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GRE General Knowledge — practice questions

Explore 16 free General Knowledge questions in GRE 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 1–16 of 16.

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 →If $\alpha + \beta = \frac{\pi}{2}$ and $\beta + \gamma = \alpha$, then $\tan \alpha$ equalsRead the answer and worked solution →In $\Delta ABC$, $b^2 \cos 2A - a^2 \cos 2B =$Read the answer and worked solution →In the equation $x^2 - \frac{15}{4}x + a^3 = 0$, one of the roots is the square of the other if $a$ is equal toRead the answer and worked solution →Consider an infinite geometric series with first term $a$ and common ratio $r$. If its sum is 4 and the second term is $3/4$, thenRead the answer and worked solution →If $\omega (\neq 1)$ be a cube root of unity and $(1 + \omega^2)^n = (1 + \omega^4)^n$, then the least positive value of $n$ isRead the answer and worked solution →The least positive integer $n$ such that $\left( \frac{2i}{1 + i} \right)^n$ is a positive integer isRead the answer and worked solution →If the term independent of $x$ in the $\left( \sqrt{x} - \frac{k}{x^2} \right)^{10}$ is 405, then $k$ equalsRead the answer and worked solution →If the $4^{\text{th}}$ term in the expansion of $\left( ax + \frac{1}{x} \right)^n$ is $\frac{5}{2}$, then the values of $a$ and $n$ areRead the answer and worked solution →If $\sin^{-1}\left(\frac{5}{x}\right) + \sin^{-1}\frac{12}{x} = \frac{\pi}{2}$, then $x =$Read the answer and worked solution →If $f(x)$ is odd function and $f(1) = a$, and $f(x + 2) = f(x) + f(2)$ then the value of $f(3)$ isRead the answer and worked solution →If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 4x + 1 = 0$, then $(\alpha + \beta)^{-1} + (\beta + \gamma)^{-1} + (\gamma + \alpha)^{-1} =$Read the answer and worked solution →

Worked practice questions

For the system $a_1x+b_1y=c_1$, $a_2x+b_2y=c_2$, Cramer's rule gives

  1. $x=\dfrac{D}{D_1},\; y=\dfrac{D}{D_2}$
  2. $x=\dfrac{D_1}{D},\; y=\dfrac{D_2}{D}$
  3. $x=\dfrac{D_2}{D_1},\; y=\dfrac{D_1}{D_2}$
  4. $x=D_1D,\; y=D_2D$
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Correct answer: B. $x=\dfrac{D_1}{D},\; y=\dfrac{D_2}{D}$

For two linear equations in two unknowns, Cramer's rule states $x=\frac{D_1}{D}$ and $y=\frac{D_2}{D}$, provided $D\ne 0$. Here $D$ is the determinant of coefficients and $D_1,D_2$ are obtained by replacing the respective variable columns by constants.

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Which property of determinants is correct?

  1. Interchanging two adjacent rows leaves the determinant unchanged
  2. If two rows are identical, the determinant is zero
  3. Multiplying one row by $k$ leaves the determinant unchanged
  4. Changing rows into columns always changes the sign of the determinant
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Correct answer: B. If two rows are identical, the determinant is zero

If two rows or two columns of a determinant are identical, its value becomes zero. Interchanging two rows changes the sign, not preserves it multiplying one row by $k$ multiplies the determinant by $k$ and transposing a determinant leaves its value unchanged.

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If $M_{ij}$ is the minor of an element $a_{ij}$, then its cofactor $C_{ij}$ is

  1. $C_{ij}=M_{ij}$ for all $i,j$
  2. $C_{ij}=(-1)^{i-j}M_{ij}$
  3. $C_{ij}=(-1)^{i+j}M_{ij}$
  4. $C_{ij}=-(i+j)M_{ij}$
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Correct answer: C. $C_{ij}=(-1)^{i+j}M_{ij}$

The cofactor of the element in the $i$th row and $j$th column is defined by $C_{ij}=(-1)^{i+j}M_{ij}$. **Thus the sign depends on whether $i+j$ is even or odd.**

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