
Also
๐ ๐ถ ๐+1
๐ ๐ถ ๐+2
=
7
42
=
1
6
โ
๐ + 2
๐ โ ๐ โ 1
=
1
6
โ 6๐ + 12 = ๐ โ ๐ โ 1
๐ โ 7๐ = 13 โฆโฆ(ii)
Solving
๐ โ 8๐ = 7 โฆ.(i)
๐ โ 7๐ = 13 โฆ..(ii)
____________
โ๐ = โ6
๐ = 6
Hence 7 ๐กโ
term is the answer.
2. The least value of the product ๐ฅ๐ฆ๐ง for which the determinant |
๐ฅ
1
1
1
๐ฆ
1
1
1
๐ง
| is non โ negative, is:
1. โ1
2. โ16โ2
3. โ8
4. โ2โ2
Answer: (3)
Solution: |
๐ฅ
1
1
1
๐ฆ
1
1
1
๐ง
| = ๐ฅ๐ฆ๐ง โ (๐ฅ + ๐ฆ + ๐ง) + 2
Since ๐ด. ๐ โฅ ๐บ. ๐
38. ๐ฅ + ๐ฆ + ๐ง
3
โฅ (๐ฅ๐ฆ๐ง)
1
3
๐ฅ + ๐ฆ + ๐ง โฅ 3(๐ฅ๐ฆ๐ง)
1
3
โด Least value of xyz will have from (when determinant non- negative terms)
๐ฅ๐ฆ๐ง โ (3)(๐ฅ๐ฆ๐ง)
1
3 + 2 โฅ 0
๐ก3
โ 3๐ก + 2 โฅ 0
(๐ก + 2)(๐ก2
โ 2๐ก + 1)
๐ก = โ2 ๐๐๐ ๐ก = +1
Least value of ๐ก3
= โ8.
3. The contrapositive of the statement โIf it is raining, then I will not comeโ, is:
1. If I will come, then it is not raining
2. If I will come, then it is raining
3. If I will not come, then it is raining
4. If I will not come, then it is not raining
Answer: (1)
Solution: Contrapositive of ๐ โ ๐ is
~๐ โ ~ ๐ So contra positive of the statement โIf it is raining, then I will not comeโ, would be
If I will come, then it is not raining.
4. lim
๐ฅโ0
๐ ๐ฅ2
โcos ๐ฅ
sin2 ๐ฅ
is equal to:
1. 2
2.
3
2
3.
5
4
4. 3
Answer: (2)
Solution:
๐ ๐ฅ2
โcos ๐ฅ
sin2 ๐ฅ
=
(1 +
๐ฅ2
โ1 +
๐ฅ4
โ2 โฆ โฆ ) โ (1 โ
๐ฅ2
โ2 +
๐ฅ4
โ4 โฆ โฆ ๐)
sin2 ๐ฅ
๐ฅ2 โ ๐ฅ2
(
+3๐ฅ2
2
+
11 ๐ฅ4
24
sin2 ๐ฅ
๐ฅ2 โ๐ฅ2
) take ๐ฅ2
common
39. [lim
๐ฅโ0
+
3
2 +
11
24 ๐ฅ2
sin2 ๐ฅ
๐ฅ2
] =
3
2
.
5. If Rolleโs theorem holds for the function ๐(๐ฅ) = 2๐ฅ3
+ ๐๐ฅ2
+ ๐๐ฅ, ๐ฅ โ [โ1, 1], at the point ๐ฅ =
1
2
,
then 2b + c equals:
1. 2
2. 1
3. -1
4. -3
Answer: (3)
Solution: If Rolleโs theorem is satisfied in the interval [-1, 1], then
๐(โ1) = ๐(1)
โ2 + ๐ โ ๐ = 2 + ๐ + ๐
๐ = โ2 also ๐โฒ(๐ฅ) = 6๐ฅ2
+ 2๐๐ฅ + ๐
Also if ๐โฒ
(
1
2
) = 0 them
6
1
4
+ 2๐
1
2
+ ๐ = 0
3
2
+ ๐ + ๐ = 0
โต ๐ = โ2,
๐ =
1
2
โด 2๐ + ๐ = 2 (
1
2
) + (โ2)
= 1 โ 2
= โ1.
6. If the points (1, 1, ๐) ๐๐๐ (โ3, 0, 1) are equidistant from the plane, 3๐ฅ + 4๐ฆ โ 12๐ง + 13 = 0,
then ๐ satisfies the equation:
1. 3๐ฅ2
+ 10๐ฅ + 7 = 0
2. 3๐ฅ2
+ 10๐ฅ โ 13 = 0
3. 3๐ฅ2
โ 10๐ฅ + 7 = 0
4. 3๐ฅ2
โ 10๐ฅ + 21 = 0
Answer: (3)
Solution: (1, 1, ๐) ๐๐๐ (โ3, 0, 1) in equidistant from 3๐ฅ + 4๐ฆ โ 12๐ง + 13 = 0 then
40. |
3 + 4 โ 12๐ + 13
โ32 + 42 + 122
| = |
โ9 + 0 โ 12 + 13
โ32 + 42 + 122
|
|20 โ 12๐| = |โ8|
|5 โ 3๐ | = |โ2|
25 โ 30๐ + 9๐2
= 4
9๐2
โ 30๐ + 21 = 0
3๐2
โ 10๐ + 7 = 0
โด Option 3๐ฅ2
โ 10๐ฅ + 7 = 0 Is correct
7. In a ฮ๐ด๐ต๐ถ,
๐
๐
= 2 + โ3 ๐๐๐ โ ๐ถ = 60 ๐
. Then the ordered pair (โ ๐ด, โ ๐ต) is equal to:
1. (105 ๐
, 15 ๐)
2. (15 ๐
, 105 ๐)
3. (45 ๐
, 75 ๐)
4. (75 ๐
, 45 ๐
)
Answer: (1)
Solution: Since
๐
๐
=
2+ โ3
1
โ ๐ด > โ ๐ต.
Hence only option 1 & 4 could be correct checking for option (1)
๐
๐
=
sin105 ๐
sin 15 ๐
=
๐ ๐๐ (60 ๐
+ 45 ๐
)
sin(60 ๐ โ 45 ๐)
=
โ3 + 1
โ3 โ 1
๐
๐
=
2 + โ3
1
Hence option (105 ๐
, 15 ๐) is correct.
8. A factory is operating in two shifts, day and night, with 70 and 30 workers respectively. If per
day mean wage of the day shift workers is Rs. 54 and per day mean wage of all the workers is
Rs. 60, then per day mean wage of the night shift workers (in Rs.) is :
1. 75
2. 74
3. 69
4. 66
Answer: (2)
Solution:
๐1 ๐ฅ1 +๐2 ๐ฅ2
๐1+๐2
= ๐ฅ
70 โ (54) + 30 (๐ฅ2)
70 + 30
= 60
42. If ๐ = ๐ + 2๐โ (๐ ร ๐โ ), then 2|๐| is equal to:
1. โ51
2. โ37
3. โ43
4. โ55
Answer: (4)
Solution: As |๐ ร ๐โ | = โ3
Squaring both the sides
|๐|2
+ |๐โ |
2
+ 2๐ โ ๐โ = 3
1 + 1 + 2 โ 1 โ 1 โ cos ๐ = 3
2๐๐๐ ๐ = 1
๐๐๐ ๐ =
1
2
๐ = 60
โด Angle between ๐ ๐๐๐ ๐โ ๐๐ 60 ๐
Now,
|๐| = |๐ + 2๐ + 3(๐ ร ๐)|
Squaring both the sides
|๐|2
= ||๐|2
+ 4|๐โ |
2
+ 9 (๐ ร ๐)2
+ 4 ๐ โ (๐) + 3๐ โ (๐ ร ๐) + 6๐ โ (๐ ร ๐)|
|๐|2
= |1 + 4 + 9 sin2
๐ + 4 ๐๐๐ ๐ + 0 + 0 |
|๐|2
= |5 + 9.
3
4
+ 4.
1
2
| =
55
4
โด 2|๐| = โ55.
11. The area (in square units) of the region bounded by the curves ๐ฆ + 2๐ฅ2
= 0 ๐๐๐ ๐ฆ + 3๐ฅ2
= 1,
is equal to:
1.
3
4
2.
1
3
3.
3
5
4.
4
3
Answer: (4)
Solution:
43. Point of intersection
Put ๐ฆ = โ2๐ฅ2
๐๐ ๐ฆ + 3๐ฅ2
= 1
๐ฅ2
= 1
๐ฅ = ยฑ 1
The desired area would be
โซ (๐ฆ1 โ ๐ฆ2) ๐๐ฅ = โซ ((1 โ 3๐ฅ2) โ (โ2๐ฅ2)) ๐๐ฅ
1
โ1
1
โ1
โซ (1 โ ๐ฅ2)๐๐ฅ
1
โ1
(๐ฅ โ
๐ฅ3
3
)
โ1
1
= ((1 โ
1
3
) โ (โ1 +
1
3
))
2
3
โ (
โ2
3
)
=
4
3
.
12. If ๐ฆ + 3๐ฅ = 0 is the equation of a chord of the circle, ๐ฅ2
+ ๐ฆ2
โ 30๐ฅ = 0, then the equation of
the circle with this chord as diameter is :
1. ๐ฅ2
+ ๐ฆ2
+ 3๐ฅ โ 9๐ฆ = 0
2. ๐ฅ2
+ ๐ฆ2
โ 3๐ฅ + 9๐ฆ = 0
3. ๐ฅ2
+ ๐ฆ2
+ 3๐ฅ + 9๐ฆ = 0
4. ๐ฅ2
+ ๐ฆ2
โ 3๐ฅ โ 9๐ฆ = 0
Answer: (2)
Solution:
44. ๐ฆ = โ3๐ฅ
4๐ฅ2
+ ๐ฆ2
โ 30๐ฅ = 0
Point of intersection
๐ฅ2
+ 9๐ฅ2
โ 30๐ฅ = 0
10๐ฅ2
โ 30๐ฅ = 0
10๐ฅ (๐ฅ โ 3) = 0
๐ฅ = 0 or ๐ฅ = 3
Therefore y = 0 if x = 0, and y =-9 if x = 3.
Point of intersection (0, 0) (3, -9)
Diametric form of circle,
๐ฅ (๐ฅ โ 3) + ๐ฆ(๐ฆ + 9) = 0
๐ฅ2
+ ๐ฆ2
โ 3๐ฅ + 9๐ฆ = 0.
13. The value of โ (๐ + 2) (๐ โ 3)30
๐=16 is equal to:
1. 7775
2. 7785
3. 7780
4. 7770
Answer: (3)
Solution: โ (๐ + 2) (๐ โ 3)30
๐=16
= โ (๐2
โ ๐ โ 6) โ โ (๐2
โ ๐ โ 6)15
1
30
1
Put r = 30
in (
๐(๐+1) (2๐+1)
6
โ
๐(๐+1)
2
โ 6๐)
30 โ (31)(61)
6
โ 15(31) โ 6(30)
9455 โ 465 โ 180
8810
And on putting ๐ = 15
We get
15โ(16) (31)
6
โ
15โ16
2
โ 6 โ (15)
= (7) โ (8) โ (31) โ
15 โ16
2
โ 6 โ (15)
45. = 1240 โ 120 โ 90
= 1030
Therefore โ (๐2
โ ๐ โ 6) โ โ (๐2
โ ๐ โ 6)15
1
30
1 = 8810 โ 1030
= 7780.
14. Let L be the line passing through the point P(1, 2) such that its intercepted segment between
the co-ordinate axes is bisected at P. If ๐ฟ1 is the line perpendicular to L and passing through the
point (-2, 1), then the point of intersection of L and ๐ฟ1 is:
1. (
3
5
,
23
10
)
2. (
4
5
,
12
5
)
3. (
11
20
,
29
10
)
4. (
3
10
,
17
5
)
Answer: (2)
Solution:
If P is the midpoint of the segment between the axes, them point A would be (2, 0) and B would be (0,
4). The equation of the line would be
๐ฅ
2
+
๐ฆ
4
= 1
That is 2๐ฅ + ๐ฆ = 4 โฆ..(i)
The line perpendicular to it would be ๐ฅ โ 2๐ฆ = ๐
Since it passes through (-2, 1) โ2 โ 2 = ๐
โ4 = ๐
โด Line will become ๐ฅ โ 2๐ฆ = โ4 โฆ..(ii)
Solving (i) and (ii) we get (
4
5
,
12
5
).
15. The largest value of r for which the region represented by the set {
๐ โ๐ถ
|๐โ4โ๐| โค ๐
} is contained in
the region represented by the set {
๐ง โ๐ถ
|๐งโ1| โค |๐ง+๐|
}, is equal to :
46. 1. 2โ2
2.
3
2
โ2
3. โ17
4.
5
2
โ2
Answer: (4)
Solution:
|๐ง โ 1| โค |๐ง + ๐|
The region in show shaded right side of the line ๐ฅ + ๐ฆ = 0
The largest value of r would be the length of perpendicular from A (4, 1) on the line ๐ฅ + ๐ฆ = 0
|
4 + 1
โ2
| =
5
โ2
=
5
2
โ2 .
16. Let the sum of the first three terms of an A.P. be 39 and the sum of its last four terms be 178. If
the first term of this A.P. is 10, then the median of the A.P. is :
1. 26.5
2. 29.5
3. 28
4. 31
Answer: (2)
Solution: Let the A.P. be a; a + d a + 2d โฆโฆโฆโฆโฆโฆโฆโ โ 3๐, โ โ 2๐, โ โ ๐, โ
Where a is the first term and โ is the last term
Sum of 1 ๐ ๐ก
3 terms is 39.
3๐ + 3๐ = 39
30 + 3๐ = 30 as ๐ = 10 (Given)
๐ =
9
3
= 3
47. Sum of last 4 terms is 178.
4โ โ 6๐ = 178
4โ โ 18 = 178
4โ = 196
โ = 49
10, 13, 16, 19โฆโฆ.46, 49
Total number of the 10 + (n โ 1) 3 - 49
n โ 1 = 13
n = 14
So the median of the series would be mean of 7 ๐กโ
๐๐๐ 8 ๐กโ
term
10+6โ(3)+10+7โ3
2
28 + 31
2
=
59
2
= 29.5
Alternate way
The median would be mean of 10 and 49, That is 29.5.
17. For ๐ฅ > 0, let ๐(๐ฅ) = โซ
log ๐ก
1+๐ก
๐๐ก.
๐ฅ
1
Then ๐(๐ฅ) + ๐ (
1
๐ฅ
) is equal to :
1.
1
2
(log ๐ฅ)2
2. log ๐ฅ
3.
1
4
log ๐ฅ2
4.
1
4
(log ๐ฅ)2
Answer: (1)
Solution:
๐(๐ฅ) = โซ
log ๐ก
1 + ๐ก
๐ฅ
1
โ ๐๐ก
And ๐ (
1
๐ฅ
) = โซ
log ๐ก
1+๐ก
โ ๐๐ก
1
๐ฅ
1
Put ๐ก =
1
๐ง
๐๐ก = โ
1
๐ง2
๐๐ก
โ
1
๐ฅ2
๐๐ฅ = ๐๐ก
๐(๐ฅ) = โซ
log ๐ง
๐ง2 (1 +
1
๐ง)
๐ง
1
โ ๐๐ง
48. ๐(๐ฅ) = โซ
log ๐ง
๐ง(1 + ๐ง)
๐๐ง
๐ง
1
๐(๐ฅ) + ๐ (
1
๐ฅ
) = โซ log ๐ง [
1
1 + ๐ง
+
1
2(1 + ๐ง)
] ๐๐ง
๐ฅ
1
= โซ
1
๐ง
log ๐ง ๐๐ง
๐ฅ
1
Put log ๐ง = ๐
1
๐ง
๐๐ง = ๐๐
โซ ๐ โ ๐๐
๐ฅ
1
(
๐2
2
)
1
๐ฅ
=
1
2
(log ๐ง)1
๐ฅ
=
(log ๐ฅ)2
2
18. In a certain town, 25% of the families own a phone and 15% own a car; 65% families own
neither a phone nor a car and 2,000 families own both a car and a phone. Consider the
following three statements:
(a) 5% families own both a car and a phone.
(b) 35% families own either a car or a phone.
(c) 40, 000 families live in the town.
Then,
1. Only (b) and (c) are correct
2. Only (a) and (b) are correct
3. All (a), (b) and (c) are correct
4. Only (a) and (c) are correct
Answer: (3)
Solution: Let set A contains families which own a phone and set B contain families which own a car.
If 65% families own neither a phone nor a car, then 35% will own either a phone or a car
โด (๐ดโ๐ต) = 35%
Also we know that
๐(๐ด โช ๐ต) = ๐(๐ด) + ๐(๐ต) โ ๐(๐ด โฉ ๐ต)
35 = 25 + 15 - ๐(๐ด โฉ ๐ต)
๐(๐ด โฉ ๐ต) = 5%
5% families own both phone and car and it is given to be 2000.
โด 5% ๐๐ ๐ฅ = 2000
5
100
๐ฅ = 2000
49. X = 40,000
Hence correct option is (a) (b) and (c) are correct.
19. IF ๐ด = [
0
1
โ1
0
], then which one of the following statements is not correct?
1. ๐ด3
+ ๐ผ = ๐ด(๐ด3
โ ๐ผ)
2. ๐ด4
โ ๐ผ = ๐ด2
+ ๐ผ
3. ๐ด2
+ ๐ผ = ๐ด(๐ด2
โ ๐ผ)
4. ๐ด3
โ ๐ผ = ๐ด(๐ด โ ๐ผ)
Answer: (3)
Solution: A = [
0 โ1
1 0
]
๐ด2
= [
0 โ1
1 0
] [
0 โ1
1 0
] = [
โ1 0
0 โ1
]
๐ด3
= [
โ1 0
0 โ1
] [
0 โ1
1 0
] = [
0 1
โ1 0
]
๐ด4
= [
0 1
โ1 0
] [
0 โ1
1 0
] [
1 0
0 1
]
Option (1) ๐ด3
+ ๐ผ = ๐ด (๐ด3
โ ๐ผ)
[
0
1
โ1
0
] [
โ1
โ1
1
โ1
] = [
1
โ1
1
1
]
[
1
โ1
1
1
] = [
1
โ1
1
1
] โฆ..Correct
Option (2) ๐ด4
โ ๐ผ = ๐ด2
+ ๐ผ
[
0 0
0 0
] = [
0 0
0 0
] โฆ.Correct
Option (3) [
0 0
0 0
] = [
0 โ1
1 0
] [
โ2 0
0 โ2
] = [
0 2
โ2 0
] โฆ..Incorrect
Option 4
๐ด3
โ ๐ผ = ๐ด(๐ด โ ๐ผ)
[
โ1 โ1
โ1 โ1
] = [
0 โ1
1 0
] [
โ1 โ1
1 โ1
] [
โ1 1
โ1 1
]
๐ด3
โ ๐ผ = ๐ด4
โ ๐ด
[
1 1
โ1 1
] = [
1 0
0 1
] โ [
0 โ1
1 0
]
= [
1 1
โ1 1
] โฆโฆCorrect.
20. Let X be a set containing 10 elements and P(X) be its power set. If A and B are picked up at
random from P(X), with replacement, then the probability that A and B have equal number of
elements, is:
1.
(210โ1)
220
50. 2.
20 ๐ถ10
220
3.
20 ๐ถ10
210
4.
(210โ1)
210
Answer: (2)
Solution: The power set of x will contain 210
sets of which
10
๐ถ0 will contain 0 element
10
๐ถ1 will contain 1 element
10
๐ถ2 will contain 2 element
โฎ
โฎ
10
๐ถ10 will contain 10 element.
So total numbers of ways in which we can select two sets with replacement is 210
ร 210
= 220
And favorable cases would be 10
๐ถ0 โ 10
๐ถ0 + 10
๐ถ1
10
๐ถ1 + โฆ โฆ 10
๐ถ10
10
๐ถ10 = 20
๐ถ10.
Hence Probability would be =
20 ๐ถ10
220
Hence
20 ๐ถ10
220 in the correct option
21. If 2 + 3๐ is one of the roots of the equation 2๐ฅ3
โ 9๐ฅ2
+ ๐๐ฅ โ 13 = 0, ๐ โ ๐ , then the real
root of this equation:
1. Exists and is equal to
1
2
2. Does not exist
3. Exists and is equal to 1
4. Exists and is equal to โ
1
2
Answer: (1)
Solution: If 2 + 3๐ in one of the roots, then 2 โ 3๐ would be other
Since coefficients of the equation are real.
Let ๐พ be the third root, then product of roots โ ๐ผ ๐ฝ ๐พ =
13
2
(2 + 3๐) (2 โ 3๐) โ ๐พ =
13
2
(4 + 9) โ ๐พ =
13
2
๐พ =
1
2
.
The value of k will come if we
Put ๐ฅ =
1
2
in the equation
2 โ
1
8
โ
9
4
+ ๐ โ
1
2
โ 13 = 0
51. ๐
2
= 15
๐ = 30.
โด Equation will become
2๐ฅ3
โ 9๐ฅ2
+ 30๐ฅ โ 13 = 0
๐ผ๐ฝ + ๐ฝ๐พ + ๐พ๐ผ =
30
2
= 15
(2 + 3๐)
1
2
+ (2 โ 3๐)
1
2
+ (2 + 3๐) (2 โ 3๐) = 15
1 +
๐
2
+ 1 โ
๐
2
+ 13 = 15
15 = 15
Hence option (1) is correct. โExists and is equal to
1
2
โ
22. If the tangent to the conic, ๐ฆ โ 6 = ๐ฅ2
at (2, 10) touches the circle, ๐ฅ2
+ ๐ฆ2
+ 8๐ฅ โ 2๐ฆ = ๐ (for
some fixed k) at a point (๐ผ, ๐ฝ); then (๐ผ, ๐ฝ) is :
1. (โ
7
17
,
6
17
)
2. (โ
8
17
,
2
17
)
3. (โ
6
17
,
10
17
)
4. (โ
4
17
,
1
17
)
Answer: (2)
Solution: The equation of tangent (T = 0) would be
1
2
(๐ฆ + 10) โ 6 = 2๐ฅ
4๐ฅ โ ๐ฆ + 2 = 0
The centre of the circle is (โ4, 1) and the point of touch would be the foot of perpendicular from
(โ4, 1) on 4๐ฅ โ ๐ฆ + 2 = 0
๐ฅ + 4
4
=
๐ฆ โ 1
โ1
= โ (
โ16 โ 1 + 2
42 + 12
)
๐ฅ+4
4
=
15
17
and
๐ฆโ1
โ1
=
15
17
๐ฅ = โ
8
17
๐ฆ =
โ15
17
+ 1 =
2
17
52. Hence option (โ
8
17
,
2
17
) is correct.
23. The number of ways of selecting 15 teams from 15 men and 15 women, such that each team
consists of a man and a woman, is:
1. 1960
2. 1240
3. 1880
4. 1120
Answer: (2)
Solution: No. of ways of selecting 1 ๐ ๐ก
team from 15 men and 15 women
15
๐ถ1
15
๐ถ1 = 152
2 ๐๐
team- 14
๐ถ1
14
๐ถ1 142
and so on.
So total number of way
12
+ 22
โฆ โฆ โฆ 152
=
15 (16) (31)
6
= (5) โ (8) โ (31)
1240
Hence option 1240 is correct.
24. If the shortest distance between the line
๐ฅโ1
๐ผ
=
๐ฆ+1
โ1
=
๐ง
1
, (๐ผ โ โ1) and ๐ฅ + ๐ฆ + ๐ง + 1 = 0 =
2๐ฅ โ ๐ฆ + ๐ง + 3 ๐๐
1
โ3
, then a value of ๐ผ is :
1. โ
19
16
2.
32
19
3. โ
16
19
4.
19
32
Answer: (2)
Solution: Let us change the line into symmetric form.
๐ฅ + ๐ฆ + ๐ง + 1 = 0 = 2๐ฅ โ ๐ฆ + ๐ง + 3
Put ๐ง = 1, so we get ๐ฅ + ๐ฆ + 2 = 0 and 2๐ฅ โ ๐ฆ + 4 = 0
We will get ๐ฅ = โ2
๐ฆ = 0
โด The point (โ2, 0, 1) lies on the line and perpendicular vector will come from
54. 25. The distance from the origin, of the normal to the curve, ๐ฅ = 2 cos ๐ก + 2๐ก sin ๐ก, ๐ฆ =
2 sin ๐ก โ 2๐ก cos ๐ก ๐๐ก ๐ก =
๐
4
, is :
1. โ2
2. 2โ2
3. 4
4. 2
Answer: (4)
Solution: at ๐ก =
๐
4
๐ฅ = 2
1
โ2
+ 2
๐
4
= (โ2 +
๐
2โ2
) = (
8 + ๐
2โ2
)
๐ฆ = 2
1
โ2
โ 2
๐
4
โ
1
โ2
= (โ2 โ
๐
2โ2
) โ (
8 โ ๐
2โ2
)
๐๐ฆ
๐๐ฅ
= 2 cos ๐ก โ 2 [cos ๐ก + ๐ก (โ sin ๐ก)] = 2๐ก sin ๐ก
๐๐ฅ
๐๐ก
= โ2 sin ๐ก + 2 [sin ๐ก + ๐ก โ cos ๐ก] = 2๐ก cos ๐ก
๐๐ฆ
๐๐ฅ
= tan ๐ก ๐๐๐ ๐ก =
๐
4
๐๐๐ tan
๐
4
= 1
๐๐ฆ
๐๐ฅ
= 1 Slope of tangent is 1 & therefore slope of normal would be -1.
Equation of normal ๐ฆ โ (
8โ๐
2โ 2
) = โ1 (๐ฅ โ (
8+๐
2โ2
))
๐ฅ + ๐ฆ = ๐ก
(8 + ๐)
2โ2
+ (
8 โ ๐
2โ2
)
๐ฅ + ๐ฆ =
16
2โ2
and distance from origin
16
2โ2
โ2 = 4
26. An ellipse passes through the foci of the hyperbola, 9๐ฅ2
โ 4๐ฆ2
= 36 and its major and minor
axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of
eccentricities of the two conics is
1
2
, then which of the following points does not lie on the
ellipse?
1. (
โ39
2
, โ3)
2. (
1
2
โ13,
โ3
2
)
3. (โ
13
2
, โ6)
4. (โ13, 0)
55. Answer: (2)
Solution: Equation of the hyperbola
๐ฅ2
4
โ
๐ฆ2
9
= 1
Focus of hyperbola (ae, 0) and (-ae, 0)
a = 2 ๐ = โ1 +
9
4
=
โ13
2
โด Focus would be (+
โ13
2
, 0) ๐๐๐ (โ
โ13
2
, 0)
Product of eccentricity would be
โ13
2
โ ๐1 =
1
2
โด ๐1 =
1
โ13
.
As the major & minor axis of the ellipse coin side with focus of the hyperbola then the value of a for
ellipse would be โ13,
๐ = โ1 โ
๐2
๐2
๐2
13
=
12
13
1
โ3
= โ1 โ
๐2
13
๐2
= 12
1
13
= 1 โ
๐2
13
โด Equation of the ellipse would be
๐ฅ2
13
+
๐ฆ2
12
= 1.
Option (i)
39
4 โ(13)
+
3
12
= 1
Satisfies the equation hence it lies on the ellipse.
Option (ii)
13
4 (13)
+
3
4.12
= 1
does not lie on the ellipse.
Option (iii)
13
2(13)
+
6
12
= 1 satisfy
Option (iv)
13
13
+ 0 = 1 satisfy
So option (
1
2
โ13,
โ3
2
) is the answer.
56. 27. The points (0,
8
3
) , (1, 3) ๐๐๐ (82, 30) :
1. Form an obtuse angled triangle
2. Form an acute angled triangle
3. Lie on a straight line
4. Form a right angled triangle
Answer: (3)
Solution: The options
A B C
(0
8
2
) (1, 3) (82, 30)
Are collinear as slope f AB is equal to slope of BC
3 โ
8
3
1 โ 0
=
30 โ 3
82 โ 1
1
3
=
27
81
=
1
3
Hence option (Lie on a straight line) is correct.
28. If ๐(๐ฅ) โ 2 tanโ1
๐ฅ + sinโ1
(
2๐ฅ
1+๐ฅ2) , ๐ฅ > 1, then ๐(5) is equal to :
1.
๐
2
2. tanโ1
(
65
156
)
3. ๐
4. 4 tanโ1 (5)
Answer: (3)
Solution:
2 tanโ1
๐ฅ + sinโ1
(
2๐ฅ
1 + ๐ฅ2
) , ๐๐๐ ๐ฅ > 1.
= 2 tanโ1
๐ฅ + ๐ โ 2 tanโ1
๐ฅ ๐๐ ๐ฅ > 1
โด ๐(5) = ๐
โด Answer is ๐
Or ๐(5) = 2 tanโ1 (5) + sinโ1
(
10
26
)
57. = ๐ โ tanโ1
(
10
24
) + tanโ1
(
10
24
)
๐ sinโ1
(
10
26
)
29. Let the tangents drawn to the circle, ๐ฅ2
+ ๐ฆ2
= 16 from the point P(0, h) meet the ๐ฅ โ ๐๐ฅ๐๐ at
points A and B. If the area of ฮ๐ด๐๐ต is minimum, then h is equal to :
1. 4โ2
2. 3โ2
3. 4โ3
4. 3โ3
Answer: (1)
Solution:
Let the equation of the tangent be (๐ฆ โ โ) = ๐ (๐ฅ โ 0)
๐๐ฅ โ ๐ฆ + โ = 0
|
โ๐
โ๐2 + 1
| = 4
โ2
= 16๐2
+ 16
๐2
=
โ2
โ 16
16
๐ =
โโ2 โ 16
4
So co-ordinate of B would be
โ
โ2 โ 16
4
๐ฅ โ ๐ฆ + โ = 0
๐ฅ =
4โ
โโ2 โ 16
Also of triangle
=
1
2
๐ต๐๐ ๐ ๐ฅ ๐ป๐๐๐โ๐ก