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JEE Advanced Mensuration: Surface Areas & Volumes — practice questions

Explore 41 free Mensuration: Surface Areas & Volumes questions in JEE Advanced Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Mensuration: Surface Areas & Volumes questions and exercises with answers

Showing questions 1–30 of 41.

The projection of vector $\vec A$ on vector $\vec B$ is given byRead the answer and worked solution →Which statement about a null vector is correct?Read the answer and worked solution →When subtracting vectors, $\vec A-\vec B$ is evaluated asRead the answer and worked solution →For nonzero vector $\mathbf{A}$, which expression gives a vector in the same direction as $\mathbf{A}$ but of magnitude $1$?Read the answer and worked solution →Let $\mathbf{A}=\langle 1,2,3\rangle$, $\mathbf{B}=\langle 2,0,1\rangle$, and $\mathbf{C}=\langle 0,1,1\rangle$. Then $\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})$ equalsRead the answer and worked solution →Which expression correctly represents the vector triple product $\mathbf{A}\times(\mathbf{B}\times\mathbf{C})$?Read the answer and worked solution →If $\mathbf{A}\cdot\mathbf{B}=0$, then the vectors $\mathbf{A}$ and $\mathbf{B}$ areRead the answer and worked solution →Which of the following equals $\mathbf{A}\times\mathbf{B}$ for $\mathbf{A}=\langle 1,2,0\rangle$ and $\mathbf{B}=\langle 3,-1,4\rangle$?Read the answer and worked solution →If $\mathbf{A}=\langle 1,2,3\rangle$, then $3\mathbf{A}$ equalsRead the answer and worked solution →If $\mathbf{A}=\langle 2,-1,4\rangle$ and $\mathbf{B}=\langle 3,5,-2\rangle$, then $\mathbf{A}+\mathbf{B}$ isRead the answer and worked solution →For the matrix equation $AX=B$, if $|A|\neq 0$, the unique solution isRead the answer and worked solution →A square matrix $A$ is orthogonal whenRead the answer and worked solution →A square matrix $A$ is symmetric ifRead the answer and worked solution →If $A$ is of order $2\times 3$ and $B$ is of order $3\times 4$, then the product $AB$ is defined and has orderRead the answer and worked solution →Which of the following is a diagonal matrix?Read the answer and worked solution →Evaluate $\displaystyle\lim_{x \to 0} \dfrac{\sin 3x}{x}$.Read the answer and worked solution →Evaluate $\displaystyle\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2}$.Read the answer and worked solution →Evaluate $\displaystyle\lim_{x \to 0} \dfrac{1 - \cos 2x}{x^2}$.Read the answer and worked solution →Evaluate $\displaystyle\lim_{x \to \infty} \left(1 + \dfrac{2}{x}\right)^x$.Read the answer and worked solution →Consider $f(x) = |x|$ at $x = 0$. Which statement is correct?Read the answer and worked solution →Find the modulus of the complex number $z = 3 + 4i$.Read the answer and worked solution →What is the principal argument of the complex number $z = 1 + i$?Read the answer and worked solution →If $\omega$ is a non-real cube root of unity, find the value of $1 + \omega + \omega^2$.Read the answer and worked solution →Using De Moivre's theorem, evaluate $\left(\cos\dfrac{\pi}{6} + i\sin\dfrac{\pi}{6}\right)^6$.Read the answer and worked solution →The locus of the point $z$ in the Argand plane satisfying $|z - 2| = |z + 2|$ is:Read the answer and worked solution →If $z = \cos\theta + i\sin\theta$, find an expression for $z^n + \dfrac{1}{z^n}$.Read the answer and worked solution →Find the eccentricity of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$.Read the answer and worked solution →The foci of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$ lie at:Read the answer and worked solution →An ellipse has foci at $(\pm 4, 0)$ and passes through the point $(5, 0)$. Find its standard equation.Read the answer and worked solution →Find the length of the latus rectum of the parabola $y^2 = 8x$.Read the answer and worked solution →

Worked practice questions

The projection of vector $\vec A$ on vector $\vec B$ is given by

  1. $\dfrac{\vec A\times\vec B}{|\vec B|}$
  2. $\dfrac{\vec A\cdot\vec B}{|\vec B|}$
  3. $\dfrac{|\vec A|}{\vec B}$
  4. $\dfrac{\vec A\cdot\vec B}{|\vec A|}$
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Correct answer: B. $\dfrac{\vec A\cdot\vec B}{|\vec B|}$

The notes explicitly give the projection of $\vec A$ on $\vec B$ as $\dfrac{\vec A\cdot\vec B}{|\vec B|}$. This is the scalar projection along $\vec B$.

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Which statement about a null vector is correct?

  1. Its direction is fixed along the positive x-axis
  2. Its dot product with any vector is always zero
  3. Adding it to a vector reverses the vector
  4. Its cross product with any vector gives a scalar
Show answer and solution

Correct answer: B. Its dot product with any vector is always zero

A null vector has zero magnitude and arbitrary direction. The notes state that its dot product with any vector is zero, and its cross product with any vector is also a null vector. Adding or subtracting it leaves the original vector unchanged.

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When subtracting vectors, $\vec A-\vec B$ is evaluated as

  1. $\vec A + \vec B$
  2. $\vec B + (-\vec A)$
  3. $\vec A + (-\vec B)$
  4. $(-\vec A) + (-\vec B)$
Show answer and solution

Correct answer: C. $\vec A + (-\vec B)$

Vector subtraction is performed by reversing the direction of the vector being subtracted and then adding. **Hence $\vec A-\vec B = \vec A + (-\vec B)$.**

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