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GRE Mensuration: Surface Areas & Volumes — practice questions

Explore 31 free Mensuration: Surface Areas & Volumes questions in GRE 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 31.

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 →Find the equation of the parabola with vertex at the origin and focus at $(0, -3)$.Read the answer and worked solution →A parabola has its vertex at the origin, its axis along the positive $x$-axis, and passes through the point $(4, 6)$. Find its equation.Read the answer and worked solution →For the parabola $y^2 = 16x$, find the equation of the directrix.Read the answer and worked solution →The standard equation of a hyperbola with centre at the origin and foci on the $x$-axis is:Read the answer and worked solution →For any hyperbola, the eccentricity $e$ satisfies:Read the answer and worked solution →What is the eccentricity of the hyperbola $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$?Read the answer and worked solution →A hyperbola has foci at $(\pm 5, 0)$ and the length of the transverse axis is $6$. Find its equation.Read the answer and worked solution →A fair coin is tossed twice. How many outcomes are there in the sample space?Read the answer and worked solution →The distance between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ in three-dimensional space is:Read the answer and worked solution →The direction cosines of a line are $l$, $m$, $n$. They satisfy:Read the answer and worked solution →

Worked practice questions

For the matrix equation $AX=B$, if $|A|\neq 0$, the unique solution is

  1. $X=AB$
  2. $X=A^{-1}$
  3. $X=A^{-1}B$
  4. $X=BA^{-1}$
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Correct answer: C. $X=A^{-1}B$

When $|A|\neq 0$, the inverse $A^{-1}$ exists. Pre-multiplying $AX=B$ by $A^{-1}$ gives $A^{-1}AX=A^{-1}B$, so $IX=A^{-1}B$, hence $X=A^{-1}B$.

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A square matrix $A$ is orthogonal when

  1. $A+A'=I$
  2. $A'=A$
  3. $AA'=I$ and $A'A=I$
  4. All diagonal elements are equal to 1
Show answer and solution

Correct answer: C. $AA'=I$ and $A'A=I$

An orthogonal matrix satisfies $AA'=I$ and also $A'A=I$. This means its transpose acts as its inverse. Merely having 1s on the diagonal does not guarantee orthogonality.

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A square matrix $A$ is symmetric if

  1. $A'=-A$
  2. $AA'=I$
  3. $A'=A$
  4. Every diagonal entry is zero
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Correct answer: C. $A'=A$

By definition, a symmetric matrix satisfies $A'=A$, equivalently $a_{ij}=a_{ji}$ for all $i,j$. **Option A is the condition for skew-symmetry, and option B is the condition for orthogonality.**

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More Mathematics topics

MathematicsStatistics and ProbabilityArithmetic ProgressionSets and RelationsQuadratic EquationsReal Numbers and PolynomialsUnit 1A1: Relations, Functions and GraphsUnit 6B: Definite Integration & Riemann Sums