[Analysis] MATH prev. yr question
📘 Functions and Models¶
🟨 171 Term
- Q1(a) Define Domain and Range with example.
- Q1(b) Sketch the graph of the function \(f(x) = |x| + |x - 1| + |x - 2|\). Find also the Domain and Range of that function.
- Q1(c) Discuss the continuity of the following function at \(x = 2\), when
$$
f(x) =
\begin{cases}
x, & 0 < x < 1 \
2 - x, & 1 \leq x \leq 2 \
x - \frac{1}{2}x^2, & x > 2
\end{cases}
$$
Also check the differentiability of \(f(x)\) at \(x = 2\).
🟨 181 Term
- Q1(a) Given \(f(x) = x\), show that
$$
\lim_{h \to 0} \frac{f(h) - f(0)}{h} \text{ does not exist}.
$$
* Q1(b) Same continuity/differentiability question as above at \(x = 2\)
🟨 191 Term
- Q1(a) Define domain and range of a function with example. Find the domain and range of \(f(x) = 3x^2 + x + 2\)
🟨 201 Term
- Q1(a) Find the domain for the function \(f(x) = \sqrt{x - 1} + \sqrt{5 - x}\)
- Q1(b) Show that \(\phi(x)\) is continuous at \(x = 0\) and discontinuous at \(x = \frac{3}{2}\)
(piecewise function given)
🟨 211 Term
- Q1(a) Define domain and range with example.
Find the domain and range of \(f(x) = |x + 1| + |x| + |x - 1|\)
🟨 221 Term
- Q1(a) Draw the graph of
\(y = \begin{cases} \frac{x^2}{2}, & x \ne 1 \\ 1, & x = 1 \end{cases}\)
Also find the domain and range. - Q1(b) Explain why \(\lim_{x \to 0} \frac{|x|}{x}\) does not exist.
- Q1(c) Test continuity of
$$
f(x) = \begin{cases}
3 + 2x, & -3 \le x < 0 \
3 - 2x, & 0 \le x < \frac{3}{2} \
-3 - 2x, & x \ge \frac{3}{2}
\end{cases}
$$
at \(x = 0\) and \(x = \frac{3}{2}\)
📘 Limits and Derivatives¶
🟨 171 Term
- Q2(a) \(v(t) = 3t - 5\), displacement over \([0, 3]\)
- Q2(b) Differentiate \(y = \sin(\sqrt{1 + \cos x})\)
- Q2(c) If \(x^y = e^{x+y}\), then prove \(\frac{dy}{dx} = \frac{\ln x}{(1 + \ln x)^2}\)
- Q2(d) Differentiate \(x^{\sin^{-1}x}\) w.r.t. \((\sin x)^x\)
🟨 181 Term
- Q1(c) Evaluate:
(i) \(\lim_{x \to 0} \frac{e^x - 1}{x^3}\)
(ii) \(\lim_{x \to 0} (\sin x)^x\)
🟨 191 Term
- Q1(b) L'Hospital's Rule
(i) \(\lim_{x \to 0} \frac{x - \tan x}{x^3}\)
(ii) \(\lim_{x \to \frac{\pi}{2}} (\sin x)^x \tan x\)
🟨 201 Term
- Q1(c) L'Hospital's Rule
(i) \(\lim_{x \to 3} \frac{x^2 - 6x + 9}{x - 3}\)
(ii) \(\lim_{x \to 0} \frac{\log(1 + x)}{\sin x}\)
🟨 211 Term
- Q1(b) \(\lim_{x \to +\infty} \frac{x}{1 - x^2}\)
- Q1(c) Same piecewise continuity/differentiability problem as 171
🟨 221 Term
-
Q2(a) Differentiate (any two):
-
\(y = \cos x^3\)
- \(y = \sin(\sqrt{1 + \cos x})\)
- \(y = \ln \frac{x^2 \sin x}{\sqrt{1 + x}}\)
- Q2(b) Find \(\frac{dy}{dx}\) if \(y e^x - 4xy = 0\)
-
Q2(c) Differentiate
\(\tan^{-1} \frac{2x}{1 - x^2}\) w.r.t. \(\sin^{-1} \frac{2x}{1 + x^2}\) -
Q4(b) Intervals where \(f(x) = x^3 - 3x^2 + 1\) is increasing/decreasing, concave up/down
- Q4(c) Max/min of
\(f(x) = 5x^6 - 18x^5 + 15x^4 - 10\)
📘 Differentiation Rules¶
🟨 171 Term
- Q3(a) Prove \(\frac{d^2y}{dx^2} = \frac{9}{y^3}\), given \(4x^2 - 2y^2 = 9\)
- Q3(b) Find \(\xi\) in MVT for \(f(x) = x^2, a = 1, b = 2\)
- Q3(c) L'Hospital’s Rule – pick any two
-
Q4(a) Differentiate any one:
-
\(3x^4 - x^2 y + 2y^3 = 0\)
- \(y = \log_e\left(\frac{1 + \sin x}{1 - \sin x}\right)\)
🟨 181 Term
- Q2(a) Differentiate:
(i) \(e^{xy} - 4xy = 2\)
(ii) \(y = x^3 \sqrt{\frac{x^4 + 4}{x^2 + 3}}\) - Q2(b) Differentiate
$$
\tan^{-1}\left(\frac{\sqrt{1 + x^2} - 1}{x}\right) \text{ w.r.t. } \tan^{-1}x
$$
🟨 191 Term
-
Q2(c) Differentiate (any two):
-
\(x = \sin p, y = \sin p\)
- \(2y^3 - 3x^2 y + \sin^2 x = 0\)
- \(y = \sin(\sqrt{1 + \cos x})\)
🟨 201 Term
-
Q2(b) Differentiate (any three):
-
\(y = \log_e \left( \frac{1 + \sin x}{1 - \sin x} \right)\)
- \(x^y = e^{x^y}\)
- \(y = (\sin x)^x\)
- \(3x^4 - x^2 y + 2y^3 = 0\)
🟨 211 Term
- Q3(a) Differentiate \(x^{\sin^{-1} x}\) w.r.t. \((\sin x)^x\)
🟨 221 Term
-
Q2(a) Differentiate (any two):
-
\(y = \cos x^3\)
- \(y = \sin(\sqrt{1 + \cos x})\)
- \(y = \ln \frac{x^2 \sin x}{\sqrt{1 + x}}\)
- Q2(b) Find \(\frac{dy}{dx}\) if \(y e^x - 4xy = 0\)
- Q2(c) Differentiate
\(\tan^{-1} \frac{2x}{1 - x^2}\) w.r.t. \(\sin^{-1} \frac{2x}{1 + x^2}\)
📘 Applications of Differentiation¶
🟨 171 Term
- Q4(b) Find \(\frac{dy}{dx}\) when
\(x = 2\sin^{-1}\left(\frac{x}{\sqrt{1 + x^2}}\right)\)
and
\(y = \cos^{-1}\left(\frac{1}{\sqrt{1 + x^2}}\right)\) - Q4(c) Find intervals where \(f(x) = 3x^3 - 4x + 6\) is concave up/down. Also find inflection point.
- Q4(d) Show that the function
\(x^5 - 5x^4 + 5x^3 - 10\)
is maximum at \(x = 1\), minimum at \(x = 3\), and neither at \(x = 0\). Also find max/min values.
🟨 181 Term
- Q3(d) Find maximum and minimum of \(x + \frac{1}{x}\)
- Q3(e) Euler's theorem — verify for \(f(x,y) = 2x^4 + 4x^2y^2 - y^4\)
- Q3(f) Prove
$$
x^2 y_{n+2} + (2n+1)xy_{n+1} + (n^2 + 1)y_n = 0
$$
for \(y = a\cos(\log x) + b\sin(\log x)\)
🟨 191 Term
- Q4(a) Same question as 171 Term Q4(d)
- Q4(c) Give geometric interpretation of \(\int_a^b f(x)dx\)
🟨 201 Term
- Q3(b) Investigate max/min of
\(f(x) = 5x^6 - 18x^5 + 15x^4 - 10\) - Q4(b) Max/min and inflection of
\(f(x) = x^3 - 9x^2 + 24x - 12\) - Q4(c) Conditions for max/min. Show that
\(f(x) = x^3 - 3x^2 + 9x - 1\)
has neither.
📘 Integrals¶
🟨 171 Term
- Q5(a) Define Antiderivative and Integration with example.
-
Q5(b) Evaluate any three:
-
\(\int \frac{e^x(1 + x)}{\cos^2 x} dx\)
- \(\int \frac{dx}{(2x + 1)\sqrt{4x + 3}}\)
- \(\int \sqrt{x^2 - 4x + 13} dx\)
- \(\int \frac{dx}{\sqrt{x + a} \cdot \sqrt{x + b}}\)
- \(\int \frac{xe^x}{(x + 1)^2} dx\)
🟨 181 Term
- Q5(a) Same as 171
-
Q5(b) Any two:
-
\(\int \frac{\cos x}{5 - 3\cos x} dx\)
- \(\int \frac{e^x(1 + x)}{\cos^2(xe^x)} dx\)
- \(\int \frac{x}{x^2 + 2x + 1} dx\)
- Q5(c) Prove:
$$
\int e^{ax} \cos bx \, dx = \frac{e^{ax}(a \cos bx + b \sin bx)}{a^2 + b^2}
$$
🟨 191 Term
-
Q5(b) Any three:
-
\(\int \frac{\cos x}{a + b\cos x} dx\)
- \(\int \cos^4 x dx\)
- \(\int \frac{xe^x}{(x + 1)^2} dx\)
- \(\int \frac{e^{m\tan^{-1} x}}{(1 + x^2)^2} dx\)
- \(\int x^3 \sqrt{b^2 - x^2} dx\)
🟨 201 Term
-
Q5(b) Any three:
-
\(\int \frac{dx}{\sqrt{(x - \alpha)(x - \beta)}}\)
- \(\int \frac{dx}{3 + 2\sin x + 3\cos x}\)
- \(\int \frac{e^{m\tan^{-1} x}}{(1 + x^2)^{3/2}} dx\)
- \(\int x^4 \sqrt{a^2 - x^{10}} dx\)
🟨 211 Term
-
Q5(b) Any four:
-
\(\int \sin^4 x dx\)
- \(\int \frac{x}{x^4 + a^4} dx\)
- \(\int \frac{xe^x}{(x + 1)^2} dx\)
- \(\int \frac{1}{x^2 - 7x + 18} dx\)
- \(\int \frac{e^{m\tan^{-1} x}}{(1 + x^2)^2} dx\)
-
\(\int \frac{e^{x^2(1 + x)}}{\cos^2(xe^x)} dx\)
🟨 221 Term -
Q5(a) Define Antiderivative and Integration with example
-
Q5(b) Evaluate (any four):
-
\(\int e^{3x} \frac{e^x + \ln x}{e^{3x} + x \ln x} dx\)
- \(\int \frac{(1 + x^2)^3}{x} dx\)
- \(\int \cos^3 2x dx\)
- \(\int \frac{1}{\sqrt{1 + \sin x}} dx\)
- \(\int \frac{e^x}{\sqrt{x}} dx\)
- \(\int \frac{x e^x}{(x + 1)^2} dx\)
📘 Applications of Integration¶
🟨 171 Term
- Q6(b) Evaluate
$$
\int_0^6 f(x) \, dx,\quad
f(x) =
\begin{cases}
x^2, & x < 2 \
3x - 2, & x \geq 2
\end{cases}
$$
* Q6(c) Prove that
$$
\int_0^1 \frac{\ln(1 + x)}{1 + x} \, dx = \frac{\pi}{8} \ln 2
$$
* Q6(d) Evaluate
$$
\int_0^{\pi} \frac{2 - x}{\sqrt{x(\pi - x)}} dx
$$
🟨 181 Term
-
Q6(b) Evaluate (any two):
-
\(\int_0^{\pi} \sin nx \sin mx dx\)
- \(\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx\)
- \(\int_0^a \sqrt{a^2 - x^2} dx\)
- Q6(c) Prove reduction relation:
$$
I_n = \int_0^{\pi/4} \tan^n \theta d\theta = \frac{1}{n - 1} - I_{n - 2},\quad
\text{find } \int_0^{\pi/4} \tan^4 x dx
$$
🟨 191 Term
- Q3(b) Show that
$$
\int_a^a \frac{xe^x}{1 + x^2} dx = 0
$$
🟨 201 Term
-
Q6(a) Evaluate any two:
-
\(\int_0^1 \frac{\ln(1 + x)}{1 + x^2} dx\)
- \(\int_0^{\pi/2} \frac{\sqrt{\tan x}}{\sqrt{\tan x} + \sqrt{\cot x}} dx\)
- \(\int_1^e \frac{\log x}{x} dx\)
- Q6(b) Same piecewise \(f(x)\) from earlier for definite integral
- Q6(c) Define Gamma and Beta function. Using Gamma, prove
$$
\int_0^\infty x^{1/2} e^{-x} dx = \frac{\sqrt{\pi}}{2}
$$
🟨 211 Term
- Q6(a) Evaluate any two (same as above)
- Q6(b) Same \(f(x)\) integral from Term 171
- Q6(c) Define Gamma and Beta. Using Gamma, prove
$$
\int_0^\infty e{-x2} dx = \frac{\sqrt{\pi}}{2}
$$
🟨 221 Term
- Q6(a) State and prove Fundamental Theorem of Integral Calculus
-
Q6(b) Evaluate (any two):
-
\(\int_0^1 \frac{dx}{1 + x^2}\)
- \(\int \frac{e^x}{x(1 + \log x)^2} dx\)
- \(\int_{1/2}^1 \frac{1}{\sqrt{1 - x^2}} dx\)
- Q6(c) Prove reduction relation:
$$
I_n = \int_0^{\pi/4} \tan^n \theta d\theta = \frac{1}{n - 1} - I_{n - 2}
$$
📘 Gamma and Beta Functions¶
🟨 181 Term
- Q7(a) Define Gamma and Beta function. Prove \(B(m, n) = B(n, m)\)
🟨 191 Term
- Q7(a) Define Gamma and Beta function, prove \(\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}\)
🟨 201 Term
- Q6(c) Same as above for \(\Gamma\left(\frac{1}{2}\right)\)
🟨 211 Term
- Q6(c) Same structure, proving \(\int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}\)
📘 Theorems: MVT, Rolle, Euler, Fundamental¶
🟨 171 Term
- Q6(a) State and prove the Fundamental Theorem of Integral Calculus
🟨 181 Term
- Q3(a) State and prove Mean Value Theorem
- Q3(b) State Rolle’s Theorem and verify for \(f(x) = x^2 - 3x + 2\) in \([1, 2]\)
- Q3(e) State & prove Euler’s theorem for homogeneous functions
🟨 191 Term
- Q3(a) State & prove Lagrange’s MVT
Find \(c\) for \(f(x) = \sqrt{x}, a = 4, b = 9\)
🟨 201 Term
- Q3(a) Leibnitz’s Theorem; find \(n^{\text{th}}\) derivative of \(y = x^3 e^{5x}\)
- Q3(c) Lagrange MVT; find \(c\) for \(f(x) = x(x - 1)(x - 2), a = 0, b = \frac{1}{2}\)
- Q4(b) Prove Fundamental Theorem again
- Q4(c) State and prove Euler’s theorem on \(f(x, y) = x^2 \log \frac{y}{x}\)
🟨 211 Term
- Q4(a) Prove Euler-type identity for
$$
u = \frac{z}{x} + \frac{z}{y} + \frac{x}{y}
$$
* Q4(c) Euler’s Theorem – verify on \(f(x, y) = x^2 \log \frac{y}{x}\)
🟨 221 Term
- Q3(b) State Rolle’s Theorem. Verify for \(f(x) = x^2 - 3x + 2\) in \((1, 2)\)
- Q3(c) Determine whether MVT applies to
$$
f(x) = \begin{cases}
x \sin\left(\frac{1}{x}\right), & x \ne 0 \
0, & x = 0
\end{cases}
$$
on interval \((-1, 1)\)
* Q4(a) For \(y = a \cos(\log x) + b \sin(\log x)\), prove:
$$
x^2 y'' + x y' + y = 0
$$
📘 Area, Volume, Surface¶
🟨 171 Term
- Q7(a) Area bounded by \(y^2 = 4x\) and \(y = 2x\)
- Q7(b) Arc length of \(y = \log \sec x\) from \(x = 0\) to \(\frac{\pi}{3}\)
- Q7(c) Area enclosed by \(r = a\cos 2\theta\)
- Q7(d) Volume between curves \(y^2 = 9x\) and \(y = 3x\)
🟨 181 Term
- Q7(b) Volume by revolving: \(y^2 = 2x\), \(y = 3x\)
- Q7(c) Area of asteroid \(x^{2/3} + y^{2/3} = a^{2/3}\)
🟨 191 Term
- Q7(b) Area by revolution: \(y = x^3\) from \(x = 0\) to \(1\)
- Q7(c) Volume of sphere from \(x^2 + y^2 = a^2\) rotating around y-axis
🟨 201 Term
- Q7(a) Area between \(x = y^2\) and \(y = x - 2\)
- Q7(b) Surface area generated by \(y = x^3\) around x-axis
- Q7(c) Volume of sphere again
🟨 211 Term
- Q7(a) Same area question as 201
- Q7(b) Surface area: \(y = x^3\), \(x = 0\) to \(1\)
- Q7(c) Volume of sphere
🟨 221 Term
- Q7(a) Area bounded by \(y = x + 6\), \(y = x\), \(x = 0\), \(x = 2\)
- Q7(b) Area between \(y^2 = x^3\) and \(y = 2x\)
- Q7(c) Volume when region under \(y = x^2\), \(x \in [0, 2]\), is rotated about \(y = -1\)