Math previous y. question
Bangladesh Open University
School of Science and Technology
B.Sc in Computer Science and Engineering Program
171 Term (1st Year 1st Semester) Final Examination
Course Code & Title: MAT1134 Differential and Integral Calculus
Time: 3 hours
Total Marks: 70
[N.B.: Answer any 5 (five) questions. The figures in the right margin indicate the full marks. All portions of each question must be answered sequentially.]
1.¶
(a) Define Domain and Range with example.
(b) Sketch the graph of the function \(f(x) = |x| + |x - 1| + |x - 2|\). Find also the Domain and Range of that function.
(c) Discuss the continuity of the following function at \(x = 2\), when
Also check the differentiability of \(f(x)\) at \(x = 2\).
(Marks: 3 + 3 + 3)
2.¶
(a) Suppose that \(v(t) = 3t - 5\) for \(0 \leq t \leq 3\) is the velocity of moving particle at time \(t\). Determine the displacement of the particle over the interval [0, 3].
(b) Evaluate \(y = \sin\left(\sqrt{1 + \cos x}\right)\) with respect to \(x\)
(c) If \(x^y = e^{x+y}\), then prove that
(d) Differentiate \(x^{\sin^{-1}x}\) with respect to \((\sin x)^x\)
(Marks: 3 + 4 + 3 + 4)
3.¶
(a) Use implicit differentiation to prove that
(b) Find value of \(\xi\) in the Mean Value Theorem \(f(b) - f(a) = (b - a)f'(\xi)\), if \(f(x) = x^2, a = 1, b = 2\).
(c) Evaluate the following using L’Hospital’s Rule (any two):
(i) \(\lim_{x \to 0} \frac{e^x - 1}{x^3}\)
(ii) \(\lim_{x \to 0} \frac{\ln x}{\csc x}\)
(iii) \(\lim_{x \to \frac{\pi}{4}} (1 - \tan x)\sec 2x\)
(Marks: 4 + 4 + 6)
4.¶
(a) Find \(\frac{dy}{dx}\) in the following cases (any one):
(i) \(3x^2 - x^2y + 2y^3 = 0\)
(ii) \(y = \log_e\left(\frac{1 + \sin x}{1 - \sin x}\right)\)
(b) Find \(\frac{dy}{dx}\) when \(x = 2\sin^{-1}\left(\frac{x}{\sqrt{1 + x^2}}\right)\) and \(y = \cos^{-1}\left(\frac{1}{\sqrt{1 + x^2}}\right)\)
(c) Find the intervals on which the function \(f(x) = 3x^3 - 4x + 6\) is concave up and concave down. Find also the inflection point.
(d) Show that the function \(x^5 - 5x^4 + 5x^3 - 10\) is maximum at \(x = 1\), minimum \(x = 3\) and neither maximum nor minimum at \(x = 0\). Find also the maximum and minimum value.
(Marks: 3 + 3 + 4 + 4)
5.¶
(a) Define Antiderivative and Integration with example.
(b) Evaluate the following integrals (any three):
(i) \(\int \frac{e^x(1 + x)}{\cos^2 x} \, dx\)
(ii) \(\int \frac{dx}{(2x + 1)\sqrt{4x + 3}}\)
(iii) \(\int \sqrt{x^2 - 4x + 13} \, dx\)
(iv) \(\int \frac{dx}{\sqrt{x + a} \cdot \sqrt{x + b}}\)
(v) \(\int \frac{xe^x}{(x + 1)^2} \, dx\)
(Marks: 3 + 9)
6.¶
(a) State and prove the Fundamental Theorem of Integral Calculus.
(b) Evaluate
(c) Prove that
(d) Evaluate
(Marks: 3 + 3 + 5 + 3)
7.¶
(a) Find the area of region bounded by the curve \(y^2 = 4x\) and the line \(y = 2x\).
(b) Find the length of the arc of the following curve \(y = \log \sec x\) from \(x = 0\) to \(x = \frac{\pi}{3}\).
(c) What is the entire area enclosed by the curve \(r = a\cos 2\theta\)
(d) Obtain the volume between the curves \(y^2 = 9x\) and \(y = 3x\)
(Marks: 4 + 3 + 3 + 4)
Bangladesh Open University
School of Science and Technology
B.Sc in Computer Science and Engineering Program
181 Term (1st Year 1st Semester) Final Examination
Course Code & Title: MAT1134 Differential and Integral Calculus
Time: 3 hours
Total Marks: 70
[N.B.: Answer any 5 (five) questions. The figures in the right margin indicate the full marks. All portions of each question must be answered sequentially.]
1.¶
(a) Given \(f(x) = x\), show that
(b) A function \(f(x)\) is defined as follows:
Discuss the continuity of the function at \(x = 2\). Also check the differentiability of \(f(x)\) at \(x = 2\).
(c) Using L’Hospital’s rule evaluate the following limits:
(i) \(\lim_{x \to 0} \frac{e^x - 1}{x^3}\);
(ii) \(\lim_{x \to 0} (\sin x)^x\)
(Marks: 4 + 3+3 + 4)
2.¶
(a) Find \(\frac{dy}{dx}\) when:
(i) \(e^{xy} - 4xy = 2\);
(ii) \(y = x^3 \sqrt{\frac{x^4 + 4}{x^2 + 3}}\)
(b) Differentiate
(c) If \(4x^2 - 2y^2 = 9\), then prove that
(Marks: 6 + 4 + 4)
3.¶
(a) State and prove Mean Value Theorem.
(b) State Rolle’s Theorem. Verify the truth of Rolle’s Theorem for the function
(c) Find \(y_3\) if \(y = \sin^{-1}x\).
(d) Find the maximum and minimum value of \(x + \frac{1}{x}\).
(e) State and prove Euler’s theorem on homogeneous function. If
(f) If \(y = a \cos(\log x) + b \sin(\log x)\), show that
(Marks: 6 + 4 + 4 + 4 + 4 + 4)
5.¶
(a) Define Anti-derivatives and Integration with example.
(b) Evaluate (any two):
(i) \(\int \frac{\cos x}{5 - 3 \cos x} dx\)
(ii) \(\int \frac{e^x(1 + x)dx}{\cos^2(xe^x)}\)
(iii) \(\int \frac{x dx}{x^2 + 2x + 1}\)
(c) Prove that
(Marks: 3 + 3×2 + 5)
6.¶
(a) Prove that
(b) Evaluate (any two):
(i) \(\int_0^{\pi} \sin nx \sin mx \, dx\)
(ii) \(\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx\)
(iii) \(\int_0^a \sqrt{(a^2 - x^2)} dx\)
(c) If
(Marks: 3 + 3×2 + 3+2)
7.¶
(a) Define Gamma and Beta function. Prove that for Beta function \(B(m, n) = B(n, m)\)
(b) Find the volume of the solid generated by revolving, about the x-axis, the area bounded by the curve \(y^2 = 2x\) and the line \(y = 3x\).
(c) Show that the area of the asteroid \(x^{2/3} + y^{2/3} = a^{2/3}\) is \(\frac{3}{8}\pi a^2\).
(Marks: 2+3 + 4 + 5)
Bangladesh Open University
School of Science and Technology
B.Sc in Computer Science and Engineering Program
191 Term (1st Year 1st Semester) Final Examination
Course Code & Title: MAT1134 Differential and Integral Calculus
Time: 3 hours
Total Marks: (5×14)=70
[N.B.: Answer any 5 (five) questions. The figures in the right margin indicate the full marks. All portions of each question must be answered sequentially.]
1.¶
(a) Define domain and range of a function with example. Find the domain and range of
(Marks: 2+3)
(b) Define limit of a function. Using L’Hospital’s Rule evaluate the following limits:
(i) \(\lim_{x \to 0} \frac{x - \tan x}{x^3}\)
(ii) \(\lim_{x \to \frac{\pi}{2}} (\sin x)^x \tan x\)
(Marks: 2+7)
2.¶
(a) A function \(f(x)\) is defined as follows:
Discuss the continuity and differentiability of \(f(x)\) at \(x = \frac{\pi}{2}\).
(Marks: 5)
(b) Differentiate
(Marks: 4)
(c) Find \(\frac{dy}{dx}\) in the following cases (any two):
(i) \(x = \sin p, y = \sin p\)
(ii) \(2y^3 - 3x^2 y + \sin^2 x = 0\)
(iii) \(y = \sin(\sqrt{1 + \cos x})\)
(Marks: 5)
3.¶
(a) State and prove Lagrange’s Mean Value Theorem. Find value of \(c\) in the Mean Value Theorem
(Marks: 7)
(b) Show that
(Marks: 4)
(c) Show that \(\sqrt{3} \sin x + \cos x\) is maximum for \(x = \frac{\pi}{6}\)
(Marks: 3)
4.¶
(a) Show that the function
is maximum at \(x = 1\), minimum at \(x = 3\) and neither maximum nor minimum at \(x = 0\). Find also the maximum and minimum value.
(Marks: 6)
(b) If \(u = \tan^{-1} \left( \frac{x^3 + y^3}{x - y} \right)\), show that
(Marks: 5)
(c) Give the geometrical interpretation of \(\int_a^b f(x) dx\)
(Marks: 3)
5.¶
(a) State and prove the fundamental theorem of integral calculus.
(Marks: 5)
(b) Evaluate (any three):
(i) \(\int \frac{\cos x}{a + b \cos x} dx\)
(ii) \(\int \cos^4 x dx\)
(iii) \(\int \frac{xe^x}{(x + 1)^2} dx\)
(iv) \(\int \frac{e^{m \tan^{-1} x}}{(1 + x^2)^2} dx\)
(v) \(\int x^3 \sqrt{b^2 - x^2} dx\)
(Marks: 3×3=9)
6.¶
(a) Obtain the reduction formula for \(\int \cos^n x dx\) and hence find \(\int \cos^6 x dx\).
(Marks: 4)
(b) Prove that
(Marks: 4)
(c) Evaluate (any two):
(i) \(\int_0^{\pi} x^7 e^{-2x} dx\)
(ii) \(\int_0^{\pi} \sin^7 x dx\)
(iii) \(\int_0^{\pi} \log(\tan x + \cot x) dx\)
(Marks: 3×2=6)
7.¶
(a) Define Gamma and Beta function, prove that \(\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}\)
(Marks: 4)
(b) Find the area bounded by the curve \(r = a(1 - \cos \theta)\)
(Marks: 5)
(c) Show that the area included between the curve \(x = y^2(1 - x)\) and the line \(x = 1\) is \(\pi\).
(Marks: 5)
Bangladesh Open University
School of Science and Technology
B.Sc in Computer Science and Engineering Program
201 Term 1st Year 1st Semester Final Examination
Course Code & Title: MAT1134 Differential and Integral Calculus
Time: 3 Hours
Total Marks (5×14): 70
[N.B.: Answer any 5 (five) questions. The figures in the right margin indicate the full marks. All portions of each question must be answered sequentially.]
1.¶
(a) Find the domain for the function
(Marks: 3)
(b) Show that \(\phi(x)\) is continuous at \(x = 0\) and discontinuous at \(x = \frac{3}{2}\), when \(\phi(x)\) is defined as follows:
(Marks: 6)
(c) Write down the fundamental theorem on limit of a function. Using L’Hospital’s Rule evaluate the following limits:
(i) \(\lim_{x \to 3} \frac{x^2 - 6x + 9}{x - 3}\)
(ii) \(\lim_{x \to 0} \frac{\log(1 + x)}{\sin x}\)
(Marks: 5)
2.¶
(a) From the first principle find the derivative of \(\log x^e\)
(Marks: 4)
(b) Find \(\frac{dy}{dx}\) in the following cases (any three):
(i) \(y = \log_e \left( \frac{1 + \sin x}{1 - \sin x} \right)\)
(ii) \(x^y = e^{x^y}\)
(iii) \(y = (\sin x)^x\)
(iv) \(3x^4 - x^2 y + 2y^3 = 0\)
(Marks: 6)
(c) Differentiate
(Marks: 4)
3.¶
(a) State Leibnitz’s theorem. Using Leibnitz’s theorem find the \(n^{th}\) derivative of \(y = x^3 e^{5x}\)
(Marks: 4)
(b) Investigate for what values of \(x\),
is a maximum or minimum
(Marks: 5)
(c) State Lagrange’s Mean Value Theorem. If the mean value theorem is
find the value of \(c\), when
(Marks: 5)
4.¶
(a) If
prove that
(Marks: 3)
(b) State and prove the fundamental theorem of integral calculus
(Marks: 4)
(c) State and prove Euler’s theorem on homogeneous function. Verify Euler’s theorem on the surface
(Marks: 4)
(d) Show that the tangent at point \((a, b)\) to the curve
is
(Marks: 3)
5.¶
(a) Define Anti-derivatives and Integration
(Marks: 2)
(b) Evaluate (any three):
(i) \(\int \frac{dx}{\sqrt{(x - \alpha)(x - \beta)}}\)
(ii) \(\int \frac{dx}{3 + 2\sin x + 3\cos x}\)
(iii) \(\int \frac{e^{m \tan^{-1} x}}{(1 + x^2)^{3/2}} dx\)
(iv) \(\int x^4 \sqrt{a^2 - x^{10}} dx\)
(Marks: 3×3=9)
(c) Prove that
(Marks: 3)
6.¶
(a) Evaluate (any two) of the following:
(i) \(\int_0^1 \frac{\ln(1 + x)}{1 + x^2} dx\)
(ii) \(\int_0^{\pi/2} \frac{\sqrt{\tan x}}{\sqrt{\tan x} + \sqrt{\cot x}} dx\)
(iii) \(\int_1^e \frac{\log x}{x} dx\)
(Marks: 3×2)
(b) Evaluate:
(Marks: 3)
(c) Define Gamma and Beta function. Using Gamma function prove that
(Marks: 2+3)
7.¶
(a) Find the area of the region enclosed by the curve \(x = y^2\) and the straight line \(y = x - 2\).
(Marks: 5)
(b) Find the area of the surface that is generated by revolving the portion of the curve \(y = x^3\) between \(x = 0\) and \(x = 1\) about the x-axis.
(Marks: 4)
(c) The circle \(x^2 + y^2 = a^2\) revolves round the y-axis, show that the volume of the whole sphere generated is
(Marks: 5)
Bangladesh Open University
School of Science and Technology
B.Sc in Computer Science and Engineering Program
211 Term (1st Year 1st Semester) Final Examination
Course Code & Title: MAT1134 Differential and Integral Calculus
Time: 3 Hours
Total Marks (5×14): 70
[N.B.: Answer any 5 (five) questions. The figures in the right margin indicate the full marks.
All portions of each question must be answered sequentially.]
1.¶
(a) Define domain and range of a function with example.
Find the domain and range of
(Marks: 2+3)
(b) A real function \(f\) is defined by
Find the value of
(Marks: 3)
(c) Discuss the continuity of the following function at \(x = 2\), when
Also check the differentiability of \(f(x)\) at \(x = 2\).
(Marks: 3+3)
2.¶
(a) From the first principle, find the derivative of \(f(x) = e^x\)
(Marks: 4)
(b) If in the rectilinear motion of a particle, distance \(s\) is
where \(u\) and \(f\) are constant, find the velocity and acceleration at time \(t\).
(Marks: 3)
(c) If \(y = e^{\sin^{-1} x}\) and \(z = e^{-\cos^{-1} x}\), then show that \(\frac{dy}{dz}\) is independent of \(x\).
(Marks: 3)
(d) Use implicit differentiation to find \(\frac{dy}{dx}\) for the Folium of Descartes:
Find the equation for the tangent line to the Folium of Descartes at the point
(Marks: 2+3)
3.¶
(a) Differentiate \(x^{\sin^{-1} x}\) with respect to \((\sin x)^x\)
(Marks: 4)
(b) Using Mean Value Theorem:
Find \(c\), when
(Marks: 4)
(c) Evaluate the following (any two) using L’Hospital’s Rule:
(i) \(\lim_{x \to 0} \frac{1 - \cos x}{x^2}\)
(ii) \(\lim_{x \to 0^+} \frac{\ln x}{\csc x}\)
(iii) \(\lim_{x \to 0} \left(\frac{1}{x} - \cot x\right)\)
(Marks: 3×2)
4.¶
(a) State Leibnitz’s Theorem. Using Leibnitz’s Theorem, find the value of the \(n^{th}\) derivative of \(y = e^{\sin^{-1} x}\) at \(x = 0\)
(Marks: 1+6)
(b) Find the maximum and minimum value of
Also find point of inflexion.
(Marks: 4)
(c) State the conditions for a function to be maximum and minimum. Show that the function
possesses neither a maximum nor a minimum.
(Marks: 1+2)
5.¶
(a) Define Antiderivative and Integration with example
(Marks: 2)
(b) Evaluate any four of the following integrals:
(i) \(\int \sin^4 x \, dx\)
(ii) \(\int \frac{x \, dx}{x^4 + a^4}\)
(iii) \(\int \frac{xe^x \, dx}{(x + 1)^2}\)
(iv) \(\int \frac{dx}{x^2 - 7x + 18}\)
(v) \(\int \frac{e^{m \tan^{-1} x}}{(1 + x^2)^2} \, dx\)
(vi) \(\int \frac{e^{x^2(1 + x)}}{\cos^2(xe^x)} \, dx\)
(Marks: 4×3)
6.¶
(a) Evaluate any two:
(i) \(\int_0^1 \frac{\ln(1 + x)}{1 + x^2} \, dx\)
(ii) \(\int_0^{\pi/2} \frac{\sqrt{\tan x}}{\sqrt{\tan x} + \sqrt{\cot x}} \, dx\)
(iii) \(\int_1^e \frac{\log x}{x} \, dx\)
(Marks: 3×2)
(b) Evaluate
(Marks: 3)
(c) Define Gamma and Beta function. Using Gamma function, prove that
(Marks: 2+3)
7.¶
(a) Find the area of the region enclosed by the curve \(x = y^2\) and the straight line \(y = x - 2\)
(Marks: 5)
(b) Find the area of the surface that is generated by revolving the portion of the curve
between \(x = 0\) and \(x = 1\) about the x-axis.
(Marks: 4)
(c) The circle \(x^2 + y^2 = a^2\) revolves round the y-axis. Show that the volume of the whole sphere generated is
(Marks: 5)
Bangladesh Open University
School of Science and Technology
221 Term 1st Year 1st Semester Final Examination
Course Code & Title: MAT1134 Differential and Integral Calculus
Time: 3 Hours
Total Marks (5×14): 70
[N.B.: Answer any 5 (five) questions. The figures in the right margin indicate the full marks. All portions of each question must be answered sequentially.]
1.¶
(a) Draw the graph of the following function,
\(y = \begin{cases} \frac{x^2}{2} & \text{for } x \ne 1 \\ 1 & \text{for } x = 1 \end{cases}\)
and also find the domain and range of this function.
3+2
(b) Explain why \(\lim_{x \to 0} \frac{|x|}{x}\) does not exist.
3
(c) Test the continuity of the following function at \(x = 0\) and \(x = \frac{3}{2}\):
3+3
2.¶
(a) Differentiating of the following with respect to \(x\) (any two):
(i) \(y = \cos x^3\)
(ii) \(y = \sin(\sqrt{1 + \cos x})\)
(iii) \(y = \ln \frac{x^2 \sin x}{\sqrt{1 + x}}\)
3×2
(b) Find \(\frac{dy}{dx}\), if \(y e^x - 4xy = 0\)
4
(c) Differentiate \(\tan^{-1} \frac{2x}{1 - x^2}\) with respect to \(\sin^{-1} \frac{2x}{1 + x^2}\)
4
3.¶
(a) Evaluate the following (any two) using L'Hospital's rule:
(i) \(\lim_{x \to 0} \frac{\sin x - \sin^3 x \cos x}{x^3}\)
(ii) \(\lim_{x \to 0} \frac{\ln x}{\csc x}\)
(iii) \(\lim_{x \to 0^+} x^x \ln x\)
3×2
(b) State Rolle’s theorem. Verify the truth of Rolle’s theorem for the function \(f(x) = x^2 - 3x + 2\) in the interval (1, 2).
1+3
(c) A function \(f(x)\) is defined as follows:
Are the conditions of the first mean-value theorem of differential calculus satisfied in this case?
4
4.¶
(a) If \(y = a \cos(\log x) + b \sin(\log x)\), Prove that \(x^2 y'' + x y' + y = 0\)
4
(b) Find the intervals on which the function \(f(x) = x^3 - 3x^2 + 1\) is increasing, decreasing, concave up, and concave down.
6
(c) Investigate for what values of \(x\),
\(f(x) = 5x^6 - 18x^5 + 15x^4 - 10\)
is a maximum or minimum.
4
5.¶
(a) Define Antiderivative and Integration with example.
2
(b) Evaluate the following integral (any four):
(i) \(\int e^{3x} \frac{e^x + \ln x}{e^{3x} + x \ln x} dx\)
(ii) \(\int \frac{(1 + x^2)^3}{x} dx\)
(iii) \(\int \cos^3 2x \, dx\)
(iv) \(\int \frac{1}{\sqrt{1 + \sin x}} dx\)
(v) \(\int \frac{e^x}{\sqrt{x}} dx\)
(vi) \(\int \frac{x e^x}{(x + 1)^2} dx\)
3×4
6.¶
(a) State and prove the fundamental theorem of Integral Calculus.
4
(b) Evaluate (any two) of the following:
(i) \(\int_0^1 \frac{dx}{1 + x^2}\)
(ii) \(\int \frac{e^x}{x(1 + \log x)^2} dx\)
(iii) \(\int_{1/2}^1 \frac{1}{\sqrt{1 - x^2}} dx\)
3×2
(c) If \(I_n = \int_0^{\pi/4} \tan^n \theta d\theta\), show that \(I_n = \frac{1}{n - 1} - I_{n - 2}\)
4
7.¶
(a) Find the area of the region bounded above by \(y = x + 6\), bounded below by \(y = x\) and bounded on the sides by the lines \(x = 0\) and \(x = 2\).
4
(b) Find the area bounded by the curve \(y^2 = x^3\) and the line \(y = 2x\).
5
(c) Find the volume of the solid generated when the region under the curve \(y = x^2\) over the interval [0, 2] is rotated about the line \(y = -1\).
5
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