How many QCAA Mathematical Methods questions cover Further integration?

AusGrader has 89 QCAA Mathematical Methods questions on Further integration, all with instant AI grading and detailed marking feedback.

QCAA (QCE) Mathematical Methods — Further integration Questions & Answers | AusGrader

Q5

2024

QCAA

Paper 1

1 mark

Q5

1 mark

Determine $\int_a^b 2\cos(x)dx$ , where $a = \frac{\pi}{3}$ and $b = \frac{\pi}{2}$

A

$1 - \frac{\sqrt{3}}{2}$

B

$\frac{\sqrt{3}}{2} - 1$

C

$2 - \sqrt{3}$

D

$\sqrt{3} - 2$

Q7

2021

QCAA

Paper 1

1 mark

Q7

1 mark

Determine $\int_{1}^{3} (2x+3)dx$

A

2

B

4

C

14

D

16

Q13

2021

QCAA

Paper 1

5 marks

Q13

Consider the functions $f(x)=x^2$ and $g(x)=4x$ .

Q13a

2 marks

Determine the $x$ -coordinates of the points of intersection of the graphs of the two functions.

Solving simultaneously
$x^2 = 4x$
Rearranging and factorising
$x(x - 4) = 0$
$\therefore x = 0$ and $x = 4$

Marking Criteria

Descriptor	Marks
correctly uses the simultaneous procedure	1
correctly determines both x-intercept ordinates	1

Q13b

3 marks

Use the results from Question 13a) to calculate the area enclosed by the graphs of $f(x)$ and $g(x)$ .

Area $= \int_0^4 (4x - x^2)dx$
$= \left[ 2x^2 - \frac{x^3}{3} \right]_0^4$
$= \left( 2 \times 4^2 - \frac{4^3}{3} \right) - 0$

$= \frac{32}{3}$ square units

Marking Criteria

Descriptor	Marks
correctly determines the integral	1
substitutes limits into integral	1
determines area	1

Q14

2022

QCAA

Paper 1

6 marks

Q14

The rate that water fills an empty vessel is given by $\frac{dV}{dt} = 0.25e^{0.25t}$ (in litres per hour), $0 \le t \le 8\ln(6)$ , where $t$ is time (in hours).

Q14a

2 marks

Determine the function that represents the volume of water in the vessel (in litres).

$V = \int 0.25e^{0.25t} dt$
$= e^{0.25t} + c$
when $t = 0, V = 0$
$\therefore 0 = e^{0.25 \times 0} + c$
$\therefore c = -1$
$\therefore V = e^{0.25t} - 1$

Marking Criteria

Descriptor	Marks
correctly determines the integral of the function V(t)	1
determines the value of c	1

Q14b

2 marks

The vessel is full when $t = 8\ln(6)$ . Determine the volume of water, to the nearest litre, the vessel can hold when full.

$V(8\ln(6)) = e^{0.25 \times 8 \ln(6)} - 1$
$= 36 - 1$

$= 35$ litres

Marking Criteria

Descriptor	Marks
determines the simplified exponential term	1
determines number of litres	1

Q14c

2 marks

Use information from the table and the trapezoidal rule to determine the approximate volume of water in the vessel after three hours.

$t$	$\frac{dV}{dt}$
0	0.25
1	0.32
2	0.41
3	0.53

Using trapezoidal rule
Volume after 3 hours $= \frac{1}{2}(0.25+0.53+2(0.32+0.41))$

Volume after 3 hours $= 1.12$ litres

Marking Criteria

Descriptor	Marks
establishes expression for approximate number of litres of water in vessel after 3 hours	1
determines approximate number of litres	1

Q17

2022

QCAA

Paper 1

4 marks

Q17

4 marks

Determine the value of $b$ given $\int_a^b 3x^2 dx = 117$ and $\int_a^{b-1} 3x^2 dx = 56$ for $b > 1$ .

Using first integral
$F(b) - F(a) = 117$
$b^3 - a^3 = 117$ ... Equation I

Using second integral
$(b-1)^3 - a^3 = 56$ ... Equation II
Equation I – Equation II
$b^3 - (b-1)^3 = 61$
$b^3 - (b^3 - 3b^2 + 3b - 1) = 61$
$3b^2 - 3b - 60 = 0$
$b^2 - b - 20 = 0$
$(b-5)(b+4) = 0$
$b = -4, 5$
Given $b > 1$
$\therefore b = 5$

Marking Criteria

Descriptor	Marks
correctly establishes a formula for one of the integrals	1
determines equation in b	1
determines values of b	1
evaluates the reasonableness of solutions	1

QCAA Mathematical Methods Further integration

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