\[\text{To draw the graph of given equations of the circles}\ x^2\ +\ y^2\ -\ 4x\ -\ 6y\ +\ 8\ =\ 0\ \hspace{7cm}\]\[and\ x^2\ +\ y^2\ -\ 10x\ -\ 6y\ +\ 14\ =\ 0\]
\[i).\ \text{Graph the equations of the circles in the Cartesian plane}\ \hspace{10cm}\]
\[ii).\ \text{To determine the coordinates of the centres and radii of the circles and mark them on the graph.}\ \hspace{5cm}\]
\[iii).\ \text{To determine the distance between the centres of the circles.}\ \hspace{10cm}\]
\[iv).\ \text{Verify whether any of the relationships}\ C_1\ C_2\ =\ r_1\ +\ r_2\ or\ C_1\ C_2\ =\ r_1\ -\ r_2\ holds\ or\ not\ \hspace{4cm}\]
\[\color {blue} {Soln:}\ \hspace{20cm}\]
Ex -2
\[\text{A pair of super gear consists of }\ Z_p\ =)\ \text{20 teeth pinion meshing with}\ (Z_g\ =)\ 120\ teeth\ gear.\ \hspace{3cm}\]
\[\text{Let the module be (m =) 4mm.}\ \hspace{5cm}\]
\[i).\ \text{Calculate the pitch circle diameters of the pinion and the gear using the formulae}\ d_p\ =\ m\ Z_p\hspace{5cm}\]\[and\ d_g\ =\ m\ Z_g\ \hspace {10cm}\]
\[ii).\ \text{Calculate the distance between the centres of the pinion and the gear using the formula}\ \hspace{5cm}\]\[\frac{1}{2}(d_p\ +\ d_g)\ \hspace {10cm}\]
\[iii).\ \text{Draw two externally touching circles to represent pinion and gear with appropriate centres and}\ \hspace{5cm}\]\[radii\ \frac{1}{2}\ d_p\ and \ \frac{1}{2}\ d_g.\ \text{Calculate the distance between the centres of the circles from the graph}\ \hspace {2cm}\]\[\text{and verify that it is equal to}\ \frac{1}{2}(d_p\ +\ d_g)\ \hspace {10cm}\]
Ex – 3
\[\text{Do the following activities for the given image of a parabolic shaped satellite dish antenna.}\ \hspace{10cm}\]
\[i).\ \text{Draw a parabola which fits the given image of the dish antenna.}\ \hspace{8cm}\]
\[ii).\ \text{Write the equation of the parabola.}\ \hspace{14cm}\]
\[iii).\ \text{Find the vertex, focus, directrix and latus-rectum and mark them on the graph.}\ \hspace{7cm}\]
\[iv).\ \text{How far from the vertex should the receiver of the antenna be placed?}\ \hspace{10cm}\]
Ex – 4
\[\text{Do the following actives for the given image of a parabolic shaped fly-over bridge..}\ \hspace{10cm}\]
\[i).\ \text{Draw a parabola which fits the given bridge.}\ \hspace{13cm}\]
\[ii).\ \text{Write the equation of the parabola.}\ \hspace{15cm}\]
\[iii).\ \text{Find the vertex, focus, directrix and lotus – rectum and mark them on the graph.}\ \hspace{10cm}\]
Ex – 5
\[\text{Do the following actives.}\ \hspace{15cm}\]
\[i).\ \text{Graph the polynomial function}\ π(π₯) =\ x^2\ -\ 5x\ +\ 6.\ \hspace{12cm}\]\[\text{Find the value of π(π₯) at π₯ = 5 and the limit of π(π₯) at π₯ = 5.}\ \hspace {8cm}\]
\[ii).\ \text{Graph the rational function}\ f(π₯) =\ \frac{x^2\ -\ 5x\ +\ 6}{x\ -\ 2}.\ \hspace{12cm}\]\[\text{Find the value of π(π₯) at π₯ = 4 and the limit of π(π₯) at π₯ = 4.}\ \hspace {8cm}\]
Ex – 6
\[\text{Two parallel straights of V = 20m apart are to be connected by reverse curve consisting}\ \hspace{7cm}\]\[\text{of arcs of same radius. The distance between the end points of the curve is L = 200m.}\ \hspace{3cm}\]
\[i).\ \text{Find the approximate value of the common radius..}\ \hspace{12cm}\]
\[ii).\ \text{Find the length of the whole curve.}\ \hspace{13cm}\]
Ex – 7
Do the following activities.
\[i).\ \text{Graph the functions c (constant)},\ x^n,\ n\ \in\ R,\ sin\ x.\ \ \hspace{10cm}\]\[\text{Find their indefinite integrals.}\ \hspace {8cm}\]
\[ii).\ \text{Evaluate the definite integral},\ \int_a^b\ f(x)\ dx\ and\ \hspace{10cm}\]\[\text{relate it to the area under the curve y = f(x) between x-axis, x = a and x = b.}\ \hspace {2cm}\]
Ex – 8
Do the following activities for the given image of a closed irregular plane figure.
i) Mark the required number of points on the boundary of the figure.
ii) Draw the boundary of the figure by joining the points.
iii) Divide the figure into trapeziums using the points on the boundary.
iv) Calculate the approximate area of the figure.
Ex – 9
1. To find the mean for the following data of size 43, 45, 47, 44, 41, 46, 48, 47, 45, 48, 45, 42, 41, 49, 42, 46, 44, 49, 47, 49, 41, 50, 45, 45, 49, 42, 46, 49, 47, 46, 45, 45, 49, 41, 45, 49, 50, 43, 45, 47, 40, 45, 48, 43, 42, 46, 43, 45, 46, 42.
2. To find the variance and standard deviation for the given data.
3. To fit the normal curve f(x) = N(, ) = , – < x <
4. To calculate the probability p = P(44 < X < 47) using the formula
5. To calculate the probability = P(44 < X < 47) using the probability calculator and verify that = p approximately.
6. To calculate the number of data points in the interval (44, 47) using the formula n = and verify it with the given data.
Ex: 10
Consider the 4 samples each of size 5 taken from the production lot of a machine.
Sample Number
6.35
6.40
6.32
6.37
6.42
6.34
6.40
6.34
6.36
6.39
6.42
6.64
6.35
6.59
6.72
6.38
6.48
6.44
6.58
6.39
1. To calculate the sample means , , , and the mean of the sample means
.
2. To calculate the sample variances , , , and =
3. To determine the central line CL = , lower control limit LCL = – and
upper control limit UCL = +
4. To draw the chart and determine the out – of – control signals.
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