JEE Advanced Calculus - Top 50 Qs

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Functions Domain & Range: Find domain of $f(x) = \sqrt{\frac{x-2}{x-3}}$ and range of $g(x) = \frac{x^2-1}{x^2+1}$. Types of Functions: Check injectivity/surjectivity for $f(x)=x^3-x$ and $g(x)=e^x$. Composite Functions: If $f(x)=x^2+1$ and $g(x)=\sin x$, find $f(g(x))$ and $g(f(x))$. Inverse Functions: Find the inverse of $f(x) = \frac{e^x-e^{-x}}{e^x+e^{-x}}$. Periodicity: Determine the period of $f(x) = \sin(\frac{x}{2}) + \cos(\frac{x}{3})$. Limits L'Hopital's Rule: Evaluate $\lim_{x \to 0} \frac{x - \sin x}{x^3}$ and $\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$. Standard Limits: Compute $\lim_{x \to 0} \frac{\sin x}{x}$, $\lim_{x \to 0} \frac{1-\cos x}{x^2}$, $\lim_{x \to 0} (1+x)^{1/x}$. Infinite Limits: Evaluate $\lim_{x \to \infty} (\sqrt{x^2+x} - x)$. Limit at a Point: Find $\lim_{x \to 1} \frac{x^{10}-1}{x-1}$. Sandwich Theorem: If $1 - \frac{x^2}{4} \le f(x) \le 1 + \frac{x^2}{2}$ for all $x \ne 0$, find $\lim_{x \to 0} f(x)$. Continuity and Differentiability Continuity: For what value of $k$ is $f(x) = \begin{cases} \frac{1-\cos 4x}{x^2} & x \ne 0 \\ k & x=0 \end{cases}$ continuous at $x=0$? Differentiability: Check differentiability of $f(x) = |x-1| + |x-2|$ at $x=1, 2$. Mean Value Theorems: Verify Rolle's Theorem for $f(x) = x^2-4x+3$ on $[1,3]$. Lagrange's MVT: Apply LMV for $f(x)=\sqrt{x}$ on $[1,4]$. Higher Order Derivatives: If $y = e^{ax} \cos(bx)$, find $\frac{d^2y}{dx^2}$. Differentiation Implicit Differentiation: If $x^y = y^x$, find $\frac{dy}{dx}$. Chain Rule: Differentiate $y = \sin(\cos(x^2))$. Logarithmic Differentiation: Find $\frac{dy}{dx}$ for $y = (\sin x)^x$. Parametric Differentiation: If $x = a(\theta - \sin\theta)$, $y = a(1-\cos\theta)$, find $\frac{dy}{dx}$. Derivatives of Inverse Trig: Differentiate $y = \tan^{-1}\left(\frac{2x}{1-x^2}\right)$. Applications of Derivatives Tangents & Normals: Find equations of tangent and normal to $y=x^3-3x^2+2$ at $(1,0)$. Monotonicity: Find intervals where $f(x)=2x^3-9x^2+12x-5$ is increasing/decreasing. Maxima & Minima: Find local max/min of $f(x) = x^3-6x^2+9x+15$. Absolute Max/Min: Find absolute max/min of $f(x) = x^3-3x^2+1$ on $[-2,2]$. Rates of Change: A ladder $5m$ long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at $2cm/s$. How fast is its height on the wall decreasing when the foot of the ladder is $4m$ from the wall? Indefinite Integration Substitution: Evaluate $\int \frac{\sin(\ln x)}{x} dx$ and $\int \frac{e^x}{e^{2x}+1} dx$. Integration by Parts: Evaluate $\int x \cos x dx$ and $\int e^x \sin x dx$. Partial Fractions: Evaluate $\int \frac{x}{(x+1)(x+2)} dx$. Trigonometric Integrals: Evaluate $\int \sin^3 x \cos^2 x dx$ and $\int \tan^4 x dx$. Special Forms: Evaluate $\int \frac{dx}{\sqrt{x^2+a^2}}$ and $\int \sqrt{a^2-x^2} dx$. Definite Integration Fundamental Theorem: Evaluate $\int_0^{\pi/2} \sin^2 x dx$. Properties of Definite Integrals: Evaluate $\int_0^a f(x) dx = \int_0^a f(a-x) dx$. Use this for $\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$. King's Rule: Evaluate $\int_0^1 x(1-x)^n dx$. Wallis' Formula: Evaluate $\int_0^{\pi/2} \sin^n x dx$. Even/Odd Functions: Evaluate $\int_{-1}^1 \frac{x^3}{e^x+e^{-x}} dx$ and $\int_{-\pi/2}^{\pi/2} \sin^4 x dx$. Area Under Curves Area between Curve & Axis: Find area bounded by $y=x^2$ and $y=x$. Area between Two Curves: Find area bounded by $y^2=4ax$ and $x^2=4ay$. Area using Integration: Find area of ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Complex Area: Find area bounded by $y=|\cos x|$, $x$-axis, $x=0$, $x=2\pi$. Area involving Modulus: Find area bounded by $y = |x^2-1|$ and $y=1$. Differential Equations Variable Separable: Solve $\frac{dy}{dx} = e^{x+y}$. Homogeneous Equations: Solve $\frac{dy}{dx} = \frac{x^2+y^2}{2xy}$. Linear Differential Equations: Solve $\frac{dy}{dx} + y \cot x = 2x \csc x$. Exact Differential Equations: Check for exactness and solve $(ax+by)dx + (cx+dy)dy = 0$. Applications: A particle moves such that its acceleration is proportional to its velocity. If velocity is $v_0$ at $t=0$, find $v(t)$. Miscellaneous Advanced Problems Functional Equations with Calculus: If $f(x+y) = f(x)f(y)$ and $f'(0)=2$, find $f(x)$. Series Expansion: Find the Maclaurin series for $f(x) = e^x$. Definite Integral as Limit Sum: Evaluate $\lim_{n \to \infty} \left[ \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n} \right]$. Properties of Functions (Advanced): If $f''(x) > 0$ for all $x \in \mathbb{R}$, prove that $f(x)+f(y) \ge 2f\left(\frac{x+y}{2}\right)$. Optimization (Advanced): A wire of length $L$ is cut into two pieces. One piece is bent into a square and the other into a circle. How should the wire be cut so that the sum of the areas is minimum?