Ex-1:

For the given equations of the circles π₯2+π¦2+2π1π₯+2π1π¦+π1=0 and π₯2+π¦2+2π2π₯+2π2π¦+π2=0 with appropriate coefficients,

i. Graph the equations of the circles in the Cartesian plane.

ii. Determine the coordinates of the centres and radii of the circles and mark them on the graph.

iii. Determine the distance between the centres of the circles.

iv. Determine whether the circles are touching each other or not.

v. If the circles are touching each other, determine whether they are touching internally or externally.

vi. Verify whether any of the relationships πΆ1πΆ2=π1+π2 or πΆ1πΆ2=|π1βπ2| holds or not.

Ex-2:

A pair of spur gears consists of (π§π=) 20 teeth pinion meshing with (π§π=) 120 teeth gear. Let the module be (π=) 4 mm.

i. Calculate the pitch circle diameters of the pinion and the gear using the formulae ππ=ππ§π and ππ=ππ§π.

ii. Calculate the distance between the centres of the pinion and the gear using the formula 12(ππ+ππ).

iii. Draw two externally touching circles to represent pinion and gear with appropriate centres and radii 12ππ and 12ππ. Determine the equations of the pinion and gear. Calculate the distance between the centres of the circles from the graph and verify that it is equal to 12(ππ+ππ).

iv. Calculate the tooth thickness using the formula π‘=1.5708π.

v. Calculate the gear ratio using the formula π=π§ππ§π

Ex-3:

Do the following activities.

(i) Draw the graphs of the parabolas (π¦βπ)2=4π(π₯ββ) and (π₯ββ)2=4π(π¦βπ) for the given values of π,π,β and π. Determine the vertex, focus, axis, directrix, latus-rectum of each parabola and mark them on the graphs.

(ii) Draw the graphs of the ellipse (π₯ββ)2π2+(π¦βπ)2π2=1 for the given values of π,π ,β and π. Determine the eccentricity, centre, foci, vertices, major axis, minor axis, directrices, and latus-rectums and mark them on the graph.

Ex-4:

Do the following activities for the given image of a parabolic shaped fly-over bridge.

(i) Draw a parabola which fits the given bridge.

(ii) Write the equation of the parabola.

(iii) Find the vertex, focus, directrix and latus-rectum and mark them on the graph.

(iv) Find the height and width of the bridge.

Ex-5:

Do the following activities.

(i) Graph the polynomial function π(π₯)=πππ₯π+ππβ1π₯πβ1+β―+π1π₯+π0, where ππ,ππβ1,β¦,π0 are real numbers and ππβ 0. Find the value of π(π₯) at π₯=π and the limit of π(π₯) at π₯=π.

(ii) Graph the rational function π
(π₯)=πππ₯π+ππβ1π₯πβ1+β―+π1π₯+π0πππ₯π+ππβ1π₯πβ1+β―+π1π₯+π0, where ππ,ππβ1,β¦,π0,ππ,ππβ1,β¦,π0 are real numbers and ππ,ππβ 0. Find the value of π(π₯) and the limit of π(π₯) at π₯=π.

(iii) Graph the functions sinππ₯ππ₯ and tanππ₯ππ₯ where π and π are real numbers and π,πβ 0. Evaluate limπ₯β0sinππ₯ππ₯ and limπ₯β0tanππ₯ππ₯.

(iv) Graph the functions π (ππππ π‘πππ‘), π₯π, sinπ₯, cosπ₯, tanπ₯, cosecπ₯, secπ₯, cotπ₯, ππ₯ and

logπ₯. Find their first derivative and second derivative.

Note: Only two functions will be given in Board Practical Examination in subdivision-(iv) of Ex-5.

Ex-6:

Two parallel straights of β²π₯β² m apart are to be connected by a reverse curve consisting of arcs of same radius. The distance between the end points of the curve is β²π¦β² m.

(i) Find the approximate value of the common radius.

(ii) Find the length of the whole curve

Ex-7:

Do the following activities.

i. Graph the functions π (ππππ π‘πππ‘), π₯π,πββ, ππ₯, sinπ₯, cosπ₯, sec2π₯, cosec2π₯, secπ₯tanπ₯ and cosecπ₯cotπ₯. Find their indefinite integrals.

ii. Evaluate the definite integral β«π(π₯)ππππ₯ and relate it to the area under the curve π¦=π(π₯) between π₯-axis, π₯=π and π₯=π.

iii. Find the volume of the solid generated by the revolution of the area bounded by π¦=π(π₯),π₯-axis, π₯=π and π₯=π about π₯-axis.

Note: Only two functions will be given in Board Practical Examination in subdivision-(i) of Ex-7.

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:

Do the following activities.

i. Find the mean π for the given data π₯1,π₯2,π₯3,β¦,π₯50 of size π=50.

ii. Find the variance π2 and standard deviation π forthe data given in (i).

iii. Fit the normal curve π(π₯)=π(π,π2)=1πβ2ππβ12(π₯βππ)2,ββ<π₯<β.

iv. Calculate the probability π=π(π1<π<π2) for some π1,π2 in the range of the data given in (i) using the formula β«π(π₯)ππ₯π2π1. Verify the answer using probability calculator.

v. Calculate the number of data points in the interval (π1,π2) using the formula π=ππ.

Ex-10:

Consider the 4 samples each of size 5 taken from the production lot of a machine

i. Calculate the sample means πΜ
1,πΜ
2,πΜ
3, πΜ
4 and the mean of the sample means πΜ
=πΜ
1+πΜ
2+πΜ
3+πΜ
44.

ii. Calculate the sample variances π£1,π£2,π£3,π£4 and π=β14Ξ£π£π4π=1.

iii. Determine the central line πΆπΏ=πΜ
, lower control limit πΏπΆπΏ=πΜ
β2.58β5π and upper control limit ππΆπΏ=πΜ
+2.58β5π.

iv. Draw the πΜ
chart and determine the out-of-control signals

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