127PRACTICE ASSESSMENT TASK 1

1. Prove that the opposite sides of a parallelogram are equal.

2. Find all values of x for which the curve y x2 1 2= −^ h is decreasing.

3. Find ( ) .x x dx3 2 12 − +#

4. Find the maximum value of the curve y x x3 42= + − in the domain .≤ ≤x1 4−

5. The area of a rectangle is 4 m 2 . Find its minimum perimeter.

6. Show that ABDE is a parallelogram, given that ACDF is a parallelogram and .BC FE=

7. Find the primitive function of .x x3 48 +

8. Sketch the curve ,y x x x3 9 23 2= − − + showing all stationary points and infl exions.

9. Find the volume of the solid of revolution formed when the curve y x 13= + is rotated about

the (a) x -axis from x 0= to x 2= the (b) y -axis from y 1= to .y 9=

10. Find the area enclosed between the curve y x 12= − and the x -axis.

11. If ( ) ,f x x x x2 5 93 2= − + − fi nd ( )f 3l and ( ) .f 2−m

12. Evaluate ( ) .x x dx6 42

1+

3#

13. ABC is an isosceles triangle with D the midpoint of BC . Show that AD is perpendicular to BC .

14. Find the volume of the solid of revolution formed, correct to 3 signifi cant fi gures, if y x 22= + is rotated about

the (a) x -axis from x 0= to x 2= the (b) y -axis from y 2= to .y 3=

15. Find the domain over which the curve y x x x3 7 3 13 2= + − − is concave upwards.

16. If ( ) ,f x x x x2 7 94 3= − − + fi nd 1) 1) .,(1), ( (f f fl m What is the geometrical signifi cance of these results? Illustrate by a sketch of ( )y f x= at .x 1=

17. A piece of wire of length 4 m is cut into 2 parts. One part is bent to form a rectangle with sides x and 3 x , and the other part is bent to form a square with sides y . Prove that the total area of the rectangle and square is given by ,A x x7 4 12= − + and fi nd the dimensions of the rectangle and square when the area has the least value.

Practice Assessment Task SET 1

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128 Maths In Focus Mathematics HSC Course

18. The gradient function of a curve is given by ( ) .f x x4 3= −l If ( ) ,f 2 3= − fi nd ( ) .f 1−

19. Evaluate ( ) .x dx2 10

3+#

20. The following table gives values for

( ) .f xx1

2=

x 1 2 3 4 5

f(x) 141

91

161

251

Use the table together with Simpson’s

rule to evaluate xdx

1 2

5# correct to 3

signifi cant fi gures.

21. Show that Δ ABC is right angled at .B+

22. Find ( ) .x dx3 5 7+#

23. Evaluate ( ) .x x dx52

0

2+ −#

24. Find the point of infl exion on the curve ( ) .f x x x x3 4 13 2= − + −

25. Find an approximation to xdx

1

3# by using

Simpson’s rule with 3 function values (a) the trapezoidal rule with (b)

2 subintervals.

26. Find the stationary points on the curve ( )f x x x2 34 2= − + and distinguish between them.

27. Evaluate x dx5 1−1

2

# as a fraction.

28. (a) Find the area enclosed between the curve y x 12= − and the y -axis between y 1= and y 2= in the fi rst quadrant, to 3 signifi cant fi gures.

(b) This area is rotated about the y -axis. Find the exact volume of the solid formed.

29. Evaluate .y dy4

9

#

30. Find the area enclosed between the curve ( )y x 1 2= − and the line .y 4=

31. Prove Δ ABC and Δ CBD are similar.

32. Show that the curve y x2 4= has a minimum turning point at ( , ) .0 0

33. Find the maximum volume of a rectangular prism with dimensions x , 2 x and y and whose surface area is 12 m 2 .

34. If a function has a stationary point at ( , )1 2− and ( )xf ,x2 4= -m fi nd ( ) .f 2

35. Find the area enclosed between the curves y x2= and .y x x2 42= − + +

36. Find the minimum surface area of a closed cylinder with volume 100 m 3 , correct to 1 decimal place.

37. A curve has a stationary point at (3, 2) . If ( )xf ,x6 8= −m fi nd the equation of the function.

38 . Sketch the function ( ) ,f x x x x3 9 53 2= − − + showing any stationary points and infl exions.

39. Prove that the area of a rhombus is

A xy21= where x and y are the lengths of

the diagonals.

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129PRACTICE ASSESSMENT TASK 1

40.

Given BG CD< and ,GF DE< prove

.ACAB

AEAF=

41. Given ( )f x x x2 12= − −l , evaluate (2)f if ( ) .f 0 5=

42. (a) Complete the statement: The primitive function of . . .xn =

(b) Explain why there is a constant C in the primitive function.

43. The graph below is the derivative of a function.

y

x

Which graph could represent the function? There may be more than one answer.

(a)

(b) y

x

(c) y

x

y

x

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130 Maths In Focus Mathematics HSC Course

(d) y

x

44. A function has dx

dy > 0. To which of the

following graphs does this apply?

(a) y

x

(b) y

x

(c) y

x

(d) y

x

45. The area of a rectangle with sides x and y is 45. The perimeter is given by:

(a) P x x45 2= +

(b) P x x45= +

(c) 2P x x90= +

(d) 2P x x45= +

46. The area enclosed between the curve – ,y x 13= the y -axis and the lines y 1= and 2y = is given by

(a) ( )x dy11

2 3 −#

(b) ( )y dy11

2+#

(c) ( )y dy11

23 +#

(d) ( )y dy11

23 +#

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131PRACTICE ASSESSMENT TASK 1

47.

D

A

B

O

C

Given O is the centre of the circle, triangles OAB and OCD can be proven to be congruent using the test

(a) SSS (b) SAS (c) AAS (d) RHS.

48. The area bounded by the curve ,y x3= the x -axis and the lines 2x = − and 2x = is

8 units (a) 2 2 units (b) 2 0 units (c) 2 4 units (d) 2 .

49. y

x

The curve has

(a) ,dx

dy

dx

d y0 0> >

2

2

(b) ,dx

dy

dx

d y0 0> <

2

2

(c) ,dx

dy

dx

d y0 0< >

2

2

(d) ,dx

dy

dx

d y0 0< <

2

2

50. A cone with base radius r and height h has a volume of 300 cm 3 . Its slant height is given by

(a) π

πlh

h 9003

= +

(b) π

lh

h 9002

= +

(c) π

lh

h 810 0002

= +

(d) π

πlh

h 9003

= +

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