Cambridge IGCSE Additional Maths 0606: thí dụ một số bài toán khó

— Pham Hoàng Minh — Đăng ngày 27/10/2025 —
IGC
Cambridge IGCSE

Xét một số bài toán khó trong đề thi Cambridge Additional Maths 0606

Dưới đây là một số bài toán trích, tương tự hóa từ đề thi IGCSE Mathematics Additional 0606 của Cambridge. Đề bài và lời giải bằng tiếng Anh.

** Circle Theorems and Angle Proof (Cyclic Quadrilateral)**

Problem:
A triangle
is inscribed in a circle with center
. Given
, and chord
extended meets the circle again at
. Let
be the second intersection point of line
with the circle. Prove that
.
Solution:
  1.  (angle at center is twice angle at circumference).
  2.  (angles in the same segment, subtended by arc
    ).
  3. In
    ,
    .
  4. Using alternate segment theorem and cyclic quadrilateral
    , deduce
    .
Answer:
.

Optimization with Calculus (Maximum Area of Rectangle)

Problem:
A rectangle has perimeter
cm. Let
be the length and
the width.
(a) Express the area
in terms of
.
(b) Find
that maximizes
.
(c) Prove the maximum occurs when the rectangle is a square.
Solution:
(a)
.
(b)
.
(c)
maximum. When
, it is a square.

Volume of Revolution (Shell Method about y-axis)

Problem:
The curve
intersects the x-axis at points
,
, and
.
(a) Find coordinates of
,
, and
.
(b) Find the volume of the solid formed when the region bounded by the curve and the x-axis is rotated about the y-axis.
Solution:
(a) Solve
(double root at
).
,
,
.
(b) Use shell method:

Combined Transformations (Matrix Multiplication and Rotation)

Problem:
Given matrices:
(a) Describe geometrically the transformations represented by
and
.
(b) Find the matrix for: “enlarge by scale factor 3 in the
-direction, then rotate
anticlockwise.”
(c) Find the image of point
.
Solution:
(a)
: rotation
anticlockwise about origin.
: enlargement scale 2 in
-direction, scale 3 in
-direction.
(b) Apply
first, then
combined matrix:
.
(c)

** Conditional Probability (Bayes’ Theorem with Replacement)**

Problem:
Three boxes:
  • Box 1: 2 red, 3 blue
  • Box 2: 4 red, 1 blue
  • Box 3: 1 red, 4 blue
A box is chosen at random, then two balls are drawn with replacement. Given both are red, find the probability it was Box 1.
Solution:
Use Bayes’ theorem:

** Recurrence Relation (Solving Linear Recursion with Quadratic Form)**

Problem:
Sequence:
,
,
, and for
,
(a) Find
and
.
(b) Prove
.
Solution:
(a)
(b) Assume
. Substitute:

** 3D Vectors (Angle Between Two Skew Lines)**

Problem:
Two lines:
Find the angle between
and
.
Solution:
Direction vectors:
,
.

** Logarithmic Inequality (Combining Logs and Solving)**

Problem:
Solve:
Solution:
Domain:
.
Solution:
.