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Practise

Question bank

Original exam-style questions, mapped to your syllabus. Answers are checked for mathematical equivalence and every attempt updates your mastery.

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14 questions

SA-9709-P3-VE3-0051 markL1 FoundationCalculatorAngles between lines
The lines with directions (2k6)\begin{pmatrix}2\\k\\6\end{pmatrix} and (1−23)\begin{pmatrix}1\\-2\\3\end{pmatrix} are parallel. Find kk.

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Salam Academy Original Practice · Proposed mark scheme
SA-9709-P3-VE3-G013 marksL1 FoundationCalculatorLine equation
The line ll has equation r=(20−1)+λ(132)\mathbf{r}=\begin{pmatrix}2\\0\\-1\end{pmatrix}+\lambda\begin{pmatrix}1\\3\\2\end{pmatrix}. Find the coordinates of the point on ll where λ=3\lambda=3.

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Salam Academy Original Practice · Proposed mark scheme
SA-9709-P3-VE3-Q011 markL1 FoundationCalculatorLine equation
A direction vector for the line through AA and BB is
Choose one option

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Salam Academy Original Practice · Proposed mark scheme
SA-9709-P3-VE3-Q021 markL1 FoundationCalculatorAngles between lines
The angle between the lines r=a1+λd1\mathbf{r}=\mathbf{a}_1+\lambda\mathbf{d}_1 and r=a2+μd2\mathbf{r}=\mathbf{a}_2+\mu\mathbf{d}_2 is found from
Choose one option

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Salam Academy Original Practice · Proposed mark scheme
SA-9709-P3-VE-0063 marksL2 StandardCalculatorLine equation
The line ll passes through A(1,−2,3)A(1,-2,3) and B(4,0,1)B(4,0,1), and has equation r=OA→+λAB→\mathbf{r}=\overrightarrow{OA}+\lambda\overrightarrow{AB}. Find the coordinates of the point on ll where λ=2\lambda=2.

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Salam Academy Original Practice · Proposed mark scheme
SA-9709-P3-VE3-0013 marksL2 StandardCalculatorLine equation
The line through A(1,2,3)A(1,2,3) and B(4,0,5)B(4,0,5) is r=OA→+λAB→\mathbf{r}=\overrightarrow{OA}+\lambda\overrightarrow{AB}. Find the xx- and yy-coordinates of the point on the line where z=11z=11.

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Salam Academy Original Practice · Proposed mark scheme
SA-9709-P3-VE3-0022 marksL2 StandardCalculatorLine equation
The point (7,a,7)(7,a,7) lies on the line r=(123)+λ(3−22)\mathbf{r}=\begin{pmatrix}1\\2\\3\end{pmatrix}+\lambda\begin{pmatrix}3\\-2\\2\end{pmatrix}. Find aa.

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Salam Academy Original Practice · Proposed mark scheme
SA-9709-P3-VE3-0033 marksL2 StandardCalculatorAngles between lines
Find the acute angle between lines with direction vectors i+2j+2k\mathbf{i}+2\mathbf{j}+2\mathbf{k} and 3j+4k3\mathbf{j}+4\mathbf{k}, in degrees correct to 1 decimal place.

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Salam Academy Original Practice · Proposed mark scheme
SA-9709-P3-VE3-0063 marksL2 StandardCalculatorLine equation
The line mm passes through A(3,−1,2)A(3,-1,2) and is parallel to r=(111)+μ(20−1)\mathbf{r}=\begin{pmatrix}1\\1\\1\end{pmatrix}+\mu\begin{pmatrix}2\\0\\-1\end{pmatrix}. Write mm as r=OA→+λd\mathbf{r}=\overrightarrow{OA}+\lambda\mathbf{d} and find the point on mm where λ=−1\lambda=-1.

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Salam Academy Original Practice · Proposed mark scheme
SA-9709-P3-VE3-0073 marksL2 StandardCalculatorLine equation
Find the point where the line r=(123)+λ(3−22)\mathbf{r}=\begin{pmatrix}1\\2\\3\end{pmatrix}+\lambda\begin{pmatrix}3\\-2\\2\end{pmatrix} crosses the plane z=0z=0. Give the xx- and yy-coordinates.

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Salam Academy Original Practice · Proposed mark scheme
SA-9709-P3-VE-0073 marksL3 ExamCalculatorLine equation
The line ll has equation r=(1−23)+λ(32−2)\mathbf{r}=\begin{pmatrix}1\\-2\\3\end{pmatrix}+\lambda\begin{pmatrix}3\\2\\-2\end{pmatrix}. The point C(−5,−6,7)C(-5,-6,7) lies on ll. Find the value of λ\lambda at CC.

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Salam Academy Original Practice · Proposed mark scheme
SA-9709-P3-VE3-0043 marksL3 ExamCalculatorAngles between lines
Find the acute angle between the lines r=(014)+λ(1−12)\mathbf{r}=\begin{pmatrix}0\\1\\4\end{pmatrix}+\lambda\begin{pmatrix}1\\-1\\2\end{pmatrix} and r=(30−2)+μ(−2−1−1)\mathbf{r}=\begin{pmatrix}3\\0\\-2\end{pmatrix}+\mu\begin{pmatrix}-2\\-1\\-1\end{pmatrix}, in degrees.

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Salam Academy Original Practice · Proposed mark scheme
SA-9709-P3-VE-0104 marksL4 ChallengingCalculatorAngles between lines
Find the acute angle between the lines r=(123)+λ(1−12)\mathbf{r}=\begin{pmatrix}1\\2\\3\end{pmatrix}+\lambda\begin{pmatrix}1\\-1\\2\end{pmatrix} and r=(631)+μ(21−2)\mathbf{r}=\begin{pmatrix}6\\3\\1\end{pmatrix}+\mu\begin{pmatrix}2\\1\\-2\end{pmatrix}, in degrees correct to 1 decimal place.

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Salam Academy Original Practice · Proposed mark scheme
SA-9709-P3-VE3-0084 marksL4 ChallengingCalculatorLine equation
The line ll is r=(102)+λ(21−2)\mathbf{r}=\begin{pmatrix}1\\0\\2\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-2\end{pmatrix}, and AA is the point (1,0,2)(1,0,2). Two points on ll are at a distance 99 from AA. Find their xx-coordinates.

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Salam Academy Original Practice · Proposed mark scheme