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IB Questionbank Mathematics Higher Level 3rd edition 1

1. Consider the function f(x) = xxln

, 0 < x < e2.

(a) (i) Solve the equation f′(x) = 0.

(ii) Hence show the graph of f has a local maximum.

(iii) Write down the range of the function f. (5)

(b) Show that there is a point of inflexion on the graph and determine its coordinates. (5)

(c) Sketch the graph of y = f(x), indicating clearly the asymptote, x-intercept and the local maximum.

(3)

(d) Now consider the functions g(x) = xxln

and h(x) = xxln

, where 0 < │x│ < e2.

(i) Sketch the graph of y = g(x).

(ii) Write down the range of g.

(iii) Find the values of x such that h(x) > g(x). (6)

(Total 19 marks)

2. The quadratic function f(x) = p + qx – x2 has a maximum value of 5 when x = 3.

(a) Find the value of p and the value of q. (4)

(b) The graph of f(x) is translated 3 units in the positive direction parallel to the x-axis. Determine the equation of the new graph.

(2) (Total 6 marks)


3. The curve C has equation y = )89(81 42 xx −+ .

(a) Find the coordinates of the points on C at which xydd

= 0.

(4)

(b) The tangent to C at the point P(1, 2) cuts the x-axis at the point T. Determine the coordinates of T.

(4)

(c) The normal to C at the point P cuts the y-axis at the point N. Find the area of triangle PTN.

(7) (Total 15 marks)

4. Consider the function f(x) = x3 – 3x2 – 9x + 10, x ∈ .

(a) Find the equation of the straight line passing through the maximum and minimum points of the graph y = f(x).

(4)

(b) Show that the point of inflexion of the graph y = f(x) lies on this straight line. (2)

(Total 6 marks)

5. The point P, with coordinates (p, q), lies on the graph of 21

21

21

ayx =+ , a > 0. The tangent to the curve at P cuts the axes at (0, m) and (n, 0). Show that m + n = a.

(Total 8 marks)


6. Consider f(x) = 4545

2

2

++

+−

xxxx .

(a) Find the equations of all asymptotes of the graph of f. (4)

(b) Find the coordinates of the points where the graph of f meets the x and y axes. (2)

(c) Find the coordinates of

(i) the maximum point and justify your answer;

(ii) the minimum point and justify your answer. (10)

(d) Sketch the graph of f, clearly showing all the features found above. (3)

(e) Hence, write down the number of points of inflexion of the graph of f. (1)

(Total 20 marks)

7. The function f is defined by f(x) = 5.122e −− xx .

(a) Find f′(x). (2)

(b) You are given that y = 1)(

−xxf

has a local minimum at x = a, a > 1. Find the value of a.

(6) (Total 8 marks)


8. The normal to the curve xe–y + ey = 1 + x, at the point (c, ln c), has a y-intercept c2 + 1.

Determine the value of c. (Total 7 marks)

9. The function f is defined by

f(x) = 21

23 )1036( −++ xxx , for x ∈ D,

where D ⊆ is the greatest possible domain of f.

(a) Find the roots of f(x) = 0. (2)

(b) Hence specify the set D. (2)

(c) Find the coordinates of the local maximum on the graph y = f(x). (2)

(d) Solve the equation f(x) = 3. (2)

(e) Sketch the graph of │y│= f(x), for x ∈ D. (3)

(f) Find the area of the region completely enclosed by the graph of │y│ = f(x). (3)

(Total 14 marks)


10. Consider the curve y = xex and the line y = kx, k ∈ .

(a) Let k = 0.

(i) Show that the curve and the line intersect once.

(ii) Find the angle between the tangent to the curve and the line at the point of intersection.

(5)

(b) Let k = 1. Show that the line is a tangent to the curve. (3)

(c) (i) Find the values of k for which the curve y = xex and the line y = kx meet in two distinct points.

(ii) Write down the coordinates of the points of intersection.

(iii) Write down an integral representing the area of the region A enclosed by the curve and the line.

(iv) Hence, given that 0 < k < 1, show that A < 1. (15)

(Total 23 marks)

11. Find the equation of the normal to the curve x3y3 – xy = 0 at the point (1, 1). (Total 7 marks)

12. The line y = m(x – m) is a tangent to the curve (1 – x)y = 1.

Determine m and the coordinates of the point where the tangent meets the curve. (Total 7 marks)


13. Let f(x) = ba

bax

x

+

+

ee , where 0 < b < a.

(a) Show that f′(x) = 2

22

)e(e)(baab

x

x

+

−.

(3)

(b) Hence justify that the graph of f has no local maxima or minima. (2)

(c) Given that the graph of f has a point of inflexion, find its coordinates. (6)

(d) Show that the graph of f has exactly two asymptotes. (3)

(e) Let a = 4 and b = 1. Consider the region R enclosed by the graph of y = f(x),

the y-axis and the line with equation y = 21

.

Find the volume V of the solid obtained when R is rotated through 2π about the x-axis. (5)

(Total 19 marks)


14. Consider the part of the curve 4x2 + y2 = 4 shown in the diagram below.

(a) Find an expression for xydd

in terms of x and y.

(3)

(b) Find the gradient of the tangent at the point ⎟⎟⎠

⎞⎜⎜⎝

⎛

52,

52 .

(1)

(c) A bowl is formed by rotating this curve through 2π radians about the x-axis. Calculate the volume of this bowl.

(4) (Total 8 marks)

15. A function is defined as f(x) = xk , with k > 0 and x ≥ 0.

(a) Sketch the graph of y = f(x). (1)


(b) Show that f is a one-to-one function. (1)

(c) Find the inverse function, f–1(x) and state its domain. (3)

(d) If the graphs of y = f(x) and y = f–1(x) intersect at the point (4, 4) find the value of k. (2)

(e) Consider the graphs of y = f(x) and y = f–1(x) using the value of k found in part (d).

(i) Find the area enclosed by the two graphs.

(ii) The line x = c cuts the graphs of y = f(x) and y = f–1(x) at the points P and Q respectively. Given that the tangent to y = f(x) at point P is parallel to the tangent to y = f–1(x) at point Q find the value of c.


16. Let f(x) = xx

+

−

11

and g(x) = 1+x , x > – 1.

Find the set of values of x for which f ′(x) ≤ f(x) ≤ g(x). (Total 7 marks)


17. The cubic curve y = 8x3 + bx2 + cx + d has two distinct points P and Q, where the gradient is zero.

(a) Show that b2 > 24c. (4)

(b) Given that the coordinates of P and Q are ⎟⎠

⎞⎜⎝

⎛−⎟

⎠

⎞⎜⎝

⎛− 20,

23 and 12,

21

, respectively, find the

values of b, c and d. (4)

(Total 8 marks)

18. The diagram below shows a sketch of the gradient function f′(x) of the curve f(x).


On the graph below, sketch the curve y = f(x) given that f(0) = 0. Clearly indicate on the graph any maximum, minimum or inflexion points.

(Total 5 marks)

19. A tangent to the graph of y = ln x passes through the origin.

(a) Sketch the graphs of y = ln x and the tangent on the same set of axes, and hence find the equation of the tangent.

(11)

(b) Use your sketch to explain why ln x ≤ ex

for x > 0.

(1)

(c) Show that xe ≤ ex for x > 0. (3)

(d) Determine which is larger, πe or eπ. (2)

(Total 17 marks)


20. Find the gradient of the curve exy + ln(y2) + ey = 1 + e at the point (0, 1). (Total 7 marks)

21. Consider the function f, defined by f(x) = x – xa , where x ≥ 0, a ∈ +.

(a) Find in terms of a

(i) the zeros of f;

(ii) the values of x for which f is decreasing;

(iii) the values of x for which f is increasing;

(iv) the range of f. (10)

(b) State the concavity of the graph of f. (1)

(Total 11 marks)

22. If f (x) = x – 32

3x , x > 0,

(a) find the x-coordinate of the point P where f ′ (x) = 0; (2)

(b) determine whether P is a maximum or minimum point. (3)

(Total 5 marks)

23. Find the area between the curves y = 2 + x − x2 and y = 2 − 3x + x2. (Total 7 marks)


24. The function f is defined by f (x) = x e2x.

It can be shown that f (n) (x) = (2n x + n 2n−1) e2x for all n∈ +, where f (n) (x) represents the nth derivative of f (x).

(a) By considering f (n) (x) for n =1 and n = 2, show that there is one minimum point P on the graph of f, and find the coordinates of P.

(7)

(b) Show that f has a point of inflexion Q at x = −1. (5)

(c) Determine the intervals on the domain of f where f is

(i) concave up;

(ii) concave down. (2)

(d) Sketch f, clearly showing any intercepts, asymptotes and the points P and Q. (4)

(e) Use mathematical induction to prove that f (n) (x) = (2nx + n2n−1) e2x for all n∈ +, where f (n) (x) represents the nth derivative of f (x).


25. Consider the curve with equation x2 + xy + y2 = 3.

(a) Find in terms of k, the gradient of the curve at the point (−1, k). (5)

(b) Given that the tangent to the curve is parallel to the x-axis at this point, find the value of k.

(1) (Total 6 marks)


26. A family of cubic functions is defined as fk (x) = k2x3 − kx2 + x, k∈ +.

(a) Express in terms of k

(i) f ′k (x) and f ′′k (x);

(ii) the coordinates of the points of inflexion Pk on the graphs of fk. (6)

(b) Show that all Pk lie on a straight line and state its equation. (2)

(c) Show that for all values of k, the tangents to the graphs of fk at Pk are parallel, and find the equation of the tangent lines.


27. Consider the curve with equation f (x) = 22e x− for x < 0.

Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks)

28. Find the equation of the normal to the curve 5xy2 – 2x2 =18 at the point (1, 2). (Total 7 marks)


indent1(a)(i);22e x− 29. A packaging company makes boxes for chocolates. An

example of a box is shown below. This box is closed and the top and bottom of the box are identical regular hexagons of side x cm.

diagram not to scale

(a) Show that the area of each hexagon is 233 2x cm2.

(1)


(b) Given that the volume of the box is 90 cm3, show that when x = 3 20 the total surface area of the box is a minimum, justifying that this value gives a minimum.

(7) (Total 8 marks)


30. Find the gradient of the normal to the curve 3x2y + 2xy2 = 2 at the point (1, –2). (Total 6 marks)


31. A curve C is defined implicitly by xey = x2 + y2. Find the equation of the tangent to C at the point (1, 0).

(Total 7 marks)


32. The function f is defined by f (x) = (ln (x – 2))2. Find the coordinates of the point of inflexion of f.

(Total 9 marks)


33. It is given that

f(x) = ∈−

=ʹ́−

=ʹ− x

xxxf

xxxf

xx

,)3(36

)( and ,)2(18

)(,)1(18

432 , x ≠ 0.


(a) Find

(i) the zero(s) of f(x);

(ii) the equations of the asymptotes;

(iii) the coordinates of the local maximum and justify it is a maximum;

(iv) the interval(s) where f(x) is concave up. (9)


(b) Hence sketch the graph of y = f(x). (5)

(Total 14 marks)


34. The function f is defined on the domain x ≥ 1 by f(x) = xxln

.

(a) (i) Show, by considering the first and second derivatives of f, that there is one maximum point on the graph of f.


(ii) State the exact coordinates of this point.


(iii) The graph of f has a point of inflexion at P. Find the x-coordinate of P. (12)


Let R be the region enclosed by the graph of f, the x-axis and the line x = 5.

(b) Find the exact value of the area of R. (6)

(Total 18 marks)


35. (a) Find the root of the equation e2–2x = 2e–x giving the answer as a logarithm. (4)


(b) The curve y = e2–2x – 2e–x has a minimum point. Find the coordinates of this minimum. (7)


(c) The curve y = e2–2x – 2e–x is shown below.

Write down the coordinates of the points A, B and C. (3)


(d) Hence state the set of values of k for which the equation e2–2x – 2e–x = k has two distinct positive roots.



36. The function f is defined on the domain x ≥ 0 by f(x) = x

xe

2

.

(a) Find the maximum value of f (x), and justify that it is a maximum. (10)


(b) Find the x coordinates of the points of inflexion on the graph of f. (3)


(c) Evaluate ∫1

0d)( xxf .
