ELITE INTERNATIONAL SCHOOL(EIS)
A CO-EDUCATIONAL SR. SECONDARY ENGLISH MEDIUM PUBLIC SCHOOL
(Affiliated to C.B.S.E)
TH
GWALISION, 6 KM STONE, JHAJJAR-DADRI ROAD, JHAJJAR
Email: [email protected] / Website: eisjhajjar.com
Summer Holidays Home Work 2016-17
Name___________________
Roll No._________________ All Subjects
Class-XI SCIENCE
Hello Kids!
‘Exciting time is here again! It’s time for Summer Vacation and fun filled activities’.
Children you are the reservoirs of potential which needs to be tapped and
channelized in diverse ways. Summer Vacation is the best and fruitful time for
learning and for nurturing creativity. It is the time when you can do many things in
your own way. Complete your Holidays’ Homework, be independent and try to
improve your basic academic skills, such as reading, writing and speaking,
develop some personal skills and learn time management too. A few suggestions
that you may keep in mind during vacation:
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Spend quality time with your family.
Go for outings and have fun time with your family.
Get yourself involved in small household activities.
Inculcate good manners, healthy habits and respect for elders.
Inculcate the feelings of empathy, affection and tolerance.
Look after your younger brothers and sisters and support your parents.
Converse with your friends and parents in English, if possible.
Read books to enhance your language skills.
Play various indoor and outdoor games.
Remember that Summer Vacation is the time to relax and enjoy. So spend these
holidays filled with fun, frolic, learning and education.
ENGLISH
Dear students, Vacations are going to start. Enjoy the vacations along with a little bit of work, it shall be a fun
and shall help you to learn more through practice as practice makes man prefect.
1. We need to introspect ourselves and think about the values which we have and what we practice in our lives. ‘Speaking
Tree’ tells us many important thoughts and ways of life. Cut 5 “Speaking Tree” columns from the news paper and
paste them in the scrap book. Then make notes for these passages (without writing the summary).
2. Thoughts are the feelings and ideas of great men who have shown us the path of life. These thoughts make us aware
that success lies in following them. Write 4 beautiful thoughts from the news paper on half chart paper each.
3. Mercy is a gentle drop which has been dropped from heaven by God. But alas! every man has not tried to receive it
into his life or use it when needed. Write a poem on a chart on the topic ‘Mercy’ and decorate it beautifully.
4. Books are the best companions in a man’s life. They give all the knowledge a man needs. Reading is the best pastime
and a fruitful one. Read novel of Premchand ‘NIRMALA’, learn about the living style and condition of the people in
that era.
5. Write a speech on the given topics on A4 size sheets in a decorated file
i) Man is the Slave of Technology ii) Young leaders of today
7. Prepare for group discussion on RELIGION CANNOT BE A PART OF POLITICS.
BIOLOGY
1. Complete the following work (spots) in the practical file.
1. Bacteria
2. Nostoc
3. Euglena, Paramoecium
4. Rhyzopus, Agaricus
5. Bacteriophage
6. Green alga
chara, Nostoc
7. Brown alga
Laminaria, Fucus
8. Red alga
Porphyra, Polysiphonia
9. Bryophyte
Marchantia, Funaria
10. Pteridophyte
Selaginella, Equisetum
11. Gymnosperm
Ginkgo, Pinus
12. Angiosperm
Dicot, Monocot, lifecycle of angiosperm
13. Phylum porifera
Spongilla, Sycon
14. Coelenterata
Hydra, Aurelia
15. Ctenophora
Pleurobrachia
16. Platyhelminthes
Tapeworm, Liver fluke
17. Aschelminthes
Ascaris- male & female
18. Annelida
Leech & Nereis
19. Arthropoda
any two
20. Mollusca
Octopus, Pila
21. Echinodermata
Starfish
22. Hemichordata
Balaenoglossus
23. Shark, Electric ray
24. Hyppocampus, Catla
25. Frog, Salamander
26. Crocodile, Turtle
27. Parrot, Peacock
28. Any two mammals
2. Work out the MCQs of all the four chapters.
CHEMISTRY
1. Write and learn all multiple choice questions by making word documents for exams.
2. Prepare a project report on any one of the following topics given below with PowerPoint slides and your
presentation must comprise text, graphics, animation and sound. The time limit is three minutes and there must be
at least ten slides.
a.
Air conditioning coolants and environment
b.
Methanol history and it’s use as a fuel.
c.
Lactic acid build up in athletes and its chemistry
3. Learn and write electronic configuration of 100 elements.
4. Revise and learn unit 1,2 and 3 for test.
5. Make a chart as per the instructions given below: Roll no 1 to 10
from unit 3
Roll no 11 to 20 from unit 2
Roll no 21 to 30 from unit 1
6. Solve all the following questions from UNIT 1 & 2:
80
7. Calculate the number of protons, neutrons and electrons in 35 Br .
8. The number of electrons, protons and neutrons in a species are equal to 18, 16 and 16 respectively. Assign the
proper symbol to the species.
9. How many neutrons and protons are there in the following nuclei?
13
16
24
56
88
6C , 8O ,
12Mg , 26Fe
38Sr
10. Write the complete symbol for the atom with the atomic number (Z) and atomic mass (A):
(i) Z = 17 , A = 35. (ii) Z = 92 , A = 233.
(iii) Z = 4 , A = 9.
11. Which of the following are iso-electronic i.e. having the same number of electrons.?
Na+,
K +,
Mg2+,
Ca2+, S2–, Ar.
+
+
Give the number of electrons in the species H2 , H2 and O2 .
Calculate the number of electrons which will together weigh one gram.
Calculate the mass and charge of one mole of electrons.
Compare Electron, Protons and Neutrons.
Describe Thomson’s Model Of Atom.
Explain Rutherford‘s scattering experiment.. What conclusions regarding the structure of atom were drawn by
Rutherford on the basis of the observations of his experiment? Give the major drawbacks of Rutherford’s
Model of atom.
18. In Rutherford‘s experiment, generally the thin foil of heavy atoms like gold, platinum etc. have been used to be
bombarded by the α-particles. If the thin foil of light atoms like aluminium etc. is used, what difference would be
observed from the above results?
19. Define the terms Atomic Number, Mass Number, Isobars, Isotopes,
12.
13.
14.
15.
16.
17.
20. Define the terms frequency wavelength & wave number (Write mathematical forms also). The Vividh Bharati
station of All India Radio, Delhi, broadcasts on a frequency of 1,368 kHz (kilohertz). Calculate the wavelength of
the electromagnetic radiation emitted by transmitter. Which part of the electromagnetic spectrum does it belong to?
21. The wavelength range of the visible spectrum extends from violet (400 nm) to red (750 nm). Express these
wavelengths in frequencies (Hz). (1nm = 10 –9 m)
22. Calculate (a) wave number and (b) frequency of yellow radiation having wavelength 5800Å.
23. Yellow light emitted from a sodium lamp has a wavelength of 580 nm. Calculate the frequency and wave number
of the yellow light.
24. Find energy of each of the photons which
(i)
correspond to light of frequency 3× 1015Hz. (ii) have wave length of 0.50 Å.
25. Calculate the wavelength, frequency and wave number of a light wave whose:
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
(i)
time period is 2.0 × 10–10s.
Electrons are emitted with zero velocity from a metal surface when it is exposed to radiation of (0 ) and work
function (W0 ) of the metal.
Explain Photoelectric effect, Emission and Absorption Spectra.
Give the postulates of Bohr‘s model of atom. Also write its Limitations.
Write a note on the Spectral Lines for Atomic Hydrogen.
What are the frequency and wavelength of a photon emitted during a transition from n = 5 state to the n = 2 state
in the hydrogen atom?
Calculate the energy associated with the first orbit of He + .What is the radius of this orbit?
Explain Dual behaviour of matter.
State de Broglie‘s relation. Give its mathematical expression.
What will be the wavelength of a ball of mass 0.1 kg moving with a velocity of 10 m s –1?
The mass of an electron is 9.1 X 10 kg. If its K.E. is 3.0 X10 –25 J, calculate its wavelength.
Calculate the mass of a photon with wavelength 3.6 Å
The velocity associated with a proton moving in a potential difference of 1000 V is 4.37 × 10 5ms–1.
If the
hockey ball of mass 0.1 kg is moving with this velocity,Calculate the wavelength associated with this velocity.
If the velocity of the electron in Bohr‘s first orbit is 2.19 × 106ms–1, calculate the De Broglie
wavelength associated with it.
39. Similar to electron diffraction, neutron diffraction microscope is also used for the determination of the structure
of molecules. If the wavelength used here is 800 pm, calculate the characteristic velocity associated with the
neutron.
40. Dual behaviour of matter proposed by de Broglie led to the discovery of electron microscope often used for the
highly magnified images of biological molecules and other type of material. If the velocity of the electron in this
microscope is 1.6 × 106ms–1 , calculate de Broglie wavelength associated with this electron.
41. Calculate the wavelength of an electron moving with a velocity of 2.05 X 107ms–1
42. The mass of an electron is 9.1 X 10–31kg. If its K.E. is 3.0 X 10–25 J, calculate its wavelength.
43. Why de Broglie‘s relation is not associated with ordinary objects.
44. State Heisenberg‘s Uncertainty Principle. Give its mathematical expression.
45. A microscope using suitable photons is employed to locate an electron in an atom within a distance
of 0.1 Å.
What is the uncertainty involved in the measurement of its velocity?
46. If the position of the electron is measured within an accuracy of + 0.002 nm, calculate the
uncertainty
in
the momentum of the electron. Suppose the momentum of the electron is h/4Π × 0.05nm, is there any problem in
defining this value?
47. Show that the circumference of the Bohr orbit for the hydrogen atom is an integral multiple of the de Broglie
wavelength associated with the electron revolving around the orbit..
4. QUESTIONS BASED ON QUANTUM NUMBERS, AUFBAU RULE, PAULI RULE,
HUND’S RULE, ELECTRONIC CONFIGURATION OF ATOMS & IONS:
1.
What information is provided by the four quantum numbers?
2.
Using s, p, d, f notations, describe the orbital with the following quantum numbers
(a) n = 2, l = 1,
(b) n = 4, l = 0,
(c) n = 5, l = 3,
(d) n = 3, l = 2
(e) n=1, l=0
(f) n = 3 l=1
(g) n = 4; l =2
(h) n= 4; l=3.
3.
What is the total number of orbitals associated with the principal quantum number n = 3?
4.
What is the lowest value of ‘n’ that allows g orbitals to exist?
5.
An electron is in one of the 3d orbitals. Give the possible values of n, l and m for this electron.
6.
(i) An atomic orbital has n = 3. What are the possible values of l and m?
(ii) List the quantum numbers (m and l ) of electrons for 3d orbital.
(iii) Which of the following orbitals are possible?
1p, 2s, 2p, 2d, 4f , 6d and 3f.
7.
Explain, giving reasons, which of the following sets of quantum numbers are not possible.
(a) n = 0, l= 0, ml = 0, ms = + ½
(b) n = 1, l = 0, ml = 0, m s = – ½
(c) n = 1, l = 1, ml = 0, ms = + ½
(d) n = 2, l = 1, ml = 0, ms = – ½
(e) n = 3, l = 3, ml = –3, ms = + ½
(f) n = 3, l = 1, ml = 0, ms = + ½
8.
(i)How many electrons in an atom may have the following quantum numbers?
(a) n = 4, m s = – ½ (b) n = 3, l = 0
(ii) How many sub-shells are associated with n = 4?
9.
10.
11.
12.
(iii) How many electrons will be present in the sub-shells having m s value of –1/2 for n = 4?
State (n+l) rule Aufbau rule & Pauli rule.
Give the electronic configuration of first 30 elements.
Explain the exceptional configuration of copper and chromium.
Give the electronic configurations of the following ions:
Cu2+ Cr3+ Fe2+ S2Fe2+ O2- Na+
PHYSICS
1. Deduce the dimensional formulae for the following physical quantities:
(a) Gravitational constant
(b) Power
(c) Young’s Modulus
(d) Coefficient of viscosity
(e) Surface tension
(f) Planck’s constant
2.
Obtain dimensions of :
(i) Impulse
(ii) Angular acceleration
(iii) force constant (iv) angular momentum
3.
4.
5.
6.
7.
If force (F), length (L) and time (T) are chosen as the fundamental quantities, then what would be the
dimensional formula for density?
Find the dimensions of linear momentum and surface tension in terms of velocity v, density
ρ and frequency υ as fundamental quantities.
-3
The density of mercury is 13.6 g cm in CGS system. Find its value in SI units.
When 1 m, I Kg and 1 min are taken as the fundamental units, the magnitude of the force is 36 units.
What will be the value of this force in CGS system?
Find the dimensions of (a/b) in the equation:
2
8.
9.
P  a  t , where P is pressure, x is distance and t is ti
bx
The viscous force ‘F’ acting on a small sphere of radius ‘r’ moving with velocity v through a
liquid is given by F = 6 η π r v. Calculate the dimensions of η.
The rate of flow (V) of a liquid flowing through a pipe of radius r and a pressure gradient
(P / L) is given by Poiseuille’s equation:
Check the dimensional consistency of this equation.
10.
11.
12.
13.
14.
15.
16.
17.
Consider a simple pendulum, having a bob attached to a string, that oscillates under the
action of the force of gravity. Suppose that the period of oscillation of the simple
pendululum depends on (i) mass m of the bob (ii) length l of the pendulum and (iii)
acceleration due to gravity g. Derive the expression for its time period using method of
dimensions
The velocity ‘v’ of water waves depends on the wavelength ‘λ’ , density of water ‘ρ’ and
the acceleration due to gravity ‘g’. Deduce by method of dimensions, the relationship
between these quantities
The length of the rod as measured in an experiment was found to be 2.48 m, 2.49 m, 2.50
m, and 2.48 m. Find the average length, the absolute error and the percentage error in
each observation
Two resistances R1 = (100 ± 3) ohm and (200 ± 4) ohm are connected in series. What is
their equivalent resistance
The error in the measurement of a sphere is 2%. What would be the error in the volume
of the sphere?
The voltage across a lamp is (6.0 ± 0.1) volt and the current passing through it is (4.0 ±
0.2) ampere. Find the power consumed
1 𝐴𝐵2
Find the percentage error in Z, if 3 𝐶𝐷3
State the number of significant figures in the following:
(a) 2.000 m (b) 5100 Kg
(c) 0.050 cm
3
18.
The mass of a body is 275.32 g and its volume is 36.41 g . Express its density upto
appropriate significant figures.
19.
9.74 g of substance occupies 1.2 cm . Express its density by keeping the significant
figures in view.
2
The displacement (in meter) of a particle along x-axis is given by x = 18 t + 5 t .
Calculate:
(i)
The instantaneous velocity at t = 2 s,
(ii)
Average velocity between t = 2 s and t = 3 s,
(iii) Instantaneous acceleration
Derive the 3 equations of motion using calculus method
20.
21.
3
Mathematics
1) Find the maximum and minimum values of cos (cos x).
2) If R is a relation on a finite set A having n elements. Find the number of relations on A and the
number of proper subsets of A.
3) One angle of a triangle is
4)
5)
6)
7)
8)
2
3
3
x grades and another is 2 x degrees while the third is
𝜋𝑥
75
radians.
Express all the angles of the triangle.
Find the symmetric difference of A = {1, 2, 3, 4} and B = {4, 5, 6, 7}.
If A and B are two sets such that n (A ∪ B) = 50, n (A) = 28 and n(B) = 32, find n(A ∩ B).
If a ∈ {-1, 2, 3, 4, 5} and b ∈ {0, 3, 6}, write the set of all ordered pairs (a, b) such that a < b.
Prove that (A U B) X C = (A X C) U (B X C).
Write the relation R = {(x, x3) : x is a prime number less than 20} in roster form.
1 − 𝑥, 𝑥 < 0
9) The function f is defined by f(x) = { 1 , 𝑥 = 0 }. Draw the graph of f(x).
𝑥 + 1, 𝑥 > 0
10) For sets A, B and C using properties of sets, prove that:
A-(B – C) = (A - B) ∪ (A ∩ C).
11) Given that (1 + cos α) (1 + cos β) (1 + cos γ) = (1 - cos α) (1 - cos β) (1 - cos γ).
Show that one of the values of each number of this equality is sin α sin β sin γ.
12) Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} D =
{5, 10, 15, 20}, Find:
(i) A – B
(ii)
A–C
(iii)
B∩ C
(iv) C ∪ D
tan A
cot A
13) Prove that:
+ 1−tan A = (secA cosecA + 1).
1−cot A
14) Sketch the graph of y=2 tan3x and cos 3x
15) Is it true that for any set A and B, P(A) U P(B) = P(AUB)? Justify your answer.
16) The numbers of sides of two regular polygons are as 5:4 and the difference between their angles is 90.
Find the numbers of sides of the polygons.
17) If f(x) = x2-3x+4, then find the values of x satisfying the equation f(x) = f(2x+1).
18) Describe the domain and range of the relation R defined by
R={(𝑥, 𝑥 + 5): 𝑥 ∈ {0,1,2,3,4,5}}.
a2 −b2
19) If sin θ = a2 +b2 , find the values of tan θ, sec θ and cosec θ.
𝑥 2 + 2𝑥 +1
20) Find the domain and range of the real valued function f(x) = 𝑥 2 − 8𝑥 + 12
21) If f, g be two real functions defined by f(x) = √𝑥 + 1 and g(x) = √9 − 𝑥 2 . Then describe
(1) f + g
(2) f x g
(3) g-f
22) If f is a real function defined by f(x) =
x−1
.
x+1
Then prove that f(2x) =
3f(x)+1
.
f(x)+3
23) In a survey of 100 students , the number of students studies the various language were found to be
English only18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and
Hindi 8, no language 24, find
(1) How many students were studying Hindi?
(2) How many students were studying English and Hindi?
24) The angles of triangle are in AP such that greatest is 5 times the least. Find the angles in radians.
25) If sin𝜃 + 𝑐𝑜𝑠𝜃 = 𝑚, 𝑡ℎ𝑒𝑛 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 ∶ sin6θ+cos6θ=
4−3(m2 −1)2
,
4
where m2≤2
26) Find the value of expression
3𝜋
𝜋
3{sin4 ( 2 − 𝜃) + sin4 (3π + θ)} - 2{𝑠𝑖𝑛6 ( 2 + 𝜃) + 𝑠𝑖𝑛6 (5𝜋 − 𝜃)}
27) Let R={(1,2),(2,3),(3,4),(4,1),(1,4)} be a relation in the set A={1,2,3,4}. Find the domain and range of R.
28) Find the domain and range of the relation R={(x,√𝑥): -4≤ 𝑥 ≤ 4, 𝑥 ∈ 𝑍}
29) If n(S)=70, n(Ac∩ 𝐵) = 30; 𝑛(𝐴) = 30 𝑡ℎ𝑒𝑛 𝑓𝑖𝑛𝑑 (1)𝑛(𝐴 ∪ 𝐵) (𝑖𝑖) 𝑛(Ac∩Bc).
30) if n(S)=700,n(A)=200,n(B)=300 and n(A∩B)=100, where S is a universal set, then find n(𝐴𝑐 ∩ 𝐵𝑐 ).
31) Let A={2,3,4,5} ,B={3,5,7}, C={2,5,8}. Verify that
(i) A-(B∪ 𝐶) = (𝐴 − 𝐵) ∩ (𝐴 − 𝐶)
(𝑖𝑖) 𝐴 − (𝐵 ∩ 𝐶) = (𝐴 − 𝐵) ∪ (𝐴 − 𝐶).
32) Prove that : A-B=A∩Bc
33) If A={x:x2-5x+6=0}; B={2,4}, C={4,5}, find the value of A× (𝐵 ∩ 𝐶)
34) Sets A and B have 3 and 6 elements each. What can be the minimum number of elements in A∪ 𝐵.
35) Two finites sets have m and n elements. Then total number of subsets of the first set is 54 more than
that of the total number of subsets of the second. Find the values of m and n .
36) In a certain town 25% families own a phone and 15% own a car, 65% families own neither a phone nor
a car, 2000 families own both a car and a phone. How many families live in the town?
37) If X={1,2,3,4}. Prove that f defined by f= { (x ,y), x + y =5} is function from A to A. Where x and y ∈ A.
38) Decide among the following sets, which sets are subset of each another:
A={𝑥: 𝑥 ∈ 𝑅 𝑎𝑛𝑑 𝑥 𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑛𝑔 𝑥 2 − 8𝑥 + 12 = 0}, 𝐵 = {2,4,6}, 𝐶 = {2,4,6,8 … … … }, 𝐷 = {6}
39) In each of the following determine whether the statement is true or false. If it is true, prove it.
If it is false, give an example.
i. If x ∈ A and A ∈ B then x ∈ B
ii. If A⊂ B and B ∈ C then A ∈ C
iii. If A⊂ B and B ⊂ C then A ⊂ C
iv. If A⊂B and x∉ 𝐶 𝑡ℎ𝑒𝑛 𝑥 ∉ 𝐴
40) Let A, B and C be the sets such that A∪ 𝐵 = 𝐴 ∪ 𝐶 𝑎𝑛𝑑 𝐴 ∩ 𝐵 = 𝐴 ∩ 𝐶, then show that B=C.
41) Show that the following four conditions are equivalent :
(𝑖𝑖)𝐴 − 𝐵 = 𝜙
(𝑖𝑖𝑖) 𝐴 ∪ 𝐵 = 𝐵
(𝑖𝑣) 𝐴 ∩ 𝐵 = 𝐴
(i) A⊂ 𝐵
42) Show that if A⊂ 𝐵, 𝑡ℎ𝑒𝑛 𝐶 − 𝐵 ⊂ 𝐶 − 𝐴.
43) Assume that P(A) =P(B), show that A=B.
44) Is it true that foe any set A and B, P(A)∪ 𝑃(𝐵) = 𝑃(𝐴 ∪ 𝐵)? 𝑗𝑢𝑠𝑡𝑖𝑓𝑦 𝑦𝑜𝑢𝑟 𝑎𝑛𝑠𝑤𝑒𝑟.
45) Show that for any sets A and B , A=(A∩ 𝐵) ∪ (𝐴 − 𝐵)𝑎𝑛𝑑 𝐴 ∪ (𝐵 − 𝐴) = (𝐴 ∪ 𝐵)
46) Using properties of sets, show that:
(𝑖𝑖) 𝐴 ∩ (𝐴 ∪ 𝐵) = 𝐴
(i) A∪ (𝐴 ∩ 𝐵) = 𝐴
47) Show that 𝐴 ∩ 𝐵 = 𝐴 ∩ 𝐶 need not imply B=C.
48) Let A and B sets. If A ∩ 𝑋 = 𝐵 ∩ 𝑋 = ∅ and A∪ 𝑋 = 𝐵 ∪ 𝑋 for some set X. show that A=B.
49) Find sets A, B and C such that A∩ 𝐵, 𝐵 ∩ 𝐶 𝑎𝑛𝑑 𝐴 ∩ 𝐶 𝑎𝑟𝑒 𝑛𝑜𝑛 𝑒𝑚𝑝𝑡𝑦 𝑠𝑒𝑡𝑠 𝑎𝑛𝑑
𝐴 ∩ 𝐵 ∩ 𝐶 = 𝜙.
50) In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking
coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor
coffee.
51) In a group of students, 100 students know hindi, 50 know English and 25 know both. Each of the
students knows either hindi or English. How many students are there in the group?
52) In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26
read newspaper I, 9 read both H and I, 11 both H and T, 8 read both T and I, 3 read all three
newspapers. Find :
(i) the number of people who read at least one of the newspaper. (ii) the number of people who
read exactly one newspaper.
53) In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C.
if 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B
and C and 8 liked all the three products . find how many liked product c only.
54) If X and Y are two sets such that n(X)=17,n(Y)=23 and n(X∪ 𝑌) = 38, 𝑓𝑖𝑛𝑑 𝑛(𝑋 ∩ 𝑌).
55) if X and y are two sets such that X∪Y has 18, X has 8 elements and Y has 15 elements; how many
elements has X∩ 𝑌?
56) In a group of 400 people, 250 can speak hindi and 200 can speak English. How many people can
speak both hindi and English?
57) If S and T are two sets such that S has 21 elements and T has 32 elements and S∩ 𝑇 has 11
elements, how many elements does 𝑆 ∪ 𝑇 ℎ𝑎𝑣𝑒?
58) If X and Y are two sets such that X has 40 elements, X∪Y has 60 elements and X∩Y has 10
elements, how many elements does Y have?
59) In a group of 65 people, 10 like both cricket and tennis. How many like tennis only and not
cricket? How many like tennis?
60) In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French.
How many speak at least one of these two languages?
61) Let A={1,2,23 … … … . ,14}. 𝑑𝑒𝑓𝑖𝑛𝑒 𝑎 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑅 𝑓𝑟𝑜𝑚 𝐴 by R={(x,y):3x-y=0, where x,y∈A}. write
down its down, co-domain and range.
62) Define a relation R on the set N of natural numbers R={(x,y):y=x+5, x is a natural number less
than 4: x,y∈N}. depict this relationship using roaster form. Write down the domain and the range.
63) A={1,2,3,5} and B={4,6,9}. Define a relation R from A to B by R={(x,y): the difference between x
and y is odd: x∈A, y∈B}. write R in roaster form.
64) Find all the values of α for which sin4x+cos4x+sin2x+α=0 is valid. Also find the general solution of the
equation.
𝜋
65) Let A,B,C be these angles such that A= 4 and tanBtanC=P. find all possible values of P such that A,B,C
are the angles of a triangle.
4
5
66) If in triangle ABC, a=6,b=3 and sin(A+B)= , Then find its area.
67) If 𝑃1 𝑃2 𝑃3 are the altitudes of a triangle from the vertices A,B,C and ∆ the area of triangle, prove that:
1
𝑃1
1
1
2
3
2𝑎𝑏
𝑐
+ 𝑃 − 𝑃 = (𝑎+𝑏+𝑐)∆cos22
68) Prove that: (1) cos 6A=32cos6A-48cos4A+18cos2A-1
4𝑡𝑎𝑛𝜃(1−𝑡𝑎𝑛2 𝜃)
(2) tan4𝜃=1−6𝑡𝑎𝑛2 𝜃+𝑡𝑎𝑛4 𝜃
𝑠𝑒𝑐8𝐴−1 𝑡𝑎𝑛8𝐴
(3) 𝑠𝑒𝑐4𝐴−1=𝑡𝑎𝑛2𝐴
𝑠𝑖𝑛2𝛽
69) If tan𝛼 = 3𝑡𝑎𝑛𝛽 show that : tan(𝛼 − 𝛽) = 5−𝑐𝑜𝑠2𝛽
70) If cos A=mcosB, then prove that cot
𝐴+𝐵
2
=
𝑚+1
(𝐵−𝐴)
tan
.
𝑚−1
2
71) Prove that: tan90-tan270-tan630+tan810=4.
𝜋
72) For all real 𝜃 in (0,2 ), show that : cos(sin𝜃) > sin(𝑐𝑜𝑠𝜃)
𝜋
𝜋
73) It is given that: tan(𝜋𝑐𝑜𝑠𝜃) = cot(√3𝜋𝑠𝑖𝑛𝜃) , 0 < 𝜃 < 2 . Find the value of sin(𝜃 + 6 )
1
74) ABC is a triangle such that : sin(2A+B)=sin(C-A)=-sin(B+2C)= 2.
If A,B and C are in arithmetical progression, determine the values of A,B and C.
𝑠𝑖𝑛𝑥𝑐𝑜𝑠3𝑥
75) Prove that values of function 𝑐𝑜𝑠𝑥𝑠𝑖𝑛3𝑥 𝑑𝑜 𝑛𝑜𝑡 𝑙𝑖𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛
1
3
𝑎𝑛𝑑 3.
76) Determine the smallest positive value of x (in degrees) for which: tan(x+1000)=tan(x+500)tanxtan(x500).
𝑠𝑖𝑛2𝛽
77) If tan(𝛼 − 𝛽) = 5−𝑐𝑜𝑠2𝛽 , 𝑡ℎ𝑒𝑛 𝑓𝑖𝑛𝑑 𝑡𝑎𝑛𝛼: 𝑡𝑎𝑛𝛽.
−𝜋 𝜋
78) Find the values of 𝜃 𝑖𝑛 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 ( 2 , 2 ) satisfying the equation:
(1-tan𝜃)(1 + 𝑡𝑎𝑛𝜃)sec2𝜃 + 2𝑡𝑎𝑛
79) In any triangle ABC, prove that:
𝐴
𝐵
𝐶
𝐴
𝐵
𝐶
2𝜃
=0
cot + cot +cot = cot cot cot .
2
2
2
2
2
2
80) If 32tan8𝜃 = 2cos2𝛼 − 3𝑐𝑜𝑠𝛼 and 3cos𝜃=1. Then find the general value of 𝛼.