Calculus PYQ 2022-2024

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 ${\lim }_{x\to 0}{x}^{\sin x}$ – 2023.
• Evaluate ${\lim }_{x\to 0}(1+\sin x{)}^{(1/x)}$ – 2024.
• Evaluate ${\lim }_{x\to -1}\frac{x^3+1}{x^2+x}$ – 2024.
• 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

• 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, 2022.
• 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 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

• Differentiate $x^{{\sin }^{-1}x}$ with respect to ${\sin }^{-1}x$ – 2023, 2024.
• Find the differential coefficient of ${\tan }^{-1}(\frac{\cos x}{1+\sin x})$ with respect to $x$ – 2023.
• Find $d/dx$ where $(\sin x{)}^{\cos x}$ – 2022.

Subtopic 1.4: Higher Order Derivatives

• Find $y_n$, when $y=\frac{e^{4x+1}}{x^2+4x+1}$ – 2024.
• Find the $n$-th derivative of $y=\frac{x-1}{x^2-5x+6}+\cos 3x$ – 2023.

Subtopic 1.5: Leibnitz’s Theorem

• 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 – 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

• 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)

• 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.

Subtopic 1.9: Maxima & Minima

• 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 maxima and minima of $u=x/y+y/x$ when $x+y=2$ – 2022.

Subtopic 1.10: Partial Derivatives & Euler’s Theorem

• 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.
• 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

• 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.
• 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 ${\rho }_1^{-2/3}+{\rho }_2^{-2/3}=(2a{)}^{-2/3}$ – 2024.
• 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 of the curve $y^2=4x$ at extremity of the latus rectum – 2022.

Subtopic 1.13: Asymptotes

• 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.

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.
• Examine concavity and convexity for the curve $f(x)=x^4-2x^3+1$. Also determine its points of inflection – 2022.

Subtopic 1.16: Jacobians (Multi-variable calculus concepts)

• Define Jacobian of two variables – 2024.

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: $\int 2{\sin }^{-1}x,dx$ – 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: (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 ${\int }_0^{\pi /2}\frac{x\sin x\cos x}{a^2{\cos }^{2}x+b^2{\sin }^{2}x},dx$ – 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.
• 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.

Subtopic 2.4: Reduction Formulas

• 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.
• 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 – 2022.

Subtopic 2.6: Gamma & Beta Functions

• 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.
• 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 ${\lim }_{n\to \infty }[\frac{1}{n+1}+\frac{1}{n+2}+\dots +\frac{1}{n+n}]$ – 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.

Major Topic 3: Applications of Calculus

Subtopic 3.1: Area

• 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 astroid $x^{2/3}+y^{2/3}=a^{2/3}$ – 2022.

Subtopic 3.2: Volume

• 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 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 $6a$ – 2023, 2024.
• 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 parabola $y^2=12x$ cut by latus rectum – 2022.

Subtopic 3.4: Surface Area

• 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.