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JEE Main General Knowledge — practice questions — page 2

Explore 58 free General Knowledge questions in JEE Main General Science. Open each question or exercise to read its content, reference answers and available explanation.

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General Knowledge questions and exercises with answers

Showing questions 31–58 of 58.

If $\mathbf{A}=\langle 2,1,2\rangle$ and $\mathbf{B}=\langle 1,0,2\rangle$, what is $\operatorname{proj}_{\mathbf{B}}\mathbf{A}$?Read the answer and worked solution →The projection of $\mathbf{A}$ onto $\mathbf{B}$ is proportional toRead the answer and worked solution →If $\mathbf{A}=\langle 3,4,0\rangle$, then the unit vector in the direction of $\mathbf{A}$ isRead the answer and worked solution →Which identity for the scalar triple product is correct?Read the answer and worked solution →For vectors $\mathbf{A}=\langle 2,-1,3\rangle$ and $\mathbf{B}=\langle 4,0,-2\rangle$, the dot product $\mathbf{A}\cdot\mathbf{B}$ isRead the answer and worked solution →The position vector of a point $P(x,y,z)$ in three-dimensional space is given byRead the answer and worked solution →The trajectory equation of a projectile shows that its path is aRead the answer and worked solution →In free fall, if the upward direction is chosen as the positive $y$-axis, the velocity after time $t$ isRead the answer and worked solution →For gases only, the molecular weight is given in the notes byRead the answer and worked solution →In volumetric analysis, the equation $N_1V_1=N_2V_2$ is used on the basis of equality ofRead the answer and worked solution →If a solute has valency factor $x$, then normality $N$ is related to molarity $M$ byRead the answer and worked solution →For a gas at NTP, the number of moles is given byRead the answer and worked solution →The number of moles present in 18 g of water isRead the answer and worked solution →One mole of any substance contains which of the following numbers of particles?Read the answer and worked solution →For the system $a_1x+b_1y=c_1$, $a_2x+b_2y=c_2$, Cramer's rule givesRead the answer and worked solution →Which property of determinants is correct?Read the answer and worked solution →If $M_{ij}$ is the minor of an element $a_{ij}$, then its cofactor $C_{ij}$ isRead the answer and worked solution →Which of the following statements about rank is given in the chapter?Read the answer and worked solution →If $A$ is skew-symmetric, which of the following must be true?Read the answer and worked solution →For a velocity-time graph, the slope representsRead the answer and worked solution →Instantaneous power delivered by a force is best expressed as:Read the answer and worked solution →Average power is defined as:Read the answer and worked solution →In a system where only conservative forces act, which quantity remains constant?Read the answer and worked solution →According to the work-energy theorem, the net work done on a particle is equal to:Read the answer and worked solution →Which of the following is an example of elastic potential energy?Read the answer and worked solution →In an ideal pendulum oscillating without air resistance, which quantity remains constant?Read the answer and worked solution →Work done by a force on a body is negative when:Read the answer and worked solution →Which statement best defines work done by a force in mechanics?Read the answer and worked solution →

Worked practice questions

If $\mathbf{A}=\langle 2,1,2\rangle$ and $\mathbf{B}=\langle 1,0,2\rangle$, what is $\operatorname{proj}_{\mathbf{B}}\mathbf{A}$?

  1. $\left\langle \frac{6}{5},0,\frac{12}{5}\right\rangle$
  2. $\langle 6,0,12\rangle$
  3. $\left\langle \frac{5}{6},0,\frac{5}{3}\right\rangle$
  4. $\left\langle \frac{4}{5},\frac{1}{5},\frac{8}{5}\right\rangle$
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Correct answer: A. $\left\langle \frac{6}{5},0,\frac{12}{5}\right\rangle$

First compute $\mathbf{A}\cdot\mathbf{B}=2\cdot 1+1\cdot 0+2\cdot 2=6$. Also, $\|\mathbf{B}\|^2=1^2+0^2+2^2=5$. Hence $\operatorname{proj}_{\mathbf{B}}\mathbf{A}=\dfrac{\mathbf{A}\cdot\mathbf{B}}{\|\mathbf{B}\|^2}\mathbf{B}=\dfrac{6}{5}\langle 1,0,2\rangle=\left\langle \frac{6}{5},0,\frac{12}{5}\right\rangle$.

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The projection of $\mathbf{A}$ onto $\mathbf{B}$ is proportional to

  1. $\mathbf{A}\times\mathbf{B}$
  2. $\dfrac{\mathbf{A}\cdot\mathbf{B}}{\|\mathbf{B}\|}\cdot\dfrac{\mathbf{B}}{\|\mathbf{B}\|}$
  3. $\dfrac{\mathbf{A}\times\mathbf{B}}{\|\mathbf{A}\|}$
  4. $\dfrac{\mathbf{B}\cdot\mathbf{B}}{\|\mathbf{A}\|}\cdot\mathbf{A}$
Show answer and solution

Correct answer: B. $\dfrac{\mathbf{A}\cdot\mathbf{B}}{\|\mathbf{B}\|}\cdot\dfrac{\mathbf{B}}{\|\mathbf{B}\|}$

From the given formula, $\operatorname{proj}_{\mathbf{B}}\mathbf{A}=\dfrac{\mathbf{A}\cdot\mathbf{B}}{\|\mathbf{B}\|}\cdot\dfrac{\mathbf{B}}{\|\mathbf{B}\|}$. This gives a vector along $\mathbf{B}$ whose magnitude equals the scalar projection of $\mathbf{A}$ on $\mathbf{B}$.

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If $\mathbf{A}=\langle 3,4,0\rangle$, then the unit vector in the direction of $\mathbf{A}$ is

  1. $\langle 3,4,0\rangle$
  2. $\left\langle \frac{3}{5},\frac{4}{5},0\right\rangle$
  3. $\left\langle \frac{5}{3},\frac{5}{4},0\right\rangle$
  4. $\left\langle \frac{1}{3},\frac{1}{4},0\right\rangle$
Show answer and solution

Correct answer: B. $\left\langle \frac{3}{5},\frac{4}{5},0\right\rangle$

A unit vector is $\hat{\mathbf{A}}=\mathbf{A}/\|\mathbf{A}\|$. Here $\|\mathbf{A}\|=\sqrt{3^2+4^2}=5$, so $\hat{\mathbf{A}}=\left\langle \frac{3}{5},\frac{4}{5},0\right\rangle$.

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