Mathematics B MEI Core Pure
questions and answers
Answer .all .the .questions.
1 (a) . . (i) Write .the .following .simultaneous .equations .as .a .matrix .equation.
x .+ . y .+.2z .= .7
2x .- .4y .- .3z . =-5
-5x .+.3y .+.5z .= .13 [1]
(ii) Hence .solve .the .equations. [2]
(b) .Determine .the .set .of .values .of .the .constant .k .for .which .the .matrix .equation
Jk .+ .1 NJ N J
K 1 OKx.O .= .K 23ON
L 2 kPLyP -17
has .a .unique .solution. [3]
2 (a) . Show .that .the .vector . i .+.4j .+ .2k . is .parallel .to .the .plane . 2x .+.y .- .3z .= .10. [3]
(b) . Determine .the .acute .angle .between .the .planes . 2x .+.y .- .3z .= .10 . and . x .-.y .-.3z .= [4]
.3.
3 The .complex .number .z .satisfies .the .equation . 5 .(z .-.i) .= .(-1 .+.2i) .z*.
Determine .z, .giving .your .answer .in .the .form . a .+ .bi, .where .a .and .b .are .real. [5]
, 4 In .this .question .you .must .show .detailed .reasoning.
The .equation . z3 .+.2z2 .+ .kz .+ .3 .= .0, .where .k .is .a .constant, .has .roots .a, and .b.
.1
.
a
Determine .the .roots .in .exact .form. [6]
© . OCR .2022 Y410/01 .Jun22