Cambridge AS Level Mathematics - Pure 1 Examination test 2024 with correct marking scheme, Exams of Mathematics

Download Cambridge AS Level Mathematics - Pure 1 Examination test 2024 with correct marking scheme and more Exams Mathematics in PDF only on Docsity! Cambridge AS Level Mathematics - Pure 1 Examination test 2024 with correct marking scheme  -(6 - 2x)/x³ - Correct Answers find die/dx given y = 6 / (2x² + 3x)  -√3 - Correct Answers Without using a calculator, find the exact value of tan(5π/3)  -√3/2 - Correct Answers Without using a calculator, find the exact value of sin(300)  -0.5 - Correct Answers Without using a calculator, find the exact value of cos(240)  -1 - Correct Answers Without using a calculator, find the exact value of sin(3π/2)  -1 - Correct Answers Without using a calculator, find the exact value of tan(135)  -1/3 - Correct Answers the nth term of a geometric progression is (-1/3) ⁿ. Find the common ratio r.  -1/32 - Correct Answers Show that the following improper integral has a finite and find that value. ⁻²₋∞∫2/x⁵ dx.  -13/3 - Correct Answers f(x) = 1/(x + 4), x ∈ ℝ, x ≠ -4. Evaluate f⁻¹(-3)  -243 - Correct Answers the 3rd term of a geometric progression is 9 and the ratio is -3. Find the 6th term.  -3 / 2 ⁴√(5 + 3x)⁵ - Correct Answers find die/dx given y = 2 / ⁴√(5 + 3x)  -3 / 2√(4 - 3x) - Correct Answers y = √(3 - 3x), find die/dx  -3i - 3k - 5k - Correct Answers Write down, using the unit vector notation, the displacement from (3, 3, 5) to (0, 0, 0)  -5 ≤ x ≤ 1/2 - Correct Answers Solve and sketch the inequality  (5 + x)(1 - 2x) ≥ 0  -7 - Correct Answers Given that x + y = 3, find the least possible value of x² + 14y.  -BA - Correct Answers what is the vector AB the same as?  (-1 - √5) / 2 < x < (-1 + √5) / 2 - Correct Answers Solve using the quadratic formula.  x² < 1 - x  (-1, -4) local minimum, (1, 4) local maximum - Correct Answers Find the coordinates of the stationary points of the curve y = 6x - 2x³ and determine their nature.  (-1/2, -3/2) local maximum, (3/2, 5/2) local minimum - Correct Answers Find the coordinates of the stationary points of the curve y = 2 / (2x - 1) + x  (-2.5, 1.5) - Correct Answers What is the midpoint of the straight line joining the sets of points (-8, 5) and (3, -2)  (-2x)ⁿ⁻¹ - Correct Answers A geometric sequence is 1 - 2x + 4x² - 8x³ + ... Find an expression for the nth term  (1/√14)(2i - 3j + k) - Correct Answers Find a unit vector in the direction of 2i - 3j + k  (1/√61)(3 4 6) - Correct Answers Find a unit vector in the direction of (3 4 6)  (100, 2.05) - Correct Answers Find the coordinates of the point on the curve y = (2x - 5 + √x) / x where the gradient is zero  (2 6) - Correct Answers Evaluate the vector 2(1 -3)  (2 9) - Correct Answers Evaluate the vector (3 4) + (-1 5)  (2, -1) - Correct Answers ABCD is a square with vertices A (-1, -2), B (1, 2), C (5, 0) and D (3, -4). Find the center coordinates of ABCD.  (2, -7) and (-2, 21) - Correct Answers Find the coordinates of the points on the curve y = 2x³ - 15x + 7 where the gradient is 9.  0 > f(x) ≥ 1 - Correct Answers Find the range of the function f(x) = 1/x, x ≥ 1  0, π/6, π/4, π/3 - Correct Answers Convert the following degree values to radians.  0°, 30°, 45°, 60°  0.0405 cm s⁻¹ - Correct Answers 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  0.0890 cm s⁻¹ - Correct Answers 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.  1 - Correct Answers Simplify (sin - coos) / (sin²θ - cos²θ)  1 - Correct Answers Without using a calculator, evaluate 0!  1 1  1 2 1  1 3 3 1  1 4 6 4 1  1 5 10 10 5 1 - Correct Answers Write the first 5 rows in pascals triangle.  Let f(x) = y.  Change x to y and y to x.  Rearrange to get x in terms of y  Write in the correct format - Correct Answers What are the steps used to find the inverse function f⁻¹(x) given f(x)?  1.57 ms⁻¹ - Correct Answers 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.  1/√3, 1, √3 - Correct Answers In degrees, without using a calculator, calculate the following; tan (30), tan (45), tan (60).  1/2 - Correct Answers a geometric progression has the sum to infinity equal to twice the 1st term. Find the common ratio, r.  1/2, 1/√2, √3/2 - Correct Answers In degrees, without using a calculator, calculate the following; sin (30), sin (45), sin (60).  1/24 - Correct Answers Find the area of the finite region bounded by y = x² and y = x - x²  1/3(-I + 2j - 2k) - Correct Answers Find a unit vector in the direction of -I + 2j - 2k  1/4 - Correct Answers Evaluate.  (-8)^(-2/3)  10 - Correct Answers Find the lengths of the straight lines joining the sets of points (-8, 4) and (-2, -4)  1024x⁵, - 3840x⁴, 5760x³ - Correct Answers Find the first three terms in decreasing powers of x for (4x - 3)⁵  10737418.23 - Correct Answers 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 fourth 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.  110 mm - Correct Answers Find the arc length of a circle with a 25 mm radius and an angle of 4.4 rad  1150 cm² - Correct Answers 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  12 - Correct Answers OA = (3 -1 2) and OB = (6 -4 -5), Calculate the scalar product of OA x OB.  120 - Correct Answers Evaluate 5! Without using a calculator.  13n - 18 - Correct Answers Find the nth term of the linear sequence -5, 8, 21, 34, 47  15 - Correct Answers Find the magnitude of the vector (10 -5 10)  15/4 - Correct Answers the curve y² = 12x intersects the line 3y = 4x + 6. Find the distance between the two points  159° - Correct Answers Find an angle to the nearest degree other than 21° that has a sine of 0.36  15y = -x - 36 - Correct Answers Find the equation of the line with the points A(9, -3) and B(-6, -2)  16π - Correct Answers 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.  1704/35 - Correct Answers Find ⁴₁∫ ((x³ + x²) / √x) dx.  172 - Correct Answers the third term of a geometric progression is 16 and the 6th term is -128. Find the sum of the first seven terms.  18° - Correct Answers Convert π/10 radians to degrees  180° - Correct Answers Find the angle between I - j + 2k and -2i + 2j - 4k  180° or π rad - Correct Answers What is the period of y = tan x  19 - Correct Answers find ¹₀∫ (3x - 2)⁸dx.  2 / sin - Correct Answers Simplify (1 + cost) / sin + sin / (1 + cost)  2/3 - Correct Answers Evaluate.  (9/4)^(-1/2)  2/3 - Correct Answers Show that the improper integral ∞₁∫1/ (x²√x) dx has a value and find that value.  20 - Correct Answers ABCD is a square with vertices A (-1, -2), B (1, 2), C (5, 0) and D (3, -4). Find the area of ABCD.  20 - Correct Answers Evaluate (2√5)²  20916 - Correct Answers find the sum of the integers between 1 and 500 that are divisible by 6.  49cm² - Correct Answers A piece of card has a length of (2x - 1) cm and a width of (x + 2) cm. A square of side cXML 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.  4πr³/3 - Correct Answers What is the volume of a sphere  5√2 - Correct Answers Find the magnitude of the vector (1 -7 0)  50.0° - Correct Answers Find the angle between 2i + j - 3k and -I +2j - 3k  51.0° - Correct Answers Find the angle between the two vectors (1 -1 1) and (2 1 4).  54.4 cm² - Correct Answers 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.  56 - Correct Answers Without using a calculator, evaluate 8C5  5i + j - 4k - Correct Answers Write down the displacement from P(2, 1, 3) to Q(7, 2, -1) in terms of the unit vectors I, j, k  5q - Correct Answers Find the lengths of the straight lines joining the sets of points (4q, -4q) and (7q, -8q)  5π/2 - Correct Answers 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.  6 + 2p - Correct Answers The position vectors of points A and B relative to an origin O are given by a = 2i + jpg + 3k and b = 3i - j + pike where p is a constant. Calculate ab.  6.51 cm - Correct Answers 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 - Correct Answers A closed box with a square base has a total surface area of 36 m². Find the maximum possible volume of the box.  625x⁴ - 1000x³ + 600x² - 160x + 16 - Correct Answers Use pascal's triangle to expand (5x - 2)⁴  64 cm² - Correct Answers A rectangle has a width of x cm. The perimeter of the rectangle is 32 cm. Find the maximum area of the rectangle  66 - Correct Answers Evaluate 12C10  7.2 cm - Correct Answers Find the arc length of a circle with a 12 cm radius and an angle of 0.6 rad  72 - Correct Answers Find the area bounded by y = x² - 4x and y = 16 - x²  8 - Correct Answers Find the area bounded by the curve with the equation y = x(x + 2)(x - 2)  8/³√(x²) + 1 / 4√(x³) - Correct Answers solve d/dx(24 ³√x - 1 / 2√x)  84 - Correct Answers Evaluate 9C3  89.2° - Correct Answers Find the angle between 2i + 5j - 2k and I + 4j + 11k  9/2 - Correct Answers 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.)  9/4 - Correct Answers The first four terms of a geometric progression are 3, -1, 1/3, -1/9 find the sum to infinity.  A / sin (A) = b / sin (B) = c / sin(C) - Correct Answers what is the sine rule?  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) - Correct Answers Prove the equation for S (n) of an arithmetic progression.  a + (n - 1) d - Correct Answers What is the formula for the nth term of an arithmetic progression?  A = 0.5 × ab sin(C) - Correct Answers What is the area of a triangle  a = 1 - Correct Answers Given that y = ax⁴ - 3x² and d²y / dx² = 42 when x = 2, work out the value of a.  A = 1/2 × r²θ - Correct Answers What is the sector area A of a circle?  a = 3 - Correct Answers Given that the coefficient of x³ in the expansion of (1 + ax + 2x²) (2 - x) ⁷ is 560, find the value of a.  a = 4  b = -1/4  c = -1/4 - Correct Answers 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 = 5  d = 7 - Correct Answers the nth term of an arithmetic progression is 7n - 2. Find a and d.  a = 5  d = 7 - Correct Answers the nth term of an arithmetic progression is 7n - 2. Find a and d.  A = 57.8°, B = 90.0°, C = 31.3° - Correct Answers 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).  A = πr² - Correct Answers What is the area of a circle?  A decreasing function - Correct Answers What is a function called when the gradient is negative?  A displacement vector represents a change in position. - Correct Answers define a displacement vector.  A function f(x) is decreasing for a < x < b if f'(x) < 0 for a < x < b - Correct Answers Define a decreasing function  A function f(x) is increasing for a < x < b if f'(x) > 0 for a < x < b - Correct Answers Define an increasing function  Horizontal - Correct Answers In the diagram used to calculate values of trigonometric functions, which line of reflection is used to find corresponding cosine values?  I + 3j + k - Correct Answers Write down, using the unit vector notation, the displacement from (1, 0, -2) to (2, 3, -1)  If y is a function of u, and u is a function of x, die/dx = die/du × du/dx - Correct Answers What is the chain rule?  k < 0  K > 8/9 - Correct Answers Find the set of values of k for which the equation 2x² + 3kx + k = 0 has distinct real roots.  k = - 1/3 - Correct Answers the functions f and g are defined for x ∈ by f(x) = 4x - 1 and g(x) = 2x + k. Find the value of k for which fog = gf.  k = -4 - Correct Answers Find the value of k for which the line y = 2kx + 7 is a tangent to the curve y = kx²  k = 24 - Correct Answers Find the value k for which the vectors (2 -1 6) and (8 -4 k) are parallel.  m = -2 - Correct Answers Find the gradient m of the straight line joining the sets of points (4, -6) and (1, 0)  m = (y2 - y1) / (x2 - x1) - Correct Answers What is the equation for the gradient m of a line?  m = 10 - Correct Answers Find the gradient of the tangent to the curve y = (4x - 1)² / x² at the point (-1, 25)  m = 4 - Correct Answers 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  m = 5 - Correct Answers Find the gradient m of the straight line joining the sets of points (-3k, -4k) and (-k, 6k)  m1 × m2 = -1 - Correct Answers How do you find out that two lines are perpendicular?  many-to-one - Correct Answers 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  Maximum at (0.5, 4)  ( (x + p)² + q and max/min cords at (-p, q) ) - Correct Answers Sketch and state the coordinates of the vertex and whether it is a maximum or a minimum for the following equation.  y = 3 + 4x - 4x²  Minimum at (-1, -1)  ( (x + p)² + q and max/min cords at (-p, q) ) - Correct Answers Sketch and state the coordinates of the vertex and whether it is a maximum or a minimum for the following equation.  y = x² + 2x  Minimum at (0.4, -2.8)  ( (x + p)² + q and max/min cords at (-p, q) ) - Correct Answers Sketch and state the coordinates of the vertex and whether it is a maximum or a minimum for the following equation.  y = 5x² - 4x - 2  N! / (n - R)! R! - Correct Answers what is the formula for nacre?  nth term = 4 + 2n  First four terms = 6, 8, 10, 12 - Correct Answers The sum of the the first n terms in terms of an arithmetic progression is n² + 5n. Find the nth term and the first 4 terms.  OB = 5i + 4j - k - Correct Answers 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.  OB = OA + AB - Correct Answers Given the vector OA and AB, find OB  OC = (2 -1 9/2) - Correct Answers 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.  OD = 4i - 2j + 9k - Correct Answers 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.  one-to-one - Correct Answers 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 - Correct Answers Sketch the function f(x) = 2x + 1, x ∈ ℝ and state weather it is a one-to-one or a many-to-one function  p = -4, q = -5 - Correct Answers the mid-point of the line joining the points A (p, -2) and B (6, -8) is (1, q). Find p and q.  p = 15/2 - Correct Answers Find the value of p if the vectors pi - 3k and 2i + j + 5k are perpendicular.  p = 2/5 - Correct Answers Find the value of p for which the quadratic equation px² - 4px + 2 - p = 0 has equal roots  p = 5 - Correct Answers Find the value of p for which the vectors I - 2j + k and 3i + 4j + pike  P (-3, 3) - Correct Answers the normal at the points (0, 0) and (1, 2) to the curve y = x + x³ meet at point P. Find the coordinates of point P.  Positive, negative, negative, negative - Correct Answers without using a calculator, state weather each of these are positive or negative.  sin 130, cos 130, sin 255, cos 255  r = 1/3  r = -1/3 - Correct Answers The first and third term of a geometric progression are 18 and 2 respectively. Find two values of r.  r = 25 m - Correct Answers A circle has an arc length of 20 m and an angle of 0.8 rad. Find the radius r.  r = u (2) / u (1) = u (3) / u (2) = u (4) / u (3) - Correct Answers How do you find the common ratio for a geometric progression?  s = or  (where θ is measured in radians and r is the radius of the circle) - Correct Answers what is the equation for arc length s of a circle?  V = πr²h - Correct Answers What is the volume of a cylinder?  Vertical - Correct Answers In the diagram used to calculate values of trigonometric functions, which line of reflection is used to find corresponding sine values?  When x = -1/2, y = 0  When x = -1, y = -1 - Correct Answers Solve the simultaneous equation finding values for y and x.  y = 1 + 2x  y² = 2x² + x  When x = 1, y = 0  When x = -3, y = -4 - Correct Answers Solve the simultaneous equation finding values for y and x.  y = 0.5(1 - x²)  y = x - 1  When x = 1, y = 2  And when x = 2, y = 4 - Correct Answers Solve the simultaneous equation finding values for y and x.  y = 2x  y = x² - x + 2  x < -4 and x > 4 - Correct Answers Find the range of values of x for which f(x) = x³ - 48x + 2 is increasing.  X < 1 and x > 2 - Correct Answers Find the range of values of x for which f(x) = 2x³ - 9x² + 12x -3 is increasing.  x = -(5/2) - Correct Answers Solve the quadratic equation  (2x² + 5x + 3) / (x² + 3x + 2) = 4  x = -0.966 or 0.901 - Correct Answers Solve. Write answer correct to 3 significant figures.  4x¹⁰ + x⁵ = 2  x = -7 or 5 - Correct Answers Solve the quadratic equation by factorization  x² + 2x - 35 = 0  x = (-5 ± √57) / 4 - Correct Answers Solve using the quadratic formula.  2x² -4 + 5x = 0  x = (-b ± √b² - 4ac) / 2a - Correct Answers Complete the square to find x  ax² + box + c = 0  x = (-b ± √b² - 4ac) / 2a - Correct Answers What is the quadratic formula?  x = ± 1.79 - Correct Answers Solve using the quadratic formula.  To 3 sig figures.  2x⁸ - 20x⁴ - 7 = 0  x = 0 or 4 - Correct Answers Solve the quadratic equation by factorization  5x² = 20x  x = 0.42 or 1.58 - Correct Answers Solve using the quadratic formula to 2 decimal places.  3x² = 6x - 2  x = 156°, 336° - Correct Answers For the equation tan x° = -0.44 find all solutions between 0° and 360° to the nearest degree.  x = 2 ± 2√3 - Correct Answers Complete the square to find x  x² - 4x - 8 = 0  x = 2, y = -2, z = 2 - Correct Answers 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.  x = 34°, 214° - Correct Answers Solve the equation 3sin x = 2cos x, giving all solutions between 0° and 360° to the nearest degree.  x = 5/2 - Correct Answers f(x) = x / (x + 3), x ∈ ℝ, x ≠ - 3. If f⁻¹(x) = - 5, find the value of x  x = 6 - Correct Answers Find x.  (6⁵ × 6ˣ) ÷ 36 = 6⁹  x = 6 - Correct Answers Find x.  7⁸ ÷ 7ˣ = 49  x = 6 ± 0.5√170 - Correct Answers Complete the square to find x  4x² - 48x - 26  x = 73°, 287° - Correct Answers Find two angles (to the nearest degree) in the range 0° < x° < 360° such that cos x° = 0.3  x = 9 - Correct Answers Find x.  (5⁷ × 5⁴) ÷ 5ˣ = 25  x = π/6, 7π/6 - Correct Answers Solve √3 sin x - cos x = 0 for 0 ≤ x ≤ 2π  X > 3 - Correct Answers Find the range of values of x for which f(x) = x² - 6x is increasing.  x ≤ - 1 - Correct Answers Solve  (4 + x) / 3 ≤ 1 - 5(1 + x)  x ≤ -7/2 or x ≥ -4/5 - Correct Answers Solve and sketch the inequality  10x² + 43x + 28 ≥ 0  x ≤ 2, x ≥ 6 - Correct Answers Solve and sketch the inequality  (x² + 12) / 2 ≥ 4x  x ≤ 2/5 - Correct Answers Solve  4x - 7(2x - 1) ≥ 3  x ≥ 2 - Correct Answers Solve  2x + 7 ≤ 8x - 5  x¹⁰ - 2x⁹ + 27x⁸ - 54x⁷ + 324x⁶ - Correct Answers Find the first five terms, in descending powers of x, in the expansion of (x -2)(x + 3/x)⁹  Xi + in - Correct Answers Convert the column vector (x y) into unit vector form.  y = -(4/x³) - (3/2x⁴) + c - Correct Answers Find y given die/dx = 12/x² + 6/x³  y = (2(1/4 x + 1)⁹ + 3368)/9 - 122x - Correct Answers d²y/ dx² = (1/4 x + 1)⁷. When x = 4, die/dx = 6 and y = 0. Find an equation for y  y = (4x - 3)⁶/24 + c - Correct Answers die/dx = (4x - 3)⁵, find y