A LevelFurther MathsEdexcelCore PureExam QuestionsFurther VectorsVector Planes

Vector Planes (Edexcel A Level Further Maths: Core Pure): Exam Questions

Exam code: 9FM0

53 mins6 questions
1a
Sme Calculator3 marks

The line l subscript 1 has equation

fraction numerator x minus 1 over denominator 2 end fraction equals fraction numerator y plus 1 over denominator negative 1 end fraction equals fraction numerator z minus 4 over denominator 3 end fraction

The line l subscript 2 has equation

bold r equals bold i plus 3 bold k plus t left parenthesis bold i minus bold j plus 2 bold k right parenthesis

where t is a scalar parameter.

Show that l subscript 1 and l subscript 2 lie in the same plane.

How did you do?

1b1 mark

Write down a vector equation for the plane containing l subscript 1 and l subscript 2.

How did you do?

2a
Sme Calculator2 marks

bold M equals open parentheses table row 2 cell negative 1 end cell 1 row 3 k 4 row 3 2 cell negative 1 end cell end table close parentheses where k is a constant

Find the values of k for which the matrix bold M has an inverse.

How did you do?

2b
Sme Calculator5 marks

Find, in terms of space p, the coordinates of the point where the following planes intersect

table row cell 2 x – y plus z end cell equals p row cell 3 x – 6 y plus 4 z end cell equals 1 row cell 3 x plus 2 y – space z end cell equals 0 end table

How did you do?

2c4 marks

(i) Find the value of q for which the set of simultaneous equations

table row cell 2 x – y plus z end cell equals 1 row cell 3 x – 5 y plus 4 z end cell equals q row cell 3 x plus 2 y – space z end cell equals 0 end table

can be solved.

(ii) For this value of q, interpret the solution of the set of simultaneous equations geometrically.

How did you do?

3a
Sme Calculator4 marks

bold M equals stretchy left parenthesis table row k 5 7 row 1 1 1 row 2 1 cell negative 1 end cell end table stretchy right parenthesis where k is a constant

Given that k not equal to 4 comma find, in terms of k, the inverse of the matrix bold M.

How did you do?

3b3 marks

Find, in terms ofspace p, the coordinates of the point where the following planes intersect.

table row cell 2 x plus 5 y plus 7 z end cell equals 1 row cell x plus y plus z end cell equals p row cell 2 x plus y minus z end cell equals 2 end table

How did you do?

3c
Sme Calculator7 marks

(i) Find the value of q for which the following planes intersect in a straight line.

table row cell 4 x plus 5 y plus 7 z end cell equals 1 row cell x plus y plus z end cell equals q row cell 2 x plus y minus z end cell equals 2 end table

(ii) For this value of q, determine a vector equation for the line of intersection.

How did you do?

47 marks

The line l subscript 1 has equation fraction numerator x minus 2 over denominator 4 end fraction equals fraction numerator y minus 4 over denominator negative 2 end fraction equals fraction numerator z plus 6 over denominator 1 end fraction

The plane capital pi has equation x minus 2 y plus z equals 6 space

The line l subscript 2 is the reflection of the line l subscript 1in the plane capital pi

Find a vector equation of the line l subscript 2

How did you do?

5a
Sme Calculator3 marks

The plane capital pi subscript 1 has vector equation

bold r. left parenthesis 3 bold i space minus space 4 bold j space plus space 2 bold k right parenthesis space equals space 5

(a)

Find the perpendicular distance from the point (6, 2, 12) to the plane capital pi subscript 1

How did you do?

5b
Sme Calculator2 marks

The plane capital pi subscript 2 has vector equation

bold r bold space equals space lambda left parenthesis 2 bold i space plus space bold j space plus space 5 bold k right parenthesis space plus space mu left parenthesis bold i space minus bold space bold j bold space minus space 2 bold k right parenthesis

where lambda and mu are scalar parameters.

(b)

Show that the vector negative bold i space minus space 3 bold j space plus space bold k is perpendicular to capital pi subscript 2

How did you do?

5c
Sme Calculator3 marks

(c)

Show that the acute angle between capital pi subscript 1 and capital pi subscript 2is 52° to the nearest degree.

How did you do?

6a4 marks

The plane capital pi subscript 1 has equation

bold r bold space equals space 2 bold i space plus space 4 bold j space – space bold k space plus space lambda left parenthesis bold i space plus space 2 bold j space – space 3 bold k right parenthesis space plus space mu left parenthesis – bold i space plus space 2 bold j space plus space bold k right parenthesis

where lambda and mu are scalar parameters.

Find a Cartesian equation for capital pi subscript 1

How did you do?

6b3 marks

The line l has equation

fraction numerator x minus 1 over denominator 5 end fraction equals fraction numerator y minus 3 over denominator negative 3 end fraction equals fraction numerator z plus 2 over denominator 4 end fraction

Find the coordinates of the point of intersection l with capital pi subscript 1

How did you do?

6c
Sme Calculator2 marks

The plane capital pi subscript 2 has equation

bold r times left parenthesis 2 bold i space – space bold j space plus space 3 bold k right parenthesis space equals space 5

Find, to the nearest degree, the acute angle between capital pi subscript 1 and capital pi subscript 2

How did you do?