aits-6
TRANSCRIPT
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GENER L INSTRUCTIONS
1. In addition to this question paper, you are given a separate answer sheet.
2. Fill up all the entries carefully in the space provided on the OMR sheet ONLY IN BLOCK CAPITALS.
Incomplete/incorrect/carelessly filled information may disqualify your candidature.
3. A student has to write his/her answers in the OMR sheet by darkening the appropriate bubble with the
help of HB Pencil as the correct answer(s) of the question attempted.
4. Paper carries 80 questions each of 3 marks.
5. Any rough work should be done only on the blank space provided at the end of question paper.
6. Foreach correct answer gets 3 marks, each wrong answer gets a penalty of 1 mark.
7. Blank papers, clip boards, log tables, slide rule, calculators, mobiles or any other electronic gadgets in
any form is "NOT PERMISSIBLE".
Time : 2 Hr. Date : 12-10-2014 Max. Marks : 240
PCCP Head Office:Address :J-2, Jawahar Nagar, Main Road, Kota (Rajasthan)-324005
Contact. No. :+91-0744-2434727, 8824078330
Website :www.pccp.resonance.ac.in E-mail :[email protected]
ALL INDIA IJSO(STAGE-I) TEST SERIES
MOCK TEST PAPER # 6
mailto:[email protected]://www.pccp.resonance.ac.in/ -
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1. If ! "#2 2a b is a prime number then ! "#2 2a b =..........
(A) ! "#a b (B) ! "$a b (C) 1 (D) 0
2. )38( nn # is always divisible by
(A) 8 (B) 3 (C) 5 (D) 2
3. If ! "#4x y is a multiple of 3, then )274( 22 yxyx #$ is divisible by
(A) 3 (B) 7 (C) 6 (D) 9
4. The highest power of 3 contained in (1000 factorial) is
(A) 490 (B) 492 (C) 498 (D) 500
5. The values of ! "%n n Z for which ! "$ $2 19 92n n is a square are
(A) ! "8,11 (B) ! "#8,11 (C) ! "#8, 11 (D) ! "# #8, 11
6. The pairs of positive integers (m, n) for which $2 3m nis a perfect square is
(A) ! "1,2 (B) ! "4,2 (C) ! "3,1 (D) ! "6,3
7. The remainder when ! "5555
2222 is divided by 7 is
(A) 4 (B) 5 (C) 6 (D) 0
8. The rational roots of # # $ &3 22 3 11 6 0x x x are
(A)' () *+ ,
1,1,2
2(B)
' (#) *+ ,
1, 2,3
2(C)
' (# #) *+ ,
1, 1,2
2(D)
' (# #) *+ ,
1, 2, 3
2
9. If # $ # $ &5 3 24 25 125 625 3125 0p qp rp sp t then the roots of the equations
tsxrxqxpxx $$$$$ 2345 are
(A) A.P. (B) G.P. (C) H.P. (D) A.G.P.
IJSO TEST P PER
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IJSO STAGE-I _MOCK TEST-4_PAGE # 3
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10. The condition that the roots of # $ # &3 2 0x px qx r be in G.P. is
(A) # &2 2 0p r q (B) # &3 3 0p r q (C) # &4 2 3 0p r q (D) # &3 2 0p r q
11. Roots of the equation # $ # $ &4 3 210 26 10 1 0x x x x are
(A) (+3, +2) (B) ! "- -3 2 2,2 3 (C) ! "- -3, 2 (D) ! "- -1 2,2 3
12. Solution of the system of equations
$ $ &
$ $ &
$ $ &
2 4 4
3 9 9
4 16 16
log log log 2
log log log 2
log log log 2
x y z
y z x
z x y
(A) & & &1 34 26
; ;3 3 7
x y z (B)3
32z;
8
27y;
3
2x &&&
(C) & # & &2 27 32
; ;3 8 3
x y z (D) & & # & #2 34 32
; ;3 3 3
x y z
13. Real (x, y) that satisfy $ & $ $ $ $ &3 3 2 27 & 4x y x y x y xy are
(A) ! " ! "# #2, 1 , 1,2 (B) ! " ! "# # # #2, 1 , 1, 2 (C) ! " ! "#2,1 , 1,2 (D) ! " ! "#2,1 , 1, 2
14. If a, b, c are +ve real numbers representing the sides of a triangle then which of the following is
correct.
(A) )(3)()(2 2
cabcabcbacabcab $$.$$.$$(B) ! " ! " ! "$ $ . $ $ . $ $
23 4ab bc ca a b c ab bc ca
(C) ! " ! " ! "$ $ . $ $ . $ $2
2ab bc ca a b c ab bc ca
(D) none of these
15. In a /ABC, the incircle touches the sides BC, CA and AB respectively at D, E and F. If the radius of
the incircle is 4 units and if BD, CE and AF are consecutive integers, the sides of /ABC
(A) ! "12,13,14 (B) ! "13,15,14 (C) )11,10,9( (D) ! "12,14,15
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IJSO STAGE-I _MOCK TEST-4_PAGE # 4
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16. A circle passes through the vertex C of a rectangle ABCD and touches its sides AB and AD at M and
N. If the distance from C to the line segment MN is equal to 5 units, find the area of the rectangle
ABCD
O
A DN
M
B C
(A) 20 (B) 36 (C) 25 (D) 15
17. In an acute angled triangle there are points D, E, F are points on BC, CA, AB such that 0AD BC ;
AE = EC and CF bisects 1Cinternally. Suppose CF meets AD and DE in M and N respectively. IfFM = 2; MN = 1; NC = 3, the perimeter of /ABCis
A
F
B D C
E
M N
(A) 6 3 (B) 18 3 (C) 12 3 (D) 24 3
18. If a transversal cuts the sides BC, CA, AB of a /ABCat X, Y, Z then . .BX CY AZ
XC YA ZB equals
(A) 1 (B) 2 (C) 3 (D) -1
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19. In a /ABC, D is the mid-point of BC. If 1 & 45ADB and 1 & 30ACD ; the value of 1BADis
(A) 60 (B) 90 (C) 30 (D) 45
20. ABCD is a cyclic quadrilateral with AC 0 DB and AC meets BD at E. Then $ $ $2 2 2 2EA EB EC ED isC
B
A
D
O
E
(A) R2 (B) 3R2 (C) 4R2 (D) 2R2
Where R is the radius of circumscribed circle]
21. A boy leaves his house at 9:30 a.m. for his school. The school is 2 km away and classes start at 10:00 a.m.
If he walks at a speed of 3 km/h for the first kilometre, at what speed should he walk the second kilometre to
reach just in time ?
(A) 6 km/h (B) 8 km/h (C) 9 km/h (D) 12 km/h
22. A cylinder of mass 10 g weighs 7 g in water. If its area of cross-section is 0.75 cm2, its length will be
(A) 4 cm (B)3
40cm
(C)3
28(D) known only in terms of its density
23. A bullet in motion hits and gets embedded in a solid resting on a frictionless table. What is conserved :
(A) Momentum and K.E. (B) Momentum only(C) K.E. alone (D) None of these
24. The statement for a given pair of forces (in figure) can be written as :
W= mg
Rn
(A) It is not an action-reaction pair (B) It is an action-reaction pair
(C) Depends on situation (D) None of these
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IJSO STAGE-I _MOCK TEST-4_PAGE # 6
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25. An object of mass m is dropped from a height h, it penetrates in sand upto a depth x. Find the average
resistance (force) exerted by the sand :
(A) mgx
h(B) mg (1 +
x
h) (C) mgh (D) none of these
26. What is the relation between FAand F
B. F
Aand F
Bare the magnitudes of magnetic field at A and B?
(A) FA= F
B(B) F
A> F
B(C) F
A< F
B(D) Nothing can be said
27. A fish looking up through the water sees the outside world contained in a circular horizon. If the refractive
index of water is 4/3 and the fish is 12 cm below the surface, the radius of the circle is :
(A) 12 3 5 cm (B) 12 3 7 cm (C) 12 2/5 cm (D) 12 7
3cm
28. A step down transformer reduces 220V to 11V. The primary coil draws 5 A current and secondary coil
supplies 90A. Efficiency of the transformer will be :
(A) 4.4% (B) 20% (C) 33% (D) 90%
29. Which of the following correctly depicts the graphical variation in case of a spherical mirror ?
(A)
v-1
u-1
y
xO
(B)
v-1
u-1
y
xO
(C)
u
-1
v-1
y
xO
(D)
u
-1
y
xO
v-1
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30. A body is moving on a circular path if its velocity (linear velocity) is incresed by 10% and angular velocity
is increased by 20%, then what will be the percent change in centripetal accleration ?
(A) 10% (B) 20% (C) 30% (D) 32%
31. The result of mixing equal masses of ice at 10C and water at 60C (the specific heat of ice = 0.5 cal g1c1) :
(A) temperature 0C,16
11of total mass of ice melts
(B) temperature 0C,11
16of total mass of Ice melts
(C) temperature 10C,16
11of total mass of ice melts
(D) data given are not sufficient
32. Water falls from a height of 100 m. Find the change in temperature of water at the bottom :
(A) 2.4C (B) 0.24C (C) 24C (D) 0C
33. A wire 88 cm long bent into a circular loop is placed perpendicular to the magnetic field of flux density2.5 Wb m2. Within 0.5 s, the loop is changed into a square and flux density is increased to 3.0 Wb m2.
The value of e.m.f. induced is :
(A) 0.018V (B) 0.016V (C) 0.020V (D) 0.012V
34. A pebble is dropped into a well of depth h. The splash is heard after time t. If cbe the velocity of sound,
then :
(A) t=h2
gc(B) t = c + gh (C) t = c v (D) t=
g
h2+
c
h
35. An infinite wire bent in the form of L carries current 2. What is the magnetic field at the point O ?
(A) zero (B)d4
0 234
5(C)
d240 234
5(D)
d
2
40 234
5
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IJSO STAGE-I _MOCK TEST-4_PAGE # 8
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36. The focal length of a concave mirror is 20 cm. Determine where an object must be placed to form an
image magnified two times when the image is real-
(A) 30cm from the mirror (B) 10cm from the mirror(C) 20cm from the mirror (D) 15cm from the mirror
37. An oscillator crosses the mean possition at t = 0. Its displacement at the end of 3s is twice that after 1s.
The period of the oscillation is :
(A) 4 second (B) 12 second (C) 4
12
second (D) 4
6
second
38. Light incident on a rotating mirror M is reflected to a fixed mirror N placed 22.5 km away from M. The fixed
mirror reflects it back to M (along the same path) which in turn reflects the light again along a direction
that makes an angle of 27 with the incident direction. The speed of rotation of the mirror is:
(A) 250 revolutions s1 (B) 500 revolutions s1
(C) 1000 revolutions s1 (D) 125 revolutions s1
39. A real inverted image in a concave mirror is represented by (u, v, f are coordinates)
(A) (B) (C) (D)
40. According to the quantum theory, a photon of electromagnetic radiation of frequency v has energy
E = hv where h is known as plancks constant. According to the theory of relativity, a particle of mass m
has equivalent energy E = mc2, where c is speed of light. Thus a photon can be treated as a particle
having effective mass m = 2C
hv. If a flash of light is sent horizonatally in earths gravitational field, then
photons while traveling a horizontal distance d would fall through a distance given by -
(A) 2
2
c2gd (B)
mch (C)
hmcd
2
(D) zero
41. Which set of quantum number is not consistent with the theory? :
(A) n = 2, != 1, m = 0, s = 1/2 (B) n = 4, != 3, m = 2 s = 1/2
(C) n = 3, != 2, m = 3, s = + 1/2 (D) n = 4, != 3, m = 3, s= + 1/2
42. Match column (A) with column (B) and select the correct option.
Column (A) Column (B)
(a) Resistant to corrosion (i) Galena
(b) Pig iron (ii) Galvanised iron
(c) Fe2O
3(iii) Impure iron
(d) PbS (iv) Haematite
(A) a(iii), b(ii),c(iv), d(i) (B) a(ii), b(iii), c(iv), d(i)(C) a(iv), b(i), c(iii), d(ii) (D) a(i), b(iv), c(ii), d(iii)
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IJSO STAGE-I _MOCK TEST-4_PAGE # 9
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43. Elements upto atomic number 103 have been synthesized and studied. if a newly discovered element is
found to have an atomic number 106. Its electronc configuration will be :
(A) [Rn] 5 f14, 6d4, 7s2 (B) [Rn] 5 f14, 6d1, 7s2,,7p3
(C) [Rn] 5 f
14, 6d6, 7s0 (D) [Rn] 5 f
14, 6d5, 7s1
44. The quantum levels upto n = 3 has :
(A) s- and p-levels (B) s, p, d, f-levels
(C) s, p, dlevels (D) slevels
45. Aluminium is not extracted by carbon reduction process. This is due to the following reason :
(A) At the temperature of the furnace, Al is oxidised by CO2
(B) Melting point at Al is very high
(C) Melting point at Al is very low
(D) Al reacts with carbon
46. Cu2S + 2Cu
2O 787 6Cu + SO
2
In which process of metallurgy of copper, above equation is involved ?
(A) Roasting (B) Reduction (C) Bessemerisation (D) Purification
47. The electron identified by quantum numbers n and !, (i) n = 4, !=1 (ii) n = 4, != 0 (iii) n = 3, != 2 (iv) n
= 3, != 1 can be placed in order of increasing energy from the lowest to highest :
(A) (iv) < (ii) < (iii) < (i) (B) (ii) < (iv) < (i) < (iii)
(C) (i) < (iii) < (ii) < (iv) (D) (iii) < (i) < (iv) < (ii)
48. In the reaction, H2O
2+ NaCO
3 787 Na
2O
2+ O
2+ H
2O, the substance underrgoing oxidation is
(A) H2O
2(B) Na
2CO
3(C) Na
2O
2(D) None of these
49. In the above question, the velocity acquired by the electron will be :
(A) ! "m/V (B) ! "m/eV (C) ! "m/eV2 (D) None of these.
50. Iodine has highest oxidation number in the compound :
(A) KIO4
(B) IF5
(C) KI2
(D) KI
51. Which is a redox reaction?
(A) 2CuI2 787 2CuI + I
2(B) NaCI + AgNO
3 787 AgCI + NaNO
3
(C) NH4CI + NaOH 787 NH
3+ NaCI + H
2O (D) Cr
2(SO
4)
3+ 6KOH 787 2Cr (OH)
3+ 3K
2SO
4
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IJSO STAGE-I _MOCK TEST-4_PAGE # 10
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52. 171 g of can sugar (mol.wt. = 342) are dissovled in 1000 g of water at 30C. If the density of solution is 1.1
g/mL, then :
(A) Molarity < Molality (B) Molarity = Molality (C) Molality < Molarity (D) None of these
53. The empirical formula of a compound is CH2O. If its VD is 30, its molecular formula is :
(A) CH2O (B) C
2H
4O
2(C) C
3H
6O
3(D) CH
3OH
54. An aqueous solution of urea containing 18 g urea in 1500 cm3of solution has a density of 1.052 g/cm3. If
the molecular weight of urea is 60, then the molality of solution is :
(A) 0.2 (B) 0.192 (C) 0.064 (D) 1.2
55. Equal volumes of 0.1 M AgNO3and 0.2 M NaCl are mixed. The concentration of NO
3ions in the mixture
will be :
(A) 0.1 M (B) 0.05 M (C) 0.2 M (D) 0.15 M
56. The species having octahedral shape is :
(A) SF6 (B)#3BF (C) PCI5 (D)
#33BO
57. How many 4- bond are there in the nitrogen molecule :
(A) One (B) Three (C) Two (D) None of these
58. The type of bonds present in CuSO4.5H
2O are.......only . :
(A) Electrovalent and covalent (B) Electrovalent and co-ordinate
(C) Electrovalent , covlent and co-ordinate (D) Covalent and co-ordinate
59. In nuclear fission and fusion:
(A) Fission refers to the absorption of a neutron by the nucleus and the emission of an alpha particle
(B) Fusion reactions be illustrated by the reaction, He2LiH42
73
11 8$ + energy
(C) The critical mass for nuclear fission depends on the temperature and pressure of the system
(D) Fusion occurs at lower temperatures than fission
60. The temperature of an ideal gas is increased from 140 K to 650 K. If at 140 K the root mean square
velocity of the gas molecules is V, at 560 K it becomes :
(A) 5V (B) 2V (C) V/2 (D) V/4
61. The organelles that contain their own genetic material are:
(A) Mitochondria, Vacuoles (B) Plastids, Golgi complex
(C) Mitochondria, Plastids (D) Ribosomes, Nucleolus
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IJSO STAGE-I _MOCK TEST-4_PAGE # 11
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62. The enclosures in which heat is trapped to maintain higher temperature, are called
(A) white houses. (B) warm houses. (C) blue houses. (D) green houses
63. The stage in which separation of sister chromatids occurs is
(A) anaphase. (B) telophase. (C) metaphase. (D) prophase.
64. Haversian canals occur in
(A) humerus. (B) pubis (C) scapula. (D) clavicle.
65. Bone marrow is absent in the bones of
(A) fish. (B) bird. (C) reptile (D) frog.
66. A fern commonly used as biofertiliser is
(A) Lycopodium. (B) Marsilea. (C) Azolla. (D) Adiantum.
67. Camel is best adapted to desert habitat as
(A) it can drink 50 litres of water at a time which is evenly distributed in all its tissues.
(B) it excretes very small amount of water during urination
(C) it can regulate its body temperature at a wider range
(D) all are correct
68. Aerenchyma is found in which plants ?
(A) sciophytes (B) hydrophytes (C) mesophytes (D) epiphytes
69. In the lunch, you ate boiled green vegetables, a piece of cooked meat, one boiled egg and a sugar candy.
Which one of these foods may have been digested first ?
(A) Boiled green vegetables (B) The piece of cooked meat
(C) Boiled egg (D) Sugar candy
70. In the given food chain 'Plants Sheep Man', 5 J of energy is available to man. The energy that
was available at producer level is
(A) 50 J. (B) 500 J. (C) 5 J. (D) 0.5 J.
71. Producers prepare their own food like
(A) blue green algae. (B) amoeba. (C) rhizopus. (D) yeast.
72. Which one does not produce any digestive enzyme ?
(A) Pancreas (B) Liver (C) Stomach (D) Duodenum
73. Tunica albuginea is the covering of
(A) lungs (B) testis (C) kidneys (D) heart
74. An organism with two identical alleles of a gene in a cell is called
(A) homozygous (B) dominant (C) heterozygous (D) hybrid
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IJSO STAGE-I _MOCK TEST-4_PAGE # 12
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75. Mendels law do not explain principle which is :
(A) Segregation of genes (B) Dominance
(C) Linkage (D) Independent assortment
76. Role of isolation in evolution is
(A) magnification (B) maintenance of species
(C) evolutionary divergence (D) extermination of species
77. Soil is composed of
(A) mineral + water + air
(B) mineral + organic matter + air
(C) mineral + organic matter + air + water
(D) organic matter + water
78. The link between krebs cycle and glycolysis is
(A) citric acid (B) acetyl - co - A (C) Succinic acid (D) Fumaric acid
79. Compensation point refers to the intensity of light at which
(A) Rate of respiration = Rate of photosynthesis
(B) Rate of respiration > Rate of photosynthesis
(C) Rate of respiration < Rate of photosynthesis (D) None of the above.
80. Prevention of a disease is more desirable than its cure because
(A) some of the body functions may be damaged during the effect of the disease.
(B) the person suffering from the disease will not be bedridden.
(C) the disease can not be communicated to others during the course of treatment.
(D) body does not look good during this condition.
IJSO Questionmaths : 1 to 20Phy : 21 to 40Che : 41 to 60Bio: 61 to 80
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SOL. IJSO STAGE-I _MOCK TEST-4_PAGE # 1
NSWER KEY
HINTS & SOLUTIONS
1. a2b2= (a b) (a + b) ..(i)
!a2b2 a prime equation= 1
!(a b) = 1 [!(a b) < (a + b)]!The only adivisor of a prime equation are 1 and itself!Equation (i) becomes (a2b2) = 1(a + b)or a2b2 = (a + b)eg. 3222= 5 (which coprime)32
22= 3 + 2, 3, 2 "N
2. 8n3n= (83) (8n1+ 8n23 + 3n1)
!(893n) is divisible by 5If n is either even or oddxnyn= (x y) (xn1+ xn2+ ... + yn1)If n is oddxn
+ yn= (x + y) (xn1xn2+ xn3y2....+ yn1)
3. !(4x y) is a multiple of 3!(4x y) = 3m!y = (4x 3m)
on putting value of y in (4x2+ 7xy
2y2)4x2+ 7xy 2y2= 4x2+ 7x(4x 3m) 2(4x 3m)2
= 4x2+28x221mx 32x 18m2+ 48mx= 27mx18m2
= 9m (3x 2m)
!4x2+ 7xy 2y2
is divisible by 9
ALL INDIA IJSO(STAGE-I) TEST SERIES
MOCK TEST PAPER # 4
DATE : 28-09-2014
Ques. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ans. B C D C D B B B A B B B A B B C C D C C
Ques. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Ans. A A B A B B D D C D A B A D A A B A A A
Ques. 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Ans. C B D C D C A D C A A A B B B A C C B B
Ques. 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
Ans. C D A A B C D B D B A B B A C C C B A A
4. !if p is a prim number and the longestexponent of p such that pe divides n! then
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SOL. IJSO STAGE-I _MOCK TEST-4_PAGE # 2
e = #$
%&
'()
*10
1ipi
n
[x] is the largest integer + xIn this case p = 3, n = 1000
%
&
'(
)
*
p
n= %
&
'(
)
*3
1000= 333
%%&
'
(()
*2p
n= %
&
'()
*23
1000= 11111
%%&
'
(()
*3p
n= %
&
'()
*33
1000= 37
%%&
'
(()
*4p
n= %
&
'()
*43
1000= 12
%%&'
(()* 5p
n = %&'(
)* 531000 = 4
%%&
'
(()
*6p
n= %
&
'()
*63
1000= 1
%%&
'
(()
*7p
n= %
&
'()
*73
1000= 0
!Highest power of 3 contained in 1000!= 333 + 111 + 37 + 12 + 4 + 1 + 0 = 498
5. n2+ 19n + 92 = m2: n is a non negative integer. Thenn2+ 19n + 92 m2= 0
solving for n =2
)7m419( 2 ,-,
!(4m27) is a square ie 4m27 = p2 when p " N!(2m p) (2m + p) = 7!(2m + p) being positive is 7 and (2mp)= 1 ; 4m = 8 .m = 2Thus we have n2+ 19n + 92 = 4n2+ 19n + 88 = 0(n + 8) (n + 11) = 0
.n = 8 or n = 116. Let 2m+ 3n= k2
since (1)m/2m/ k2/ 1 (med3) [!31k]m is , say 2pnow (k2p) (k + 2p) = 3n
.k 2p= 1 andk + 2p= 3n;
.k 2p= 3n
.2p+1+ 1 = 3n
since (1)n/ 3n(mod4)= 2p+1+ 1n is even, say 2qNow (3q1) (3q+1) = 2p+1
.3q1 = 2
.3q= 3.q = 1 and hence p = 12
so we have only one solution (4, 2)
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SOL. IJSO STAGE-I _MOCK TEST-4_PAGE # 3
7. 2222 /3 mod7(2222)3/27 mod 7and 27 / 1!(2222)3=1 mod 7.(2222)5553=1 mod 7(2222)2= 9 mod 7
.(2222)5555=9 mod 7But 9 / 5 mod 7 ; !(2222)5555/ 5 mod 7
8. Let the runs be if the form2
p(where (p q) = 1 and q > 0)
since q/2, q must be 2 or 1and p/6
.*1, - 2, - 3, - 6
f 01
234
52
1= f(2) = f(3) = 0
Hence root are2
1,2 and 3
9. Let the root bea 2d, a
d, a, a + d, a + 2d
sum of roots = 5a = p
!a =5
p,
But a is a root of the given equation hence
5
5
p01
234
5,+ p
4
5
p01
234
5,+ q
3
3
p01
234
5,+ r
2
5
p01
234
5,+ t = 0
or 4p525ap2+ 125 rp2625p + 3125 t = 0
10. Let the root of the given equation be ap, a, pa
!# 678 = 678 = ap :a : pa
= r
a3= r ; a = r1/3
But a is root of the given equation
!(r1/3) p(82/3) + qr1/3r = 0!pr2/3= qr1/3
.p2r2= q2r
.p3rq3= 0 is required condition
11.Dividing throughtout by x
2
01
234
5;
2
2
x
1x 10 0
1
234
5;
x
1x + 26 = 0
Let x +x
1= 9
!(y22) 10y + 26 = 0y2 10y + 24 = 0(y6) (y 4) = 0y = 6 OR y = 4
x +
x
1= 6 , x3 6x + 1 = 0
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SOL. IJSO STAGE-I _MOCK TEST-4_PAGE # 4
x =2
326-= 3 - 22
and x +x
1= 4 ; x2 4x + 1 = 0
x =
2
124-= 2 - 3
12. Log2x + log
4y + log
4z = 2
log3y + log
9z + log
9x
=
2
log4z + log
16y + log
16z = 2
.(log4x) log
24 + log
6y + log
4z = 2
x2yz = 16 (i)similarly the other equation riduce to(ii) y2zx = 81 andz2xy = 256 ...(iii)solvingdivide (i) by (iii)
yx =
8116 ; x =
81y16
Dividing (ii) by (iii)
z
y=
256
81; z =
81
256y
! 2
2
)81(
)16(y2. y.
81
256y = 16
y4= 3
3
)16(
)81(, y =
8
27;
x =818
2716
::
=3
2; z =
3
32
13. Let x + y = 6, xy = 7and hence x2+ y2= 6227x2+ y3= (x + y) (x2xy + y2)
6(6237) = 7, 633a7= 7and x2+ y2+ x + y + xy = 4
62 27+ 6+ 7 = 4 = 627+ 6= 47 = 62 + 64!623636+ 12 6= 7262362126+ 7 = 0
f(1) = 262
+ 362
126+ 7 = 0hence (61) (factor of f(6))f(6) = (61)(2a2+ 5a7)f(6) = (61) (2a + 7a 2a7)f(6) = (61) {a(2d + 7) 1 (2a + 7)}f(6) = (61) (a 1) (2a + 7)6= 1 OR 6= 7/2when 6= 1, 7=2. thenx + y = 1 and xy = 2Hence xy are roots oft2 t 2 = 0 ; t = 2 or t = 1x = 2; x = 1 .y = 1, y = 2
If we take d =27 and f =
419 then
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SOL. IJSO STAGE-I _MOCK TEST-4_PAGE # 5
x,y are root of 4t + 14t + 19 = 0whose discriminent < 0. Hence no real roots.hence solution are (2,1) and (1, 2)
14.2
ba 22 ; > ab ,
2
cb 22 ;> bc
and2
ac 22 ;> ca [A.M. > G.M.]
a2+ b2> 2ab ; b2+ c2> 2bc ; c2+ a2> 2ca ;
.a2+ b2+ b2+ c2+ c2+ a2> 2 (ab + bc + ca)
.a2+ b2+ c2> ab + bc + caab + bc + ca < a2+ b2+ c2 ..(i)In
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SOL. IJSO STAGE-I _MOCK TEST-4_PAGE # 6
x2+ 8x 6x 48 = 0x(x + 8) 6(x + 8) = 0(x + 8) (x 6) = 0x = 6; hence a = 13, b = 15, c = 14
16. Let O be the centre of the circle and P be the root of the perpendicular from C to MN.Then OM is perpendicular to AB , ON is perpendicular to AD. Hence OM = ON = radius of the circle.so AMON is a square
=MNC =2
1=MON = 45
=CMP + =CNP = 135 = =CMP + =CMB= =CNP + =CNDHence=CNP ==CMB and =CMP ==CNDHence in
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SOL. IJSO STAGE-I _MOCK TEST-4_PAGE # 7
18.
A
Qz
B C x
YR
P
h2
h1
h3
Let h1, h
2, h
3be the length of 1sAP, BQ, CR respectively from A, B, C on the transersal
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SOL IJSOSTAGE I MOCKTEST 4 PAGE# 9
52.
d1000
)1711000(342
171m
:;
:$ = 0.429 1.4 = 0.47
m =1342
171
:= 0.5
! M = 0.1853. Mol. wt. of (CH2O)n= 30 2 = 60
! n = 230
60$
Empirical formula wt. (CH2O) = 30
54. m =)18052.11500(60
100018
,:::
= 0.19
55. mM of AgNO3= 0.1 V
mM of NaCl = 0.2 V! mM of NO3
= 0.1 V and total V = 2V
! [NO3] =
V2
V1.0 := 0.05
56. S atom in SF6in sp3d2-hybridized state and shows octahedral shape
57. N /N or N2has 1Aand 2B-bonds.
59. Fusion involves joining of two lighter nuclei.
60. urms
=M
RT3=
M
140R3 :at 140 K
uCrms
=M
560R3 :at 560 K
uCrms
= 2 urms
.