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GRE General Science — practice by topic

Browse 16 questions across 1 General Science topics. Choose what you want to revise, then open a question to check its answer and worked solution.

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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 →

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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