Calculus Term Exam Questions

math-1109 MATH-1109 Differential and Integral Calculus

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Major Topic 1: Differential Calculus

Subtopic 1.1: Limits

• Discuss the distinction between $\lim f(x)$ and $f(a)$. Describe the nature of the function – 2023.
• Evaluate ${\lim }_{x\to 0}\frac{a(1+\cos x)-b\sin x}{x^3}$ exists. Find also the limit value – 2021.
• Evaluate ${\lim }_{x\to 0}{x}^{\sin x}$ – 2023.
• Evaluate ${\lim }_{x\to 0}(1+\sin x{)}^{(1/x)}$ – 2024, 2019.
• Evaluate ${\lim }_{x\to \pi /2}(\tan x{)}^{(1-2x)}$ – 2016.
• Evaluate $L{t}_{x\to 0}(\sin x{)}^{(1/x)}$ – 2017, 2018.
• Evaluate ${\lim }_{x\to -1}\frac{x^3+1}{x^2+x}$ – 2024.
• Using $(ϵ-\delta )$ definition show that ${\lim }_{x\to 2}\frac{x^2-4}{x-2}=4$ – 2016.
• What are necessary and sufficient conditions for existence of limit of a function? Applying $(ϵ,\delta )$ definition of limit, show that ${\lim }_{x\to 2}\frac{x^2-4}{x-2}=8$. Also find $\delta$, if $ϵ=0.1$ – 2018 (Note: The limit for $\frac{x^2-4}{x-2}$ as $x\to 2$ is 4, not 8, as stated in the 2018 question.)
• Apply the $\delta -ϵ$ definition to show ${\lim }_{x\to 3}(x^2-3x-7)=9$ – 2022.
• Define limit of a function – 2022.

Subtopic 1.2: Continuity & Differentiability

• Discuss the continuity and differentiability of the function $f(x)$ at $x=-3/2$ – 2020.
• Discuss the continuity and differentiability of the function $f(x)$ at $x=0$ – 2021.
• A function $f(x)$ is defined as follows: $f(x)=...$. Discuss the continuity of $f(x)$ at $x=1$ and the continuity of $f(x)$ at $x=1/2$ – 2023.
• A function $f(x)$ is defined as follows: $f(x)=...$. Discuss the differentiability of $f(x)$ at $x=\pi /2$ and $x=0$ – 2023.
• Discuss about the existence of $f^{\prime }(x)$ at $x=\pi /2$ and $x=0$ – 2023.
• What do you mean by continuity of function at a particular point and at an interval? Also, define the differentiability of a function at that cases – 2024.
• Given, $f(x)=...$ for $x<0$, $x\ge 0$. Evaluate the continuity and differentiability of the function $f(x)=...$ for $x\le x$ – 2024.
• Test the differentiability of the function at $x=0$ and $x=\pi /2$ – 2024.
• Define limit and continuity of a function – 2024.
• Discuss the differentiability at $x=\pi /2$ of the function $f(x)=(1+\sin x)$ for $x<\pi /2$, $(z+(x-\pi /2{)}^2)$ when $\pi /2\le x$ – 2024.
• Examine continuity for the function $f(x)=\frac{x^2-4}{x-2}$ at $x=2$ – 2016 (Note: Function is likely $\frac{x^2-4}{x-2}$ when $x\ne 2$, and 4 when $x=2$ for continuity; as given, it’s ambiguous. If $f(2)=4$, it’s continuous.)
• Discuss the differentiability at $x=0$ for the function $f(x)=...$ (piecewise) – 2016.
• What is meant by the continuity of a function? A function $f(x)=...$. Discuss the continuity of $f(x)$ and the existence of $f^{\prime }(x)$ at $x=0$ – 2017.
• Let $f(x)=...$. Is $f(x)$ continuous at $x=9$? Determine whether $f(x)$ is differentiable at $x=9$ – 2018.
• Define continuity and differentiability of a function. Let $f(x)=...$. Is $f(x)$ continuous at $x=9$? Determine whether $f(x)$ is differentiable at $x=9$ or not. If possible find the value of the derivative at that point – 2019.
• Define limit and continuity of a function – 2022.
• Examine the continuity and differentiability of the function $f(x)=|x-1|+|x+3|$ at $x=-3$ and $x=1$ – 2022.

Subtopic 1.3: Differential Coefficients

• If $y=\frac{x^{x-3}}{x^x}$, then find $y_x$ – 2020.
• Differentiate $x^{{\sin }^{-1}x}$ with respect to ${\sin }^{-1}x$ – 2021, 2023, 2024.
• Find the differential coefficient of ${\tan }^{-1}(\frac{\cos x}{1+\sin x})$ with respect to $x$ – 2023.
• Find $dy/dx$: (i) $y=x^3-4xy=2$. (ii) $y={\sin }^{2}t$, $x=\tan t$ – 2016.
• Differentiate $x^{\sin x}$ with respect to $\ln x$ – 2017.
• Find $dy/dx$ for the following cases: i) $x^3-4xy=2$. ii) $x=a(\cos t+t\sin t)$, $y=a(\sin t-t\cos t)$ – 2018.
• Find the differential coefficient of $(\sin x{)}^{\tan x}+(\cos x{)}^{\cot x}$ – 2019.
• Differentiate ${\tan }^{-1}(\frac{\sqrt{1+x^2}-1}{x})$ with respect to ${\tan }^{-1}x$ – 2019.
• Find $d/dx$ where $(\sin x{)}^{\cos x}$ – 2022.

Subtopic 1.4: Higher Order Derivatives

• Find $y_n$, when $y=\frac{e^{4x+1}}{x^2+4x+1}$ – 2021, 2024.
• Find the $n$-th derivative of $y=\frac{x-1}{x^2-5x+6}+\cos 3x$ – 2023.
• If $y={\cos }^{2}x{\sin }^{2}x$ then find $y_n$ – 2016.
• If $y=\frac{x^2+x-1}{x^3+x^2-6x}$, find $y_n$ – 2018.
• If $y=e^{ax}\sinh x$. then show that $\frac{{\partial }^{2}v}{\partial x^2}+(a^2+1)v=0$ – 2018 (Note: There is a typo, $y$ should be $v$ to match the equation.)

Subtopic 1.5: Leibnitz’s Theorem

• State Leibnitz’s theorem. If $y=e^{a{\cos }^{-1}x}$, then show that $(1-x^2)y_{n+2}-(2n+1)x{y}_{n+1}-(n^2+1)y_n=0$ – 2021.
• If $y={\tan }^{-1}x$, then show that $(1+x^2)y_{n+2}+(2n+1)x{y}_{n+1}+n^2{y}_n=0$. Find also the value of $(y_n{)}_0$ – 2023.
• Write the statement of Leibnitz’s theorem – 2024.
• If $y=({\sin }^{-1}x{)}^2$, then construct the relations between $y_{n+2}$, $y_{n+1}$ and $y_n$ – 2024.
• State Leibnitz’s theorem and Rolle’s theorem – 2024.
• If $y=({\sin }^{-1}x{)}^2$, then prove that $(1-x^2)y_{n+2}-(2n+1)x{y}_{n+1}-(n^2+m^2)y_n=0$ (typo, probably $n^2{y}_n$ for original ${\sin }^{-1}x$) – 2024.
• State Leibnitz’s theorem. If $y=\ln (x+\sqrt{1+x^2})$ find the relation among $y_{n+2}$, $y_{n+1}$ and $y_n$ – 2017.
• If $y={\sin }^{-1}x$, show that $(1-x^2)y_{n+2}-(2n+1)x{y}_{n+1}-n^2{y}_n=0$. Find also the value of $(y_n{)}_0$ – 2019.
• State Leibnitz’s theorem – 2022.
• If $y={\tan }^{-1}x$ then find the relation $y_n,y_{n+1}$ and $y_{n+2}$ – 2022.
• If $y=\sin (m{\sin }^{-1}x)$ then prove that $(1-x^2)y_{n+2}-(2n+1)x{y}_{n+1}-(n^2-m^2)y_n=0$ and hence evaluate $(y_n{)}_0$ – 2022.

Subtopic 1.6: Rolle’s Theorem

• State Rolle’s theorem. Verify the truth of the Rolle’s theorem for the function $f(x)=x^2-5x+6$ in the interval $$$$ – 2021.
• Verify whether Rolle’s theorem is applicable for the function $f(x)=\ln x+4x+3$ in the interval $(1,3)$ – 2016.
• State Rolle’s theorem. Is Rolle’s theorem applicable to the function $f(x)=3+(2x-1{)}^{(2/3)}$ in $$$$? Justify your answer – 2017.
• Discuss the applicability of the Rolle’s theorem to the function $f(x)=|x|$ in $[-1,1]$ – 2022.

Subtopic 1.7: Mean Value Theorem (Lagrange’s)

• Justify the validity of the Lagrange’s mean value theorem for $f(x)=3+2x-x^2$ at $(0,1)$ – 2016.
• State mean value theorem. Differentiate $x^{\sin x}$ with respect to $\ln x$ – 2017 (Note: Differentiating part is not related to MVT).
• State Mean Value Theorem. Use this theorem to evaluate the value of $c$, if $a<h<c$ – 2018.
• State mean value theorem. Find the point $C$ (if exist), where the tangent line is parallel to the secant line on $$$$ of $f(x)=x^3/4$ – 2019.
• State Lagrange’s mean value theorem – 2022.
• Justify the validity of the Lagrange’s mean value theorem for the function $f(x)=a{x}^2+bx+c$ in the interval $$$$, where $A,B,C$ are real constant – 2022.

Subtopic 1.8: Taylor/Maclaurin Series & Approximations

• Expand $\cos x+x$ in Taylor’s series in powers of $(x-\pi /2)$ – 2023.
• Expand $\log (1+x)$ in ascending powers of $x$ – 2024.
• Evaluate $\sin 33^{\circ }$ using Taylor’s series – 2024.
• Expand $f(x)=x^5-2x^4+9x^3-3$ in a Taylor’s series in power $x-2$ – 2024.
• Expand $\cos x$ in powers of $(x-\pi /4)$ – 2016.
• Expand $\ln x$ in the powers of $(x-2)$ – 2017.
• Find the value of $\sin 31^{\circ }$ without using calculator correct up to two decimal places – 2018.
• Find the Taylor finite series of $f(x)={\cos }^{2}x$ about $x=\pi /4$ – 2019.

Subtopic 1.9: Maxima & Minima

• Given $2xy=4$, find the maximum and minimum values of $4x+9y$ – 2020.
• Discuss the maxima and minima of $f(x,y)=x^3+y^3-3x-12y+20$, where $x,y>0$ – 2021.
• Define maximum of a function – 2023.
• Find the maximum and minimum value of the function $f(x)=\sin x(1+\cos x)$ in the interval $0\le x\le \pi$ – 2023.
• Determine the maximum and minimum values of $u$ where, $u=x/y+y/x$ – 2024.
• Discuss the maximum and minimum of the function $\sin x(1+\cos x)$. If possible, find the point of inflexion – 2024.
• Find the maximum and minimum values of $12(\log x+1)+x^2-10x+3$ – 2024.
• Find the maximum and minimum of $1+2\sin x+3{\cos }^{2}x$ for $0\le x\le \pi /2$ – 2016.
• Determine the maximum or minimum value for $f(x)=x^3-6x^2+9x+1$ in the interval $0\le x\le 1$ – 2017.
• The cost $c(x)=5+48/x+3x^2$. Find the minimum cost – 2018.
• Find the relative extrema (if exist) of the following function: $f(x)=4/x+36/(2-x)$ – 2019.
• Find maxima and minima of $u=x/y+y/x$ when $x+y=2$ – 2022.

Subtopic 1.10: Partial Derivatives & Euler’s Theorem

• Verify Euler’s theorem for the function $u=y^2\ln (x/y)$ – 2020.
• State Euler’s theorem. If $u=f(x^2+2yz,y^2+2zx)$, then prove that $(y-z{)}^2(\partial u/\partial x)+(z-x{)}^2(\partial u/\partial y)+(x-y{)}^2(\partial u/\partial z)=0$ – 2021.
• Give an example of homogeneous function. If $u=f(y-x/z,x-y/z)$, then estimate the value of $x\partial u/\partial x+y\partial u/\partial y+z\partial u/\partial z$ – 2023.
• Write the statement of Euler’s theorem on homogeneous function – 2024.
• If $u=x^2/y+y^2/x=1$, then show that $u_{xx}+u_{yy}=0$ – 2024.
• Define limit and continuity of a function. State Euler’s theorem – 2024.
• If $u=f(x^2+y^2+z^2)$, then prove that $(y-z)\partial u/\partial x+(z-x)\partial u/\partial y+(x-y)\partial u/\partial z=0$ – 2024.
• If $u=\log (x^2+y^2+z^2)$, then prove that $x{\partial }^{2}u/\partial x\partial y=y{\partial }^{2}u/\partial y\partial z=z{\partial }^{2}u/\partial z\partial x$ – 2016 (Note: This equality seems to be incorrect as stated. Usually it’s about Laplacian or sums of mixed partials.)
• Define homogeneous function. If $u={\tan }^{-1}(\frac{x^3+y^3}{x-y})$, then show that $x\partial u/\partial x+y\partial u/\partial y=\sin 2u$ – 2017.
• If $u=1/r$ and $r^2=x^2+y^2+z^2$, then prove that $u_{xx}+u_{yy}+u_{zz}=0$ – 2017.
• If $v=z{\tan }^{-1}(y/x)$, then evaluate the value of ${\partial }^{2}v/\partial x^2+{\partial }^{2}v/\partial y^2+{\partial }^{2}v/\partial z^2$ – 2018.
• If $u=f(x^2+2yz,y^2+2zx)$, then find the value of $(y-z{)}^2\partial u/\partial x+(x-yz{)}^2\partial u/\partial y+(z-xy{)}^2\partial u/\partial z$ – 2019.
• State Euler’s theorem for a $n$-th degree homogeneous function of three variables – 2022.
• If $u=f(y-z/x,z-x/y)$, then calculate $x\partial u/\partial x+y\partial u/\partial y+z\partial u/\partial z$ – 2022.

Subtopic 1.11: Tangents & Normals

• Prove that the curve $(x/a{)}^n+(y/b{)}^n=3$ touches the straight line $x/a+y/b=2$ at the point $(a,b)$, whatever $n$ be the value of $n$ – 2020.
• Define tangent and normal. Find the condition that the conics $a{x}^2+b{y}^2=1$ and $a^{\prime }{x}^2+b^{\prime }{y}^2=1$ cut orthogonally – 2021.
• Define sub-tangent and subnormal. For the curve $x^m{y}^n=a^{m+n}$, show that the sub-tangent at any point varies as the abscissa of the point – 2021.
• Find the equation of the tangent and normal at the point $(1,-1)$ to the curve $x^3+x{y}^2-3x^2+4x+5y+2=0$ – 2023.
• If $(x+y{)}^m=1$ touch the curve $(ax{)}^n+(by{)}^n=1$, then show that $(...)$ – 2024.
• Define tangent and normal state Taylor’s theorem – 2024.
• If $x\cos \alpha +y\sin \alpha =P$ touches the curve $x^m{y}^n=1$, then show that $(a\cos \alpha {)}^{m-1}+(b\sin \alpha {)}^{n-1}=P^{m+n-1}$ – 2024.
• In the curve $x^m{y}^n=a^{m+n}$, then show that the subtangent at any point varies as the abscissa of the point – 2024.
• Find the equation of the tangent and normal to the curve $x{y}^2=4(4-x)$ at the point where it is cut by the line $y=x$ – 2017.
• Find the tangent and normal to the curve $y(x-2)(x-3)-x+7=0$ – 2018.
• Show that for the curve $b{y}^2=(x+a{)}^3$ the square of the subtangent varies as the subnormal – 2019.
• Prove that the segment (between the coordinate axes) of a tangent to the astroid $x^{2/3}+y^{2/3}=a^{2/3}$ is of constant length. Find the area formed by the tangent and the normal to the above curve – 2022.

Subtopic 1.12: Radius of Curvature

• Define radius of curvature. If ${\rho }_1$ and ${\rho }_2$ be the radii of curvature at the end of the focal chord of the parabola $y^2=4ax$, then show that ${\rho }_1^{-2/3}+{\rho }_2^{-2/3}=(2a{)}^{-2/3}$ – 2021, 2024, 2019.
• What is meant by curvature? – 2023.
• Find the radius of curvature at $(0,1)$ of the curve $y=x^3+2x^2+x+1$ – 2023.
• Find the radius of curvature of the curve $y^2=2x$ at $x=0$. Evaluate the radius of curvature at the point $(2,2)$ for the curve $x^3+y^3-2xy+4=0$ – 2024.
• Define asymptote, radius of curvature, subtangent and subnormal – 2024.
• Find the radius of curvature at the point $(r,\theta )$ on the condition $r=a(1-\cos \theta )$ – 2016.
• Find the radius of curvature of the parabola $y^2=4x$ at the vertex – 2017.
• Find the radius and centre of curvature at the point $(3,1)$ on the curve $y=x^2-6x+10$ – 2018.
• Find the radius of curvature of the curve $y^2=4x$ at extremity of the latus rectum – 2022.

Subtopic 1.13: Asymptotes

• Find all the asymptotes of the curve $x^3-y^2+2xy+x-y+5=0$ – 2020.
• Define asymptotes. Prove the asymptotes of the cubic $(x^2-y^2)x+(x^2+y^2)y=3axy+7=0$ form a triangle of area $a^2$ – 2021.
• Determine the asymptotes of $x^3-y^3-x^{2}y-x{y}^2-x^2-y^2+2xy+3y-x+7=0$ – 2024.
• Prove that the asymptotes of the cubic $(x^2-y^2)x-2a{y}^2+5x-7=0$ form a triangle of area $a^2$ – 2024.
• Find all the asymptotes of the curve $y^2{x}^2-3y{x}^2-5x{y}^2+i{x}^2+6y^2-x-3y+2=0$ – 2017.
• Find the asymptotes of the curve $x^3+y^3=3axy$ – 2019.

Subtopic 1.14: Point of Inflection / Concavity & Convexity

• Discuss the maximum and minimum of the function $\sin x(1+\cos x)$. If possible, find the point of inflexion – 2024.
• Prove that $(a-2,-1/2)$ is a point of inflection of the curve $y=(x-a)e^{(x-a)}$ – 2018.
• Examine concavity and convexity for the curve $f(x)=x^4-2x^3+1$. Also determine its points of inflection – 2022.

Subtopic 1.15: Related Rates

• The adiabatic law for the expression of air is $P{V}^{1.4}=k$, where $k$ is a constant. If at a given time the volume is observed to be 20 cu.ft. and the pressure is 50 lbs per square inch, at what rate is the pressure changing if the volume is decreasing at the rate of 2 cu.ft. per sec – 2018.

Subtopic 1.16: Jacobians (Multi-variable calculus concepts)

• Define Jacobian of two variables – 2024.

Subtopic 1.17: Other Differential Calculus Concepts

• What is the physical meaning of $d^{2}y/d{x}^2$? – 2018.

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Major Topic 2: Integral Calculus

Subtopic 2.1: Basic Concepts of Integration

• What is meant by integration? – 2024.
• What is integration? Write the name of four applications of integration – 2022.

Subtopic 2.2: Indefinite Integrals

• Integrate $\int \frac{dx}{(x+1)\sqrt{1+2x-x^2}}$ – 2020.
• Integrate $\int \frac{dx}{2\cos x+3\sin x}$ – 2020.
• Integrate $\int \frac{dx}{(2x+3)\sqrt{3x^2+3x+2}}$ – 2021.
• Integrate $\int \sqrt{3x-2}\sqrt{x^3-x+1},dx$ – 2021 (Note: This integral looks unusual; it might be a typo in the source or a complex substitution.)
• Integrate $\int \frac{\sqrt{x(1+a\cos x)}}{1-\cos x},dx$ – 2021 (Note: This also looks like a typo in the source, with ‘dx’ repeated.)
• Integrate: $\int 2{\sin }^{-1}x,dx$ – 2023.
• Integrate: $\int \frac{dx}{5x^2-8x+7}$ – 2023.
• Find $\int x\ln \sin x,dx$ – 2023.
• Integrate $\int \frac{2\cos x+3\sin x}{4\sin x-\cos x+5},dx$ – 2024.
• Integrate $\int \frac{x^2-1}{x^4+x^2+1},dx$ – 2024.
• Integrate $\int \frac{{\sin }^{-1}x}{\sqrt{1+x}},dx$ – 2024.
• Evaluate $\int \log (x+\sqrt{1+x^2}),dx$ by using differentiation under the sign of integration – 2024.
• Find $\int \frac{dx}{\sqrt{x^2+x-1}}$ – 2024.
• Find $\int \frac{dx}{3+3\cos x+4\sin x}$ – 2024.
• Find $\int \frac{dx}{\sqrt{x^2+2x+3}}$ – 2024.
• Integrate any three of the followings: (a) $\int \frac{dx}{(x^2-1)\sqrt{x^2+1}}$ (b) $\int \frac{x,dx}{(x-1)(x^2+4)}$ (c) $\int \frac{e^x}{2+\cos x-2\sin x},dx$ (d) $\int \frac{dx}{x^4\sqrt{x^2-1}}$ – 2016.
• Integrate ANY THREE of the followings: (a) $\int e^{2x}\sin 2x,dx$ (b) $\int (x+2)\sqrt{x^2+x+1},dx$ (c) $\int \frac{5\cos x+6}{2\cos x+\sin x+3},dx$ (d) $\int \frac{dx}{(x^2-1)\sqrt{x^2-1}}$ – 2017.
• Answer any Three of the followings: (a) $\int \frac{e^{{\tan }^{-1}x}}{1+x^2},dx$ (b) $\int \frac{dx}{x^4+1}$ (c) $\int x{\sin }^{2}x,dx$ (d) $\int \frac{1-\sin x+\cos x}{1+\sin x-\cos x},dx$ – 2018.
• Evaluate $\int \frac{dx}{x\sqrt{1+x^3}}$ – 2018.
• Integrate ANY THREE of the followings: (a) $\int \frac{dx}{(2x+1{)}^2\sqrt{3x^2+2}}$ (b) $\int \frac{dx}{x^4+3x^2+1}$ (c) $\int \frac{e^x\sqrt{x^2+1}}{(x+1{)}^2},dx$ (d) $\int \frac{1-\sin x+\cos x}{1+\sin x-\cos x},dx$ – 2019.
• Integrate: (i) $\int \frac{\sin 2x-\sin x}{1+{\sin }^{2}x},dx$ (ii) $\int \frac{1+x\ln x}{x},dx$ (iii) $\int \frac{1-\cos x-\cos 2x}{1+\cos x-\cos 2x},dx$ – 2022.

Subtopic 2.3: Definite Integrals

• Evaluate ${\int }_0^{\pi /2}\frac{\sin x}{1+{\cos }^{2}x},dx$ – 2020.
• Evaluate ${\int }_0^{\pi /2}x{\sin }^{2}x,dx$ – 2020.
• Evaluate ${\int }_0^{\pi /2}\frac{x\sin x\cos x}{a^2{\cos }^{2}x+b^2{\sin }^{2}x},dx$ – 2021, 2024.
• Write down some properties of definite integrals – 2023.
• Evaluate: ${\int }_0^{\pi /2}{\sin }^{5}x{\cos }^{4}x,dx$ – 2023.
• Evaluate ${\int }_0^{\pi /2}\frac{\sqrt{1-x}}{1+x},dx$ – 2023.
• Evaluate ${\int }_0^1\frac{\ln (1+x^2)}{1+x^2},dx$ – 2024.
• Find the value of the improper integral ${\int }_0^{\infty }{e}^{-x^2},dx$ – 2024.
• Write down some general properties of definite integral – 2024.
• Evaluate ${\int }_1^2\sqrt{(x-1)(2-x)},dx$ – 2024.
• Evaluate ${\int }_0^{\pi /2}\log \sin x,dx$ – 2024.
• Integrate any three of the followings: (a) ${\int }_0^{\pi /2}\frac{5\cos x+6}{2\cos x+\sin x+3},dx$ (b) ${\int }_0^1\frac{dx}{(1+x)\sqrt{1+2x-x^2}}$ (c) ${\int }_0^2(\sqrt{x}+\sqrt{2-x}),dx$ (d) $\int x{\cos }^{n}x,dx$ – 2016 (Note: (d) is an indefinite integral, others are definite.)
• Evaluate ANY THREE of the following definite integrals: (a) ${\int }_0^{\pi /2}{\sin }^6\theta {\cos }^3\theta ,d\theta$ (b) ${\int }_0^{\pi }\frac{dx}{1-2a\cos x+a^2}$, ($0<a<1$) (c) ${\int }_0^1\frac{\ln (1+x)}{1+x^2},dx$ (d) ${\int }_0^1{x}^3(1-x^2),dx$ – 2017.
• Answer any Three of the followings: (a) ${\int }_0^{\pi }\frac{dx}{1-2a\cos x+a^2}$ (b) $\int \frac{x\tan x}{\sec x+\tan x},dx$ (c) ${\int }_0^1\frac{\ln x}{\sqrt{1-x^2}},dx$ (d) ${\lim }_{n\to \infty }[\frac{n^2}{n^3+1}+\frac{n^2}{n^3+2}+...+\frac{n^2}{n^3+n}]$ – 2018 (Note: (d) is a limit of sum, (b) & (c) are definite integrals, (a) is a definite integral.)
• Evaluate ${\int }_0^{\pi /2}x{\sin }^{6}x{\cos }^{4}x,dx$ – 2019.
• Evaluate ${\int }_0^1\frac{x^3{\sin }^{-1}x}{\sqrt{1-x^2}},dx$ – 2019.
• Evaluate ${\int }_0^{\pi /2}\frac{dx}{a^2{\cos }^{2}x+b^2{\sin }^{2}x}$ – 2022.
• Show that ${\int }_0^{\pi /2}\log \sin (x/2),dx=\log 2$ – 2022 (Likely typo in question, should be $\log (\sin x)$ or similar.)

Subtopic 2.4: Reduction Formulas

• Obtain a reduction formula for $\int {\cos }^{n}x\cos nx,dx$ and hence find the value of ${\int }_0^{\pi /2}{\cos }^{5}x\cos 2x,dx$ – 2020.
• If $n$ be a positive integer, then show that ${\int }_0^{\pi /2}{\sin }^{n}x,dx={\int }_0^{\pi /2}{\cos }^{n}x,dx=\frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdot \dots \cdot (1/2\text{ or }2/3)$ depending on $n$ – 2021.
• Find the reduction formula for ${\int }_0^{\pi /2}{\sin }^m\theta {\cos }^n\theta ,d\theta$ and $n>1$ – 2021.
• Obtain the reduction formula for $\int {\cot }^{m}x{\csc }^{n}x,dx$ and hence evaluate $\int {\cot }^{5}x{\csc }^{3}x,dx$ – 2024.
• Obtain the reduction formula for $\int e^{ax}{\cos }^{n}x,dx$ – 2024.
• If $I_{m,n}={\int }_0^{\pi /2}{\sin }^{m}x{\cos }^{n}x,dx$ then show that $I_{m,n}=\frac{m-1}{m+n}\cdot I_{m-2,n}$ and hence evaluate ${\int }_0^{\pi /2}{\sin }^{4}x{\cos }^{5}x,dx$ – 2016.
• Obtain the reduction formula for $\int {\tan }^{n}x,dx$ and hence find $\int {\tan }^{6}x,dx$ – 2017.
• Determine the reduction formula for $\int {\sin }^{m}x{\cos }^{n}x,dx$ and hence find the value of $\int {\sin }^{4}x{\cos }^{5}x,dx$ – 2018.
• Evaluate ${\int }_0^{\pi /2}{\sin }^{m}x{\cos }^{n}x,dx$ – 2019.
• Find the reduction formula for ${\int }_0^{\pi /2}{e}^{ax}{\sin }^{n}x,dx$, $n>1$. Hence evaluate ${\int }_0^{\pi /2}{e}^{2x}{\sin }^{3}x,dx$ – 2022.

Subtopic 2.5: Wallis’s Formula

• State Wallis formula – 2023.
• State Wallis’s formula, write down some general properties of definite integral – 2024.
• State Wallis’s formula. Use this formula to evaluate ${\int }_0^{\pi /2}{\sin }^{4}x,dx$ – 2017.
• State and prove Walli’s formula – 2019.
• State Wallis’s formula – 2022.

Subtopic 2.6: Gamma & Beta Functions

• Show that ${\int }_0^{\pi /2}{\sin }^{m-1}\theta {\cos }^{n-1}\theta ,d\theta =\frac{\Gamma (m/2)\Gamma (n/2)}{2\Gamma ((m+n)/2)}$ and hence find ${\int }_0^{\pi /2}{\sin }^9\theta {\cos }^4\theta ,d\theta$ – 2021.
• Define Gamma and Beta function – 2023.
• Show that ${\int }_0^{\pi /2}{\sin }^{2m-1}\theta {\cos }^{2n-1}\theta ,d\theta =\frac{\Gamma (m)\Gamma (n)}{2\Gamma (m+n)}$. Hence find the value of ${\int }_0^{\pi /2}{\sin }^9\theta {\cos }^4\theta ,d\theta$ and ${\int }_0^{\pi /2}{\sin }^3\theta {\cos }^5\theta ,d\theta$ – 2023.
• Define Beta and Gamma functions. Write down the relationship between them – 2024.
• Determine the value of ${\int }_0^1{x}^2(1-x^3{)}^{3/2},dx$ by using Beta and Gamma functions – 2024.
• Define Beta and Gamma functions. Find the relation between the Beta and Gamma functions – 2024.
• Establish the relation between Gamma and Beta function – 2016.
• Define Gamma function and Beta function. Prove that $\Gamma (1/2)=\sqrt{\pi }$ – 2017.
• Establish the relationship between Gamma and Beta function – 2018.
• Show that $B(m,n)={\int }_0^{\infty }\frac{y^{m-1}}{(1+y{)}^{m+n}},dy$ – 2019.
• Evaluate ${\int }_0^a{x}^{\alpha -1}(a-x{)}^{\beta -1},dx$ where $\alpha$ and $\beta$ are constants – 2022.
• Define Gamma and Beta function. Write down the different form of Gamma function – 2022.
• Show that ${\int }_0^1{x}^{\alpha -1}(\beta -x{)}^{\gamma },dx=\frac{B(\alpha ,\gamma +1){\beta }^{\alpha +\gamma }}{\Gamma (\alpha +\gamma +1)}$ (Typo likely, should be $\int x^{\alpha -1}(\beta -x{)}^{\gamma -1},dx={\beta }^{\alpha +\gamma -1}B(\alpha ,\gamma )$ ) – 2022.

Subtopic 2.7: Limits of Sums (as definite integrals)

• Evaluate ${\lim }_{n\to \infty }[(1+1/n^2{)}^{(1/n)}\cdot (1+2^2/n^2{)}^{(2/n)}\cdot \dots \cdot (1+n^2/n^2{)}^{(n/n)}]$ – 2023.
• Evaluate ${\lim }_{n\to \infty }[\frac{n^2}{(n^2+1{)}^2}+\frac{n^2}{(n^2+2^2{)}^2}+\dots +\frac{n^2}{(n^2+(2n{)}^2{)}^2}]$ – 2016.
• Evaluate ${\lim }_{n\to \infty }[\frac{n^2}{n^3+1}+\frac{n^2}{n^3+2}+\dots +\frac{n^2}{n^3+n}]$ – 2018.
• Evaluate ${\lim }_{n\to \infty }[\frac{1}{n+1}+\frac{1}{n+2}+\dots +\frac{1}{n+n}]$ – 2022.

Subtopic 2.8: Multiple Integrals

• Evaluate $\iint (x^2+y^2),dx,dy$ over the positive quadrant to the circle $x^2+y^2=a^2$ – 2024.
• Evaluate ${\int }_0^{\pi /2}{\int }_0^{x\sin \theta }\frac{x\sin \theta }{\sin \theta },d\theta ,dx$ – 2017.

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Major Topic 3: Applications of Calculus

Subtopic 3.1: Area

• Find the area of the region bounded by the curve $y^2=2ax-x^2$ and $y^2=ax$ – 2020.
• Find the area of the loop of the curve $3a{y}^2=x^2(a-x)$ – 2021.
• Find the area of the loop of the curve $3a{y}^2=x(x-a{)}^2$ – 2023, 2024.
• Find the area of the region that is enclosed between the curve $y=x^2$ and the line $y=x+5$ – 2023.
• Determine the area bounded by the curve $y^2=\frac{(a-x{)}^2}{a+x}$ – 2024.
• Find the area bounded by the curve $r=a(1+\cos \theta )$ – 2016, 2018.
• Find the area above the $x$-axis, included between the parabola $y^2=ax$ and the circle $x^2+y^2=2ax$ – 2017.
• Find the area of the region included between the curve $x^2(2a-x)=x^3$ and its asymptote – 2018 (Assuming the curve is $y^2(2a-x)=x^3$ (Cissoid of Diocles)).
• Find the area of the region bounded by the parabolas $y^2=4ax$ and $x^2=4ay$ – 2019.
• Find the area bounded by the astroid $x^{2/3}+y^{2/3}=a^{2/3}$ – 2022.

Subtopic 3.2: Volume

• Find the volume and the surface area of the solid generated by revolving the cycloid $x=a(\theta +\sin \theta )$, $y=a(1+\cos \theta )$ about its base – 2021.
• Find the volume and surface area of the solid generated by revolving the cardioid $r=a(1+\cos \theta )$ about the initial line ($\theta =0$) – 2023.
• Find the volume of the solid generated by revolving of the parabola $y^2=4ax$ between $x=0$ and $x=a$ – 2024.
• Find the volume of the solid generated by the revolution of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ about $x$-axis – 2016.
• Find the volume of the solid produced by the revolution of the loop of the curve $a{y}^2=x^3(2a-x)$ – 2019.
• Find the volume and surface area of the solid generated by involving of the parabola $y^2=4ax$ between $x=0$ and $x=a$ about $x$-axis – 2022.

Subtopic 3.3: Arc Length

• Show that the entire length of the curve $x^{2/3}+y^{2/3}=a^{2/3}$ is $6a$ – 2021, 2023, 2024, 2016, 2018.
• Find the length of the arc of the parabola $y^2=4ax$ measured from the vertex to one extremity of the latus rectum – 2024.
• Find the arc length of the curve $y=x^{3/2}$ from $(1,1)$ to $(2,2\sqrt{2})$ – 2017.
• Find the whole perimeter of the curve $x^{2/3}+y^{2/3}=a^{2/3}$ – 2018.
• Find the length of the perimeter of the circle $x^2+y^2=4$ – 2019.
• Find the arc length of parabola $y^2=12x$ cut by latus rectum – 2022.

Subtopic 3.4: Surface Area

• Find the volume and the surface area of the solid generated by revolving the cycloid $x=a(\theta +\sin \theta )$, $y=a(1+\cos \theta )$ about its base – 2021.
• Find the volume and surface area of the solid generated by revolving the cardioid $r=a(1+\cos \theta )$ about the initial line ($\theta =0$) – 2023.
• Find the surface area of the solid formed by revolving the curve $r=a(1-\cos \theta )$ about the $x$-axis – 2024.
• Find the volume and surface area of the solid generated by involving of the parabola $y^2=4ax$ between $x=0$ and $x=a$ about $x$-axis – 2022.