Showing all your working and without the use of a calculator, find the square roots of the complex number 7 - ( 6 2 ) i . Give your answers in the form x + i y , where x and y are real and exact.

Let 7 - 6 2 i = x + i y Squaring to both the sides ⇒ 7 - 6 2 i = ( x + i y ) 2 ⇒ 7 - 6 2 i = x 2 - y 2 + 2 i x y ⇒ 7 + ( - 6 2 ) i = x 2 - y 2 + i ( 2 x y ) Equating real and imaginary part both sides, we get x 2 - y 2 = 7 . . . . 1 and 2 x y = - 6 2 ⇒ y = - 3 2 x Substitute y in equation 1 ⇒ x 2 - - 3 2 x 2 = 7 ⇒ x 2 - 18 x 2 = 7 ⇒ x 4 - 18 = 7 x 2 ⇒ x 4 - 7 x 2 - 18 = 0 ⇒ x 4 - 9 x 2 + 2 x 2 - 18 = 0 ⇒ x 2 x 2 - 9 + 2 x 2 - 9 = 0 ⇒ x 2 - 9 x 2 + 2 = 0 Therefore, x 2 - 9 = 0 or x 2 ≠ - 2 ⇒ x 2 = 9 ⇒ x = ± 3 , ⇒ y = - 3 2 3 or - 3 2 3 ∴ y = - 2 or 2 The roots are - 3 + i 2 and 3 - i 2

The line L 1 has vector equation r = - 2 3 0 + λ 1 - 1 2 . The line L 2 has vector equation r = - 4 5 m + μ - 1 0 1 . In the case where m = 2 , show that L 1 and L 2 do not intersect.

Let r 1 = - 2 3 0 + λ 1 - 1 2 and r 2 = - 4 5 2 + μ - 1 0 1 Consider r 1 ( λ ) = r 2 ( μ ) - 2 + λ = - 4 - μ . . . . 1 3 - λ = 5 . . . . 2 2 λ = 2 + μ . . . . 3 From equation 2 , we get λ = - 2 Substitute λ value in equation 3 , we get μ = - 6 Now, substitute the obtained values in the equation 1 , we get - 2 + ( - 2 ) = - 4 - ( - 6 ) - 4 = 2 is not valid Hence, L 1 and L 2 do not intersect.

The line L 1 has vector equation r = - 2 3 0 + λ 1 - 1 2 . The line L 2 has vector equation r = - 4 5 2 + μ - 1 0 1 . Find the acute angle between L 1 and L 2 . .

If θ is the acute angle between the lines r → = a → 1 + λ b → 1 and r → = a → 2 + λ b → 2 then θ is given by cos θ = b → 1 · b → 2 b → 1 b → 2 Here, b 1 → = 1 - 1 2 and b → 2 = - 1 0 1 cos θ = - 1 + 0 + 2 1 + 1 + 4 1 + 0 + 1 cos θ = 1 12 θ = cos - 1 1 12 θ = 73 . 2 °

MEDIUM

AS and A Level

IMPORTANT

Earn 100

The diagram shows a cuboid $O A B C D E F G$ with a horizontal base $O A B C$ . The cuboid has a length $O A$ of $10 cm,$ a width $A B$ of $4 cm$ and a height $B F$ of $h cm .$ The point $M$ is the midpoint of $C B$ and the point $N$ is the point on $D G$ such that $\vec{O N} = j + 3 k .$ The unit vectors $i, j$ and $k$ are parallel to $O A, O C$ and $O D,$ respectively.

Find angle $N M E .$

Important Questions on Cross-Topic Review Exercise 4

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 17 - Cross-Topic Review Exercise 4>CROSS-TOPIC REVIEW EXERCISE 4>Q 24

Question Image

The diagram shows a cuboid $O A B C D E F G$ with a horizontal base $O A B C$ . The cuboid has a length $O A$ of $10 cm,$ a width $A B$ of $4 cm$ and a height $B F$ of $h cm .$ The point $M$ is the midpoint of $C B$ and the point $N$ is the point on $D G$ such that $\vec{O N} = j + 3 k .$ The unit vectors $i, j$ and $k$ are parallel to $O A, O C$ and $O D,$ respectively.

Find, in the form $r = a + λ b,$ a vector equation of the line $M F .$

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 17 - Cross-Topic Review Exercise 4>CROSS-TOPIC REVIEW EXERCISE 4>Q 25

The variables $x$ and $y$ are related by the differential equation $\frac{d y}{d x} = \frac{6 x e^{3 x}}{y^{2}} .$

It is given that $y = 2$ when $x = 0 .$ Solve the differential equation and hence find the value of $y$ when $x = 0.5,$ giving your answer correct to $2$ decimal places.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 17 - Cross-Topic Review Exercise 4>CROSS-TOPIC REVIEW EXERCISE 4>Q 26

Showing all your working and without the use of a calculator, find the square roots of the complex number

7 - (6 \sqrt{2}) i

. Give your answers in the form

x + i y,

where

x

and

y

are real and exact.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 17 - Cross-Topic Review Exercise 4>CROSS-TOPIC REVIEW EXERCISE 4>Q 26

On an Argand diagram, sketch the loci of points representing complex numbers

w

and

z

such that

| w - 1 - 2 i | = 1

and

\arg (z - 1) = \frac{3}{4} π .

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 17 - Cross-Topic Review Exercise 4>CROSS-TOPIC REVIEW EXERCISE 4>Q 26

On an Argand diagram, sketch the loci of points representing complex numbers $w$ and $z$ such that $| w - 1 - 2 i | = 1$ and $\arg (z - 1) = \frac{3}{4} π .$

Calculate the least value of $|w - z|$ for points on these loci.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 17 - Cross-Topic Review Exercise 4>CROSS-TOPIC REVIEW EXERCISE 4>Q 27

The line $L_{1}$ has vector equation $r = (\begin{array}{r} - 2 \\ 3 \\ 0 \end{array}) + λ (\begin{array}{r} 1 \\ - 1 \\ 2 \end{array}) .$
The line $L_{2}$ has vector equation $r = (\begin{array}{r} - 4 \\ 5 \\ m \end{array}) + μ (\begin{array}{r} - 1 \\ 0 \\ 1 \end{array}) .$

In the case where $m = 2,$ show that $L_{1}$ and $L_{2}$ do not intersect.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 17 - Cross-Topic Review Exercise 4>CROSS-TOPIC REVIEW EXERCISE 4>Q 27

Find the value of $m$ in the case where $L_{1}$ and $L_{2}$ intersect.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 17 - Cross-Topic Review Exercise 4>CROSS-TOPIC REVIEW EXERCISE 4>Q 27

The line $L_{1}$ has vector equation $r = (\begin{array}{r} - 2 \\ 3 \\ 0 \end{array}) + λ (\begin{array}{r} 1 \\ - 1 \\ 2 \end{array}) .$
The line $L_{2}$ has vector equation $r = (\begin{array}{r} - 4 \\ 5 \\ 2 \end{array}) + μ (\begin{array}{r} - 1 \\ 0 \\ 1 \end{array}) .$

Find the acute angle between $L_{1}$ and $L_{2} .$ .

Important Questions on Cross-Topic Review Exercise 4

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