MAT01A2
PAPER A
2023
Question 1 [5 marks]
For questions 1.1 – 1.5, choose one correct answer, and make a cross (X) in the correct block.
Question
1.1
1.2
1.3
1.4
1.5
a
b
c
d
e
2n
1.1 Determine whether the sequence defined by an = e n+1 converges or diverges. Pick the
correct answer below.
(1)
(a)
The sequence converges to 2
(b)
The sequence converges to 1
(c)
The sequence converges to e2
(d)
The sequence diverges
(e)
None of the above
1.2
Consider the series and select the correct option below.
8−1+
1
1
−
+ ···
8 64
(d)
64
65
64
The series converges to
9
65
The series converges to
9
The series diverges
(e)
None of the above
1.3
∞
X
1
n
Consider the series
(−1)
. Pick the correct option below.
2n
n=1
(a)
The series diverges
(b)
The series is absolutely convergent but not convergent
(c)
The series is absolutely convergent
(d)
The series is conditionally convergent
(e)
None of the above
(a)
(b)
(c)
(1)
The series converges to
1/8
(1)
MAT01A2
1.4
PAPER A
2023
Determine the limit below.
(1)
lim h8 sin t,
t→0+
24 cos t,
t ln ti
(a)
8i - k
(b)
k
(c)
24j
(d)
8i
(e)
None of the above.
1.5
Determine the set of real numbers for which the function
√
1
2
, t−4
r(t) =
16 − t ,
t+6
is defined.
(a)
(−∞, 4)
(b)
[0, 4]
(c)
(−∞, ∞)
(d)
[−4, 4]
(e)
None of the above
(1)
2/8
MAT01A2
PAPER A
2023
Question 2 [6 marks]
(a) State the Remainder Estimate for the Integral Test.
(3)
(b) Determine, with full reasons given, whether the series below converges or diverges.
(3)
∞
X
n=1
n3
1
+5
3/8
MAT01A2
PAPER A
2023
Question 3 [6 marks]
(a) Determine whether the series below converges or diverges. Justify all the steps in your
reasoning. If you use a test to support your reasoning, you must show, in detail, that the series
satisfies the conditions of the test.
(4)
∞
X
(−1)n
n=1
n2
n3 + 1
(b) Prove that if a series is absolutely convergent then it is convergent.
4/8
(2)
MAT01A2
PAPER A
2023
Question 4 [9 marks]
(a) Find the radius of convergence and the interval of convergence of the series below.
∞ √
X
n
n=1
8n
(5)
(x + 6)n
(b) Find a power series representation for the function below and determine its radius of convergence.
(4)
f (x) =
x
3−x
5/8
3
MAT01A2
PAPER A
2023
Question 5 [5 marks]
Find the Maclaurin series of f (x) = ln(1 + x) using the definition of Maclaurin series. Assume
that f has a power series expansion. Do not show that Rn (x) → 0. Also find the associated radius
of convergence.
6/8
MAT01A2
PAPER A
2023
Question 6 [7 marks]
(a) Find a vector equation and parametric equations for the line segment that joins P (1, 0, 1) to
Q(2, 3, 1).
(3)
(b) Suppose that u and v are differentiable vector functions. Prove that
d
u(t) · v(t) = u0 (t) · v(t) + u(t) · v0 (t).
dt
7/8
(4)
MAT01A2
PAPER A
2023
Question 7 [7 marks]
(a) Find the length of the curve
r(t) = ht, 3 cos t, 3 sin ti
−5 6 t 6 5.
(3)
(b) Find the curvature of the curve given by
r(t) = ht, 2t2 , 2ti.
(4)
8/8