Questions - Edexcel C4 - A-Level Maths

Find the equation of the normal to $C$ at $M$ in the form $a x + b y = c$, where $a , b$ and $c$ are integers. This normal to $C$ at $M$ crosses the $x$-axis at the point $N ( n , 0 )$.
Show that $n = 14$. The point $P ( \ln 4,13 )$ lies on $C$. The finite region $R$ is bounded by $C$, the axes and the line $P N$, as shown in Fig. 3.
Find the area of $R$, giving your answers in the form $p + q \ln 2$, where $p$ and $q$ are integers to be found.

Edexcel C4 Q8

8. Referred to an origin $O$, the points $A , B$ and $C$ have position vectors ( $9 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$ ), $( 6 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k } )$ and $( 3 \mathbf { i } + p \mathbf { j } + q \mathbf { k } )$ respectively, where $p$ and $q$ are constants.

Find, in vector form, an equation of the line $l$ which passes through $A$ and $B$. Given that $C$ lies on $l$,
find the value of $p$ and the value of $q$,
calculate, in degrees, the acute angle between $O C$ and $A B$. The point $D$ lies on $A B$ and is such that $O D$ is perpendicular to $A B$.
Find the position vector of $D$.

Edexcel C4 Q1

A curve has the equation

$$x ^ { 2 } ( 2 + y ) - y ^ { 2 } = 0 .$$ Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.

Edexcel C4 Q2

2. $$f ( x ) = \frac { 3 } { \sqrt { 1 - x } } , | x | < 1$$

Show that $\mathrm { f } \left( \frac { 1 } { 10 } \right) = \sqrt { 10 }$.
Expand $\mathrm { f } ( x )$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
Use your expansion to find an approximate value for $\sqrt { 10 }$, giving your answer to 8 significant figures.
Find, to 1 significant figure, the percentage error in your answer to part (c).

Edexcel C4 Q3

3. Relative to a fixed origin, $O$, the line $l$ has the equation $$\mathbf { r } = ( \mathbf { i } + p \mathbf { j } - 5 \mathbf { k } ) + \lambda ( 3 \mathbf { i } - \mathbf { j } + q \mathbf { k } ) ,$$ where $p$ and $q$ are constants and $\lambda$ is a scalar parameter.
Given that the point $A$ with coordinates $( - 5,9 , - 9 )$ lies on $l$,

find the values of $p$ and $q$,
show that the point $B$ with coordinates $( 25 , - 1,11 )$ also lies on $l$. The point $C$ lies on $l$ and is such that $O C$ is perpendicular to $l$.
Find the coordinates of $C$.
Find the ratio $A C : C B$
3. continued

Edexcel C4 Q4

4. During a chemical reaction, a compound is being made from two other substances. At time $t$ hours after the start of the reaction, $x \mathrm {~g}$ of the compound has been produced. Assuming that $x = 0$ initially, and that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 2 ( x - 6 ) ( x - 3 )$$

show that it takes approximately 7 minutes to produce 2 g of the compound.
Explain why it is not possible to produce 3 g of the compound.
4. continued

Edexcel C4 Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e877dc80-4cfc-4c8b-9640-9b186cd7ab13-08_617_917_146_475} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = 4 x ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { - x }$.
The shaded region is bounded by the curve, the $x$-axis and the line $x = 2$.

Use the trapezium rule with four intervals of equal width to estimate the area of the shaded region. The shaded region is rotated through $2 \pi$ radians about the $x$-axis.
Find, in terms of $\pi$ and e, the exact volume of the solid formed.
5. continued

Edexcel C4 Q6

6. (a) Find $$\int 2 \sin 3 x \sin 2 x d x$$ (b) Use the substitution $u ^ { 2 } = x + 1$ to evaluate $$\int _ { 0 } ^ { 3 } \frac { x ^ { 2 } } { \sqrt { x + 1 } } \mathrm {~d} x$$ 6. continued

Edexcel C4 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e877dc80-4cfc-4c8b-9640-9b186cd7ab13-12_556_860_246_452} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the curve with parametric equations $$x = \cos 2 t , \quad y = \operatorname { cosec } t , \quad 0 < t < \frac { \pi } { 2 } .$$ The point $P$ on the curve has $x$-coordinate $\frac { 1 } { 2 }$.

Find the value of the parameter $t$ at $P$.
Show that the tangent to the curve at $P$ has the equation $$y = 2 x + 1$$ The shaded region is bounded by the curve, the coordinate axes and the line $x = \frac { 1 } { 2 }$.
Show that the area of the shaded region is given by $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } k \cos t \mathrm {~d} t$$ where $k$ is a positive integer to be found.
Hence find the exact area of the shaded region.
7. continued
7. continued

Edexcel C4 Q1

Use integration by parts to find

$$\int x ^ { 2 } \sin x d x$$

Edexcel C4 Q2

Given that $y = - 2$ when $x = 1$, solve the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ^ { 2 } \sqrt { x }$$ giving your answer in the form $y = \mathrm { f } ( x )$.

Edexcel C4 Q3

3. A curve has the equation $$4 x ^ { 2 } - 2 x y - y ^ { 2 } + 11 = 0$$ Find an equation for the normal to the curve at the point with coordinates $( - 1 , - 3 )$. (8)
3. continued

Edexcel C4 Q4

4. (a) Expand $( 1 + a x ) ^ { - 3 } , | a x | < 1$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$. Give each coefficient as simply as possible in terms of the constant $a$. Given that the coefficient of $x ^ { 2 }$ in the expansion of $\frac { 6 - x } { ( 1 + a x ) ^ { 3 } } , | a x | < 1$, is 3 ,
(b) find the two possible values of $a$. Given also that $a < 0$,
(c) show that the coefficient of $x ^ { 3 }$ in the expansion of $\frac { 6 - x } { ( 1 + a x ) ^ { 3 } }$ is $\frac { 14 } { 9 }$.
4. continued

Edexcel C4 Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b71c9832-e502-4a25-85fb-a49c03ea9209-08_663_899_146_495} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = \frac { 1 } { \sqrt { 3 x + 1 } }$.
The shaded region is bounded by the curve, the $x$-axis and the lines $x = 1$ and $x = 5$.

Find the area of the shaded region. The shaded region is rotated completely about the $x$-axis.
Find the volume of the solid formed, giving your answer in the form $k \pi \ln 2$, where $k$ is a simplified fraction.
5. continued

Edexcel C4 Q6

6. $$f ( x ) = \frac { 15 - 17 x } { ( 2 + x ) ( 1 - 3 x ) ^ { 2 } } , \quad x \neq - 2 , \quad x \neq \frac { 1 } { 3 }$$

Find the values of the constants $A , B$ and $C$ such that $$\mathrm { f } ( x ) = \frac { A } { 2 + x } + \frac { B } { 1 - 3 x } + \frac { C } { ( 1 - 3 x ) ^ { 2 } }$$
Find the value of $$\int _ { - 1 } ^ { 0 } f ( x ) d x$$ giving your answer in the form $p + \ln q$, where $p$ and $q$ are integers.
6. continued

Edexcel C4 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b71c9832-e502-4a25-85fb-a49c03ea9209-12_495_784_246_461} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the curve with parametric equations $$x = - 1 + 4 \cos \theta , \quad y = 2 \sqrt { 2 } \sin \theta , \quad 0 \leq \theta < 2 \pi$$ The point $P$ on the curve has coordinates $( 1 , \sqrt { 6 } )$.

Find the value of $\theta$ at $P$.
Show that the normal to the curve at $P$ passes through the origin.
Find a cartesian equation for the curve.
7. continued

Edexcel C4 Q8

8. The line $l _ { 1 }$ passes through the points $A$ and $B$ with position vectors $( - 3 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } )$ and ( $7 \mathbf { i } - \mathbf { j } + 12 \mathbf { k }$ ) respectively, relative to a fixed origin.

Find a vector equation for $l _ { 1 }$. The line $l _ { 2 }$ has the equation $$\mathbf { r } = ( 5 \mathbf { j } - 7 \mathbf { k } ) + \mu ( \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } )$$ The point $C$ lies on $l _ { 2 }$ and is such that $A C$ is perpendicular to $B C$.
Show that one possible position vector for $C$ is $( \mathbf { i } + 3 \mathbf { j } )$ and find the other. Assuming that $C$ has position vector $( \mathbf { i } + 3 \mathbf { j } )$,
find the area of triangle $A B C$, giving your answer in the form $k \sqrt { 5 }$.
8. continued
8. continued

Edexcel C4 Q1

Use integration by parts to show that

$$\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x = 2 \ln 2 - \frac { 3 } { 4 }$$

Edexcel C4 Q2

(a) Use the trapezium rule with two intervals of equal width to find an approximate value for the integral

$$\int _ { 0 } ^ { 2 } \arctan x \mathrm {~d} x$$ (b) Use the trapezium rule with four intervals of equal width to find an improved approximation for the value of the integral.

Edexcel C4 Q3

3. A curve has the equation $$3 x ^ { 2 } - 2 x + x y + y ^ { 2 } - 11 = 0$$ The point $P$ on the curve has coordinates $( - 1,3 )$.

Show that the normal to the curve at $P$ has the equation $y = 2 - x$.
Find the coordinates of the point where the normal to the curve at $P$ meets the curve again.
3. continued

Edexcel C4 Q4

4. The points $A$ and $B$ have coordinates $( 3,9 , - 7 )$ and $( 13 , - 6 , - 2 )$ respectively.

Find, in vector form, an equation for the line $l$ which passes through $A$ and $B$.
Show that the point $C$ with coordinates $( 9,0 , - 4 )$ lies on $l$. The point $D$ is the point on $l$ closest to the origin, $O$.
Find the coordinates of $D$.
Find the area of triangle $O A B$ to 3 significant figures.
4. continued

Edexcel C4 Q5

5. A bath is filled with hot water which is allowed to cool. The temperature of the water is $\theta ^ { \circ } \mathrm { C }$ after cooling for $t$ minutes and the temperature of the room is assumed to remain constant at $20 ^ { \circ } \mathrm { C }$. Given that the rate at which the temperature of the water decreases is proportional to the difference in temperature between the water and the room,

write down a differential equation connecting $\theta$ and $t$. Given also that the temperature of the water is initially $37 ^ { \circ } \mathrm { C }$ and that it is $36 ^ { \circ } \mathrm { C }$ after cooling for four minutes,
find, to 3 significant figures, the temperature of the water after ten minutes. Advice suggests that the temperature of the water should be allowed to cool to $33 ^ { \circ } \mathrm { C }$ before a child gets in.
Find, to the nearest second, how long a child should wait before getting into the bath.
5. continued

Edexcel C4 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0f2d48ab-1f61-4fb9-b35a-25d684dbd50f-10_454_602_255_479} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with parametric equations $$x = 3 \sin t , \quad y = 2 \sin 2 t , \quad 0 \leq t < \pi .$$ The curve meets the $x$-axis at the origin, $O$, and at the point $A$.

Find the value of $t$ at $O$ and the value of $t$ at $A$. The region enclosed by the curve is rotated through $\pi$ radians about the $x$-axis.
Show that the volume of the solid formed is given by $$\int _ { 0 } ^ { \frac { \pi } { 2 } } 12 \pi \sin ^ { 2 } 2 t \cos t \mathrm {~d} t$$
Using the substitution $u = \sin t$, or otherwise, evaluate this integral, giving your answer as an exact multiple of $\pi$.
6. continued

Edexcel C4 Q7

7. $$f ( x ) = \frac { 8 - x } { ( 1 + x ) ( 2 - x ) } , \quad | x | < 1$$

Express $\mathrm { f } ( x )$ in partial fractions.
Show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x = \ln k$$ where $k$ is an integer to be found.
Find the series expansion of $\mathrm { f } ( x )$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
7. continued
7. continued

Edexcel C4 Q1

(a) Find the binomial expansion of $( 2 - 3 x ) ^ { - 3 }$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
(b) State the set of values of $x$ for which your expansion is valid.
A curve has the equation

$$x ^ { 2 } + 3 x y - 2 y ^ { 2 } + 17 = 0$$ (a) Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
(b) Find an equation for the normal to the curve at the point $( 3 , - 2 )$.

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