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JEE Main Problem Solver Diagnostic Quiz

Ten advanced multiple choice questions covering calculus, coordinate geometry, rotational mechanics, and thermodynamics.

10 questões · Deck: JEE Main Master Class: Calculus, Mechanics, Coordinate Geometry · Redaktion: Ada

Frage 1

If $F(x) = \int_0^{x^2} \cos(\sqrt{t})\,dt$, what is the derivative $F'(x)$?

  1. $\cos(x)$
  2. $2x\cos(x)$
  3. $x\cos(x)$
  4. $-2x\sin(x)$

Frage 2

What is the value of the definite integral $I = \int_0^{\pi/2} \frac{\sin^3(x)}{\sin^3(x) + \cos^3(x)}\,dx$?

  1. $\frac{\pi}{4}$
  2. $\frac{\pi}{2}$
  3. $1$
  4. $0$

Frage 3

What is the distance between the parallel lines $3x + 4y - 9 = 0$ and $3x + 4y + 6 = 0$?

  1. $1$
  2. $2$
  3. $3$
  4. $5$

Frage 4

What is the eccentricity of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$?

  1. $\frac{3}{5}$
  2. $\frac{4}{5}$
  3. $\frac{9}{25}$
  4. $\frac{5}{3}$

Frage 5

A uniform thin rod has mass $M$, length $L$, and moment of inertia $I_{\text{cm}} = \frac{1}{12}ML^2$ about its center. What is its moment of inertia about an axis through one end?

  1. $\frac{1}{6}ML^2$
  2. $\frac{1}{4}ML^2$
  3. $\frac{1}{3}ML^2$
  4. $\frac{1}{2}ML^2$

Frage 6

If satellite orbital velocity near Earth's surface is $v_o$, what is the minimum launch velocity required for an object to escape Earth's gravity?

  1. $v_o$
  2. $\sqrt{2}\,v_o$
  3. $2v_o$
  4. $4v_o$

Frage 7

An ideal monoatomic gas ($\gamma = \frac{5}{3}$) expands adiabatically to $8$ times its initial volume ($V_2 = 8V_1$). The final pressure $P_2$ is what fraction of initial pressure $P_1$?

  1. $\frac{1}{8}$
  2. $\frac{1}{16}$
  3. $\frac{1}{32}$
  4. $\frac{1}{64}$

Frage 8

In a hydrogen atom, the ground state energy is $-13.6\text{ eV}$. What is the energy of the second excited state ($n = 3$)?

  1. $-3.40\text{ eV}$
  2. $-1.51\text{ eV}$
  3. $-0.85\text{ eV}$
  4. $-4.53\text{ eV}$

Frage 9

What is the value of the limit $\lim_{x \to 0} \frac{1 - \cos(2x)}{x^2}$?

  1. $0$
  2. $1$
  3. $2$
  4. $4$

Frage 10

Given vectors $\vec{a} = 2\hat{i} + 3\hat{j}$ and $\vec{b} = \hat{i} - 2\hat{j}$, what is the scalar projection of $\vec{a}$ onto $\vec{b}$?

  1. $-\frac{4}{\sqrt{5}}$
  2. $\frac{4}{\sqrt{5}}$
  3. $-4$
  4. $\frac{2}{\sqrt{13}}$