Date | May 2019 | Marks available | 6 | Reference code | 19M.2.AHL.TZ2.H_9 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 2 |
Command term | Sketch | Question number | H_9 | Adapted from | N/A |
Question
Consider the polynomial P(z)≡z4−6z3−2z2+58z−51,z∈C.
Sketch the graph of y=x4−6x3−2x2+58x−51, stating clearly the coordinates of any maximum and minimum points and intersections with axes.
Hence, or otherwise, state the condition on k∈R such that all roots of the equation P(z)=k are real.
Markscheme
shape A1
x-axis intercepts at (−3, 0), (1, 0) and y-axis intercept at (0, −51) A1A1
minimum points at (−1.62, −118) and (3.72, 19.7) A1A1
maximum point at (2.40, 26.9) A1
Note: Coordinates may be seen on the graph or elsewhere.
Note: Accept −3, 1 and −51 marked on the axes.
[6 marks]
from graph, 19.7 ≤ k ≤ 26.9 A1A1
Note: Award A1 for correct endpoints and A1 for correct inequalities.
[2 marks]
Examiners report
Syllabus sections
-
22M.3.AHL.TZ1.2g.i:
Show that the x-coordinate of P is 13(2a+r).
You are not required to demonstrate a change in concavity.
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18M.2.SL.TZ2.S_9a:
Find the initial velocity of P.
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18M.2.SL.TZ2.S_9d:
Find the acceleration of P when it changes direction.
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22M.1.SL.TZ1.7e.ii:
Find the values of x for which the graph of f is concave-down. Justify your answer.
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22M.1.SL.TZ1.7e.i:
Write down the value of a.
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19M.2.AHL.TZ2.H_9c:
Hence, or otherwise, state the condition on k∈R such that all roots of the equation P(z)=k are real.
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18N.2.AHL.TZ0.H_9c.i:
sketch the graph of y=f(x), showing clearly any axis intercepts and giving the equations of any asymptotes.
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17M.1.AHL.TZ2.H_9a.i:
Showing any x and y intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.
y=f(x);
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18M.1.AHL.TZ1.H_9c:
Sketch the graph of y=f(x) showing clearly the position of the points A and B.
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18M.2.SL.TZ1.S_10a:
Find the coordinates of A.
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18M.2.SL.TZ1.S_10b.i:
For the graph of f, write down the amplitude.
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19M.2.SL.TZ1.S_4a:
Sketch the graph of f″ on the grid below:
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18M.2.SL.TZ1.T_6c:
Find the height of the cylinder, h , of the new trash can, in terms of r.
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19M.2.SL.TZ2.S_5c:
Find the value of t for which the population of fish is increasing most rapidly.
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18M.1.SL.TZ2.T_13a:
Find the cost of producing 70 shirts.
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18M.1.SL.TZ2.T_13c:
Find the number of shirts produced when the cost of production is lowest.
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18M.2.SL.TZ1.T_6b:
Find the total volume of the trash can.
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18N.2.SL.TZ0.T_6a:
Calculate the area of cloth, in cm2, needed to make Haruka’s bag.
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17M.1.AHL.TZ2.H_9a.ii:
Showing any x and y intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.
y=1f(x);
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19N.2.SL.TZ0.S_8d:
Let R be the region enclosed by the graph of f, the x-axis and the lines x=p and x=b. The region R is rotated 360º about the x-axis. Find the volume of the solid formed.
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18M.2.SL.TZ1.T_6e:
Using your graphic display calculator, find the value of r which maximizes the value of V.
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19M.2.SL.TZ2.T_5b:
Write down the y-intercept of the graph of y=f(x).
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SPM.1.SL.TZ0.8b:
Find the value of a and the value of b.
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19N.2.SL.TZ0.S_8c.ii:
Find the rate of change of f at B.
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17M.2.SL.TZ2.T_6f:
Write down the number of possible solutions to the equation f(x)=5.
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17M.2.SL.TZ2.S_8b.i:
Write down the coordinates of A.
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17M.2.SL.TZ2.S_8c.i:
Find the coordinates of B.
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18M.2.SL.TZ2.S_9e:
Find the total distance travelled by P.
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SPM.1.SL.TZ0.8c.i:
Sketch the graph of y=f′(x).
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17N.2.AHL.TZ0.H_11a.i:
Determine an expression for f′(x) in terms of x.
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16N.1.AHL.TZ0.H_11h:
Find the value κ for x=π2 and comment on its meaning with respect to the shape of the graph.
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18M.2.SL.TZ1.S_10e:
Find the first time when the ball’s speed is changing at a rate of 2 cm s−2.
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17M.2.SL.TZ1.S_10b.i:
Find ∫3.310.111(h(x)−x)dx.
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19N.1.SL.TZ0.S_8a:
Write down an expression for y in terms of x.
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19N.2.SL.TZ0.S_8b.ii:
Find the equation of the tangent to the graph of f at A.
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18N.1.AHL.TZ0.H_10b:
Hence, show that ∫excos2xdx=ex5sin2x+ex10cos2x+ex2+c, c∈R.
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21N.1.SL.TZ0.9c.ii:
Find the values of x where the graph of f has points of inflexion. Justify your answer.
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19N.1.SL.TZ0.S_8d.i:
Find the value of x for which V is a maximum.
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19N.2.SL.TZ0.S_8a:
Find the value of p.
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16N.1.AHL.TZ0.H_11c:
Show that the function f has a local maximum value when x=3π4.
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17M.1.AHL.TZ2.H_9b:
Find ∫f(x)cosxdx.
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SPM.1.SL.TZ0.8a:
Find f′(x).
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18N.2.AHL.TZ0.H_9c.ii:
sketch the graph of y=f−1(x), showing clearly any axis intercepts and giving the equations of any asymptotes.
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17N.1.SL.TZ0.S_7:
Consider f(x)=logk(6x−3x2), for 0<x<2, where k>0.
The equation f(x)=2 has exactly one solution. Find the value of k.
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17M.2.SL.TZ2.S_8b.ii:
Write down the rate of change of f at A.
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19M.2.SL.TZ2.T_5c:
Sketch the graph of y=f(x) for −3 ≤ x ≤ 3 and −4 ≤ y ≤ 12.
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17N.1.SL.TZ0.T_15a:
Write down how many kilograms of cheese Maria sells in one week if the price of a kilogram of cheese is 8 EUR.
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18N.1.SL.TZ0.S_10b.i:
Find f′(x).
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18M.2.SL.TZ1.S_10d:
Find the maximum speed of the ball.
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18M.1.SL.TZ2.S_9b:
Show that C=20πr2+320πr.
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17M.1.AHL.TZ2.H_9a.iii:
Showing any x and y intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.
y=|1f(x)|.
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18M.2.SL.TZ2.S_9b:
Find the maximum speed of P.
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18M.2.SL.TZ1.S_7a:
Given that xk + 1 = xk + a, find a.
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17M.2.SL.TZ2.S_8a:
Find the value of p.
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18N.1.SL.TZ0.S_10c:
The graph of f has a local minimum at the point Q. The line L passes through Q.
Find the value of a.
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18M.1.AHL.TZ1.H_9b.ii:
The coordinates of B can be expressed in the form B(2a,b×2−3a) where a, b∈Q. Find the value of a and the value of b.
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17N.2.AHL.TZ0.H_10b:
Sketch the graph of y=f(x) showing clearly the minimum point and any asymptotic behaviour.
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19N.1.SL.TZ0.S_8b:
Find an expression for V in terms of x.
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19M.2.SL.TZ1.S_4c:
Hence find the values of x for which the graph of f is concave-down.
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19M.2.SL.TZ2.S_5a:
Find the population of fish at t = 10.
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19N.2.SL.TZ0.S_8b.i:
Write down the coordinates of A.
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18N.1.AHL.TZ0.H_10a:
Use integration by parts to show that ∫excos2xdx=2ex5sin2x+ex5cos2x+c, c∈R.
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17M.2.SL.TZ2.S_8d:
Let R be the region enclosed by the graph of f , the x-axis, the line x=b and the line x=a. The region R is rotated 360° about the x-axis. Find the volume of the solid formed.
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22M.3.AHL.TZ1.2g.ii:
Hence describe numerically the horizontal position of point P relative to the horizontal positions of the points R and A.
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22M.3.AHL.TZ2.1d.ii:
Hence deduce that the curve y2=x3+x has no local minimum or maximum points.
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EXN.1.SL.TZ0.7c:
Find an expression for A in terms of x.
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EXN.1.SL.TZ0.7e:
Determine the maximum area of rectangle ORST.
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22M.1.SL.TZ2.7e:
Find the values of x for which the graph of g is concave-up.
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22M.1.AHL.TZ2.6b:
The range of f is a≤y≤b, where a, b∈ℝ.
Find the value of a and the value of b.
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SPM.1.SL.TZ0.8c.ii:
Hence explain why the graph of f has a local maximum point at x=a.
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22M.2.AHL.TZ1.12c.iii:
Using the graph of y=8x+x44-x3, suggest a reason why the approximation given by Euler’s method in part (a) is not a good estimate to the actual value of y at x=1.5.
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22M.2.AHL.TZ2.11e:
Let D(t) represent the distance between airplane A and airplane B for 0≤t≤2.5.
Find the minimum value of D(t).
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22M.2.AHL.TZ2.12d:
Hence determine the maximum value of dPdt in terms of k and N.
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17N.2.AHL.TZ0.H_10a.ii:
Determine the values of x for which f(x) is a decreasing function.
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20N.2.SL.TZ0.S_10a:
Show that f''(x)=24-6x2(x2+4)2.
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EXN.2.AHL.TZ0.12c:
The curve has a point of inflexion at (x1, y1) where e-π2<x1<eπ2. Determine the coordinates of this point of inflexion.
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SPM.1.SL.TZ0.8d.i:
Find f″(b).
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17N.2.AHL.TZ0.H_11b.iii:
Hence show that f(x)=0 can be expressed as u3−7u2+15u−9=0.
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17M.1.AHL.TZ2.H_9c:
By finding g′(x) explain why g is an increasing function.
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18N.2.AHL.TZ0.H_9d:
Hence, or otherwise, solve the inequality f(x)>f−1(x).
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16N.1.AHL.TZ0.H_11f:
Find the area of the region enclosed by the graph of f and the x-axis.
The curvature at any point (x, y) on a graph is defined as κ=|d2ydx2|(1+(dydx)2)32.
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21M.2.AHL.TZ1.11c:
Find the x-coordinate of the point of inflexion.
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17M.2.SL.TZ2.S_8c.ii:
Find the the rate of change of f at B.
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20N.2.SL.TZ0.S_10b:
Find the least value of n.
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20N.2.SL.TZ0.T_4a:
Write down an equation in r and h that shows this information.
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18M.2.SL.TZ1.S_10b.ii:
For the graph of f, write down the period.
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19M.1.AHL.TZ2.H_8a:
Show that the volume of the cone may be expressed by V=π3(2Rh2−h3).
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17N.1.SL.TZ0.T_11b:
Write down the value of c.
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21M.3.AHL.TZ1.1c.i:
a point of inflexion with zero gradient.
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21M.3.AHL.TZ1.1c.iii:
no points where the gradient is equal to zero.
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21M.3.AHL.TZ1.1c.ii:
one local maximum point and one local minimum point.
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21M.3.AHL.TZ1.1e.i:
exactly one x-axis intercept.
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20N.2.SL.TZ0.T_4f:
To prevent leaks, a sealant is applied to the interior surface of the bird bath.
Find the surface area to be covered by the sealant, given that the bird bath has maximum volume.
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19N.2.SL.TZ0.S_8c.i:
Find the coordinates of B.
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SPM.1.SL.TZ0.9d:
The region R is enclosed by the graph of f, the x-axis, and the vertical lines through the maximum point P and the point of inflexion Q.
Given that the area of R is 3, find the value of k.
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16N.1.AHL.TZ0.H_11e:
Sketch the graph of f, clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts.
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17N.2.AHL.TZ0.H_10c:
Find the coordinates of the point on the graph of f where the normal to the graph is parallel to the line y=−x.
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19M.2.SL.TZ2.S_5b:
Find the rate at which the population of fish is increasing at t = 10.
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SPM.1.SL.TZ0.9b:
Find the x-coordinate of P.
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18N.1.AHL.TZ0.H_10d:
Find the area enclosed by the curve and the x-axis between B and D, as shaded on the diagram.
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18M.2.SL.TZ2.S_9c:
Write down the number of times that the acceleration of P is 0 m s−2 .
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EXN.1.SL.TZ0.7b:
Find the dimensions of rectangle ORST that has maximum perimeter and determine the value of the maximum perimeter.
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18M.1.SL.TZ2.S_9a:
Express h in terms of r.
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19N.1.SL.TZ0.S_8e:
Find the maximum volume.
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19N.1.SL.TZ0.S_8d.ii:
Justify your answer.
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19N.1.SL.TZ0.S_8c:
Find dVdx.
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19M.1.SL.TZ2.T_15c:
Hence, find the number of vases that will maximize the profit.
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EXN.1.SL.TZ0.7a:
Show that P=-2x2+2x+8.
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21M.3.AHL.TZ2.1f:
Hence, or otherwise, show that fn'(a4)>0, for n∈ℤ+.
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17N.1.SL.TZ0.S_6:
Let f(x)=15−x2, for x∈R. The following diagram shows part of the graph of f and the rectangle OABC, where A is on the negative x-axis, B is on the graph of f, and C is on the y-axis.
Find the x-coordinate of A that gives the maximum area of OABC.
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21M.3.AHL.TZ2.1g.ii:
a point of inflexion with zero gradient for odd values of n, where n>1 and a∈ℝ+.
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21M.3.AHL.TZ2.1h:
Consider the graph of y=xn(a-x)n-k, where n∈ℤ+, a∈ℝ+ and k∈ℝ.
State the conditions on n and k such that the equation xn(a-x)n=k has four solutions for x.
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SPM.1.SL.TZ0.9c:
Show that the x-coordinate of Q is 15e32.
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21M.3.AHL.TZ1.1d.i:
the y-coordinate of the local maximum point is 2c32+2.
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SPM.1.SL.TZ0.8e:
The normal to the graph of f at x=a and the tangent to the graph of f at x=b intersect at the point (p, q) .
Find the value of p and the value of q.
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20N.2.SL.TZ0.S_10c:
Find ∫6xx2+4dx.
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19M.1.AHL.TZ2.H_8b:
Given that there is one inscribed cone having a maximum volume, show that the volume of this cone is 32πR381.
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EXN.1.SL.TZ0.7d:
Find the dimensions of rectangle ORST that has maximum area.
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17M.2.SL.TZ1.S_10a.i:
Write down the value of q;
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21N.1.SL.TZ0.9a:
Find all the values of x where the graph of f is increasing. Justify your answer.
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21N.1.SL.TZ0.9c.i:
Find the value of x where the graph of f has a local minimum. Justify your answer.
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21N.1.SL.TZ0.9d:
The total area of the region enclosed by the graph of f', the derivative of f, and the x-axis is 20.
Given that f(p)+f(t)=4, find the value of f(0).
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22M.1.AHL.TZ2.11a:
Sketch the curve y=f(x), clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.
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22M.3.AHL.TZ2.1b.i:
Write down the coordinates of the two points of inflexion on the curve y2=x3+1.
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22M.3.AHL.TZ2.1e:
Find the value of this x-coordinate, giving your answer in the form x=√p√3+qr, where p, q, r∈ℤ.
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21N.1.SL.TZ0.9b:
Find the value of x where the graph of f has a local maximum.
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19M.1.SL.TZ1.T_15a:
Write down an equation for the area, A, of the curved surface in terms of r.
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17N.1.SL.TZ0.T_11a:
Find the equation of the axis of symmetry of the graph of y=f(x).
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19M.1.SL.TZ2.T_15a:
Find the value of P if no vases are sold.
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17M.2.SL.TZ2.T_6d.i:
Write down the x-coordinates of these two points;
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17M.2.SL.TZ2.T_5c:
Find the value of x that maximizes the area of the plot.
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16N.2.SL.TZ0.T_6b:
Express this volume in cm3.
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16N.2.SL.TZ0.T_6c:
Write down, in terms of r and h, an equation for the volume of this water container.
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16N.2.SL.TZ0.T_6e:
Find dAdr.
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16N.2.SL.TZ0.T_6f:
Using your answer to part (e), find the value of r which minimizes A.
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18M.2.SL.TZ1.T_6a:
Write down the height of the cylinder.
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18M.1.SL.TZ2.T_13b:
Find the value of s.
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16N.1.AHL.TZ0.H_11g:
Find the value of the curvature of the graph of f at the local maximum point.
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18N.1.SL.TZ0.S_10b.ii:
Hence, find the equation of L in terms of a.
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18M.1.SL.TZ2.S_9c:
Given that there is a minimum value for C, find this minimum value in terms of π.
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17N.2.AHL.TZ0.H_11a.ii:
Sketch a graph of y=f′(x) for 0⩽x<π2.
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18M.2.SL.TZ1.S_10c:
Hence, write f(x) in the form pcos(x+r).
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SPM.1.SL.TZ0.8d.ii:
Hence, use your answer to part (d)(i) to show that the graph of f has a local minimum point at x=b.
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17N.1.SL.TZ0.T_15d:
Find the price, p, that will give Maria the highest weekly profit.
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EXN.1.AHL.TZ0.11e:
Show that the maximum value of m is 2732√23.
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20N.2.SL.TZ0.T_4d:
Using your answer to part (c), find the value of h for which V is a maximum.
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17N.2.AHL.TZ0.H_11c:
Solve the equation f(x)=0, giving your answers in the form arctank where k∈Z.
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SPM.1.SL.TZ0.9a:
Show that f′(x)=1−ln5xkx2.
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17N.2.AHL.TZ0.H_10a.i:
Show that the x-coordinate of the minimum point on the curve y=f(x) satisfies the equation tanx=2x.
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17N.1.SL.TZ0.T_15b:
Find how much Maria earns in one week, from selling cheese, if the price of a kilogram of cheese is 8 EUR.
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21M.3.AHL.TZ1.1e.iii:
exactly three x-axis intercepts.
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17N.2.AHL.TZ0.H_11a.iii:
Find the x-coordinate(s) of the point(s) of inflexion of the graph of y=f(x), labelling these clearly on the graph of y=f′(x).
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17N.2.AHL.TZ0.H_10d:
This region is now rotated through 2π radians about the x-axis. Find the volume of revolution.
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18N.1.AHL.TZ0.H_10c:
Find the x-coordinates of A and of C , giving your answers in the form a+arctanb, where a, b∈R.
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17N.2.AHL.TZ0.H_11b.i:
Express sinx in terms of u.
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16N.1.AHL.TZ0.H_11d:
Find the x-coordinate of the point of inflexion of the graph of f.
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20N.2.SL.TZ0.S_10d:
Let R be the region enclosed by the graph of f, the x-axis and the lines x=1 and x=3. The area of R is 19.6, correct to three significant figures.
Find f(x).
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18N.2.AHL.TZ0.H_9b:
Hence, or otherwise, find the coordinates of the point of inflexion on the graph of y=f(x).
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17N.2.AHL.TZ0.H_11b.ii:
Express sin2x in terms of u.
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EXN.2.AHL.TZ0.12d:
Use the differential equation dydx=y2+3xy+2x2x2 to show that the points of zero gradient on the curve lie on two straight lines of the form y=mx where the values of m are to be determined.
-
19M.2.SL.TZ1.S_4b:
Find the x-coordinates of the points of inflexion of the graph of f.
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21M.2.AHL.TZ1.12e:
Find the value of t when the rate of change of the population is at its maximum.
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21M.3.AHL.TZ1.1d.ii:
the y-coordinate of the local minimum point is -2c32+2.
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21M.1.SL.TZ1.8c:
Given that f''(x)=20 ln x-9x6, show that P is a local maximum point.
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21M.3.AHL.TZ1.1e.ii:
exactly two x-axis intercepts.
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21M.3.AHL.TZ2.1g.i:
a local minimum point for even values of n, where n>1 and a∈ℝ+.
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17M.2.SL.TZ1.S_10a.iii:
Write down the value of k.
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17M.2.SL.TZ1.S_10c:
Let d be the vertical distance from a point on the graph of h to the line y=x. There is a point P(a, b) on the graph of h where d is a maximum.
Find the coordinates of P, where 0.111<a<3.31.
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17M.2.SL.TZ1.S_10a.ii:
Write down the value of h;
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17M.2.SL.TZ1.S_10b.ii:
Hence, find the area of the region enclosed by the graphs of h and h−1.
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18N.2.SL.TZ0.T_6g:
Use your answer to part (f) to show that the width of Nanako’s bag is 12 cm.
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17N.1.SL.TZ0.T_11c:
Find the value of a and of b.
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19M.1.SL.TZ1.T_15c:
Find the value of r when the area of the curved surface is maximized.
-
18N.2.SL.TZ0.T_6d:
Write down and simplify an expression in x and y for the area of cloth, A, used to make Nanako’s bag.
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19M.1.SL.TZ2.T_15b:
Differentiate P(x).
-
17N.1.SL.TZ0.T_15c:
Write down an expression for W in terms of p.
-
20N.2.SL.TZ0.T_4b:
Show that V=2500πh3-πh33.
-
17M.2.SL.TZ2.T_6a:
Write down the y-intercept of the graph.
-
17M.2.SL.TZ2.T_6c.i:
Show that a=8.
-
17M.2.SL.TZ2.T_6b:
Find f′(x).
-
20N.2.SL.TZ0.T_4e:
Find the maximum volume of the bird bath.
-
17M.2.SL.TZ2.T_5b:
Write down the area of the plot in terms of x.
-
19M.2.SL.TZ2.T_5h:
Determine the range of f(x) for p ≤ x ≤ q.
-
17M.2.SL.TZ2.T_5a:
Show that the width of the plot, in metres, is given by 100−x.
-
17M.2.SL.TZ2.T_6g:
The equation f(x)=m, where m∈R, has four solutions. Find the possible values of m.
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20N.2.SL.TZ0.T_4c:
Find dVdh.
-
16N.2.SL.TZ0.T_6d:
Show that A=πr2+1000000r.
-
16N.2.SL.TZ0.T_6g:
Find the value of this minimum area.
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19M.1.SL.TZ1.T_15b:
Find dAdr.
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18N.2.SL.TZ0.T_6e:
Use your answers to parts (c) and (d) to show that
A=3x2+10368x.
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18M.2.SL.TZ1.T_6d:
Show that the volume, V cm3 , of the new trash can is given by
V=110πr3.
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16N.2.SL.TZ0.T_6a:
Write down a formula for A, the surface area to be coated.
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16N.2.SL.TZ0.T_6h:
Find the least number of cans of water-resistant material that will coat the area in part (g).
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18M.2.SL.TZ1.T_6f:
The designer claims that the new trash can has a capacity that is at least 40% greater than the capacity of the original trash can.
State whether the designer’s claim is correct. Justify your answer.
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17M.2.SL.TZ2.T_5d:
Show that Violeta earns 5000 BGN from selling the flowers grown on the plot.
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18M.2.SL.TZ1.T_4e:
Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.
Clearly indicate the minimum point P and the x-intercepts on your graph. -
17M.2.SL.TZ2.T_6c.ii:
Find f(2).
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17M.2.SL.TZ2.T_6d.ii:
Write down the intervals where the gradient of the graph of y=f(x) is positive.
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17M.2.SL.TZ2.T_6e:
Write down the range of f(x).
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18N.2.SL.TZ0.T_6b:
Calculate the volume, in cm3, of the bag.
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18N.2.SL.TZ0.T_6c:
Use this value to write down, and simplify, the equation in x and y for the volume of Nanako’s bag.
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18N.2.SL.TZ0.T_6f:
Find dAdx.