Podcast
Questions and Answers
Given that $\int_0^1 (ax+b) dx = 1$ and $\int_0^1 x(ax+b) dx = 1$, find the value of a + b.
Given that $\int_0^1 (ax+b) dx = 1$ and $\int_0^1 x(ax+b) dx = 1$, find the value of a + b.
- 2
- 1
- 4 (correct)
- 5
- -1
- 3
- 0
The graphs of $y = x^2 + 5x + 6$ and $y = mx - 3$, where m is a constant, are plotted on the same set of axes.
Given that the graphs do not meet, what is the complete range of possible values of m?
The graphs of $y = x^2 + 5x + 6$ and $y = mx - 3$, where m is a constant, are plotted on the same set of axes. Given that the graphs do not meet, what is the complete range of possible values of m?
- m < -11, m > 1
- $m < -\sqrt{11}, m > \sqrt{11}$
- $-\sqrt{11} < m < \sqrt{11}$
- -1 < m < 11 (correct)
- m < -1, m > 11
- -11 < m < 1
For any integer $n \ge 0$, $\int_n^{n+1} f(x) dx = n+1$ Evaluate $\int_0^3 f(x)dx + \int_1^3 f(x)dx +\int_2^3 f(x)dx + \int_3^3 f(x)dx + \int_4^5 f(x)dx $
For any integer $n \ge 0$, $\int_n^{n+1} f(x) dx = n+1$ Evaluate $\int_0^3 f(x)dx + \int_1^3 f(x)dx +\int_2^3 f(x)dx + \int_3^3 f(x)dx + \int_4^5 f(x)dx $
- 27
- 1
- 18 (correct)
- -2
- 0
- 4
Evaluate $\sum_{n=0}^{\infty} \frac{\sin(n\pi + \frac{\pi}{3})}{2^n}$
Evaluate $\sum_{n=0}^{\infty} \frac{\sin(n\pi + \frac{\pi}{3})}{2^n}$
The following shape has two lines of reflectional symmetry. MNOP is a square of perimeter 40 cm.
The vertices of rectangle RSTU lie on the edge of square MNOP. MR has length x cm.
What is the largest possible value of x such that RSTU has area 20 cm²?
The following shape has two lines of reflectional symmetry. MNOP is a square of perimeter 40 cm. The vertices of rectangle RSTU lie on the edge of square MNOP. MR has length x cm. What is the largest possible value of x such that RSTU has area 20 cm²?
In the simplified expansion of $(2+3x)^{12}$, how many of the terms have a coefficient that is divisible by 12?
In the simplified expansion of $(2+3x)^{12}$, how many of the terms have a coefficient that is divisible by 12?
P(x) and Q(x) are defined as follows:
P(x) = $2^x + 4$
Q(x) = $2^{(2x-2)} - 2^{(x+2)} + 16$
Find the largest value of x such that P(x) and Q(x) are in the ratio 4:1, respectively.
P(x) and Q(x) are defined as follows: P(x) = $2^x + 4$ Q(x) = $2^{(2x-2)} - 2^{(x+2)} + 16$ Find the largest value of x such that P(x) and Q(x) are in the ratio 4:1, respectively.
A triangle XYZ is called fun if it has the following properties:
angle YXZ = $30^\circ$
XY = $\sqrt{3}a$
YZ = $a$
where a is a constant.
For a given value of a, there are two distinct fun triangles S and T, where the area of S is greater than the area of T.
Find the ratio area of S : area of T
A triangle XYZ is called fun if it has the following properties: angle YXZ = $30^\circ$ XY = $\sqrt{3}a$ YZ = $a$ where a is a constant. For a given value of a, there are two distinct fun triangles S and T, where the area of S is greater than the area of T. Find the ratio area of S : area of T
How many solutions are there to $(1 + 3\cos 3\theta)^2 = 4$ in the interval $0^\circ \le \theta \le 180^\circ$?
How many solutions are there to $(1 + 3\cos 3\theta)^2 = 4$ in the interval $0^\circ \le \theta \le 180^\circ$?
The trapezium rule with 4 strips is used to estimate the integral: $\int_{-2}^{2} \sqrt{4-x^2} dx$
What is the positive difference between the estimate and the exact value of the integral?
The trapezium rule with 4 strips is used to estimate the integral: $\int_{-2}^{2} \sqrt{4-x^2} dx$ What is the positive difference between the estimate and the exact value of the integral?
It is given that f(x) = $x^2$ – 6x
The curves y = f(kx) and y = f(x – c) have the same minimum point, where k > 0 and c > 0
Which of the following is a correct expression for k in terms of c?
It is given that f(x) = $x^2$ – 6x The curves y = f(kx) and y = f(x – c) have the same minimum point, where k > 0 and c > 0 Which of the following is a correct expression for k in terms of c?
How many solutions are there to the equation $\frac{2^{\tan^2 x}}{4^{\sin^2 x}} = 1$ in the range $0 \le x \le 2\pi$?
How many solutions are there to the equation $\frac{2^{\tan^2 x}}{4^{\sin^2 x}} = 1$ in the range $0 \le x \le 2\pi$?
Point P lies on the circle with equation $(x – 2)^2 + (y − 1)^2 = 16$
Point Q lies on the circle with equation $(x – 4)^2 + (y + 5)^2 = 16$
What is the maximum possible length of PQ?
Point P lies on the circle with equation $(x – 2)^2 + (y − 1)^2 = 16$ Point Q lies on the circle with equation $(x – 4)^2 + (y + 5)^2 = 16$ What is the maximum possible length of PQ?
The function $f(x) = \frac{2}{3}x^3 + mx^2 + nx$, $m, n > 0$ has three distinct real roots.
What is the complete range of possible values of n, in terms of m?
The function $f(x) = \frac{2}{3}x^3 + mx^2 + nx$, $m, n > 0$ has three distinct real roots. What is the complete range of possible values of n, in terms of m?
The difference between the maximum and minimum values of the function f(x) = $a^{\cos x}$, where a > 0 and x is real, is 3.
Find the sum of the possible values of a.
The difference between the maximum and minimum values of the function f(x) = $a^{\cos x}$, where a > 0 and x is real, is 3. Find the sum of the possible values of a.
A right-angled triangle has vertices at (2, 3), (9, −1) and (5, k).
Find the sum of all the possible values of k.
A right-angled triangle has vertices at (2, 3), (9, −1) and (5, k). Find the sum of all the possible values of k.
A circle $C_i$ is defined by $x^2 + y^2 = 2n(x + y)$ where n is a positive integer.
$C_1$ and $C_2$ are drawn and the area between them is shaded.
Next, $C_3$ and $C_4$ are drawn and the area between them is shaded.
This process continues until 100 circles have been drawn.
What is the total shaded area?
A circle $C_i$ is defined by $x^2 + y^2 = 2n(x + y)$ where n is a positive integer. $C_1$ and $C_2$ are drawn and the area between them is shaded. Next, $C_3$ and $C_4$ are drawn and the area between them is shaded. This process continues until 100 circles have been drawn. What is the total shaded area?
You are given that $S = 4 - \frac{8k}{7} + \frac{16k^2}{49} + \frac{32k^3}{343} + ... + 4(-\frac{2k}{7})^n + ...$
The value for k is chosen as an integer in the range -5 ≤ k ≤ 5
All possible values for k are equally likely to be chosen.
What is the probability that the value of S is a finite number greater than 3?
You are given that $S = 4 - \frac{8k}{7} + \frac{16k^2}{49} + \frac{32k^3}{343} + ... + 4(-\frac{2k}{7})^n + ...$ The value for k is chosen as an integer in the range -5 ≤ k ≤ 5 All possible values for k are equally likely to be chosen. What is the probability that the value of S is a finite number greater than 3?
The solution to the differential equation $\frac{dy}{dx} = |-6x|$ for all x is y = f(x) + c, where c is a constant.
Which one of the following is a correct expression for f(x)?
The solution to the differential equation $\frac{dy}{dx} = |-6x|$ for all x is y = f(x) + c, where c is a constant. Which one of the following is a correct expression for f(x)?
The diagram shows the graph of y = f(x). The function f attains its maximum value of 2 at x = 1, and its minimum value of –2 at x = −1
Find the difference between the maximum and minimum values of $(f(x))^2 - f(x)$
The diagram shows the graph of y = f(x). The function f attains its maximum value of 2 at x = 1, and its minimum value of –2 at x = −1 Find the difference between the maximum and minimum values of $(f(x))^2 - f(x)$
Flashcards
Trapezium Rule
Trapezium Rule
The trapezium rule approximates the area under a curve by dividing it into trapeziums.
Even Function
Even Function
A function where f(-x) = f(x) for all x in the domain.
Odd Function
Odd Function
A function where f(-x) = -f(x) for all x in the domain.
Study Notes
- The "Test of Mathematics for University Admission" Paper 1 is in 2023; the duration is 75 minutes.
- Candidates must read the instructions carefully and are not permitted to open the question paper until instructed.
- A separate answer sheet is provided, along with the need for a soft pencil and an eraser.
- Candidates must complete the answer sheet with their candidate number, centre number, date of birth, and full name.
- This paper is the first of two.
- There are 20 questions; candidates should choose one correct answer per question on the answer sheet, erasing mistakes thoroughly.
- There are no penalties for incorrect answers; each question is worth one mark, encouraging attempts on all questions.
- You can use the question paper for rough work, but no extra paper is permitted.
- The answer sheet must be completed within the time limit.
- Calculators, dictionaries, and formula booklets are not allowed.
Question 1
- Given ∫01(ax + b) dx = 1 and ∫01x(ax + b) dx = 1, the task is to find the value of a + b.
Question 2
- For the graphs y = x² + 5x + 6 and y = mx - 3, where m is a constant.
- The range of m is required such that the graphs do not meet.
Question 3
- For any integer n ≥ 0, ∫n^(n+1) f(x) dx = n + 1.
- The Expression is evaluated
- ∫03 f(x) dx + ∫13 f(x) dx + ∫23 f(x) dx + ∫43 f(x) dx + ∫53 f(x) dx
Question 4
- The expression is evaluated: ∑n=0∞ sin(nπ/3 + π/3) / 2^n.
Question 5
- A shape has two lines of reflectional symmetry.
- MNOP is a square with a perimeter of 40 cm.
- Vertices of rectangle RSTU lie on the edge of square MNOP.
- MR is x cm long.
- The largest possible value of x is found where RSTU = 20 cm².
Question 6
- From the simplified expansion of (2 + 3x)^12
- The number of terms with a coefficient divisible by 12 is to be found
Question 7
- P(x) = 2^x + 4 and Q(x) = 2^(2x-2) - 2^(x+2) + 16.
- The largest value of x such that P(x) and Q(x) are in a 4:1 ratio is determined.
Question 8
- Triangle XYZ is "fun" if angle YXZ = 30°, XY = √3a, and YZ = a, where a is a constant.
- For a given a, there exist two distinct "fun" triangles, S and T. S's area is greater.
- The ratio of the area of S to the area of T is calculated.
Question 9
- The number of solutions to (1 + 3cos(3θ))^2 = 4 within the interval 0° ≤ θ ≤ 180° is sought
Question 10
- The trapezium rule with 4 strips is used to estimate the integral ∫-22 √(4 - x²) dx.
- The positive difference between the estimate and the exact integral is to be found.
Question 11
- Given f(x) = x² - 6x, curves y = f(kx) and y = f(x - c) share the same minimum point, and k > 0 and c > 0.
- A correct expression for k in terms of c is required.
Question 12
- Find the number of solutions to equation 2^(tan²x) / 4^(sin²x) = 1 in range 0 ≤ x ≤ 2π.
Question 13
- Point P lies on the circle (x – 2)² + (y – 1)² = 16.
- Point Q lies on the circle (x – 4)² + (y + 5)² = 16.
- The maximum possible length of PQ is calculated.
Question 14
- The function f(x) = (2/3)x³ + mx² + n, with m > 0, is defined to have three distinct real roots.
- The complete range of possible values for n, in terms of m, is to be determined.
Question 15
- For the function f(x) = a*cos(x), where a > 0 and x is real.
- Find the sum of the possible values of a, given that the difference between the max and min is 3.
Question 16
- A right-angled triangle has vertices at (2, 3), (9, -1), and (5, k).
- The sum of all possible values of k is found.
Question 17
- A circle Cₙ is defined by x² + y² = 2n(x + y).
- C₁ and C₂ are drawn, and the area is shaded, followed by C₃ and C₄.
- The total shaded area after 100 circles is determined.
Question 18
- Given S = 4 - (8k/7) + (16k²/49) + (32k³/343) + ... + 4(-2k/7)^n + ...
- The value for k is chosen as an integer in the range -5 ≤ k ≤ 5.
- Find the probability S > 3
Question 19
- Given the differential equation dy/dx = |6x| for all x. the solution is y = f(x) + c, where c is a constant.
- Which one is a correct expression for f(x)?
Question 20
- Function f attains a maximum of 2 at x = 1 and a minimum of -2 at x = -1.
- Find the difference between the min and max values of ( f(x) )² - f(x).
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