Applied Mathematics – II (Practicals-for circuit)

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Regulation 2023

 

Ex-1:
Do the following activities.
(i) Draw the graphs of the parabolas (π‘¦βˆ’π‘˜)2=4π‘Ž(π‘₯βˆ’β„Ž) and (π‘₯βˆ’β„Ž)2=4π‘Ž(π‘¦βˆ’π‘˜) for the given values of π‘Ž,𝑏,β„Ž 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‘₯βˆ’β„Ž)2π‘Ž2+(π‘¦βˆ’π‘˜)2𝑏2=1 for the given values of π‘Ž,𝑏 ,β„Ž and π‘˜. Determine the eccentricity, centre, foci, vertices, major axis, minor axis, directrices, and latus-rectums and mark them on the graph.

Ex-2:
Do the following activities for the given image of a parabolic shaped satellite dish antenna.
(i) Draw a parabola which fits the given image of the dish antenna.
(ii) Write the equation of the parabola.
(iii) Find the vertex, focus, directrix and latus-rectum and mark them on the graph.
(iv) How far from the vertex should the receiver of the antenna be placed?

Ex-3:

Do the following activities.
i. Draw the graphs of sinπ‘₯,cosπ‘₯,tanπ‘₯,cosecπ‘₯,secπ‘₯ and cotπ‘₯ and write their domain and range. Find the maximum value, minimum value, amplitude and periodicity of sinπ‘₯,cosπ‘₯,tanπ‘₯,cosecπ‘₯,secπ‘₯ and cotπ‘₯.
ii. Draw the graphs of sinβˆ’1π‘₯,cosβˆ’1π‘₯,tanβˆ’1π‘₯,cosecβˆ’1π‘₯,secβˆ’1π‘₯ and cotβˆ’1π‘₯ and write their domain and range. Find the maximum value and minimum value of sinβˆ’1π‘₯,cosβˆ’1π‘₯,tanβˆ’1π‘₯,cosecβˆ’1π‘₯,secβˆ’1π‘₯ and cotβˆ’1π‘₯.

Ex-4:
The alternating current passing through a circuit is 𝑖(𝑑)=πΌπ‘šsinπœ”π‘‘, where πΌπ‘š is the maximum value of current and πœ” is the angular velocity. Let 𝑅 be the resistance and π‘‰π‘š be the maximum voltage.
i. Graph the sinusoidal waveform of 𝑖(𝑑) for the given values of πΌπ‘š and πœ”.
ii. Calculate the maximum voltage π‘‰π‘š using the formula π‘‰π‘š=πΌπ‘šπ‘… for the given value of 𝑅.
iii. Graph the sinusoidal waveform of voltage using the formula 𝑣(𝑑)=π‘‰π‘šsin(πœ”π‘‘+πœ‹2).
iv. Determine the value of root mean square (r.m.s) current.
v. Determine the frequency.
vi. Calculate the instantaneous value of the current at 𝑑 sec.

Ex-5:
Do the following activities.
i. Mark the given complex number 𝑧 on the Argand plane. Find the real and imaginary parts of 𝑧. Find the distance of 𝑧 from π‘₯-axis and relate it to the real part of 𝑧. Find the distance of 𝑧 from 𝑦-axis and relate it to the imaginary part of 𝑧.
ii. Find the conjugate of 𝑧. Mark 𝑧̅ on the Argand plane. Find the reflection of 𝑧 on π‘₯-axis and relate it to 𝑧̅.
iii. Find the modulus of 𝑧. Find the distance between 𝑧 and origin of the Argand plane and relate it to the modulus of 𝑧. Find the modulus of 𝑧̅ and relate it to the modulus of 𝑧.
iv. Find the argument of 𝑧. Find the angle between the line segment 𝑂𝑧 and π‘₯ axis and relate it to the argument of 𝑧. Find the argument of 𝑧̅ and relate it to the argument of 𝑧.

Ex-6:
Do the following activities.
i. The representation of apparent power as phasor sum of active power and reactive power is given by 𝑆=550+952.63𝑗. Draw the phasor diagram of the system. Find the numerical value of the apparent power. Also calculate the phase angle and power factor of the system.
ii. A machine takes 10KW (real power, P) at a power factor of 0.6 from 400V supply. Calculate the total load in KVA and KVAR. Represent the apparent power as a phasor sum of active power and reactive power.

Ex-7:
Do the following activities.
(i) Graph the polynomial function 𝑓(π‘₯)=π‘Žπ‘›π‘₯𝑛+π‘Žπ‘›βˆ’1π‘₯π‘›βˆ’1+β‹―+π‘Ž1π‘₯+π‘Ž0, where π‘Žπ‘›,π‘Žπ‘›βˆ’1,…,π‘Ž0 are real numbers and π‘Žπ‘›β‰ 0. Find the value of 𝑓(π‘₯) at π‘₯=π‘Ž and the limit of 𝑓(π‘₯) at π‘₯=π‘Ž.
(ii) Graph the rational function 𝑅(π‘₯)=π‘Žπ‘›π‘₯𝑛+π‘Žπ‘›βˆ’1π‘₯π‘›βˆ’1+β‹―+π‘Ž1π‘₯+π‘Ž0π‘π‘šπ‘₯π‘š+π‘π‘šβˆ’1π‘₯π‘šβˆ’1+β‹―+𝑏1π‘₯+𝑏0, where π‘Žπ‘›,π‘Žπ‘›βˆ’1,…,π‘Ž0,π‘π‘š,π‘π‘šβˆ’1,…,𝑏0 are real numbers and π‘Žπ‘›,π‘π‘šβ‰ 0. Find the value of 𝑓(π‘₯) and the limit of 𝑓(π‘₯) at π‘₯=π‘Ž.
(iii) Graph the functions sinπ‘Žπ‘₯𝑏π‘₯ and tanπ‘Žπ‘₯𝑏π‘₯ where π‘Ž and 𝑏 are real numbers and π‘Ž,𝑏≠0. Evaluate limπ‘₯β†’0sinπ‘Žπ‘₯𝑏π‘₯ and limπ‘₯β†’0tanπ‘Žπ‘₯𝑏π‘₯.
(iv) Graph the functions 𝑐 (π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘), π‘₯𝑛, sinπ‘₯, cosπ‘₯, tanπ‘₯, cosecπ‘₯, secπ‘₯, cotπ‘₯, 𝑒π‘₯ and logπ‘₯. Find their first derivative and second derivative.

Ex-8:
The alternating current passing through a circuit is 𝑖(𝑑)=πΌπ‘šsinπœ”π‘‘ where, πΌπ‘š is the maximum value of current and πœ” is the angular velocity. Let 𝐿 be the inductance.
(i) Graph the sinusoidal wave form of 𝑖(𝑑) for the given values of πΌπ‘š and πœ”.
(ii) Graph the voltage using the formula 𝑣(𝑑)=𝐿𝑑𝑖(𝑑)𝑑𝑑 for the given value of 𝐿.
(iii) Determine the values of 𝑖(𝑑) and 𝑣(𝑑) for a fixed 𝑑 and different values of πœ”.
(iv) Determine the values of 𝑖(𝑑) and 𝑣(𝑑) for fixed value of πœ” and different values of 𝑑.
(v) Determine the values of 𝑑 for which 𝑖(𝑑) and 𝑣(𝑑) are equal.

Ex-9:
Do the following activities.
i. Graph the functions 𝑐 (π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘), π‘₯𝑛,π‘›βˆˆβ„, 𝑒π‘₯, sinπ‘₯, cosπ‘₯, sec2π‘₯, cosec2π‘₯, secπ‘₯tanπ‘₯ and cosecπ‘₯cotπ‘₯. Find their indefinite integrals.
ii. Evaluate the definite integral βˆ«π‘“(π‘₯)π‘π‘Žπ‘‘π‘₯ and relate it to the area under the curve 𝑦=𝑓(π‘₯) between π‘₯-axis, π‘₯=π‘Ž and π‘₯=𝑏.
iii. Find the volume of the solid generated by the revolution of the area bounded by 𝑦=𝑓(π‘₯),π‘₯-axis, π‘₯=π‘Ž and π‘₯=𝑏 about π‘₯-axis.

Ex-9:
Do the following activities.
i. Graph the functions 𝑐 (π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘), π‘₯𝑛,π‘›βˆˆβ„, 𝑒π‘₯, sinπ‘₯, cosπ‘₯, sec2π‘₯, cosec2π‘₯, secπ‘₯tanπ‘₯ and cosecπ‘₯cotπ‘₯. Find their indefinite integrals.
ii. Evaluate the definite integral βˆ«π‘“(π‘₯)π‘π‘Žπ‘‘π‘₯ and relate it to the area under the curve 𝑦=𝑓(π‘₯) between π‘₯-axis, π‘₯=π‘Ž and π‘₯=𝑏.
iii. Find the volume of the solid generated by the revolution of the area bounded by 𝑦=𝑓(π‘₯),π‘₯-axis, π‘₯=π‘Ž and π‘₯=𝑏 about π‘₯-axis.

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