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IB Mathematics AA HL - Questionbank

Systems of Equations

Solving 3 x 3 Systems of Linear Equations, Row Operations, Unique/No/Infinite Solutions…

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Paper 2

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Question 1

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easy

[Maximum mark: 6]

The system of equations given below represents three planes in space.

x+5z=22x+y6z=12y+8z=6\begin{aligned} x + 5z &= 2 \\[6pt] -2x + y - 6z &= -1 \\[6pt] 2y + 8z &= 6\end{aligned}
  1. Show that this system of equations has an infinite number of solutions. [3]

  2. Find the parametric equations of the line of intersection of the three planes. [3]

easy

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Question 2

no calculator

medium

[Maximum mark: 5]

The system of equations given below represents three planes in space.

x+2y2z=43x+5y4z=9[where a,bR]4x+6y+az=b\begin{aligned} x + 2y - 2z &= 4 \\[6pt] 3x + 5y - 4z &= 9 \hspace{3em} [\text{where $a,b \in \mathbb{R}$}] \\[6pt] \hspace{8em} 4x + 6y + az &= b\end{aligned}

Find the set of values of aa and bb such that the three planes have no points of
intersection.

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Question 3

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medium

[Maximum mark: 6]

The system of equations given below represents three planes in space.

x3y+2z=53x+5y+az=b[where a,bR]4x+2y3z=7\begin{aligned} x - 3y + 2z &= 5 \\[6pt] 3x + 5y + az &= b \hspace{3em} [\text{where $a,b \in \mathbb{R}$}] \\[6pt] \hspace{8em} 4x + 2y - 3z &= 7\end{aligned}

Find the set of values of aa and bb such that the three planes have exactly one intersection point.

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Question 4

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medium

[Maximum mark: 9]

Consider the following system of equations:

x3z=23x+y+6z=32x2y+(a4)z=b3\begin{aligned} x - 3z &= -2 \\[6pt] -3x + y + 6z &= 3 \\[6pt] 2x - 2y + (a-4)z &= b-3\end{aligned}

where a,bRa,b \in \mathbb{R}.

  1. Find conditions on aa and bb for which

    1. the system has no solutions;

    2. the system has only one solution;

    3. the system has an infinite number of solutions. [6]

  2. In the case where the number of solutions is infinite, find the general
    solution of the system of equations in Cartesian form. [3]

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Question 5

no calculator

medium

[Maximum mark: 9]

Consider the following system of equations:

x+y+4z=13x+2y+16z=54x+2y+(a1)z=b4\begin{aligned} x + y + 4z &= 1 \\[6pt] 3x + 2y + 16z &= 5 \\[6pt] 4x + 2y + (a-1)z &= b-4\end{aligned}

where a,bRa,b \in \mathbb{R}.

  1. Find conditions on aa and bb for which

    1. the system has no solutions;

    2. the system has only one solution;

    3. the system has an infinite number of solutions. [6]

  2. In the case where the number of solutions is infinite, find the general
    solution of the system of equations in Cartesian form. [3]

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Question 6

no calculator

medium

[Maximum mark: 8]

Consider the following system of equations:

x+y+z=14x+2y+z=39x+3y=p\begin{aligned} x + y + z &= -1 \\[6pt] 4x + 2y + z &= 3 \\[6pt] 9x + 3y &= p\end{aligned}

where pRp \in \mathbb{R}.

  1. Show that this system does not have a unique solution for any value of pp. [4]

    1. Determine the value of pp for which the system is consistent.

    2. For this value of pp, find the general solution of the system. [4]

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Question 7

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hard

[Maximum mark: 11]

Consider three planes represented by the following system of equations:

Π1:2x+ay+4z=2Π2:x+by2z=1Π3:2xy+cz=3\begin{align*} \Pi_1: -2x + ay + 4z &= 2 \\[6pt] \Pi_2: x + by - 2z &= -1 \\[6pt] \Pi_3: 2x - y + cz &= 3 \end{align*}

Where a,b,cRa,b,c \in \mathbb{R}.

  1. State the values of aa, bb and cc for which

    1. The system has infinite solutions.

    2. The system is inconsistent. [7]

  2. In the case where the system has infinite solutions, describe the geometric relationship between the three planes. [2]

  3. In the case where the system is inconsistent, identify one of the geometric relationships that could exist between the three planes. [2]

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Frequently Asked Questions

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