Activities for Class XIIThe basic principles of learn (a) learning should be related (b) the need for mathe intimate acquaintance wit should be active and ir wide variety of illustrati: process (e) unde stage of acq broadly base ee various brancl adequi mathematical usage should be ing mathematics ech child indi hot velog es an mi ‘the child terial and the learning raged at each content should be iation of the links thematics, (g) correct encouraged at all stages. —Ronwill © scanned ih oven scamerctvity OpyectivE Materiat REQUIRED To verify that the relation R in the set A piece of plywood, some pieces of L of all lines in a plane, defined by wires (8), nails, white paper, glue etc. R= {(/,m):11L m)} is symmetric but neither reflexive nor transitive. METHOD OF CONSTRUCTION Take a piece of plywood and paste a white paper on it. Fix the wires randomly on the plywood with the help of nails such that some of them are parallel, some are perpendicular to each other and some are inclined as shown in Fig.1. I, i fe 4 L oe ie h — 4 ¥ Fig. 1 DEMONSTRATION 1, Let the wires represent the lines /,, 1,, .... ly. 2. 1, is perpendicular to each of the lines 1, /,, 1,. [see Fig. © scanned wth oKEnscamer3. J, is perpendicular to /,. 4. 1, is parallel to /,, /, is parallel to J, and J, is parallel to /,. Sis Ls Uy Ids Up bys ly L) RK OBSERVATION 1. In Fig. 1, no line is perpendicular to itself, so the relation R= {(l,m): 11m} _____ reflexive (is/is not). 2. InFig. 1, hip. Is, 11,2 (Yes/No) (1, L)¢ R= (1, |.) ___R (€/e) Similarly, J, 11, .Is J, 11,2 (Yes/No) (yh) € R= (1,1) R (gle) Also, /,11.Isi,1/,? — (Yes/No) (1L,lye R=>(L,1) — _R_(€/e) The relation R .... symmetric (isfis not) 3. In Fig. 1,1, L1,and I, J, . Is 1, 11,2 .1. (Yes/No) ie, le Rand (,,. 1) R> (1) R(¢/e) The relation R ..... transitive (is/is not). APPLICATION This activity can be used to check whether a 1. In this case, the relation is given relation is an equivalence relation or not an equivalence relation. not. 2. The activity can be repeated by taking some more wire in different positions. 102 Laboratory Manual © scanned wth oKEnscameromy § OBJECTIVE Materia REQUIRED To verify that the relation R in the set A piece of plywood, some pieces of L of all lines in a plane, defined by wire (8), plywood, nails, white paper, R= {(/, m): Illm) isan equivalence glue. relation. METHOD OF CONSTRUCTION Take a piece of plywood of convenient size and paste a white paper on it. Fix the wires randomly on the plywood with the help of nails such that some of them are parallel, some are perpendicular to each other and some are inclined as shown in Fig. 2. Fig.2 DEMONSTRATION 1. Let the wires represent the lines /,, 1,, .... ,- 2. 1, is perpendicular to each of the lines /,, 1, , (see Fig. 2). © scanned wth oKEnscamer3. J, is perpendicular to 1. 4. 1, is parallel to /,, J, is parallel to J, and I, is parallel to /,. 5. (yp yds (yy Ls (ley Hye © R OBSERVATION 1, In Fig. 2, every line is parallel to itself. So the relation R = {(/, m): Ill m} reflexive relation (is/is not) 2, In Fig. 2, observe that Jy. Is, .. 2 WII) So, (lye R= (L,1)..R le) Similarly, IM Lye IS Uy oe? OMI) So, Golye R= (I, 1)..R ele) and (ide R= (ly l).R (le) ~. The relation R ... symmetric relation (is/is not) 3. In Fig. 2, observe that /, II, and J, dj. Is J, 00,2 (V/IL) So, (1, 1) © Rand (/,, lL) € R = (1, L,) ... R(E/e) Similarly, 1,Wd,andd, Il /,. Ist, ... 1,2 OI) So, (L, Lye R, (LL) € R= (L, L) .. RE.) Thus, the relation R ... transitive relation (is/is not) Hence, the relation R is reflexive, symmetric and transitive. So, R is an equivalence relation. APPLICATION This activity is useful in understanding the | This activity can be repeated concept of an equivalence relation. by taking some more wires in different positions. Laboratory Manual © scanned wth oKEnscamermonly OBJECTIVE Materia REQUIRED To demonstrate a function which is Cardboard, nails, strings, adhesive not one-one but is onto. and plastic strips. MErHOp OF CONSTRUCTION 1. Paste a plastic strip on the left hand side of the cardboard and fix three nails on it as shown in the Fig.3.1. Name the nails on the strip as 1, 2 and 3. . Paste another strip on the right hand side of the cardboard and fix two nails in 2 the plastic strip as shown in Fig.3.2. Name the nails on the strip as a and b. 3. Join nails on the left strip to the nails on the right strip as shown in Fig. 3.3. x el Y de. ae ee a 02 a} “KO ° P>.5 3 Y 3e— I Fig. 3.1 Fig. 3.2 Fig. 3.3 DEMONSTRATION 1, Take the set X = {1, 2, 3} 2. Take the set Y = (a, b} 3. Join (correspondence) elements of X to the elements of Y as shown in Fig. 3.3 OBSERVATION 1. The image of the element 1 of X in Y is The image of the element 2 of X in Y is © scanned wth oKEnscamerThe image of the element 3 of X in Y is : So, Fig. 3.3 represents a : N . Every element in X has a _________(one-one/not one-one). 3. The pre-image of each element of Y in X So, the function is APPLICATION This activity can be used to demonstrate the concept of one-one and onto function. ___ image in Y. So, the function is (exists/does not exist). (onto/not onto). Demonstrate the same activity by changing the number of the elements of the sets X and Y. Laboratory Manual © scanned wth oKEnscamerctvity OBJECTIVE Materia ReQuirep To demonstrate a function which is Cardboard, nails, strings, adhesive one-one but not onto. and plastic strips. METHOD OF CONSTRUCTION 1. Paste a plastic strip on the left hand side of the cardboard and fix two nails in it as shown in the Fig. 4.1. Name the nails as a and b. 2. Paste another strip on the right hand side of the cardboard and fix three nails on it as shown in the Fig. 4.2. Name the nails on the right strip as 1,2 and 3. 3. Join nails on the left strip to the nails on the right strip as shown in the Fig. 4.3. y x oI | ] : be. Fig. 41 Fig. 4.2 Fig. 4.3 DEMONSTRATION 1. Take the set X = {a, b} 2. Take the set Y = {1, 2, 3}. 3. Join elements of X to the elements of Y as shown in Fig. 4.3. © scanned wth oKEnscamerOBSERVATION 1. The image of the element a of X in Y is The image of the element b of X in Y is So, the Fig. 4.3. represents a 2. Every element in X has a image in Y. So, the function is (one-one/not one-one). 3. The pre-image of the element 1 of Y in X (exists/does not exist). So, the function is (onto/not onto). Thus, Fig. 4.3 represents a function which is but not onto. APPLICATION This activity can be used to demonstrate the concept of one-one but not onto function. 108 Laboratory Manual © scanned wth oKEnscamersctvity § ObgECTIVE To draw the graph of sin“! x, using the graph of sin x and demonstrate the concept of mirror reflection (about the line y = x). METHOD OF CONSTRUCTION Marerist REQUIRED Cardboard, white chart paper, ruler, coloured pens, adhesive, pencil, eraser, cutter, nails and thin wires. 1. Take a cardboard of suitable dimensions, say, 30 cm x 30 cm. 2. On the cardboard, paste a white chart paper of size 25 cm x 25 cm (say). 3. On the paper, draw two lines, perpendicular to each other and name them X’OX and YOY’ as rectangular axes [see Fig. 5]. © scanned wth oKEnscamer4. Graduate the axes approximately as shown in Fig. 5.1 by taking unit on X-axis = 1.25 times the unit of Y-axis. 5. Mark approximately the points uk) (m0 Tin). . ome) get gp | g78it g | im the coordinate plane and at each point fix a nail. 6. Repeat the above process on the other side of the x-axis, marking the points —m . -m) (- . —0 a. 0 ° sin Losing} | asin approximately and fix nails on these points as N,’, N,’, N,’, Ny’. Also fix a nail at O. 7. Join the nails with the help of a tight wire on both sides of x-axis to get the h of sin x from — to = graph of sin x from * to 2, 8. Draw the graph of the line y = x (by plotting the points (1,1), (2, 2), (3, 3), ... etc. and fixing a wire on these points). 9. From the nails N,, N,,N,.N,, draw perpendicular on the line y =x and produce these lines such that length of perpendicular on both sides of the line y = x are equal. At these points fix nails, I,.L,l1,. 10. Repeat the above activity on the other side of X- axis and fix nails at L.L/.1/1/. 11. Join the nails on both sides of the line y = x by a tight wire that will show the DEMONSTRATION Put a mirror on the line y = x. The image of the graph of sin x in the mirror will represent the graph of sin“'x showing that sin”! x is mirror reflection of sin x and vice versa. Ho) Laboratory Manual © scanned wth oKEnscamerOBSERVATION The image of point N, in the mirror (the line y = x) is . The image of point N, in the mirror (the line y = x) is . The image of point N, in the mirror (the line y = x) is The image of point N, in the mirror (the line y = x) is The image of point Nj in the mirror (the line y = x) is . The image point of N%, in the mirror (the line y = x) is . The image point of N’, in the mirror (the line y = x) is The image point of Ni, in the mirror (the line y = x) is . ‘The image of the graph of six x in y x is the graph of , and the image of the graph of sin"'x in y = x : the graph of APPLICATION Similar activity can be performed for drawing the graphs of cos~'x, tan” x, ete. © scanned wth oKEnscamerra Activity § OBJECTIVE Mareriar ReQuireD To explore the principal value of Cardboard, white chart paper, rails, the function sin-'x using a unit tuler, adhesive, steel wires and circle. needle. MEetHop OF CONSTRUCTION Take a cardboard of a convenient size and paste a white chart paper on it. 2. Draw a unit circle with centre O on it. . Through the centre of the circle, draw two perpendicular lines X’OX and YOY’ representing x-axis and y-axis, respectively as shown in Fig. 6.1. . Mark the points A, C, B and D, where the circle cuts the x-axis and y-axis, respectively as shown in Fig. 6.1. . Fix two rails on opposite sides of the cardboard which are parallel to y-axis, Fix one steel wire between the rails such that the wire can be moved parallel to x-axis as shown in Fig. 6.2. Fig. 6.1 © scanned wth omen camer6. Take a needle of unit <—Rail length. Fix one end of it at the centre of the < - €Steel wire . \ Needle circle and the other end to move freely along the circle Fig. 6.2. DEMONSTRATION 1. Keep the needle at an arbitrary angle, say x, with the positive direction of x-axis. Measure of angle in radian is equal to the length of intercepted arc of the unit circle. Yy Fig. 6.2 2. Slide the steel wire between the rails, parallel to x-axis such that the wire meets with free end of the needle (say P,) (Fig. 6.2). 3. Denote the y-coordinate of the point P, as y,, where y, is the perpendicular distance of steel wire from the x-axis of the unit circle giving y, = sin x,. 4. Rotate the needle further anticlockwise and keep it at the angle 1 — x,. Find the value of y-coordinate of intersecting point P, with the help of sliding steel wire. Value of y-coordinate for the points P, and P, are same for the different value of angles, y, = sinx, and y, = sin (1 — x,). This demonstrates that sine function is not one-to-one for angles considered in first and second quadrants. 5. Keep the needle at angles — x, and (— % + x,), respectively. By sliding down the steel wire parallel to x-axis, demonstrate that y-coordinate for the points P, and P, are the same and thus sine function is not one-to-one for points considered in 3rd and 4th quadrants as shown in Fig. 6.2. Naeraes 1B © scanned wth omen camer6. However, the y-coordinate of the points P, and P, are different. Move the needle in anticlockwise direction £m 22 look at the behaviour of X y-coordinates of points P,, P,, P, and P, by sliding the steel wire parallel to x-axis accordingly. y-co- ordinate of points P,, P,, P. and P, are different (see Fig. 6.3). Hence, sine function is one-to-one in starting from — and y Fig. 6.3 nt the domian [-z. | and its range lies between — 1 and 1. x . Keep the needle at any arbitrary angle say @ lying in the interval [ and denote the y-coordi- nate of the intersecting point P, as y. (see Fig. 6.4). Then y = sin 8 or 6 = are sinr'y) as sine function is one-one and onto in the aT domain [: 5] and x . range [-l, 1]. So, its inverse arc sine function exist. The domain of arc sine function is [-1, 1] and nd Laboratory Manual © scanned wth omen camernt range is [-5 £], This range is called the principal value of arc sine function (or sin” function). OBSERVATION 1. sine function is non-negative in and quadrants. 2. For the quadrants 3rd and 4th, sine function is T 3. @=are siny>y = 8 where 5 SOS 4. The other domains of sine function on which it is one-one and onto provides for are sine function. APPLICATION This activity can be used for finding the principal value of arc cosine function (cosy). © scanned wth oKEnscamerActivity 7 OBJECTIVE Mareriar ReQuireD To sketch the graphs of a* and log x, Drawing board, geometrical instru- a>0,a# Land to examine that they ments, drawing pins, thin wires, are mirror images of each other. sketch pens, thick white paper, adhesive, pencil, eraser, a plane mirror, squared paper. MEetHOp OF CONSTRUCTION 1. On the drawing board, fix a thick paper sheet of convenient size 20 cm x 20 cm (say) with adhesive. Fig.7 © scanned wth oKEnscamer2. On the sheet, take two perpendicular lines XOX’ and YOY’, depicting coordinate axes. 3. Mark graduations on the two axes as shown in the Fig. 7. 4. Find some ordered pairs satisfying y = a and y = log,x. Plot these points corresponding to the ordered pairs and join them by free hand curves in both the cases. Fix thin wires along these curves using drawing pins. 5. Draw the graph of y = x, and fix a wire along the graph, using drawing pins. DEMONSTRATION 1. For a’, take a = 2 (say), and find ordered pairs satisfying it as nis i) and plot these ordered pairs on the squared paper and fix a drawing pin at each point. 2. Join the bases of drawing pins with a thin wire. This will represent the graph of 2", 3. log,x= y gives x=2". Some ordered pairs satisfying it are: 1 2] ifa - 8 1 * 2 4 8 y 0 1 -1 2 -2 3 -3 Plot these ordered pairs on the squared paper (graph paper) and fix a drawing pin at each plotted point. Join the bases of the drawing pins with a thin wire. This will represent the graph of log,x. Mathematics a © scanned wth oKEnscamer4. Draw the graph of line y = x on the sheet. 5. Place a mirror along the wire representing y = x. It can be seen that the two graphs of the given functions are mirror images of each other in the line y = x. OBSERVATION 1, Image of ordered pair (1, 2) on the graph of y = 2" in y = xis - It lies on the graph of y = . 2. Image of the point (4, 2) on the graph y = log,x in y = xis which lies on the graph of y = Repeat this process for some more points lying on the two graphs. APPLICATION This activity is useful in understanding the concept of (exponential and logarithmic functions) which are mirror images of each other in y = x. us) Laboratory Manual © scanned wth oKEnscamerctvity ObgECTIVE Materia REQUIRED To establish a relationship between Hardboard, white sheet, graph common logarithm (to the base 10) paper, pencil, scale, log tables or and natural logarithm (to the base e) calculator (graphic/scientific). of the number x. METHOp OF CONSTRUCTION 1. Paste a graph paper on a white sheet and fix the sheet on the hardboard. 2. Find some ordered pairs satisfying the function y = log,,x. Using log tables/ calculator and draw the graph of the function on the graph paper (see Fig. 8) © scanned wth oKEnscamer3. Similarly, draw the graph of y’ = log.x on the same graph paper as shown in the figure (using log table/calculator). DEMONSTRATION . Take any point on the positive direction of x-axis, and note its x-coordinate. 2. For this value of x, find the value of y-coordinates for both the graphs of y= log,,« and y’ = log,x by actual measurement, using a scale, and record them as y and y’, respectively. 7? 3. Find the ratio y 4, Repeat the above steps for some more points on the x-axis (with different values) and find the corresponding ratios of the ordinates as in Step 3. 5. Each of these ratios will nearly be the same and equal to 0.4, which is 1 approximately equal to log, 10° OBSERVATION z * ao S.No. | Points on y=log, x y’=log,x Ratio +7 the x-axis (approximate) 2 yee J 3. ye | 4. ye | 5. a 6. ye | —— (20 Taberaony Manual © scanned wth oKEnscamerThe value of y for each point x is equal to approximately. y The observed value of +7 in each case is approximately equal to the value of 1 Tog, 10-%esNo) 4. Therefore, logy, x ~ log, 10° APPLICATION that number in another base. This activity is useful in converting log of a number in one given base to log of Let, y = logy, Le. x= 10, Taking logarithm to base ¢ on both the sides, we get log, x= ylog, 10 ——(I toe, 10 "8°4) login X log,x log, 10 > = 0.434294 (using log tables/calculator). ‘Mathemati io © scanned wth omen camerctvity OBJECTIVE To find analytically the limit of a function f(x) at x= and also to check the continuity of the function at that point. MetHop OF CONSTRUCTION MarertAt REQUIRED Paper, pencil, calculator. x =16 1. Consider the function given by f=} -4 10, x#4 x=4 2. Take some points on the left and some points on the right side of ¢ (= 4) which are very near to c. 3, Find the corresponding values of f (x) for each of the points considered in step 2 above. 4. Record the values of points on the left and right side of c as x and the corresponding values of f (x) in a form of a table. DEMONSTRATION 1. The values of x and f (x) are recorded as follows: Table 1: For points on the left of c (= 4). x |3.9] 3.99 | 3.999 3.9999 3.99999 3.999999] 3.9999999 f (7.9 | 7.99 | 7.999 7.9999 7.99999 7.999999 7.9999999 © scanned wth oKEnscamer2. Table 2: For points on the right of ¢ (= 4). x 4.1 | 4.01}4.001 | 4.0001] 4.00001 | 4.000001 | 4.0000001 Ff (x)/8.1 | 8.01) 8.001 | 8.0001} 8.00001 | 8.000001 | 8.0000001 OBSERVATION 1. The value of f (x) is approaching to » as x4 from the left. 2. The value of f (x) is approaching to , as x—»4 from the right. 3. So, lim f(x) = and lim f(x)= 4. Therefore, lim f (x)= fAa= 5. 1s lim s(x) f (4) _____? (Yes/No) 6. Since f (c)#lim f(x), so, the function is at x = 4 (continuous/ xe not continuous). APPLICATION This activity is useful in understanding the concept of limit and continuity of a function at a point. Mathematics a © scanned wth oKEnscamerro Activity 10 OBJECTIVE Mareriar ReQuireD To verify that for a function f to be Hardboard, white sheets, pencil, continuous at given point x,, scale, calculator, adhesive. Ay=|f (a + Ax)= f (%)] is arbitrarily small provided. Avis sufficiently small. MEtuop oF ConstRUCTION 1. Paste a white sheet on the hardboard. 2. Draw the curve of the given continuous function as represented in the Fig. 10. 3. Take any point A (x,, 0) on the positive side of x-axis and corresponding to this point, mark the point P (x,, y,) on the curve. Yy © scanned wth omen camerDEMONSTRATION 1. Take one more point M, (x, + Ax, 0) to the right of A, where Ax, is an increment in x. N . Draw the perpendicular from M, to meet the curve at N,. Let the coordinates of N, be (x, + Ax, y, + Ay,) . Draw a perpendicular from the point P (x,, y,) to meet N,M, at T,. w 4, Now measure AM, = Ax, (say) and record it and also measure N,T, = Ay, and record it. 5. Reduce the increment in x to Ax, (i.e., Av, < Ax,) to get another point M, (xy +Ax,,0). Get the corresponding point N, on the curve 6. Let the perpendicular PT, intersects NM, at T,. 7. Again measure AM, =Ax, and record it. Measure N,T,=Ay, and record it. 8, Repeat the above steps for some more points so that Ax becomes smaller and smaller. OBSERVATION S.No. | Value of increment Corresponding in x, increment in y | [aul ——— | [a 2 | Jasle —— | lle 3 | [ays —— ] Janis 4 | fani= —— ] lanl= 5. | [Axsl= Mathematics 1s © scanned wth oKEnscamer2. So, Ay becomes when Ax becomes smaller. 3. Thus Jim y= 0 for a continuous function. APPLICATION This activity is helpful in explaining the concept of derivative (left hand or right hand) at any point on the curve corresponding to a function. yyy pe Laboratory Manual © scanned wth oKEnscamer