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Title Balbharati
List of Practicals
Subject: Mathematics and Statistics (Arts and Science)
Standard : XI
Following are the guidelines for conducting practicals in Mathematics and Statistics.
(1) Total 18 practical sessions should be conducted in the academic year, 9 from paper
I and 9 from paper II.
(2) All the practical sessions are compulsory.
(3) Every practical session must contain at least 4 problems.
(4) These are specimen problems. Teachers may use these or can give similar problems
in practical session.
(5) Practical sessions conducted during the year should be maintained in the form of
journal (file).
(6) Teacher in charge must sign the journal at the end of every practical session.
(7) Journal should be certified before the practical examination.
(8) The practical examination will be of 20 marks with duration of1 hour. Two problems
from paper I and two problems from paper II will be given. Student will have to solve
3 problems out of given 4 problems.
(9) In the journal, on the first page of each of the new practical session; definition,
formulae, rules, laws related to the topic are to be written as introduction to the
session.
Practical Session No. 1
Angle and its measurement
(1) Find the radian measure of the angle between hour-hand and minute-hand of a clock
at : (i) twenty minutes past two (ii) ten minutes past eleven
(iii) thirty minutes past six (iv) five minutes past one
(2) Find the exact time when the angle between hour-hand and the minute-hand of a
clock will be :
(i) 440 for the first time after 12 Oโclock. (ii) 590 for the first time after 6 Oโclock.
(3) A wire of length 10 cm is bent to form an arc of a circle of radius 4 cm. Find the
radian and degree measure of the angle subtended by the arc at the center of the circle.
Also find the area of the corresponding sector.
(4) Arrange following angles in ascending order : ๐๐๐๐, ๐๐0, 3๐๐, 1000, ๐๐2
๐๐
(5) ABCDEFGH is a regular octagon inscribed in a circle of radius 1 unit. O is the center
of the circle. Find
(i) radian measure of โ AOB (ii) ๐๐(๐๐โ๐๐๐๐๐๐ ๐ด๐ด๐ด๐ด) (iii) ๐๐(๐๐๐๐๐๐ ๐ด๐ด๐ด๐ด)
(iv) area of the region enclosed between chord AB and arc AB.
Practical Session No. 2 Trigonometry 1
(1) Construct an angle in standard position whose terminal arm passes through ๐ด๐ด(โ6, 8).
Hence find all trigonometric ratios of this angle.
(2) Construct angle of measure 22.50 by bisecting angle of measure 450. Hence find the
value of ๐ก๐ก๐๐๐ก๐ก22.50.
(3) Draw and measure the angle in standard position whose tangent ratio is 713
.
(4) Draw and estimate the angle in standard position whose sine ratio is 0.57.
(5) Construct a triangle, the measures of whose angles are 320, 580 and 900. Measure
the lengths of sides of this triangle. Find ๐ ๐ ๐ ๐ ๐ก๐ก320 and ๐๐๐๐๐ ๐ 320. Hence find the value of
sin2 320 + cos2 320.
(6) (a) Which is greater, sin(18560)or sin(20190) ? Give reason.
(b) Which of the following is positive ? sin(โ3100) and sin 3100. Give reason.
Practical Session No. 3 Trigonometry 2
(1) Show that (cos๐๐ + ๐ ๐ sin๐๐)3 = cos 3๐๐ + ๐ ๐ sin 3๐๐
(2) Show that cos 3๐๐ โ sin 3๐๐ = (cos๐๐ + sin๐๐)(1โ 2 sin 2๐๐)
(3) Prove that cos2 ๐๐ + cos2(๐๐ + 1200) + cos2(๐๐ โ 1200) = 32
(4) If ๐ด๐ด + ๐ด๐ด + ๐ถ๐ถ = ๐๐ then prove that
cos2 ๐ด๐ด + cos2 ๐ด๐ด + cos2 ๐ถ๐ถ = 1 โ 2 cos๐ด๐ด cos๐ด๐ด cos๐ถ๐ถ.
Hence prove that cos 600 = 12.
(5) Find the values of sin 180 and cos 180.
(6) If tan๐ด๐ด = 56 and tan๐ด๐ด = 1
11 then show that ๐ด๐ด + ๐ด๐ด = ๐๐
4 ๐๐๐๐ 5๐๐
4.
Practical Session No. 4
Determinants and Matrices
(1) If ๐ด๐ด = ๏ฟฝ cos๐ผ๐ผ sin๐ผ๐ผโ sin๐ผ๐ผ cos๐ผ๐ผ๏ฟฝ then show that
(i) ๐ด๐ด2 = ๏ฟฝ cos 2๐ผ๐ผ sin 2๐ผ๐ผโ sin 2๐ผ๐ผ cos 2๐ผ๐ผ๏ฟฝ (ii) ๐ด๐ด
3 = ๏ฟฝ cos 3๐ผ๐ผ sin 3๐ผ๐ผโ sin 3๐ผ๐ผ cos 3๐ผ๐ผ๏ฟฝ
(2) If ๐ด๐ด = ๏ฟฝ3 โ41 โ1๏ฟฝ then prove that ๐ด๐ด๐๐ = ๏ฟฝ1 + 2๐ก๐ก โ4๐ก๐ก
๐ก๐ก 1 โ 2๐ก๐ก๏ฟฝ, for all ๐ก๐ก โ ๐๐.
(3) Construct a 3 ร 3 matrix ๐ด๐ด = [๐๐๐๐๐๐] whose elements are given by :
(i) ๐๐๐๐๐๐ = ๐ ๐ ร ๐๐ (ii) ๐๐๐๐๐๐ = (๐ ๐ + ๐๐)2
(iii) ๐๐๐๐๐๐ = ๏ฟฝ0 , ๐ ๐ ๐๐ ๐ ๐ โ ๐๐1 , ๐ ๐ ๐๐ ๐ ๐ = ๐๐ (iv) ๐๐๐๐๐๐ = ๏ฟฝ
0 , if ๐ ๐ = ๐๐1 , if ๐ ๐ > ๐๐โ1 , if ๐ ๐ < ๐๐
(4) Show that ๏ฟฝ1 ๐๐ ๐๐๐๐1 ๐๐ ๐๐๐๐1 ๐๐ ๐๐๐๐
๏ฟฝ = (๐๐ โ ๐๐)(๐๐ โ ๐๐)(๐๐ โ ๐๐)
(5) Show that ๏ฟฝ1 + ๐๐ 1 1
1 1 + ๐๐ 11 1 1 + ๐๐
๏ฟฝ = ๐๐๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐
(6) If ๐ฅ๐ฅ โ ๐ฆ๐ฆ โ ๐ง๐ง and ๏ฟฝ๐ฅ๐ฅ ๐ฅ๐ฅ2 1 + ๐ฅ๐ฅ3๐ฆ๐ฆ ๐ฆ๐ฆ2 1 + ๐ฆ๐ฆ3
๐ง๐ง ๐ง๐ง2 1 + ๐ง๐ง3๏ฟฝ = 0 then show that 1 + ๐ฅ๐ฅ๐ฆ๐ฆ๐ง๐ง = 0.
(7) Show that ๏ฟฝ๐๐ โ ๐๐ โ ๐๐ 2๐๐ 2๐๐
2๐๐ ๐๐ โ ๐๐ โ ๐๐ 2๐๐2๐๐ 2๐๐ ๐๐ โ ๐๐ โ ๐๐
๏ฟฝ = (๐๐ + ๐๐ + ๐๐)3
Practical Session No. 5
Straight Lines
(1) Show that the equation of the line passing through ๐ด๐ด(๐ฅ๐ฅ1,๐ฆ๐ฆ1) and parallel to the line
๐๐๐ฅ๐ฅ + ๐๐๐ฆ๐ฆ + ๐๐ = 0 is ๐๐(๐ฅ๐ฅ โ ๐ฅ๐ฅ1) + ๐๐(๐ฆ๐ฆ โ ๐ฆ๐ฆ1) = 0. Hence find the equation of the line
passing through (1, 1) and parallel to the line 15๐ฅ๐ฅ + 8๐ฆ๐ฆ + 1947 = 0.
(2) Show that the equation of the line having slope ๐๐ and making X - intercept ๐๐ is given
by ๐ฆ๐ฆ = ๐๐(๐ฅ๐ฅ โ ๐๐). Find the Y - intercept of this line.
(3) A line makes intercepts โ and ๐๐ on the co-ordinate axes. If ๐๐ is the length of the
perpendicular drawn from the origin to the line then show that 1โ2
+ 1๐๐2
= 1๐๐2
.
(4) Show that there are two lines which pass through the point ๐ด๐ด(3, 7) and the sum of
whose intercepts on the co-ordinate axes is zero. Draw the rough sketch of these two
lines.
(5) Find the number of lines which pass through the point ๐ด๐ด(5 ,5) and the sum of whose
intercepts on the co-ordinate axes is zero.
(6) Find the co-ordinates of the orthocenter of the triangle formed by lines
2๐ฅ๐ฅ โ ๐ฆ๐ฆ โ 9 = 0, ๐ฅ๐ฅ โ 2๐ฆ๐ฆ + 9 = 0 and ๐ฅ๐ฅ + ๐ฆ๐ฆ โ 9 = 0.
Practical Session No. 6
Circle and parabola
1) Find the center and radius of the circle ๐ฅ๐ฅ2 + ๐ฆ๐ฆ2 โ ๐ฅ๐ฅ + 2๐ฆ๐ฆ โ 3 = 0
2) Find the equation of circle passing through the point of intersection of the lines
๐ฅ๐ฅ + 3๐ฆ๐ฆ = 0 and 2๐ฅ๐ฅ โ 7๐ฆ๐ฆ = 0 and whose centre is the point of intersection of the lines
๐ฅ๐ฅ + ๐ฆ๐ฆ + 1 = 0 and ๐ฅ๐ฅ โ 2๐ฆ๐ฆ + 4 = 0.
3) Find the equation of circle, the end points of whose diameter are the centers of circles
๐ฅ๐ฅ2 + ๐ฆ๐ฆ2 + 6๐ฅ๐ฅ โ 14๐ฆ๐ฆ โ 1 = 0 and ๐ฅ๐ฅ2 + ๐ฆ๐ฆ2 โ 4๐ฅ๐ฅ + 10๐ฆ๐ฆ โ 2 = 0.
4) Find the equation of the circle passing through points (5, 7), (6, 6) and (2,โ2).
5) Consider a circle with center at origin and radius r. Let P(x, y) be any point on the
circle making an angle ฮธ with positive direction of the X - axis then verify that
P(x, y) = P (r cosฮธ , r sinฮธ ). By taking r = 5 and ฮธ = 1350 verify the above result.
6) Find the equations of tangents to the circle ๐ฅ๐ฅ2 + ๐ฆ๐ฆ2 = 4 drawn from the point
(2, โ1).
7) Find the co-ordinates of the focus, equation of the directrix, length of Latus
Rectum, and the co-ordinates of the end points of the Latus Rectum of the parabola
๐ ๐ ) 5 ๐ฆ๐ฆ2 = 24๐ฅ๐ฅ , ๐ ๐ ๐ ๐ ) ๐ฅ๐ฅ2 = 12 ๐ฆ๐ฆ , ๐ ๐ ๐ ๐ ๐ ๐ ) 3 ๐ฆ๐ฆ2 = โ16 ๐ฅ๐ฅ .
8) Find the co-ordinates of the focus, equation of the directrix, length of Latus
Rectum, and the co-ordinates of the end points of the Latus Rectum of the parabola
๐ฅ๐ฅ2 + 4๐ฅ๐ฅ + 4๐ฆ๐ฆ + 16 = 0.
9) Find the area of triangle formed by the lines joining the vertex of the parabola
๐ฅ๐ฅ2 = 12๐ฆ๐ฆ to the ends of its Latus rectum.
10) For which point of the parabola ๐ฆ๐ฆ2 = 18๐ฅ๐ฅ is the ordinate equal to 3 time the
abscissa?
Practical Session No. 7
(Ellipse and hyperbola)
1) Find the lengths of the major and minor axes, coordinates of vertices, eccentricity,
co-ordinates of the foci, equations of directrices and the length of the Latus Rectum
of the following conics
๐ ๐ ) 9๐ฅ๐ฅ2 + 16๐ฆ๐ฆ2 = 144, ๐ ๐ ๐ ๐ ) 4๐ฅ๐ฅ2 + 25๐ฆ๐ฆ2 = 100
๐ ๐ ๐ ๐ ๐ ๐ ) ๐ฅ๐ฅ2
25โ ๐ฆ๐ฆ2
9= 1 ๐ ๐ ๐๐) ๐ฆ๐ฆ
2
4โ ๐ฅ๐ฅ2
9= 1
2) Find the equation of ellipse referred to its principal axes with eccentricity 34 and
passing through the point (6,4).
3) An ellipse has OB as a semi-major axis, S and Sโ are its foci and โ ๐๐๐ด๐ด๐๐โฒ is a right-
angle, then find the eccentricity of the ellipse.
4) Find focal distances of the point P๏ฟฝ5, 4โ3๏ฟฝ on the ellipse 16๐ฅ๐ฅ2 + 25๐ฆ๐ฆ2 = 1600.
5) If ๐๐1 and ๐๐2 are eccentricities of hyperbolas ๐ฅ๐ฅ2
๐๐2โ ๐ฆ๐ฆ2
๐๐2= 1 and ๐ฆ๐ฆ
2
๐๐2โ ๐ฅ๐ฅ2
๐๐2= 1
then show that 1๐๐12
+ 1๐๐22
= 1
6) Find the equation of hyperbola whose
i) directrix is 2๐ฅ๐ฅ + ๐ฆ๐ฆ = 1 , focus at (1,2) and eccentricity โ3.
ii) foci are at (ยฑ4 , 0) and the length of its latus rectum is 12 unit.
iii) vertices are ( 0 , ยฑ 2 ) and the foci are at ( 0 , ยฑ 3 ).
7) An interesting property of rectangular hyperbola
Equation of rectangular hyperbola is x y = k (k is non zero constant) โฆ(I)
Tangent is drawn to the curve at point on it whose abscissa is โaโ.
Therefore point of contact is (a, โฆ )
Slope of tangent = ๏ฟฝ ๐๐๐๐๐ฅ๐ฅ๏ฟฝ๐๐๐ฅ๐ฅ๏ฟฝ๏ฟฝ at point (a, โฆ )
= ๏ฟฝโ๐๐๐ฅ๐ฅ2๏ฟฝ at point (a, โฆ )
= โ๐๐๐๐2
โฆ(II)
Equation of tangent to xy = k at point (a, โฆ ) is
(by slope point form)
y - โฆ.. = ๏ฟฝโ๐๐๐๐2๏ฟฝ(x - a)
i.e. a2 (y - โฆ) = -k (x โ a)
i.e. โฆโฆโฆโฆ
i.e. โฆโฆโฆโฆ
this equation of tangent in terms of double intercept form is
๐ฅ๐ฅ2๐๐
+ ๐ฆ๐ฆ2๐๐๐๐
= 1.
tangent cuts the x โ axis at point P and the Y โ axis at point Q.
clearly P = (โฆ , 0) and Q = (0, โฆ)
Area of โ POQ = 12(OP)(OQ)
= 12(โฆ.)(โฆ.)
= 2k Verify this interesting result for different values of k.
i) xy = 4 at (2, 2) ii) xy = 12 at (-2, -6)
Practical Session No. 8
Measures of Dispersion
(1) For the following data : 710, 635, 423, 221, 971, 843, 307, 289. Which are the
extreme values in this data ? and compute the range.
(2) A die is rolled 30 times and the following distribution is obtained. Find the
variance and the standard deviation.
Score 1 2 3 4 5 6
Frequency 2 6 2 5 9 6
Repeat the experiment and construct the frequency table as shown above. Find its
mean and standard deviation.
(3) Find the mean and the standard deviation of the first ๐ก๐ก natural numbers. Hence find
the mean and the standard deviation of the first 200 natural numbers.
(4) The following table shows weight of students of two classes. Calculate the
coefficient of variation of the two distributions.
Weight in Kg. Class A Class B 30-40 8 9 40-50 16 12 50-60 12 18
Also find the mean and the standard deviation for both the classes.
Practical Session No. 9
Probability
(1) The turnout of spectators at the world cup cricket match is dependent upon the
weather. On a rainy day the probability of a big turnout is 0.3. If it doesnโt rain,
then the probability of big turnout increases by 0.6. The weather forecast gives a
probability of 0.75 that it will rain on the day of the match.
Find the probability that (i) there is a big turn out and it rains (ii) there is a big turn
out.
(2) A bag contains 7 red and 5 blue balls. A ball is taken at random from the bag, its
color is noted and not replaced into the bag. Now a second ball is taken from the
bag and its color is noted. Find the probability that one is red and the other is blue.
(3) If ๐๐(๐ด๐ด) = 14
,๐๐(๐ด๐ด) = 25
๐๐๐ก๐ก๐๐ ๐๐(๐ด๐ด โช ๐ด๐ด) = 12 then find
1) ๐๐(๐ด๐ด โฉ ๐ด๐ด) 2) ๐๐(๐ด๐ด โฉ ๐ด๐ดโฒ) 3) ๐๐(๐ด๐ดโฒ โฉ ๐ด๐ด) 4) ๐๐(๐ด๐ดโฒ โช ๐ด๐ดโฒ) 5) ๐๐(๐ด๐ดโฒ โฉ ๐ด๐ดโฒ)
(4) 2% of the population have a certain blood disease in a serious form. 10% have it in
a mild form and 88% donโt have it at all. A new blood test is developed, the
probability of testing positive is 9/10 if the subject has the serious form, 6/10 if the
subject has the mild form, and 1/10 if the subject doesnโt have the disease. A
subject is tested positive. What is the probability that the subject has serious form
of the disease?
(5) A coin is tossed until a head appears or until it has been tossed three times. Given
that head does not occur on the first toss, what is the probability that coin is tossed
three times ?
(6) From a group of 8 boys and 5 girls, a committee of 5 is to be formed. Find the
probability that a committee contains i) 3 boys and 2 girls ii) at least 3 boys.
Practical Session No. 10
Complex Numbers
1. If 3 4z i= + then show the following on the Argand diagram
a) z b) zโ c) z d) zโ
2. If 1 2 3z i= + and 2 2 4z i= โ then show the following on the Argand diagram
a) 1z b) 2z c) 1 2z z+ d) 1 2z zโ
3. If 1 1 2z i= + and 2 3z i= + . Show the following on the Argand diagram
a) 1z b) 2z c) 1 2z zโ d) 1
2
zz
4. By means of Argand diagram, for 1 0z i= + and 2 2 0z i= โ + ,verify the following
a) 1 2 1 2| | | | | |z z z zโ = โ b) 11
2 2
zzz z
= c) ( )1 2 1 2arg arg argz z z zโ = + d) 11 2
2
arg arg argz z zz
= โ
5. Show the roots of the equation a) 2 6 10 0x xโ + = b) 3 1 0x โ = on the Argand diagram.
Practical Session No. 11
Sequence and series
1. A sequence is generated by the formula 2nM an bn c= + + , where a ,b and c are
constants. If 1 2 34, 10 and 18M M M= = = , find the values of a ,b and c.
2. Soham starts a new job on an annual salary of Rs. 2,00,000. He is given an annual rise of Rs. 5000 at the end of every year until he reaches his annual salary of Rs. 2,50,000.Then his annual rise will be Rs 7000 per year. Find the total amount he earns a) in the first 9 years b) over 15 years
3. Sanvi decided to save some money during the two week holiday. She saved Rs. 2 on the first day, Rs. 5 on the second, Rs. 8 on the third and so on. How much did she have at the end of the vacation? If she continuessaving in the same way, how long would it take to exceed the total saving Rs. 610?
4.Suhani is offered a job. The starting annual salary is Rs.12 Lacs. She is given 5% increment per year. What will be her annual salary after 10th year? Also find her total earnings in 10 years.
(Given 9 101.05 1.55,1.05 1.63โ โ )
5. Find a) ( )6
1
2 35
n
n=โ b) 1729 243 81 ...
3โ
โ + โ c) ( )18 0.25 n
n
โ
=โ
6. If the ratio of H.M. and G.M. of two quantities is 12 :13, then show that the ratio of the numbers is 9 : 4
7. If , ,a b c are in H.P., then show that , ,a b cb c c a a b+ + +
are in H.P.
8. The sum of three decreasing numbers in A.P. is 27. If 1, 1,3โ โ are added to them respectively, the resulting series is a G.P. Find the numbers.
Practical Session No. 12
Permuations and Combinations
1. How many three-digit numbers can be formed using digits 0, 2,3,5,6,8 ,9 with no digit is repeated in each of the following?
a) there are no restrictions b) number must be multiple of 5.
c) the number is greater than 600. d) even number is less than 400.
2. A shipment of 12 cell phones contains three defective units. In how many ways can a buyer purchase four of these units and receive (a) all good units (b) two good units (c) at least two good units.
3. In how many ways can we arrange 3 red flowers, 5 yellow flowers and 7 white flowers in a row? In how many arrangement the yellow flowers are to be separated (flowers of same color are identical ).
4. There are 10 persons among whom two are friends. Find the number of ways in which they can be arranged round a circle (a) if there is exactly one person between the two friends. (b) the two friends are always separated.
5. Diagonals are formed by joining vertices of a polygon count all the diagonals. Consider triangle whose sides are diagonals or sides of the polygon. How many such triangles are there in a (a) Hexagon (b) Octagon (c) Decagon
6. A committee of 4 is chosen from 8 men and 6 women. Determine the number of ways of selecting the committee if (a) there are no restrictions (b) it must contain 2 men and 2 women (c) it must contain all men (d) it must contain at least 3 men (e) it must contain at least one of each gender.
7. A class has 25 students. The class teacher has been asked to make groups of m students each and go to museum taking one group at a time. Find the size of group for which the teacher goes the maximum number of times to the museum.
8. Veer has 8 friends and wants to invite some of them on his birthday party. In how many ways this can be done if
(a) any number of friends can be invited? (b) at least two friends must be invited ?
Practical Session No. 13
Mathematical induction and Binomial theorem
1. Prove the following statements using principle of Mathematical Induction.
a) 13 3n n+<
b) 2 3n n+ is divisible by 2
c) ( )
1 1 1 1...1 2 2 3 3 4 1 1
nn n n
+ + + + =โ โ โ + +
d) ( )10 1 2 3 10 1 10 2 10 3 10log ... log log log ... logn na a a a a a a a= + + + +
2. Using Binomial Theorem expand ( )53x + .
3. Expand each expression using Binomial Theorem
a) ( )62x y+ b) ( )42 3x + c) ( )31/3 2/3x y+ d) ( )4
1/43x y+
4. Find the 5th term in the Binomial expansion of a) ( )73x y+ b) ( )42 7x โ c) ( )92 3x y+
5. Find the coefficient A of the given term in each of the binomial expansion
Binomial Term a ( )75x + A 6x b ( )823 1yโ โ A 8yโ
c 64xx
โ
A 0x
d ( )42x y+ A 2 2x y 6. Use Binomial theorem to express in the form of a+ ib
a) ( )51 i+ b) ( )43 4+ โ c) ( )6
2 7iโ
Practical Session No. 14
Sets and Relations
1. For the following sets, if possible i) list the elements of A ii) Find n(A)
iii) write using interval notation iv) sketch on number line
a) { }: 3 9A x N x= โ โ โค โค b) { }: 3 9A x Z x= โ โ โค โค c) { }: 3 9A x R x= โ โ โค โค
d) { }: 4 7A x R x= โ โค < e) { }: 2A x Q x= โ โ โค f) { }: 7A x Z x= โ <
2. Use separate Venn Diagrams and shade the regions for the following:
a) A b) 'B c) A Bโช d) 'A Bโช e) 'A Bโช f) ' 'A Bโช g) ( ) 'A Bโช h) ( )' 'A Bโช i) A Bโฉ j) 'A Bโฉ k) 'A Bโฉ l) ' 'A Bโฉ m) ( ) 'A Bโฉ n) ( )' 'A Bโฉ
3. Use separate Venn Diagrams and shade the regions for the following:
a) A b) 'B c) A Cโช d) 'B Cโช e) 'A B Cโช โช
f) ' 'B C Cโช โช g) ( ) 'A B Cโช โฉ h) ( )' 'A B Cโช โช i) B A Cโฉ โฉ j) ( )'C B Aโฉ โช
4. In a group of 50 students, 20 study subject A, 25 study subject B and 20 study subject C. 10 study both A and B, 5 study both B and C, 7 study both A and C. 2 study all three subjects.
a) show this information on Venn diagram
b) find the number of students who study i) A only ii) B or C iii) A but not C iv) none of A , B or C.
5. Show all possible relations from { },A a b= to { }5,6B = using Arrow diagram. Find among them the relations that are i) One-One relation ii) Onto relations iii) Null relation iv) Universal relation.
6. Plot A Bร and B Aร on XY plane, if
a) { } { }: ,1 5 ; : , 3A x x N x B y y W y= โ โค โค = โ <
b) { } { }: ,1 4 ; : , 1 3A x x R x B y y Z y= โ < โค = โ โ โค <
7. Let R be the relation defined by ( ){ }, : 2 6, , 10,R x y x y x W x y Z= + = โ โค โ
Find i) R ii) Domain of R iii) Co-domain of R iv) Range of R
8. Given the relation ( ) ( ){ }2,3 , 3,4R = on set of natural numbers N, add a minimum number of ordered pairs so that the relation is i) reflexive ii) symmetric iii) transitive iv) equivalence.
Practical Session No. 15
Functions
Theory: Vertical shift : For 0c > , ( )f x c+ shifts graph of ( )f x to c units upward , and ( )f x cโ shifts graph of ( )f x to c units downward.
Horizontal shift : For 0c > , ( )f x c+ shifts graph of ( )f x to c units to left , and ( )f x cโ
shifts graph of ( )f x to c units to right.
Reflection about -axisX : ( )f xโ reflects the graph about -axisX
Reflection about -axisY : ( )f xโ reflects the graph about
Note: If ( ) ( )f x f xโ = , then ( )f x is called Even Function, where the graph is symmetric about x -axis (i.e. Reflection about -axisy ) , and if ( ) ( )f x f xโ = โ , then
( )f x is called Odd function, where the graph is symmetric about Origin.
Activity:
1) Draw the graph of ( ) 2xf x = and ( ) ( )42 xh x โ= on the same graph paper
2) Draw the graph of ( ) logf x x= and ( ) ( )log 1h x x= + on the same graph paper
3) Draw the graph of ( )f x x= and ( ) 3 2h x x= โ + on the same graph paper
4) Draw the graph of ( ) 3f x x= and ( ) 3h x x= โ on the same graph paper.
5)Draw the graph of ( ) [ ]cos , 0,2f x x x ฯ= โ and ( ) [ ]cos , 0,2h x x x ฯ= โ โ on the same graph paper.
6) Draw the graph of ( )f x x= and ( )h x x= โ on the same graph paper.
Practical Session No. 16
Limits
Practical Session No. 17
Continuity
Practical Session No. 18
Differentiation