P
P
h
h
y
y
s
s
i
i
c
c
s
s
PART – I
SECTION – A
This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE option is correct.
1. Consider the shown circuit. The net current supplied as a function of time is (A) (2 2 sin t) A (B) (2 2 cos t) A (C) ( 2 2 sin t) A (D) ( 2 2 cos t) A 1 H ~ ~ 0.2 mF 100 1002 sin 100t 1002 cos 100t 2. The figure shows a system of 2 concentric spheres of radii r1 and
r2 and kept at temperatures T1 and T2 respectively. The radial rate of flow of heat in a substance between the two co-centric spheres is proportional to:
(A) 1 2 2 1 r r (r r ) (B) (r2r )1 (C) 2 1 1 2 (r r ) r r (D) 2 1 r log r
T
1r
1r
2T
23. The rod is free to slide along the two horizontal parallel rails; the two rails being maintained at a constant potential difference by the ideal battery. The only resistance is in the rod itself. A student performs an experiment with uniform magnetic field B. In first case when rod is kept at rest, the rod is flow to move with uniform velocity after a sufficient time. Now in second case when a force of F0 is applied along with first case, so as to retard the rod, its velocity changes and finally decreases by 20 cm/sec. B
Now if above experiment is repeated with magnetic field B/3 then velocity would have
(A) increased by 60 cm/s (B) decreased by 180 cm/s
(C) decreased by 60 cm/s (D) increased by 20/9 cm/s
4. The graph between two temperature scales A and B is shown in the figure. Between upper fixed point and lower fixed point there are 150 equal divisions on scale A and 100 on scale B. The relationship for conversion between the two scales is given by (A) tA 180 tB 100 150 (B) tA 30 tB 150 100 (C) tB 180 tA 150 100 (D) tB 40 tA 100 180 Temperature (B) T e m p e ra tu re ( A ) tA = 1 5 0 tB = 100
5. A particle of mass m and charge q is dropped from a height h. In whole space, there exists a constant magnetic field B as shown in figure. Find the minimum value of B such that the particle just grazes the ground. (A) m g 2q h (B) 2m g q h (C) m g 4q h (D) m 2g q h h B m, q
6. In the RC circuit shown, switch is closed at t = 0. The resistor in the circuit is very sensitive to heat and as current is flowing, its resistance increases linearly as R = R0 + kt. Find the ratio of heat developed to the work done by the battery upto the moment the resistance becomes thrice its original value. (kC = 1) S R C V (A) 1/3 (B) 2/3 (C) 5/3 (D) 7/3
7. A solid conducting cylinder (radius a) is rotated about its symmetric axis at an angular speed . Find the magnitude of induced emf across A(a/2, 0, 0) and B(a/2, a/2, a/3) (m: mass of electron, e : charge of electron) (A) 2 2 ma 16e (B) 2 2 ma 24e (C) 2 2 ma 32e (D) 2 2 ma 8e A B x y z
8. An infinitely long wire with triangular cross section (equal sides a) carries a current I uniformly. The magnitude of the magnetic field at O is (A) 0I 6a (B) 0I 4a (C) 0I 12a (D) 0I 3a O
9. The potential due to point charge at a point A is 7V and the electric field there is 3 V/m. There is another point B such that electric field at this point has smaller magnitude as compared to that at point A. However, if magnitude of the charge is tripled, the electric field at B becomes 3 V/m. The potential at B now is closest to
(A) 7/3 V (B) 7 V
(C) 12 V (D) 21 V
10. Three identical thin uniform charged filaments are fixed along the sides of a cube as shown in the figure. Length of each filament is and line charge density on each of them is . Determine the magnitude of electric field at the centre of the cube. The cube is a geometrical construct and not made of any matter.
(A) 0 2 2 3 (B) 0 2 4 3 (C) zero (D) 0 1 4 3
Comprehension Type
This section contains 3 groups of questions. Each group has 2 multiple choice question based on a paragraph. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which only ONE is correct
Paragraph for Questions 11 & 12 One mole of helium gas follows cycle 1-2-3-1 shown in the diagram. During process 3-1, the internal energy (U) of the gas depends on its volume (V) as U = bV2, where b is a positive constant. If gas releases the amount of heat Q1 during process 3-1 and gas absorbs the amount of heat Q2 during process 123. 4U0 V1 V 2 3 U U0 1
Read the passage carefully and answer the following questions
11. The value of ratio of volume of gas in state-3 (V3) to that of gas in state –1(V1) is (A) 1 2 (B) 2 1 (C) 1 3 (D) none of these 12. The value of Q1 / Q2 is (A) 8 9 (B) 9 8 (C) 12 13 (D) none of these
Paragraph for Questions 13 & 14 An Infinite sheet having surface charge density is
placed in front of a metallic charged sphere of charge Q and radius R, as shown in the figure such that surface of metallic plate is perpendicular to the line CO, where O is the centre of the plate. The separation between the sheet and the centre of the metallic sphere C is 2R. Answer the following questions based on the given situations.
+ P1 R C P2 2R Metallic sphere haveing charge Q O + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
13. Electric field at point P2 (just outside the sphere) (A) 0 2 (B) 0 (C) 2 0 kQ 2 R (D) greater than 2 0 14. Electric field due to the metallic sphere at point C
(A) zero (B) 2 kQ R towards left (C) 0 2 towards left (D) 2 0 towards right
Paragraph for Questions 15 & 16
When a current flows through the medium, an electric field exists as well as a potential which varies in space. Suppose that there is a break in a high – voltage transmission line and the free end of a wire of length L is lying on the ground. An electric current flows through the regions of soil adjoining the conductor. If a man happens to be walking near by a potential difference, which is called the step voltage appears between the points where his feet touch the ground. Consequently, an electric current whose strength depends on this potential difference flows through the man.
Let us calculate the step voltage. Since the conductor is quite long, we assume that the current flows from it to the ground in a direction perpendicular to the conductor. The equipotential surfaces are the surfaces of semi-cylinders whose axes coincide with the conductor. Suppose that the man is walking in a direction perpendicular to the conductor with a step of length ‘b’ the distance between the conductor and the foot closer to it being d. Assuming that the current flows uniformly from the conductor over the semi cylindrical region we obtain the following expression for the current density at a distance r from the conductor: I j rL
In this case, the field strength along the radii perpendicular to the conductor is
r j I E rL
Consequently, the step voltage is
d b st r d I d b V E dr ln L d
.For example, If I = 500A, d = 1 m, b = 65 cm and L = 30 m we find that Vst = 270 V. Much higher voltages may appear under other conditions and other shapes of conductors.
15. When a part of a high – voltage transmission line falls on the ground, it creates a hazard (A) because there can be a direct contact between the cable and a human being
(B) because of the emergence of step voltages. (C) both (A) and (B)
(D) none of (A) and (B)
16. A hemispherical earth plate is buried into the earth in level with its surface. Find the voltage which may be applied to a woman approaching this earth plate (step voltage). The current passing through the earth plate is equal to 1 amp, the length of the step is b = 1m, and the distance between the plate, the foot closer to it is r0 = 2m and = 102 /m. (A) 0.9 Volt (B) 1.8 Volt (C) 2.7 Volt (D) 3.6 Volt j r0 b I
Match List Type (Only One Option Correct)
This section contains four questions, each having two matching lists. Choices for the correct combination of elements from List-I and List-II are given as option (A), (B), (C) and (D) out of which one is correct. 17. List – I contains four possibilities after removing an infinite cylinder of radius R/2 from an infinite
cylinder of radius R and carrying current I of uniform density (I = jR2). List – II contains the magnitude of magnetic field at the centre (O) or a point at the perimeter (P) due to system in List – I. Match list-I with list –II and select the correct answer using the code given below the list
List I List II
(P) Axis of cavity lies on the axis of cylinder. The magnetic field on the axis of
cylinder (BO) is O
1. Zero
(Q) Axis of cavity lies on the axis of cylinder. The magnetic field on the
surface of cylinder (BP) is P
2. 0I
4 R
(R) Cavity’s axis is R/2 away from the original cylinder’s axis. The magnetic field on
the axis of cylinder (BO) is O
3. 3 I0 8 R
(S) Cavity’s axis is R/2 away from the original cylinder’s axis. The magnetic field on the surface of cylinder at P as shown P 4. 0 5 I 12 R Codes: P Q R S (A) 1 3 4 2 (B) 1 3 2 4 (C) 3 1 2 4 (D) 1 2 3 3
18. There are two types of rods:
Rod 1: Length L, Thermal conductivity K, Area of cross section A. Rod 2: Length 2L, Thermal conductivity K, Area of cross section A.
Four possible arrangements of these rods in steady state are shown in List – I and List – II gives the temperature of the junction. Match the List – I to List – II.
List – I List - II (P) L L L 100C 10C 10C (1) 20C (Q) 2L L L 100C 10C 10C (2) 27.5C (R) L L L L 10C 10C 10C 80C (3) 28C (S) L L L 2L 10C 10C 10C 80C (4) 40C Codes: P Q R S (A) 4 3 3 1 (B) 3 3 2 4 (C) 4 4 2 1 (D) 4 3 2 1
19. From an insulating solid hemisphere (radius R) carrying a charge Q, a part of it is removed in the possibilities shown in List-I. List-II gives the magnitude electric field at the centre of the original solid hemisphere for the four combinations shown. Match List I with List II and select the correct answer using the code given below the lists.
0 1 given K 4 List – I List - II (P) O R (1) 0 (Q) O R (2) 2 3 KQ 4 R (R) O R R/2 (3) 2 11 KQ 18 R (S) O R R/2 (4) 2 1 KQ 2 R Codes: P Q R S (A) 4 2 3 1 (B) 4 3 2 1 (C) 4 3 1 2 (D) 4 1 2 3
20. In the circuit shown, initially switch S1 is closed while S2 and S3 are open. After a long time, S1 is opened while S2 is closed. Further, after a long time, S2 is opened while S3 is closed. Consider the following parameters
a: Total heat generated in 2R0 b : Total heat generated in 4R0 c : Final energy of C0 d : Final energy of 2C0 e : Final energy of 4C0 R0 2R0 4R0 4C0 2C0 C0 V S1 S2 S3 List – I List - II (P) a/b (1) 2/9 (Q) b/c (2) 1/2 (R) c/d (3) 5 (S) d/e (4) 1 Codes: P Q R S (A) 4 2 3 1 (B) 3 2 1 4 (C) 1 4 2 3 (D) 3 4 2 1
C
C
h
h
e
e
m
m
i
i
s
s
t
t
r
r
y
y
PART – IISECTION – A
This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE option is correct.
1. In the following sequence of reaction
2 o ZnBr BuLi PhCOCl THF THF 100 C Major product A B C NC Br
the major product (C) is: (A) NC C O Ph (B) Br C Ph O (C) C C Ph O O Ph (D) NC CN
2. In the following sequence of reaction
2 H /Pd Major product A B NO2 OH O O O
the major product (B) is: (A) NH2 O C O CH3 (B) N H OH C O CH3 (C) NH2 OH C CH3 O (D) NH2 OH C CH3 O
3. In the following sequence of reaction
3 FeCl Major product A C CH3 CH3 CH2 C H3 Clthe major product (A) is: (A) CH2 C CH3 CH3 CH3 (B) C CH3 CH3 CH2 CH3 (C) CH CH3 CH CH3 CH3 (D) CH2 CH2 HC CH3 CH3
4. In which of the following reaction the major product formed is incorrect? (A) H C 3 CH2 CH F CH3
2 5 2 5 C H O 3 2 2 C H OH Major product CH CH CH CH (B) C H3 C C CH3 Na/ Liq.NH3 C C C H3 H CH3 H
Major product
(C) C C H C H3 CH3 H 2 4 Br / CCl
Major product
H Br H Br CH3 CH3 (D) 2 2 2 3 1 CrO Cl / Cs 2 H O
Major product
CH3 COOH5. In the following reaction
CN Major product A N Me Me C O H C O H the major product (A) is:(A) N Me Me C O C O (B) N Me Me CH OH C O (C) N Me Me CH CH (D) N Me Me C O CH OH
6. In the following reaction
N+ C Ph O CH3 OH Major productA
the major product (A) is: (A) N C Ph O CH3 (B) N C Ph O CH3 (C) O C Ph (D) NH C Ph O
7. In the following sequence of reactions OMe 3 3 CH OH/H H O mCPBA Major product A B C
the major product (C) is:
(A) OMe OH (B) OH (C) O O (D) O OH 8. O H C H3 H
2 SOCl Major p r oduct A Which of the following statement is correct regarding this reaction? (A) The reaction takes place by SN1 mechanism.
(B) Retention of configuration takes place.
(C) In presence of pyridine retention of configuration takes place. (D) The major product (A) of the reaction is
9. In the following sequence of reaction O CH3 O 2 2 3 i I KOH Ph P CH ii H A B Major productC
the major product (C) is (A) CH3 (B) CH2 (C) O CH2 (D) CH3 CH3 O
10. The major product (P) in the following sequence of reaction is:
Ph O i LDA, THF ii OH , Major product A P H O (A) Ph O (B) Ph O (C) Ph O (D) Ph O CH3
Comprehension Type
This section contains 3 groups of questions. Each group has 2 multiple choice question based on a paragraph. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which only ONE is correct
Paragraph for Question Nos. 11 to 12
Addition of water to carbonyl compounds is a nucleophilic addition reaction. The reaction is reversible in nature. Most of the carbonyl compounds form unstable hydrates. However, certain carbonyl compounds form stable hydrates. The stability of hydrates depends upon certain factors like steric effect, intramolecular hydrogen bonding, ring stain, inductive effect, mesomeric effect etc.
11. Which of the following compound form stable hydrate in aqueous solution?
(A) O (B) O (C) CH3 C H3 O (D) C O H
12. Which of the following compounds does not form stable hydrate?
(A) Cl3C – CHO (B) CF3 C O CF3 (C) Ph C O C O C O Ph (D) Ph C O Ph
Paragraph for Question Nos. 13 to 14
Nitrous acid is a weak acid and unstable. It is always prepared in situ, usually by treating NaNO2 with an aqueous solution of strong acid.
2
HCl aq NaNO aq HONO aq NaCl aq
2 4 2 2 4
H SO aq 2NaNO aq 2HONO aq Na SO aq
Nitrous acid reacts with amines. The product that is obtained from these reaction depends upon the nature of amines.
13. Which of the following compounds reacts with nitrous acid to form N-Nitrosoamine?
(A) NH2 (B) HN CH3 (C) NH2 (D) N CH3 C H3
14. The major product (P) formed in the following reaction is:
2 NaNO ,HCl Major productP N CH3 C H3 (A) N CH3 C H3 N O (B) N CH3 C H3 N O (C) N+ CH3 CH3 N O (D) N CH3 C H3 N2+Cl
Paragraph for Question Nos. 15 to 16
Elimination reactions are the most important means for synthesising alkenes. In Saytzeff’s elimination the most substituted alkene is formed as major product where as in Hoffmann’s elimination the least substituted alkene is the major product.
15.
2 5 2 5 C H ONa C H OH, Major productA CH3 Cl C H3 CH3the major product (A) is:
(A) CH3 (B) CH3
(C) CH3 (D) CH3
16. In which of the following reaction 2-Butene is the major product? (A) H3C CH2 CH Br CH3 KOH alc. Major pr oducts (B) H3C CH2 CH Br CH3
CH3 3C OH 3 3 CH C O Major p r oducts (C) H3C CH2 CH N+ CH3 CH3 C H3 CH3 OH Major pr oducts (D) H3C CH2 CH N+ CH3 C H3 C H3 O Major pr oducts (Matching List Type)
This section contains 4 multiple choice questions. Each question has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which ONLY ONE is correct.
17. List-I List-II
(Reaction) (Reagent used in the reaction)
(P) O COCH3 OH
COCH3
(1) m-CPBA
(Q) O
O
O (2) Cold. Aq. KMnO4/OH
(R) NOH NH O (3) Anhydrous AlCl3 (S) O H OH (4) PCl5 Codes: P Q R S (A) 1 2 3 4 (B) 2 1 4 3 (C) 2 4 1 3 (D) 3 1 4 2
18. List-I List-II
(Reaction) (Reagents used in the reaction)
(P) OH (1) (i) BH3 – THF
(ii) H O / OH2 2
(Q) OH (2) (i) OsO4/Pyridine
(ii) NaHSO3/H2O (R) OH OH (3) Dilute H2SO4 (S) OH (4) (i) Hg(OAc)2 – THF, H2O (ii) NaBH4/OH Codes: P Q R S (A) 1 2 3 4 (B) 4 1 2 3 (C) 2 4 1 3 (D) 3 2 4 1
19. List – I List – II
(P) Terylene (1) Hexamethylene diamine and adipic acid
(Q) Nylon – 6, 6 (2) Ethylene glycol and terephthalic acid
(R) Teflon (3) 1, 3-Butadiene and acrylonitrile
(S) Buna – N (4) Tetrafluoroethene Codes: P Q R S (A) 1 2 3 4 (B) 2 1 4 3 (C) 2 4 1 3 (D) 3 2 4 1 20. List – I List – II (Hydrolysis product)
(P) Sucrose (1) Glucose & Galactose
(Q) Maltose (2) Glucose, Galactose & Fructose
(R) Lactose (3) Glucose & Fructose
(S) Raffinose (4) Only Glucose
Codes: P Q R S (A) 1 2 3 4 (B) 2 1 4 3 (C) 3 4 1 2 (D) 3 2 4 1
M
M
a
a
t
t
h
h
e
e
m
m
a
a
t
t
i
i
c
c
s
s
PART – III SECTION – A Straight Objective TypeThis section contains 10 multiple choice questions numbered 1 to 10. Each question has 4 choices (A), (B), (C) and (D), out of which only ONE is correct
1. Let A be the point (8, 6) and D be the point (6, 4). What is the length of the shortest path ABCD, where B is point (x, 3) and C is a point (x, 0). This path consist of three connected segments, with the middle one vertical
(A) 53 (B) 53 3
(C) 50
533 (D) none of these
2. Suppose points F1 and F2 are the left and right foci of the ellipse
2 2
x y
1
16 4 , respectively and point P is on the line x 3y82 30. When F1PF2 reaches the maximum then the value of the ratio 1 2 PF PF is equal to (A) 1 2 (B) 2 1 (C) 3 1 (D) 3 1
3. Let A be the point (–2, 2) and B be the point (3, 3). Number of the integral values of m if the line x + my + m = 0 intersect the line segment AB?
(A) 0 (B) 2
(C) 3 (D) none of these
4. The graph of y = sin x, y = cos x, y = tan x and y = cosec x are drawn on the same axes from x = 0 to x
2
. A vertical line is drawn through the point where the graph of y = cos x and y = tan x cross, intersecting the other two graphs at point A and B. The length of line segment AB is
(A) 0 (B) 1
2
(C) 1 (D) 2
5. Tangent are drawn from (–4, 0) the circle x2 + y2 – 6x – 10y + 32 = 0 which touches it at point A having integral coordinate (both abscissa and ordinate are integer). Two points B and C are taken on the circle whose distance from A is 2 units, then the common chord of member of the system of circles passing through points B and C and the circle x2 + y2 – 2x + 4y – 14 = 0 passes through a fixed point, that fixed point is
(A) (2, 3) (B) (2, 4)
(C) (1, 3) (D) 4, 6)
6. A parabola touches the bisectors of the angles between the lines 7x – y + 3 = 0 and x + y – 3 = 0 at the points (3, 4) and (2, –3), then the length of latus rectum of the parabola is
(A) 16 2 (B) 16 2
5
(C) 4 2 (D) 4 2
5
7. P lies inside the triangle ABC, right angle at B and each side subtend an angle 120º at P. If PA = 10, PB = 6, then PC is equal to
(A) 11 (B) 22
(C) 33 (D) 44
8. If p, q, r are the length of perpendicular from the vertices of a ABC upon a straight line meeting the sides AB, AC and CB externally in D, E and F respectively (a, b, c are length of sides BC, AC and AB), then a2(p – q)(p – r) + b2(q – r)(q – p) + c2(r – p)(r – q) is equal to
(A) 2 (B) 22
(C) 32 (D) 42
9. The number of x-intercept on the graph of y sin 1 x in the interval (0.001, 0.01), is (A) 283 (B) 285 (C) 287 (D) cannot be determined
10. If the lines 2x + y + 5 = 0 and 2px + y + 1 = 0 are the asymptotes of a hyperbola, the possible value of p are
(A) R (B) R – {1}
(C) R – {2} (D) R – {3}
(Where R is set of real numbers)
Comprehension Type
This section contains 3 groups of questions. Each group has 2 multiple choice question based on a paragraph. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which only ONE is correct
Paragraph for Question Nos. 11 to 12 Read the following write up carefully and answer the following questions:
Let A(4, –2) and B(0, 1) are two points on the parabola, normals at which intersect at the point P(2, 2) 11. Equation of directrix of the parabola is
(A) y + x + 1 = 0 (B) x = 2
(C) y = –3 (D) 2x + y – 1= 0
12. Focus of the parabola is (A) 6, 6 5 5 (B) 16 7 , 5 5 (C) 12 6, 5 5 (D) 6 6 , 5 5
Paragraph for Question Nos. 13 to 14 Read the following write up carefully and answer the following questions:
Let ABCD be any quadrilateral and E, F, G, H are mid-points of the sides AB, BC, CD and DA respectively. The bisectors of the internal angles of quadrilateral EFGH form a quadrilateral PQRS. If coordinates of E, F and G are (1, 2), (2, 1) and (3, 8) respectively, then
13. Area of quadrilateral PQRS is (A) 61 5 sq. units (B) 62 5 sq. units (C) 63 5 sq. units (D) 64 5 sq. units 14. Equation of the circumcircle of quadrilateral PQRS is
(A) x2 + y2 – 4x – 10y + 21 = 0 (B) x2 + y2 – 4x – 8y + 12 = 0 (C) x2 + y2 – 4x – 10y + 25 = 0 (D) x2 + y2 – 4x – 8y + 16 = 0
Paragraph for Question Nos. 15 to 16 Read the following write up carefully and answer the following questions:
Let D, E, F be the mid-points of the sides BC, CA and AB respectively of a ABC and C1, C2 be the circumcircles of ABC and DEF respectively
15. If C1 and C2 touch each other then cos 2A + cos 2B + cos 2C is equal to
(A) 0 (B) 1
2
(C) 1 (D) –1
16. If C1 and C2 intersect orthogonally, then cos 2A + cos 2B + cos 2C is equal to
(A) 0 (B) 1
2
(C) 1 (D) –1
(Match List Type)
This section contains 4 multiple choice questions. Each question has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which only ONE is correct
17. Match the following List-I with List-II
List – I List – II
(P) Normals are drawn at points P and Q to the parabola y2 = 4x, which meet at a point R(9, 6) on the parabola. Tangents at P and Q to the parabola meet at T. Then the length of the tangents drawn from (–1, –1) to the circumcircle of the quadrilateral PTQR is
1. 1
(Q) The radius of smaller circle that touches the parabola 75y2 = 64(5x – 3) at the point P 6 8,
5 5
and the x-axis is
2. 2
(R) If P(, ) lies inside or on the triangle formed by the lines 2x + y = 2, x – y = 1 and x + 2y = 4, then the least value of 3 + 2 is
3. 3
(S) A tangent is drawn at any point (x1, y1) on the parabola y 2
= 4ax from any point on this tangent, tangents are drawn to the circle x2 + y2 = a2. The chord of contact pass through a fixed point (x2, y2) if 2 1 1 2 2 y x 2 y x
, > 0, then the value of is
4. 4 Codes: P Q R S (A) 4 1 3 2 (B) 1 2 3 4 (C) 4 3 2 1 (D) 4 1 2 3
18. Match the following List-I with List-II
List – I List – II
(P) If the circle x2 + y2 + 2gx + 2fy + c = 0 is touched by the line y = x at point p such that OP = 6 2 , where O is origin then the value of c is
1. 72
(Q) Number of integers in the range of
y = cos3 – 6 cos2 + 11 cos – 6 is 2. 25
(R) If tangent at point P(4t2 , 8t3) to the curve x3 = y2 is also normal to the curve at point Q, then the value of 62
t is
3. 27
(S) Let (sin x – cos x)(tan x + cot x) = 2 and
(sin x + cos x)(tan x – cot x) = a b, (a, b N), then |a2 – b2| is equal to 4. 24 Codes: P Q R S (A) 4 3 2 1 (B) 1 2 4 3 (C) 2 3 4 1 (D) 1 2 3 4
19. Match the following List-I with List-II
List – I List – II (P) If in a triangle ABC, 1 1 2 3 r r 1 1 2 r r
the ABC is 1. equilateral
(Q) If in a ABC, c a
b cos
B b a
c cos
C2 2
, then ABC is 2. right-angled
(R) If in a ABC,
2 2 2 2 sin A B a b sin A B a b , then ABC is 3. isosceles
(S) If in a ABC,
2 A 2B 2C
acos bcos c cos
3 2 2 2 a b c 4 , then ABC is 4. either right angle or isosceles Codes: P Q R S (A) 4 3 2 1 (B) 1 2 3 4 (C) 2 3 1 4 (D) 2 3 4 1
20. Consider the hyperbola 9x2 – 4y2 – 126x – 24y + 369 = 0. A variable point P(t + 7, t2 – 4), t R exists in xy-plane. Let HL and HR be the left and right branches of given hyperbola, then match the following List-I with List-II
List – I List – II
(P) The set of values of t for which two distinct real tangents can be drawn to HL from P
1.
, 2
1 1,
2,
2 2 (Q) The set of values of t for which real tangents can be drawn to both HL and HR from P
2. 2, 1 2
(R) The set of values of t for which only one real tangent can be drawn to HL only from point P
3. 1, 2 2
(S) The set of values of t for which two real and distinct tangents can be drawn to hyperbola from point P 4. R 2, 1 1, , 2 2 2 Codes: P Q R S (A) 2 1 4 3 (B) 1 2 3 4 (C) 2 1 3 4 (D) 4 3 2 1
