JEE Main Master Class: Calculus, Mechanics, Coordinate Geometry · 公開単語カード

問題 1: What is the standard examination format and scoring scheme for JEE Main Paper 1 (B.E./B.Tech)?

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解答

90 questions total across Math, Physics, and Chemistry (30 each). Attempt 75 questions (20 MCQ + 5 numerical per subject). Scoring: +4 for correct, -1 for incorrect.

Conducted twice annually by the National Testing Agency (NTA). Paper 1 serves as the qualifier for JEE Advanced and admissions to NITs, IIITs, and CFTIs.

問題 2: What is Leibniz's rule for differentiating an integral with variable limits $\frac{d}{dx} \int_{u(x)}^{v(x)} f(t)\,dt$ ?

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解答

$\frac{d}{dx} \int_{u(x)}^{v(x)} f(t)\,dt = f(v(x)) \cdot v'(x) - f(u(x)) \cdot u'(x)$ .

Widely tested in JEE Main calculus questions involving limits of the form $\frac{0}{0}$ evaluated using L'Hôpital's Rule.

問題 3: What is King's Property of definite integrals for $\int_a^b f(x)\,dx$ ?

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解答

$\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$ . In particular, $\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx$ .

Adding the original integral $I$ to the transformed integral often eliminates transcendental or trigonometric denominators.

問題 4: What is the general solution to the first-order linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ ?

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解答

$y \cdot e^{\int P(x)dx} = \int Q(x) \cdot e^{\int P(x)dx}\,dx + C$ , where $\text{IF} = e^{\int P(x)dx}$ is the integrating factor.

Multiplying by the integrating factor transforms the left side into the exact derivative $\frac{d}{dx}(y \cdot \text{IF})$ .

問題 5: What is the perpendicular distance from point $(x_1, y_1)$ to the straight line $Ax + By + C = 0$ ?

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解答

$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$ .

The distance between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$ .

問題 6: What is the equation of the tangent line to the circle $x^2 + y^2 = r^2$ at point $(x_1, y_1)$ ?

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解答

$x x_1 + y y_1 = r^2$ .

In slope form, the line $y = mx + c$ is tangent to the circle $x^2 + y^2 = r^2$ if and only if $c = \pm r\sqrt{1 + m^2}$ .

問題 7: What is the eccentricity $e$ and condition for standard parabola, ellipse, and hyperbola?

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解答

Parabola: $e = 1$ ; Ellipse: $e = \sqrt{1 - \frac{b^2}{a^2}} < 1$ ( $a > b$ ); Hyperbola: $e = \sqrt{1 + \frac{b^2}{a^2}} > 1$ .

Eccentricity measures the degree of deviation of a conic section from circularity ( $e = 0$ ).

問題 8: What is the Parallel Axis Theorem for moment of inertia $I$ of a rigid body of mass $M$ ?

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解答

$I = I_{\text{cm}} + M d^2$ , where $I_{\text{cm}}$ is the moment of inertia about the parallel center-of-mass axis and $d$ is the distance between axes.

Used to shift rotational axes from the center of mass to hinges or rolling contact points.

問題 9: What is the moment of inertia of a uniform solid cylinder/disk of mass $M$ and radius $R$ about its central symmetry axis?

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解答

$I = \frac{1}{2} M R^2$ .

For a thin hoop/ring, $I = M R^2$ ; for a uniform solid sphere, $I = \frac{2}{5} M R^2$ ; for a thin spherical shell, $I = \frac{2}{3} M R^2$ .

問題 10: What are the orbital velocity $v_o$ and escape velocity $v_e$ from the surface of Earth ( $M, R$ )?

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解答

$v_o = \sqrt{\frac{GM}{R}} = \sqrt{gR}$ ; Escape velocity $v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2} v_o \approx 11.2\text{ km/s}$ .

Escape velocity is obtained by equating total mechanical energy $E = \frac{1}{2}mv^2 - \frac{GMm}{R}$ to zero.

問題 11: What is the period $T$ of a simple harmonic oscillator with mass $m$ attached to a spring of spring constant $k$ ?

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解答

$T = 2\pi \sqrt{\frac{m}{k}}$ and angular frequency $\omega = \sqrt{\frac{k}{m}}$ .

For a simple pendulum of length $L$ in uniform gravity $g$ , $T = 2\pi \sqrt{\frac{L}{g}}$ for small amplitude oscillations ( $\sin\theta \approx \theta$ ).

問題 12: What is the adiabatic gas relation for an ideal gas undergoing a reversible quasi-static process?

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解答

$P V^\gamma = \text{constant}$ , $T V^{\gamma - 1} = \text{constant}$ , and $P^{1-\gamma} T^\gamma = \text{constant}$ , where $\gamma = \frac{C_p}{C_v}$ .

For a monoatomic ideal gas, $\gamma = \frac{5}{3} \approx 1.67$ ; for a diatomic gas, $\gamma = \frac{7}{5} = 1.40$ .

問題 13: What is the expression for work done in an isothermal expansion of $n$ moles of ideal gas from $V_1$ to $V_2$ at temperature $T$ ?

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解答

$W = nRT \ln\left(\frac{V_2}{V_1}\right) = nRT \ln\left(\frac{P_1}{P_2}\right)$ .

Since temperature is constant, $\Delta U = 0$ for an ideal gas, meaning all absorbed heat equals work done ( $Q = W$ ).

問題 14: What is the Bohr formula for the energy of an electron in the $n$ -th orbit of a hydrogen-like atom of atomic number $Z$ ?

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解答

$E_n = -13.6 \frac{Z^2}{n^2}\text{ eV}$ .

The orbital radius scales as $r_n = 0.529 \frac{n^2}{Z}\text{ \AA}$ , and electron velocity scales as $v_n = 2.18 \times 10^6 \frac{Z}{n}\text{ m/s}$ .

問題 15: What is radioactive decay law for the number of undecayed nuclei $N(t)$ with decay constant $\lambda$ ?

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解答

$N(t) = N_0 e^{-\lambda t}$ , with half-life $T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$ .

Activity is defined as $A = |\frac{dN}{dt}| = \lambda N = A_0 e^{-\lambda t}$ , measured in Becquerels (Bq).

問題 16: What is the standard limit $\lim_{x \to 0} \frac{\sin x}{x}$ and $\lim_{x \to 0} \frac{e^x - 1}{x}$ ?

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解答

$\lim_{x \to 0} \frac{\sin x}{x} = 1$ and $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$ .

Also $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$ and $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$ .

問題 17: What is Rolle's Theorem and the Lagrange Mean Value Theorem (LMVT)?

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解答

LMVT states that if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , there exists $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$ .

Rolle's theorem is the special case where $f(a) = f(b)$ , guaranteeing at least one stationary point where $f'(c) = 0$ .

問題 18: How is vector cross product $\vec{a} \times \vec{b}$ computed using components?

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解答

$\vec{a} \times \vec{b} = \det \begin{pmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{pmatrix}$ , resulting in a vector perpendicular to both $\vec{a}$ and $\vec{b}$ .

Magnitude is $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$ , representing the area of the parallelogram formed by the vectors.

問題 19: What is the magnetic field $B$ at the center of a circular current loop of radius $R$ carrying current $I$ ?

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解答

$B = \frac{\mu_0 I}{2R}$ .

Along the central axis at distance $x$ from the loop plane, the field is $B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$ .

問題 20: What is the condition for an extremum (local minimum or maximum) of a function $f(x)$ using second derivatives?

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解答

If $f'(c) = 0$ and $f''(c) > 0$ , $f(c)$ is a local minimum; if $f'(c) = 0$ and $f''(c) < 0$ , $f(c)$ is a local maximum.

If $f''(c) = 0$ , the test is inconclusive and higher derivatives or sign changes of $f'(x)$ must be examined.