Maths Lab Manual
Maths Lab Manual
Maths Lab Manual
Aepie
Equivalence Relation.
Objective defined
by R {(, m): /||m} isis an.
an equivalence
all lines in a plane,
that the relation R in the set L
of
lo verity
relation.
Pre-requisite Knowledge
(a) Definition of an equivalence relation. lines etc.
lines and perpendicular
(b) Properties of parallel
Fig. 2.2
Fig. 2.1
Observation
(a), is parallel tol, l , is parallel to
Result
i.e. .1,)
R
e R and
{(I, m) : 7 ||m}
((2 !)
is
e R
transitive.
((, 1) e R
111
Since R m} relation.
i(, m) : 'I| Is
reflexive, symmetric and transitive, therefore it is an cquivac
APPLICATION
(a) lake sOme more colourful threads in different positions and repeat this activity to clear the concept o
different types of relations.
.
(6)Take few triangles in such a way that some are each other and put them in a collection
similar to
Define R =
{(A1, A,): A A,} in A. Check Whether R is an
equivalence relation or no
Sol. {2, 3, 5, 7}
(ii) If R, and R, are two equivalence relation on a set A, then R, n R, is an .
Sol. Equivalence relation.
(iv) Relation R defined in set A = {1, 2, 3 , . 13, 14} as R {(r,y) : 3x-y = 0} is not an
ViVA VoCE
() Define empty relation?
elements of A is related to any element of A i.e.
Sol. Relation R in a set A is called an empty relation, if no
R .
Is the relation 'is less than to' R equivalence relation?
(i) or equal on an
Sol. No.
Tell the domain and range of R.
(ii) Let R (a.:aE N and 1 <a<5.
=
Activities 11
Activity 6
Topic
Inverse Trigonometrie Punction
properties of inverse
Concept of trigonometric ratios and
Demonstration
drawing sheet.
(a) Draw a circle of radius 1 unit on a
the centre O.
Draw XOX and YOY wo mutually
perpendicular lines passing through
) as shown in Fig. 6.1.
(c)Mark the points P. Q. R, S
where the circle meets the r-axis and y-axis
shown in Fig. 6.1.
(d) Fix two bars on Q and S as
its other end at the circumference ofthe
th
(e) Takc a needle of length 1 unit and
fix its one end at 0 and keep
same circle to move along it. See Fig. 6.1.
freely
Bar | Bar
jeed
Thin wire
Bar R Bar
Fig. 6.1
Fig. 6.2
)Also fix
thin wire between the bars and
a
parallel to x-axis. See Fig. 6.1.
(g) Find the coordinate of A as (x, sin
(h) Rotate the needle
a,).
anticlockwise and mark the
points B, C, D
() y-coordinate of B issin(n u,) =
sin
respectively (Fig. 6.2).
) Since
a
y-coordinate of A and B are same.
Since function
is nol one-one in first and second
(k) Similarly
keeping the
needle at quadrants.
Therefore, ine function is also ungles u, and (- n t a,) the
not one-one in y-coordinate of the points C and D a same
3in
arc
sin
As
Y Y
Fig. 6.3 Fig. 6.4
Observation
sin's: 1, 11
(b) This range of sinx is called the principal value of sine function.
APPLICATION
(a) This activity helps the students to understand the concept of principal values of different inverse
trigonometric functions.
Activities
Activity 11
Tople
ARpliatiol Derivative
0bjective
Rvlle's Theonem
8 veri
aMaln Statement of Rolle's Theorem
u) is:
aa)runctin
contununus on the closed interval (a, b1.
derivalble on the open interval (a, b) and
() Aa) )
theorem is applicable.
then Rolle s
Pre-requisite Knowledge
ie) Functiom is continuous and ditfferentiable in the given interval.
of tangent.
() Slope
Materials Required
Apiece of cardboard, thin wires or threads (coloured) of different lengths, white paper, sketch pen, scale.
Demonstration
(a) ihe a cardboard of a convenient size and paste a white paper on it (as in Fig. 11.1).
() Draw two lines which are perpendicular to each other named as x-axis and y-axis, on the white paper pasted
ou cardboard (Fig. 11.1) (Take convenient scale on x-axis and y-axis).
Ay-axis
AC 8 cm
(a) = 8| |8 f(b) BD 8 cm
OA 2 cm
OB 2 cm
B
O(c) (2, 0) x-axis
(-2, 0)
(b 0)
(a, 0)
Fig. 11.1
Activities/37
Akea piece of wire or thread (about 15 cm) and bend it in the shape of curve (parabola) and.
white sheet as shown in the x it
Fig. 11.1. on
(d) Now, take he
nwo straight wires or threads of the same length and paste them in susuch a way
Perpendicular to v-axis at the points A and B and meeting the curve at the points C and D that
(Fi y a
Fig. 11.J).
Observation
(a) Take function
a y
=fAu), such that y = x* + 4 =
fW)
(6) Here, 04 = a units (-2 units) and OB =b units (2 units). So, coordinates of the point Ais (-2,( nd
(c)As there is no break in the curve Bis
ntinuous (a,(2,0b)
in the interval la, b] i.e. [-2, 2]. So, the function is
(d) At each point, a
conti on
tangent canbe drawn, so function is differentiable in (a, b) i.e. (-2,
2).
(e) As the wires or
threads at A and B are of equal lengths i.e. AC BD = 8 units. This s
A-2) = A2) = 8. = hows fla)
)From above, three conditions of
fb =
(ii) If ffx) =
*-2, atx =
-3, the value of f(-3) is
Sol. 9-2 7
(iv) If a function is continuous in la,
b] and differentiable in
then Rolle's theorem is
(a, b), then third condition is,
applicable.
Sol. fa) =
fb)
() For any curve, flr) =
x*- 2, f(«) =
_
Sol. 2x 2
Activity 12
Topic
Application of Derivative
Objective
To verity Lagrange's Mean Value theorem.
Pre-requisite Knowledge
(b) Slope of tangent = f't).
(a) Continuity and ditfferentiability of function.
(d) Condition of parallelism.
(c) Slope of chord
=
Materials Required
A cardboard. thin wires or threads (coloured) of different lengths, white sheet of A size, sketch pens, scale.
Demonstration
white sheet it.
piece of cardboard and fix a on
(a) Take a
y-axis
B (b, ftb))
AM 8 cm
BN 15 cm Wire (a, fla))
OR 2.5
PR 11.2
>X-axis
M(a, R(c, 0) N(b, 0)
Fig. 12.1
the white sheel, grauud
perpendicular to each other named asr-axis and y-axis,
on
(6) Draw two lines which are
r-axis and y-axis (Fig. 12.1). IN t
the
bend it in the shape of a
curve and o
C) Take a piece of wire or thread (about 10 cm -15 em) and
white sheet as shown in the Fig. 12.1.
(b) Let OM =
a units
(2 units) and ON b
=
=
units =
(3 units).
So. coordinates of M(2. 0) andN is (3, 0).
(c) And cordinates of A(a. fla) and B(b. fb)) > fla) = f(2) = 8 and f(b) = f3) = 15
d) AB is a chord joining two points A(a. fla)) & B(b. sb) > (3. 15).
(e) CD is a tangent to curve at
the point P(c. flc)) in the interval (a. b).
Slope of tangent at P = f(c) = 2c + 2
)
f6)-fa)
)-fa)
b-a
= =
7 =
slope of AB (chord)
(h) As chord is parallel to the tangent
Slope of tangent =
fc) slope =
of chord
f(c) = (6)- f(a) 2c + 2 7
b- a
2c 5 c
=2.5 (2, 3) R(c, 0) => R(2.5, 0) (in figure)
Result
Hence, Lagrenge's mean value theorem is verified. It is evident from the figure that there is at least one point P
between A and B on the curve, the tangent at which is parallel to the chord AB in the given interval.
APPLICATION
This activity can be done for other curves also. e.g. (i) y = f«) = r on [0, 1] (i) y = cosx on [r/3, Sn/3]
Sol.2
fu) is
(ü) Slope of tangent at any point 'ce' on the curvey =
Sol. flc).
the chord the curve, then slope of tangent is
(i) If a tangent to the curve is parallel to on
Activities /41
Activity 15
Topic minimum values of a function.
maximum and absolute
af absolute
Pre-requisite Knowledge
on graph.
)Representation of a curve
intervals.
closed intervals and open
Meaning of minimum values.
absolute maximum and absolute
icà Meaning of
Materials Required
calculator.
paper, adhesive, pen,
pencil, eraser, ruler,
Waite sheet. graph
graph: Here,
curve calculation).
(e) Plotting of curve
on
find corresponding values offir) (using
different value ofr in [-2.5, 2.5] and
Drawa table: Take
(i)
fr)
4
0
0
-1
0
-2
-1.58
-2.25
-2.25
1.58
11.82
-2.5
2.5 11.82
42.5. 111
40
4)
Fig. 15.1
Observation
(a) Largest value of f) is 11.82 atx =
2.5 and x -2.5. So, absolute
=
Objective
given rectangular sheet by cutting equal squares from.
COnstruct an open box of maximum volume from a
each
corner.
Pre-requisite Knowledge
Volume of cuboid = l xb xh
Materials Required
Ditterent coloured chart papers, scissors, cellotape, glue stick, pencil, scale.
Demonstration
(a)Take a rectangular sheet or chart paper of size 20 cm x 10 cm of any colour and name it as PQRS.
(6) Cut four squares of equal size of side x cm (1 cm) from each corner P, 0, R and S. (Fig. 16.1)
= 1 cm
x= 1 cmR
X 1 Cmn
10cm
X=1 cm
20 cm
Fig. 16.1
paperssize of chart and different values ofx (1.5 cm, 2 cm, 2.1 cm, 2.5 cm, 3 cm).
(c) Repeat this process with the
same
18 cm B
Fig. 16.2
2) (10 -
2)(1) =
18 x 8 x 1 =144 cm*
(a) r
c)Whenx
= 2 (for third cuboid)
Volume of box = (20- 4)(10- 4)(2) = 16 x 6 x 2 = 192 cm
id Whenx
= 2.1 (for fourth cuboid)
volume of box = (20 - 4.2)(10- 4.2)(2.1) = (15.8)(5.8)(2.1) = 192.4 cm
(e)Whenx
= 2.5 cm (tor fifth cuboid)
Volume of box (20 - 5)(10 5)(2.5)
= -
= 15 x 5 x 2.5 =
187.5 cm
sixth cuboid)
When x 3 cm (for
=
Volume of box
=
(20 -
6)(10 -
Result
all the boxes, the volume of
box is maximum when x =2.1 cm
Clearly among
APPLICATION
more clear.
to make the concept
of students with different size of rectangles
can be done in a group
This activity
BLANKSs
FLL IN THE
breadth b, height h is
)Volume of cuboid of
length 1,
Sol. Ix b xh
is square of
side x and height y is
whose base a
a cuboid
i) Volume of by cutting equal
squares
cuboid from
a
to make a
) l s it possible will get
sheet? rectangular flaps we
Sol. Yes rectangular fold the
cuboid from a
sheet, then
will you make a of the
rectangular
How corner
squares
from each increases? decreasing.
Dy Cutting the value of x
volume
start
r,and
conditions
i Activities 53
Activity19 4Parsieiur
Topic
Geometrical meaning of definite integration (By limit of a sum).
Objective
To evaluate the definite integralVI-rdr as the limit of a sum and verify it by actual integration.
a
Pre-requisite Knowledge
Area of trapezium
(a)
(b) Representation of curve on graph
c)Knowledge of integration.
Materials Required
Cardboard, white drawing sheet, wires (coloured), scale, pencils, graph paper, glue, nails.
Demonstration
(a) Take a cardboard of a 30 x 20 cm and paste a white drawing sheet on it.
(6) Now paste a graph paper on white drawing sheet and draw two perpendicular lines representing r-atis and
y-axis (XOX and YOY)
(c) Draw a quadrant of a circle with O (0, 0) as centre and radius1 unit (10 cm) (as shown in Fig 19.1) in Ist
quadrant. It represent graphs of the function 1 - r in [0, 1].
B,B B Ba
1 unit
A
o AA, A As As A, A A
0.1 0.1 1unit
Fig. 19.1
Activities/61
to each other) meet the
lines (parallel
Om the points A,, A,s . A, draw vertical urve at B,B
(Fig. 19.1)
8) Measure the length of OB, A,B,, A, B2 ... A59:
=x0.1(1 + 0.99)
(ii) Area of A,A,B,B, =
x0.1 (0.99 + 0.97)
Area of AA,B,B, x0.1 (0.97 + 0.95) and so on.
(ii)
(v) Area of A,AB,B, = x 0.1 (0.43 0)
(where A and B10 are very close to each other which is very small nearto
Now, Total Area of the quadrant of the circle (area bounded by the curve and two axes) = sum of the ares
(d)
of all trapezium.
= x0.1[(1 + 0.99) + (0.99 + 0.97) + (0.97 + 0.95) +. + (0.43)]
, 0.97+0.97
=
0.1/0.5+ 0.99+0,99
2 2 +0.43
=
0.1[0.5 + 0.99 + 0.97 + 0.95 + 0.92 + . . + 0.43]
= 0.1 x 7.74 = 0.774 sq. units (approx)
(e) Using actual integration
x= = =0.785 sq units
qual o
From (d) and (e), we, observethat area of the quadrant ( 1 - b y as a limit of sum is approx egqua
the area obtained by actual integration.
Result
In both cases area of a quadrant are approximately same.
Objective
To verify that angle in a semicircle is a right angle, using vector metnod.
Pre-requisite Knowledge
(a) Pythagoras theorem
(6) Triangle law of vector addition
OA AB =OB
(c) Negative vectors
(d) Definition of dot product.
Materials Required
A cardboard, white drawing sheet, wires, adhesive, sketch pen, scale, pins(nails), Paper arrows about 20 pieces
Demonstration
(a) Take a thick cardboard and paste a white drawing sheet on it.
(6) Draw three circles of radius 4 cm each (or 4.5 cm) Fig. 21.1 shade them or colour the circles.
(c) In first circle, mark centre O and fix three small nails at the circumference named as P, 0, R where P and
Q are extreme ends of the diameter of circle (Fig 21.1)
d) Join P and R, Q and R, O and R using wires or threads (Fig 21.1)
e) Now, put arrow OP, 00, PR, QR and OR to represent them as vectors (using coloured paper arrows or wires)
(Fig. 21.1)
Rep this process in another two replicas of circle with different namings as shown in Fig. 21.2 and
Fig. 21.3.
AAA
Paperamows
G
Fig. 21.1 Fig. 21.2 Fig. 21.3
ie) Diamefer
PO. | PO|= 2(radius) Cm
PR | + |QR |?
B y pythagoras, |
POI
(using converse of Pythagoras theorem)
So, P R 0
method, we know that,
olNow. By vector
OP +PR OR
In AOPR
PX OR - OP =r +
OQ +QR OR
In AOOR,
OR =
OR -00 =r -a
|PX ||QX |cos ZPRO
Dot product,
PR QR =
APPLICATION
done onthe basis of dot product e.g. that side.
Another activity can be is perpendicular to
that side bisector of an isosceles triangle
(a) to verify rhombus bisect each
other at right angles.
that the diagonals
of a
(b) to verify
Sol. 0 or
or b| =
.
(v) a b =
Objective
To demonstrate the cquation of plane in normal form.
Pre-requisite Knowledge
(a) Equation of a plane on different forms- Cartesian form, Normal form.
(b) Position vector of a point.
Materials Required
Two pieces of cardboard sheets, thin plastic sticks, paper pins, etc.
Demonstration
(a) Take two pieces of cardboard sheets to represent two planes and fix a plastic stick in such a way that it i
perpendicular to both as shown in Fig. 23.1.
(6) Take O as origin and fix two arbitrary nails named as A and P with position vectors a and with respect
to origin O see Fig. 23.1.
() Join OA, AP and OP with the help of thin wires and stick paper arrowheads on ON, OA, OP and AP see
Fig. 23.1.
NA a P
plane 1
AAA Arrow heads
Plastic stick
plane 2
Fig. 23.1
Observation
(a) ON is normal to the plane.
(b) O =
ä, OF =
Result
Therefore (Y - a). n = 0 is the required equation o f the plane in normal form.
0bjective
angle between twO planes IS t h e s a m e as the angle between their normals
rifv that the
Pre-requisite Knowledge
and its equations.
Concept of plane
(a) plane.
Normal to the
(b) b e t w e e n two
planes.
Angle
(c)
Materials Required
cardboard or thermocol sheets, few pieces of thin wire, fevicol, pencil etc.
or
thick
Tiwo
sheets
Plane 1
Fig. 24.1
Observation is 90°.
angle at O
> 90° after
measuring.
Two planes are at right angle and the angle
at P is
(a) normals to the plane
90°.
(6) Thin wires , l, are
at O angle at P=
=
as the angle
planes is
same
Activities
Activity 26
Topic Dimensional Geometry
Objective
distance between two skew lines and verify it analytically
the
the shortest
shortest
measure
oPre-requisite K n o w l e d g e
lines.
straight
) E q u a t i o n of
lines.
of skew
Concept
between skew lines.
) distance
)Shortest
D6,2,4)
A(2,2,0)
Fig. 26.1
26.1.
as shown
in Fig.
help of thin wire actual
with the the
Join AC and BD
measure
c) BD and
two skew
lines.
the lines AC and
with
(4) AC and BD
represent perpendicularly
thread and join it
a thin wire BD.
or
along the thread
) lake
of
the right-angle touches the other piece
distance. side forming
its one the right-angle
such that forming
F l a c e a set square its other side
line AC till betweenAC
Move the set-square along the shortest
distance
required
6 position to get the
thread. threads in this
between two
the distance
Measure
Result
Thus from the above activity it is
verified that the shortest distance between tWO skew lines obtained by
measurement and obtained actual
analytically is same.
APPLICATION
It helps the students to understand the concept of skew lines and the shortest distance betwecn two lines
in space
3 =
15y + 7 =
3- 10z. Then its direction cosines
Sol are
VIVA VOcCE
5 +4j -5k).+
Objective
To explain the computation of conditional probability of a given event A, when event B has already ocu
Pre-requisite Knowledge
(a) Meaning of Basic terms used in probability like sample space, event etc.
Materials Required
A piece of cardboard, white drawing sheet, sketch pen, scale, pencil, a pair of dice.
Demonstration
(a) Paste a white drawing sheet on a cardboard of size 20x20 cm.
(b) Make a square and divide it into 36 unit squares of size 1 unit each (see Fig 27.1)
(C) Write a pair of numbers as we throw a pair of dice. Like (1, 1) (1, 2), (1, 3) .. (1, 6) etc. On the 36
squares drawn on white sheet. (as shown in Fig. 27.2)
(d) Fig. 27.2 represents the sample space of the experiment, which is also known as total outcomes.
1.11.21.31. 41.51.6
.13,23,313. 43.63.6
4.114.24.34,44. 54.6
5.5. 25.54s.s.
e..6.6.4s.ss.o
Fig. 27.1 Fig. 27.2
Experiments
Here we have taken two experiments:
1. To find the conditional probability of an event A if an event B has already ocurred where A is the event "a
number 5 appears on both the dice and B Is event "5 has appeared atleast one of the dice" i.e. we have
find P (AB).
2. Reverse of the above i.e. if the event A has already occurred, to find P(B/A).
5)}
((5. 27.2 shaded square)
A
=
in fig.
Fvent
I
(showing in Fig. 27.2
squares
a) =
ighlighted
{ A )
c o u n t
higl
event
B
For
n ( B ) = 11 B) = 1
) outcomes
in (A n
)No. offavourable
able
Eyperiment No. 1.
= 36
outcomes
)Total
no.
of
= , P(A n B) =
36
P(A)= P(B) 36
1
P(A n B) 36 = (using formula)
11 11
) P(A/B) P(B) =(where
event B act as sample space
36 respect to B
P(A/B)
= Probability of Awith
experiment
Through
(ii) event A)
for the
Experiment No. 2.
outcomes
= 36
Total
0) 11 (A nB)
P(B): 36 PP (A nB) =
36
P(A) = 36
P(BnA) = 1. (using formulas)
(i) P(B/A) P(A) space for
event B)
A act as a sample
36 A =
= 1 (where event
with respect to
= Probability of B
(in) P(B/A)
formula and using B as
doublet
APPLICATION a sum of 10 when a
such as the
probability of getting
other events
done for any
(a) This activity c a n be
further
has already occurred. which is
three cons. conditional probability
ot
can be done using the concept
(6) This activity understanding
useful or helpful in
C)This activity is
used in Baye's theorem. --*
BLANKS
FILLIN THE dice
are
rolled".
two
"when
based,
All estions are
die
thrown
twice
is at least once then number of event
Sample points
for a
th
number 4
t numbe
aat
appears
) .tth
event
Sol. 36 then E be
twice
and number of
When two die thrown 4), (3, 5), (4, 2).
(5, 1)}
(ü) are {(1, 5), ((2,
{(1,
is 66
are
in E n u m b e r
appearing
eing
Sol. 11 sum
of
that
F be event nt
ii)
events in F is .
Sol. 5.