Cambridge AS Level Mathematics - Pure 1
θ = 1.67 rad
A circle has an arc length of 10 cm and a radius of 6 cm. Find the angle θ in radians
r = 25 m
A circle has an arc length of 20 m and an angle of 0.8 rad. Find the radius r.
6.51 cm
A circle has an area of 50 cm² and a sector angle of 3π/4 rad. Find the radius r to 3 significant figures.
6√6
A closed box with a square base has a total surface area of 36 m². Find the maximum possible volume of the box.
y = 3x - x³
A curve is such that d²y/dx² = -6x and the curve has a maximum point at (1, 2). Find the equation of the curve.
f(x) = -5/x² - 4x + 3
A curve is such that f'(x) = 10/x³ - 4 and the point (-1, 2) lies on the curve. Find the equation of the curve.
(3, 0)
A diagram with a straight line AB and a perpendicular line PC. The point A is (-3, -4), the point B is (6, 2) and the point C is (5, -3). Find the coordinates for point P.
1/2
A geometric progression has the sum to infinity equal to twice the 1st term. Find the comon ratio, r.
(-2x)ⁿ⁻¹
A geometric sequence is 1 - 2x + 4x² - 8x³ + ... Find an expression for the nth term
54.4 cm²
A hexagon is cut out of a circle with radius of 10 cm so all of the corners touch the circumference of the circle. Find the area of the remaining shape to 3 significant figures.
OD = 4i - 2j + 9k
A is a point with coordinates (1, 0, 3) and B is a point with coordinates (3, -2, 6) OADB is a parallelogram. Find using vector notation the position vector of D.
OC = (2 -1 9/2)
A is a point with coordinates (1, 0, 3) and B is a point with coordinates (3, -2, 6). Find using column vectors the position of the point C, which is the midpoint of AB.
49cm²
A piece of card has a length of (2x - 1)cm and a width of (x + 2)cm. A square of side xcm is removed from the card. The area of the card that is left is 68cm². Find the area of the card that has been removed.
64 cm²
A rectangle has a width of x cm. The perimeter of the rectangle is 32 cm. Find the maximum area of the rectangle
24.5 cm²
A right angled triangle has a width of x cm. The length of the hypotenuse is 10 cm. The perimeter of the triangle is 24 cm. Find the maximum area of the triangle.
0.0405 cm s⁻¹
A spherical balloon is being blown up so that its volume increases at a rate of 3 cm³ s ⁻¹. Find the rate of increase of the radius of the balloon when the volume of the balloon is 60 cm³
m = 4
A straight line L passes through the point (1, 0), has a gradient m and is a tangent to the curve y = x² + 2x - 3. Find the value of m
(3, -1)
A(1, -2), B(-1, -3), C(-2, 14) and D(2, 2) are four points. The lines AB and CD meet at P. Find the coordinates of P.
20
ABCD is a square with vertices A(-1, -2), B(1, 2), C(5, 0) and D(3, -4). Find the area of ABCD.
(2, -1)
ABCD is a square with vertices A(-1, -2), B(1, 2), C(5, 0) and D(3, -4). Find the centre coordinates of ABCD.
(x + b/2)² - (b/2)² + c
Complete the square x² + bx + c
x = 6 ± 0.5√170
Complete the square to find x 4x² - 48x - 26
x = (-b ± √b² - 4ac) / 2a
Complete the square to find x ax² + bx + c = 0
x = 2 ± 2√3
Complete the square to find x x² - 4x - 8 = 0
3π/4
Convert 135° to radians.
π/12
Convert 15° to radians.
40
Convert 2π/9 radians to degrees
240°
Convert 4π/3 radians to degrees
3i + j - 2k
Convert the column vector (3 1 -2) into unit vector form.
xi + yj
Convert the column vector (x y) into unit vector form.
0, π/6, π/4, π/3
Convert the following degree values to radians. 0°, 30°, 45°, 60°
π/2, π, 3π/2, 2π
Convert the following degree values to radians. 90°, 180°, 270°, 360°
18°
Convert π/10 radians to degrees
A function f(x) is decreasing for a < x < b if f'(x) < 0 for a < x < b
Define a decreasing function
A displacement vector represents a change in position.
Define a displacement vector.
A position vector describes the displacement from the origen to a point.
Define a position vector.
A quantity which has magnitude (size) but no direction
Define a scalar
A quantity which has both magnitude (size) and direction.
Define a vector
A function f(x) is increasing for a < x < b if f'(x) > 0 for a < x < b
Define an increasing function
tan x = sin x / cos x
Define tan x
20
Evaluate (2√5)²
66
Evaluate 12C10
120
Evaluate 5! without using a calculator.
84
Evaluate 9C3
(2 9)
Evaluate the vector (3 4) + (-1 5)
(4 -1)
Evaluate the vector (3 4) - (-1 5)
(2 6)
Evaluate the vector 2(1 -3)
1/4
Evaluate. (-8)^(-2/3)
4
Evaluate. (-8)^(2/3)
2/3
Evaluate. (9/4)^(-1/2)
(1/√61)(3 4 6)
Find a unit vector in the direction of (3 4 6)
1/3(-i + 2j - 2k)
Find a unit vector in the direction of -i + 2j - 2k
(1/√14)(2i - 3j + k)
Find a unit vector in the direction of 2i - 3j + k
159°
Find an angle to the nearest degree other than 21° that has a sine of 0.36
f⁻¹(x) = (2x + 3)/2, x ∈ ℝ
Find an expression for f⁻¹(x) for the function f(x) = (2x - 3)/2, x ∈ ℝ
35x⁴
Find the 5 term in ascending powers of x, in the expansion of (1 + x)⁷
89.2°
Find the angle between 2i + 5j - 2k and i + 4j + 11k
50.0°
Find the angle between 2i + j - 3k and -i +2j - 3k
180°
Find the angle between i - j + 2k and -2i + 2j - 4k
51.0°
Find the angle between the two vectors (1 -1 1) and (2 1 4).
A = 57.8°, B = 90.0°, C = 31.3°
Find the angles in the triangle ABC where the coordinates of A, B and C are (1, 0, 1), (2, 2, -2) and (-3, 0, -5).
7.2 cm
Find the arc length of a circle with a 12 cm radius and an angle of 0.6 rad
110 mm
Find the arc length of a circle with a 25 mm radius and an angle of 4.4 rad
4/3
Find the area bounded by the curve with the equation x = y² - 2y
22/3
Find the area bounded by the curve with the equation y = (2 - x)(3 + x), the positive x-axis and the y-axis.
8
Find the area bounded by the curve with the equation y = x(x + 2)(x - 2)
32/3
Find the area bounded by the curve with the equation y = x(x-4) and the x-axis
72
Find the area bounded by y = x² - 4x and y = 16 - x²
1150 cm²
Find the area of a sector of a circle with radius 20 cm and an angle of 11π/6 rad. Write your answer to 3 significant figures
2645 mm²
Find the area of a sector of a circle with radius 46 mm and an angle of 2.5 rad
1/24
Find the area of the finite region bounded by y = x² and y = x - x²
(100, 2.05)
Find the coordinates of the point on the curve y = (2x - 5 + √x) / x where the gradient is zero
(2, -7) and (-2, 21)
Find the coordinates of the points on the curve y = 2x³ - 15x + 7 where the gradient is 9.
(-1/2, -3/2) local maximum, (3/2, 5/2) local minimum
Find the coordinates of the stationary points of the curve y = 2 / (2x - 1) + x
(-1, -4) local minimum, (1, 4) local maximum
Find the coordinates of the stationary points of the curve y = 6x - 2x³ and determine their nature.
y = 2√x⁵ + 2√x - 69
Find the equation of the curve that passes through the point (4, -1) and where dy/dx = (5x² + 1)/√x
2y = x - 10
Find the equation of the line with the points A(-8, -1) and B(2, -6)
y = x + 4
Find the equation of the line with the points A(1, 5) and B(3, 7)
15y = -x - 36
Find the equation of the line with the points A(9, -3) and B(-6, -2)
y = 7x - 28 y + 7x + 21 = 0
Find the equations of the tangents to the curve y = x² - x - 12 at each of the points where the curve crosses the x-axis.
x¹⁰ - 2x⁹ + 27x⁸ - 54x⁷ + 324x⁶
Find the first five terms, in descending powers of x, in the expansion of (x -2)(x + 3/x)⁹
1024x⁵, - 3840x⁴, 5760x³
Find the first three terms in decreasing powers of x for (4x - 3)⁵
m = 5
Find the gradient m of the straight line joining the sets of points (-3k, -4k) and (-k, 6k)
m = -2
Find the gradient m of the straight line joining the sets of points (4, -6) and (1, 0)
m = 10
Find the gradient of the tangent to the curve y = (4x - 1)² / x² at the point (-1, 25)
g⁻¹(x) = 2/x - 5, x ∈ ℝ, x ≠ 0
Find the inverse of g(x) = 2/(x + 5), x ∈ ℝ, x ≠ -5
10
Find the lengths of the straight lines joining the sets of points (-8, 4) and (-2, -4)
5q
Find the lengths of the straight lines joining the sets of points (4q, -4q) and (7q, -8q)
5√2
Find the magnitude of the vector (1 -7 0)
15
Find the magnitude of the vector (10 -5 10)
√61
Find the magnitude of the vector (3 4 6)
√34
Find the magnitude of the vector 5i + 3j.
13n - 18
Find the nth term of the linear sequence -5, 8, 21, 34, 47
0 > f(x) ≥ 1
Find the range of the function f(x) = 1/x, x ≥ 1
f(x) ≤ 1
Find the range of the function f(x) = x -2, x ≤ 3
h(x) > -4
Find the range of the function h(x) = x³ + 4, x > -2
x < 1 and x > 2
Find the range of values of x for which f(x) = 2x³ - 9x² + 12x -3 is increasing.
x > 3
Find the range of values of x for which f(x) = x² - 6x is increasing.
x < -4 and x > 4
Find the range of values of x for which f(x) = x³ - 48x + 2 is increasing.
k < 0 k > 8/9
Find the set of values of k for which the equation 2x² + 3kx + k = 0 has distinct real roots.
20916
Find the sum of the integers between 1 and 500 that are divisible by 6.
k = 24
Find the value k for which the vectors (2 -1 6) and (8 -4 k) are parallel.
k = -4
Find the value of k for which the line y = 2kx + 7 is a tangent to the curve y = kx²
p = 2/5
Find the value of p for which the quadratic equation px² - 4px + 2 - p = 0 has equal roots
p = 5
Find the value of p for which the vectors i - 2j + k and 3i + 4j + pk
p = 15/2
Find the value of p if the vectors pi - 3k and 2i + j + 5k are perpendicular.
t = -2
Find the value t for which the vectors (5+t 4 3-3t) and (3 4 9) are equal
c = 15/2
Find the values of c for which (2t -4t 5t) and (3t -6t ct) are parallel
c = 1
Find the values of c for which i - 2j + 4k and 0.5i + cj + 2k are parallel.
16π
Find the volume obtained for y = √(2x + 1) above the curve between x = 0, x = 4 and y = 3 when rotated 360° about the x-axis.
3π/4
Find the volume when y = 1/x is rotated between x = 1 and x = 4 through 360°
x = 73°, 287°
Find two angles (to the nearest degree) in the range 0° < x° < 360° such that cos x° = 0.3
x = 9
Find x. (5⁷ × 5⁴) ÷ 5ˣ = 25
x = 6
Find x. (6⁵ × 6ˣ) ÷ 36 = 6⁹
x = 6
Find x. 7⁸ ÷ 7ˣ = 49
y = -(4/x³) - (3/2x⁴) + c
Find y given dy/dx = 12/x² + 6/x³
1704/35
Find ⁴₁∫((x³ + x²) / √x)dx.
424/3
Find ⁹₁∫(2 + √x)²dx
3√(4x + 9) + c
Find ∫(6/√(4x + 9)dx
4/√a³ - 4/√n³
Find, in terms of a and b, the value of the integral ₐⁿ∫6/xˆ(5/2) dx.
√(x² + y² + z²)
For a vector AB = (x y z), find |AB|
x = 156°, 336°
For the equation tan x° = -0.44 find all solutions between 0° and 360° to the nearest degree.
The Chain Rule
For the equation y = 2(5x + 2)³ would you use the chain rule or the product rule to differentiate?
The Product Rule
For the equation y = x²(5x + 2)³ would you use the chain rule or the product rule to differentiate?
y = 3x³ - 2x² + 2√x³ - 7x + c
Given dy/dx = 9x² - 4x + 3x^(1/2) -7, find y
f(x) = (9/4)x⁴ - 8x³ + 8x² + c
Given f'(x) = x(3x - 4)², find f(x)
a = 3
Given that the coefficient of x³ in the expansion of (1 + ax + 2x²)(2 - x)⁷ is 560, find the value of a.
-7
Given that x + y = 3, find the least possible value of x² + 14y.
a = 1
Given that y = ax⁴ - 3x² and d²y / dx² = 42 when x = 2, work out the value of a.
OB = OA + AB
Given the vector OA and AB, find OB
ff⁻¹(x) = f⁻¹f(x) = x
How do you check that your answer for the inverse f⁻¹(x) is correct?
m1 × m2 = -1
How do you find out that two lines are perpendicular?
r = u(2) / u(1) = u(3) / u(2) = u(4) / u(3)
How do you find the common ratio for a geometric progression?
d²y/dx² ≤ 0
How do you know if the point is a local maximum point?
d²y/dx² ≥ 0
How do you know if the point is a local minimum point?
The scalar product is 0.
How do you know if two vectors are perpendicular?
3
How many significant figures should your final answer in the exam have?
AB = (2 -1 -2), BA = (-2 1 2)
If A has coordinates A(3, -2, 4) and B has coordinates B(5, -3, 2), find the column vector AB and BA.
AB = (3 -1), BA = (-3 1)
If A is the point (2, 1) and B is the point (5, 0). Find AB and BA
378
In an arithmetic progression the 8th term is 39 and the 4th term is 19. Find the sum of the first 12 terms.
√3/2, 1/√2, 1/2
In degrees, without using a calculator, calculate the following; cos(30), cos(45), cos(60).
1/2, 1/√2, √3/2
In degrees, without using a calculator, calculate the following; sin(30), sin(45), sin(60).
1/√3, 1, √3
In degrees, without using a calculator, calculate the following; tan(30), tan(45), tan(60).
The x values
In functions, what is the domain?
The y values
In functions, what is the range?
All, Sine, Tangent Cosine
In the diagram used to calculate exact values of trigonometric functions. Which functions are positive in order from quadrant 1 to quadrant 4?
Horizontal
In the diagram used to calculate values of trigonometric functions, which line of reflection is used to find corresponding cosine values?
Vertical
In the diagram used to calculate values of trigonometric functions, which line of reflection is used to find corresponding sine values?
10737418.23
Kwame asks his father for some money. He asks for 1 cent on the first day, 2 cents on the second day, 4 cents on the third day, 8 cents on the forth day. He wants his father to continue to double the money each day. Calculate how much money he would get from his father after 30 days, if, of course, his father agrees to pay him. Give your answer in dollars.
12
OA = (3 -1 2) and OB = (6 -4 -5), Calculate the scalar product of OA x OB.
0.0890 cm s⁻¹
Paint is poured onto a table, forming a circle which increases at a rate of 2.5 cm² s⁻¹. Find the rate the radius is increasing when the area of the circle is 20π cm².
a + (a + d) + (a + 2d) + ... + (a + (n - 2)d) + (a + (n - 1)d) (swap the direction of the sum and add them together) (a + (n - 1)d) + (a + (n - 2)d) + ... + (a + 2d) + (a + d) + a a + (a + d) + (a + 2d) + ... + (a + (n - 2)d) + (a + (n - 1)d)
Prove the equation for S(n) of an arithmetic progression.
The integral has no finite value.
Show that the following improper integral has a finite and find that value. ⁰₋₂∫2/x⁵dx
-1/32
Show that the following improper integral has a finite and find that value. ⁻²₋∞∫2/x⁵ dx.
2/3
Show that the improper integral ∞₁∫1/(x²√x)dx has a value and find that value.
ab = 0
Show that the vectors qi - j + k and i + (q + 1)j + k are perpendicular
(delta) = -55/4 There are no real roots
Show that there are no real solutions to the simultaneous equations y = 1 + 2x - x² and y = (1/2)x + 5
(draw the graph)
Show that x² + 8x + 16 ≥ 0 for all values of x
2 / sinθ
Simplify (1 + cosθ) / sinθ + sinθ / (1 + cosθ)
1
Simplify (sin⁴θ - cos⁴θ) / (sin²θ - cos²θ)
Maximum at (0.5, 4) ( (x + p)² + q and max/min coords at (-p, q) )
Sketch and state the coordinates of the vertex and wheather it is a maximum or a minimum for the following equation. y = 3 + 4x - 4x²
Minimum at (0.4, -2.8) ( (x + p)² + q and max/min coords at (-p, q) )
Sketch and state the coordinates of the vertex and wheather it is a maximum or a minimum for the following equation. y = 5x² - 4x - 2
Minimum at (-1, -1) ( (x + p)² + q and max/min coords at (-p, q) )
Sketch and state the coordinates of the vertex and wheather it is a maximum or a minimum for the following equation. y = x² + 2x
one-to-one
Sketch the function f(x) = (x - 3)², x ∈ ℝ, x ≥ 4 and state weather it is a one-to-one or a many-to-one function
one-to-one
Sketch the function f(x) = 2x + 1, x ∈ ℝ and state weather it is a one-to-one or a many-to-one function
many-to-one
Sketch the function f(x) = x² - 4, x ∈ ℝ, -3 < x < 3 and state weather it is a one-to-one or a many-to-one function
f(x) > 1
Sketch the graph defined by f(x) = 5 - 2x, x < 2 and f(x) = x² + 1 and find the range
x ≤ - 1
Solve (4 + x) / 3 ≤ 1 - 5(1 + x)
x ≥ 2
Solve 2x + 7 ≤ 8x - 5
x ≤ 2/5
Solve 4x - 7(2x - 1) ≥ 3
-5 ≤ x ≤ 1/2
Solve and sketch the inequality (5 + x)(1 - 2x) ≥ 0
x ≤ 2, x ≥ 6
Solve and sketch the inequality (x² + 12) / 2 ≥ 4x
x ≤ -7/2 or x ≥ -4/5
Solve and sketch the inequality 10x² + 43x + 28 ≥ 0
θ = 54.7°, 125.3°, 234.7°, 305.3°
Solve the equation 3cos 2θ = -1 for 0°≤ θ ≤ 360°
x = 34°, 214°
Solve the equation 3sin x = 2cos x, giving all solutions between 0° and 360° to the nearest degree.
4.6°, 55.4°, 94.6°, 145.4°
Solve the equation sin²(2x - 60°) = 0.6 for 0° ≤ x ≤ 180° to one decimal place
x = -(5/2)
Solve the quadratic equation (2x² + 5x + 3) / (x² + 3x + 2) = 4
x = 0 or 4
Solve the quadratic equation by factorisation 5x² = 20x
x = -7 or 5
Solve the quadratic equation by factorisation x² + 2x - 35 = 0
When x = 1, y = 0 When x = -3, y = -4
Solve the simultaneous equation finding values for y and x. y = 0.5(1 - x²) y = x - 1
When x = -1/2, y = 0 When x = -1, y = -1
Solve the simultaneous equation finding values for y and x. y = 1 + 2x y² = 2x² + x
When x = 1, y = 2 and when x = 2, y = 4
Solve the simultaneous equation finding values for y and x. y = 2x y = x² - x + 2
x = 0.42 or 1.58
Solve using the quadratic formula to 2 decimal places. 3x² = 6x - 2
x = (-5 ± √57) / 4
Solve using the quadratic formula. 2x² -4 + 5x = 0
x = ± 1.79
Solve using the quadratic formula. To 3 sig figures. 2x⁸ - 20x⁴ - 7 = 0
(-1 - √5) / 2 < x < (-1 + √5) / 2
Solve using the quadratic formula. x² < 1 - x
x = π/6, 7π/6
Solve √3 sin x - cos x = 0 for 0 ≤ x ≤ 2π
x = -0.966 or 0.901
Solve. Write answer correct to 3 significant figures. 4x¹⁰ + x⁵ = 2
sin(θ) / cos(θ) = tan(θ) and sin²(θ) + cos(θ)² = 1
State the two trigonometric identities
ab = |a||b|cosθ, ab = a1b1 + a2b2 + a3b3
State two equations for the scalar product.
tan θ = sin θ/cos θ sin²θ + cos²θ = 1
State two trigonometric identities.
-243
The 3rd term of a geometric progression is 9 and the ratio is -3. Find the 6th term.
5π/2
The curve y = 3x² is rotated about the y-axis 360° between y = 1 and y = 4. Find the exact value of the volume of revolution obtained.
9/2
The curve y = x² + 2x and the line y = -x intersect twice. Find the area between both the line and curve. (A real question would have a diagram to show what area but adding diagrams cost money and i'm cheap.)
15/4
The curve y² = 12x intersects the line 3y = 4x + 6. Find the distance between the two points
u(18) = 73
The fifth term of an arithmetic progression is 21 and the sum of the first six terms is 90. Find the 18th term.
32
The first 3 terms of an arithmetic progression are 2, 6, 10. Find the least number of terms so that the sum of the terms are greater than 2000.
r = 1/3 r = -1/3
The first and third term of a geometric progression are 18 and 2 respectively. Find two values of r.
9/4
The first four terms of a geometric progression are 3, -1, 1/3, -1/9 find the sum to infinity.
u(4) = 9, u(n) = 5n - 11
The first term of an arithmetic progression is -6 and the common difference is 5. Find the 4th and the nth term
a = 4 b = -1/4 c = -1/4
The functions f and g are defined for x ∈ ℝ by f(x) = 2x - 1 and g(x) = x² + x. Express gf(x) in the form a(x + b)² + c, where a, b and c are constants. State the values of a, b and c.
a) 5x² -1 b) (ans in book = 25x - 6) c) 256 d) 1
The functions f and g are defined for x ∈ ℝ by f(x) = 5x - 1 and g(x) = x². a) find fg(x) b) find f(x) c) find gf(-3) d) find gg(1)
k = - 1/3
The functions f and g are defined for x ∈ ℝby f(x) = 4x - 1 and g(x) = 2x + k. Find the value of k for which fg = gf.
y = 3x² - 2x
The gradient of a curve is given by dy/dx = ax + b. Given that the curve passes through (0, 0), (1, 1) and (-2, 16), find the equation of the curve
4.5
The line y = 2x and the curve y = x(5 - x) intersect at the origen. Find the area between the line and the curve. (hint, sketch)
p = -4, q = -5
The mid-point of the line joining the points A(p, -2) and B(6, -8) is (1, q). Find p and q.
P(-3, 3)
The normals at the points (0, 0) and (1, 2) to the curve y = x + x³ meet at point P. Find the coordinates of point P.
-1/3
The nth term of a geometric progression is (-1/3)ⁿ. Find the common ratio r.
a = 5 d = 7
The nth term of an arithmetic progression is 7n - 2. Find a and d.
C(0, 2, 4)
The points A(-1, 0, 3), B(1, -2, 2), C(x, y, z) and D(-2, 4, 5) form a parallelogram. Find the coordinates of C.
(Show proof lol, they have to be scalar multiples of each other)
The points A, B and C are such that OA = 2i + 6j + 3k, OB = i + 2j + 7k and OC = 4i + 14j - 5k. Show that the vectors BA and AC are in the same direction and hence that A, B, C lie on the same line.
x = 2, y = -2, z = 2
The position vector of point A is (x y z). The position vector of B is (5 1 -3). Given that AB = (3 3 -5), find the values of x, y and z.
OB = 5i + 4j - k
The position vector of point A is 3i - j + 2k and AB = 2i + 5j - 3k. Find the position vector of B as a unit vector.
6 + 2p
The position vectors of points A and B relitive to an origen O are given by a = 2i + pj + 3k and b = 3i - j + pk where p is a constant. Calculate ab.
1.57 ms⁻¹
The radius r, of a circle is increasing at the rate of 2/r² ms⁻¹. Find the rate at which the area, A, is increasing when r = 8.
u(n) = 4 + 2n
The sum of the first n terms of an arithmetic progression is n² + 5n. Find the nth term.
nth term = 4 + 2n First four terms = 6, 8, 10, 12
The sum of the the first n terms n terms of an arithmetic progression is n² + 5n. Find the nth term and the first 4 terms.
324
The sum of two numbers is 36. Find the maximum product of the two numbers.
30 cm³ s⁻¹
The surface area, A, of a cube is increasing at the rate of 12 cm² s⁻¹. Find the rate of increase of the volume, V, of the cube when each edge is 10 cm.
172
The third term of a geometric progression is 16 and the 6th term is -128. Find the sum of the first seven terms.
625x⁴ - 1000x³ + 600x² - 160x + 16
Use pascal's triangle to expand (5x - 2)⁴
21°
Use your calculator to find the angle (to the nearest degree) between 0° and 360° whose sine is 0.36.
(nC0 × xⁿ) + (nC1 × xⁿ⁻¹ × y) + (nC2 × xⁿ⁻² × y²)
What are the first 3 terms of (x + y)ⁿ in decreasing powers of x?
1. Let f(x) = y. 2. Change x to y and y to x. 3. Rearrange to get x in terms of y 4. Write in the correct format
What are the steps used to find the inverse function f⁻¹(x) given f(x)?
S(n) = (n/2)(a + l) S(n) = (n/2)(2a + (n - 1)d) (Where l is the last term of an AP)
What are the two equations used to calculate the sum of an arithmetic progression?
The coefficient of (r + 1)th term.
What does the equation nCr signify?
A decreasing function
What is a function called when the gradient is negative?
An increasing function
What is a function called when the gradient is positive?
The modulus of a vector.
What is another way of saying the magnitude of a vector?
A = πr²
What is the area of a circle?
A = 0.5 × ab sin(C)
What is the area of a triangle
If y is a function of u, and u is a function of x, dy/dx = dy/du × du/dx
What is the chain rule?
a² = b² + c² - 2bc cos(A)
What is the cosine rule?
s = rθ (where θ is measured in radians and r is the radius of the circle)
What is the equation for arc length s of a circle?
(delta) = b² - 4ac
What is the equation for the discriminant?
m = (y2 - y1) / (x2 - x1)
What is the equation for the gradient m of a line?
u(n) = a + (n - 1)d
What is the equation for the nth term?
V = ∫πy²dx (with b at the top of ∫ and a at the bottom)
What is the equation for the volume of a solid revolution that is generated by rotating the curve y = f(x) between x = a and x = b through 360° about the x-axis?
(y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)
What is the equation of a straight line given two coordinates?
(x1 + x2) / 2, (y1 + y2) / 2
What is the equation of the midpoint of a line?
θ = tan⁻¹m2 - tan⁻¹m1
What is the equation used to calculate the angle between two tangents?
n! / (n - r)!r!
What is the formula for nCr?
a + (n - 1)d
What is the formula for the nth term of an arithmetic progression?
√((x2 - x1)² + (y2 - y1)²)
What is the length of a segment of line?
√(x² + y²+ z²)
What is the magnitude of the vector xi + yj + zk?
(-2.5, 1.5)
What is the midpoint of the straight line joining the sets of points (-8, 5) and (3, -2)
arⁿ⁻¹
What is the nth term of a geometric progression
360°/a
What is the period of the graph y = sin ax°?
360° or 2π rad
What is the period of y = cos x
360° or 2π rad
What is the period of y = sin x
180° or π rad
What is the period of y = tan x
x = (-b ± √b² - 4ac) / 2a
What is the quadratic formula?
y ≤ a ≤ a
What is the range of the graph y = asin x°
A = 1/2 × r²θ
What is the sector area A of a circle?
a / sin(A) = b / sin(B) = c / sin(C)
What is the sine rule?
S(n) = a(1 - rⁿ) / (1 - r)
What is the sum of a geometric progression when -1 < r < 1.
S(n) = a(rⁿ - 1) / (r - 1)
What is the sum of a geometric progression when r > 1 or r < -1.
s(∞) = a / (1 - r)
What is the sum to infinity?
(delta)
What is the symbol for the discriminant?
The graph moves -c in the x axis.
What is the translation of the graph y = sin (x + c)°
-BA
What is the vector AB the same as?
V = πr²h
What is the volume of a cylinder?
4πr³/3
What is the volume of a sphere
The domain of the inverse will take the value of the range from the original function
When finding the inverse of a function what value will the domain of inverse take?
The range of the inverse will take the value of the domain from the original function
When finding the inverse of a function what value will the range of inverse take?
1
Without using a calculator, evaluate 0!
56
Without using a calculator, evaluate 8C5
-0.5
Without using a calculator, find the exact value of cos(240)
-√3/2
Without using a calculator, find the exact value of sin(300)
-1
Without using a calculator, find the exact value of sin(3π/2)
-1
Without using a calculator, find the exact value of tan(135)
-√3
Without using a calculator, find the exact value of tan(5π/3)
positive, negative, negative, negative
Without using a calculator, state weather each of these are positive or negative. sin 130, cos 130, sin 255, cos 255
(delta) = -23 There are no real roots
Work out the discriminant and determine how many roots the equation has. 2x² + 4 = 3x
(delta) = 12 There are two distinct real roots
Work out the discriminant and determine how many roots the equation has. 6x² = 6x - 1
(delta) = 0 There are equal roots (one repeated root)
Work out the discriminant and determine how many roots the equation has. x² + 12x + 36 = 0
(3 1)
Write down the displacement from A(2, 6) to B(5, 7) as a column vector.
5i + j - 4k
Write down the displacement from P(2, 1, 3) to Q(7, 2, -1) in terms of the unit vectors i, j, k
(5 -5)
Write down, using the column vector notation, the displacement from (-3, 4) to (2, -1)
(4 0 5)
Write down, using the column vector notation, the displacement from (1, 0, 3) to (5, 0, 8).
i + 3j + k
Write down, using the unit vector notation, the displacement from (1, 0, -2) to (2, 3, -1)
-3i - 3k - 5k
Write down, using the unit vector notation, the displacement from (3, 3, 5) to (0, 0, 0)
(x - 5)² - 5
Write in the form (x + p)² + q x² - 10x + 20
1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1
Write the first 5 rows in pascals triangle.
Use pascals triangle to expand (a - b)⁵
a⁵ - 5a⁴b + 10a³b² - 10a²b³ + 5ab⁴ - b⁵
y = (4x - 3)⁶/24 + c
dy/dx = (4x - 3)⁵, find y
y = (2(1/4 x + 1)⁹ + 3368)/9 - 122x
d²y/ dx² = (1/4 x + 1)⁷. When x = 4, dy/dx = 6 and y = 0. Find an equation for y
-13/3
f(x) = 1/(x + 4), x ∈ ℝ, x ≠ -4. Evaluate f⁻¹(-3)
x = 5/2
f(x) = x / (x + 3), x ∈ ℝ, x ≠ - 3. If f⁻¹(x) = - 5, find the value of x
-3 / 2 ⁴√(5 + 3x)⁵
find dy/dx given y = 2 / ⁴√(5 + 3x)
-(6 - 2x)/x³
find dy/dx given y = 6 / (2x² + 3x)
19
find ¹₀∫(3x - 2)⁸dx.
(x - 1) / (2 - x)
h(x) = 1 / (x - 1) for x ≠ 1, x ∈ ℝ. Find hh(x)
8/³√(x²) + 1 / 4√(x³)
solve d/dx(24 ³√x - 1 / 2√x)
3/2
y = 2/√x, find the area between between y = 8 and y = 2.
-3 / 2√(4 - 3x)
y = √(3 - 3x), find dy/dx