TMUA 2016 · Paper 2 · interactive paper

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University Admissions Tests UK (Pearson VUE) · United Kingdom

TMUA 2016 Paper 2

2016

1h15

20 questions. Each part is annotated: check the hint if you are stuck, or reveal the worked solution once you have committed to an approach.

Integration
Differentiation
Trigonometry
Logic & Proof
Proof & Number Theory
Sequences & Series
Exponentials & Logarithms
Algebra & Inequalities
Logic & Proof
Logic & Proof
Functions & Graphs
Sequences & Series
Proof & Number Theory
Functions & Graphs
Algebra & Inequalities
Coordinate Geometry & Circles
Trigonometry
Logic & Proof
Logic & Proof
Proof & Number Theory

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This is the full TMUA 2016 Paper 2 sitting. Open any question to check its hint or reveal the worked solution.

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Algebra & Inequalities2
Coordinate Geometry & Circles1
Differentiation1
Exponentials & Logarithms1
Functions & Graphs2
Integration1
Logic & Proof5
Proof & Number Theory3
Sequences & Series2
Trigonometry2

Statements are open to everyone. Hints and worked solutions are reserved for registered accounts.

Find the value of

\int_{1}^{2} (x^{2} - \frac{4}{x ^{2}})^{2} d x

A $\frac{43}{15}$
B $3$
C $\frac{97}{15}$
D $\frac{103}{15}$

f (x) = \frac{( x ^{2} + 5 ) ( 2 x )}{4 x ^{3}}, x > 0

Which one of the following is equal to

f^{'} (x)

?

A $8 x^{\frac{9}{4}} + \frac{40}{3} x^{\frac{1}{4}}$

What is the value, in radians, of the largest angle

x

in the range

0 \leq x \leq 2 π

that satisfies the equation

8 sin^{2} x + 4 cos^{2} x = 7

?

A $\frac{2 π}{3}$
B $\frac{5 π}{6}$
C $\frac{4 π}{3}$
D $\frac{5 π}{3}$
E $\frac{7 π}{4}$
F $\frac{11 π}{6}$

Five sealed urns, labelled P, Q, R, S, and T, each contain the same (non-zero) number of balls. The following statements are attached to the urns.

Urn P: This urn contains one or four balls. Urn Q: This urn contains two or four balls. Urn R: This urn contains more than two balls and fewer than five balls. Urn S: This urn contains one or two balls. Urn T: This urn contains fewer than three balls.

Exactly one of the urns has a true statement attached to it.

Which urn is it?

AUrn P
BUrn Q
CUrn R
DUrn S
EUrn T

Consider the statement:

(*)

A whole number

n

is prime if it is 1 less or 5 less than a multiple of 6.

How many counterexamples to

(*)

are there in the range

0 < n < 50

?

A2
B3
C4
D5
E6

The sequence of functions

f_{1} (x), f_{2} (x), f_{3} (x), \dots

is defined as follows:

f_{1} (x) = x^{10}

f_{n + 1} (x) = x f_{n}^{'} (x) for n \geq 1

where

f_{n}^{'} (x) = \frac{d f _{n} ( x )}{d x}

.

Find the value of

n = 1 \sum 20 f_{n} (x)

A $\frac{x ^{10} ( x ^{20} - 1 )}{x - 1}$
B $\frac{x ^{10} ( x ^{21} - 1 )}{x - 1}$

The four real numbers

a, b, c,

and

d

are all greater than 1.

Suppose that they satisfy the equation

lo g_{c} d = (lo g_{a} b)^{2}

.

Use some of the lines given to construct a proof that, in this case, it follows that

(*) lo g_{b} d = (lo g_{a} b) (lo g_{a} c) .

The available lines are:
- (1) Let

x = lo g_{a} b

and

y = lo g_{a} c

- (2)

d = (c^{x})^{2}

- (3)

d = c^{(x^{2})}

- (4)

d = b^{x y}

- (5)

d = (a^{y})^{(x^{2})}

- (6)

d = ((a^{y})^{x})^{2}

- (7)

d = (a^{x})^{x y}

- (8)

d = a^{(y^{2 x})}

- (9)

d = a^{(x^{2} y)}

Which sequence of lines gives a correct proof?

A(1). Then (2), so (6), so (8), so (7), and therefore (4), hence (*) as required.
B(1). Then (2), so (7), so (8), so (6), and therefore (4), hence (*) as required.
C(1). Then (3), so (5), so (9), so (7), and therefore (4), hence (*) as required.
D(1). Then (3), so (7), so (9), so (5), and therefore (4), hence (*) as required.
E(1). Then (4), so (5), so (9), so (7), and therefore (3), hence (*) as required.
F(1). Then (4), so (6), so (8), so (7), and therefore (2), hence (*) as required.
G(1). Then (4), so (7), so (8), so (6), and therefore (2), hence (*) as required.
H(1). Then (4), so (7), so (9), so (5), and therefore (3), hence (*) as required.

A region is defined by the inequalities

x + y > 6

and

x - y > - 4

.

Consider the three statements:
1.

x > 1

2.

y > 5

3.

(x + y) (x - y) > - 24

Which of the above statements is/are true for every point in the region?

Anone
B1 only
C2 only
D3 only
E1 and 2 only
F1 and 3 only
G2 and 3 only
H1, 2 and 3

Triangles

A B C

and

X Y Z

have the same area.

Which of these extra conditions, taken independently, would imply that they are congruent?

(1)

A B = X Y

and

B C = Y Z

(2)

A B = X Y

and

∠ A B C = ∠ X Y Z

(3)

∠ A B C = ∠ X Y Z

and

∠ B C A = ∠ Y Z X

ADoes not imply congruent / Does not imply congruent / Does not imply congruent
BDoes not imply congruent / Does not imply congruent / Implies congruent
CDoes not imply congruent / Implies congruent / Does not imply congruent
DDoes not imply congruent / Implies congruent / Implies congruent
EImplies congruent / Does not imply congruent / Does not imply congruent
FImplies congruent / Does not imply congruent / Implies congruent
GImplies congruent / Implies congruent / Does not imply congruent
HImplies congruent / Implies congruent / Implies congruent

In this question

x

and

y

are non-zero real numbers.

Which one of the following is sufficient to conclude that

x < y

?

A $x^{4} < y^{4}$
B $y^{4} < x^{4}$
C $x^{- 1} < y^{- 1}$
D $y^{- 1} < x^{- 1}$
E $x^{\frac{3}{5}} < y^{\frac{3}{5}}$
F $y^{\frac{3}{5}} < x^{\frac{3}{5}}$

f (x)

is a polynomial with real coefficients.

The equation

f (x) = 0

has exactly two real roots,

x = - p

and

x = p

, where

p > 0

.

Consider the following three statements:
1.

f^{'} (x) = 0

for exactly one value of

x

between

- p

and

x^{4} < y^{4}

2. The area between the curve

y = f (x)

, the

x

-axis and the lines

x = - p

and

x = p

is given by

2 \int_{0}^{p} f (x) d x

3. The graph of

y = - f (- x)

intersects the

x

-axis at the points

x = - p

and

x = p

only

Which of the above statements must be true?

Anone
B1 only
C2 only
D3 only
E1 and 2 only
F1 and 3 only
G2 and 3 only
H1, 2 and 3

The first term of an arithmetic sequence is

a

and the common difference is

d

.

The sum of the first

n

terms is denoted by

S_{n}

.

If

S_{8} > 3 S_{6}

, what can be deduced about the sign of

a

and the sign of

d

?

Aboth $a$ and $d$ are negative
B $a$ is positive, $d$ is negative
C $a$ is negative, $d$ is positive
D $a$ is negative, but the sign of $d$ cannot be deduced
E $d$ is negative, but the sign of $a$ cannot be deduced
Fneither the sign of $a$ nor the sign of $d$ can be deduced

In this question,

a

,

b

, and

c

are positive integers.

The following is an attempted proof of the false statement:

"If

a

divides

p

, then

a

divides

b

or

a

divides

c

."

[

a

divides

b c

means

a

is a factor of

b c

.]

Which line contains the error in this proof?

1. The statement is equivalent to: if

a

does not divide

b

and

a

does not divide

c

, then

a

does not divide

y = - f (- x)

.
2. Suppose

a

does not divide

b

and

x

does not divide

x = - p

. Then the remainder when dividing

b

by

a

is

r

, where

0 < r < a

, and the remainder when dividing

c

by

a

is

s

, where

0 < s < a

.
3. So

b = a x + r

and

c = a y + s

for some integers

x

and

y

.
4. Thus

b c = a (a x y + x s + y r) + r s

.
5. So the remainder when dividing

b c

by

a

is

r s

.
6. Since

r > 0

and

s > 0

, it follows that

r s > 0

.
7. Hence

a

does not divide

b c

.

ALine 1
BLine 2
CLine 3
DLine 4
ELine 5
FLine 6

f (x) = a x^{4} + b x^{3} + c x^{2} + d x + e

, where

c

and

e

are real numbers.

Suppose

f (x) = 1

has

p

distinct real solutions,

f (x) = 2

has

q

distinct real solutions,

f (x) = 3

has

s

distinct real solutions, and

0 < s < a

has

s

distinct real solutions.

Which one of the following is not possible?

A $p = 1, q = 2, r = 4$ and $s = 3$
B $p = 1, q = 3, r = 2$ and $s = 4$

Consider the quadratic

f (x) = x^{2} - 2 p x + q

and the statement:

(*)

a

has two real roots whose difference is greater than 2 and less than 4.

Which one of the following statements is true if and only if

(*)

is true?

A $q < p^{2} < q + 4$
B $q + 1 < p < q + 4$
C $q - 3 \leq p^{2} - 4 \leq q$
D $q < p^{2} - 1 < q + 3$
E $q - 2 < p^{2} - 3 < q + 2$

In the figure,

P QR S

is a trapezium with

P Q

parallel to

S R

.

The diagonals of the trapezium meet at

X

.

U

lies on

S P

and

T

lies on

R Q

such that

U T

is a line segment through

X

parallel to

P Q

.

The length of

P Q

is 12 cm and the length of

S R

is 3 cm.

What, in centimetres, is the length of

U T

?

Trapezium PQRS with PQ at the bottom (length 12 cm, horizontal) and SR at the top (length 3 cm, horizontal). The vertices are: P bottom-left, Q bottom-right, R top-right, S top-left. Both diagonals PR and SQ are drawn and intersect at point X. A horizontal line segment UT passes through X parallel to PQ, with U on side SP and T on side RQ. Arrows indicate PQ = 12 cm and SR = 3 cm. The label [diagram not to scale] appears to the right.

A4.2
B4.5
C4.8
D5.25
E6

Consider these simultaneous equations, where

c

is a constant:

y = 3 sin x + 2

y = x + c

Which of the following statements is/are true?

1. For some value of

c

: there is exactly one solution with

0 \leq x \leq π

and there is at least one solution with

- π < x < 0

.
2. For some value of

c

: there is exactly one solution with

0 \leq x \leq π

and there are no solutions with

- π < x < 0

.
3. For some value of

c

: there is exactly one solution with

0 \leq x \leq π

and there are no solutions with

x > π

.

Anone
B1 only
C2 only
D3 only
E1 and 2 only
F1 and 3 only
G2 and 3 only
H1, 2 and 3

Consider this statement about a function

f (x)

:

(*)

If

(f (x))^{2} \leq 1

for all

P Q

, then

\int_{- 1}^{1} (f (x))^{2} d x \leq \int_{- 1}^{1} f (x) d x

.

Which one of the following functions provides a counterexample to

(*)

?

A $f (x) = x + \frac{1}{2}$
B $f (x) = x - \frac{1}{2}$

Some identical unit cubes are used to construct a three-dimensional object by gluing them together face to face.

Sketches of this object are made by looking at it from the right-hand side, from the front and from above. These sketches are called the side elevation, the front elevation, and the plan view respectively.

The side elevation shows a 2-wide by 3-tall grid (top-right cell occupied, then 2 wide rows below). The front elevation shows a 3-wide by 2-tall grid (top-right cell occupied, then a full 3-wide bottom row). The plan view shows an L-shaped 3-wide by 2-tall grid with the top-right cell absent (2 cells in top row, 3 cells in bottom row).

How many cubes were used to construct the object?

Three grid diagrams showing the three views of a 3D object made from unit cubes. Side elevation (view from right): a 2-column by 3-row grid where the top row has only the right cell filled, the middle row has both cells filled, and the bottom row has both cells filled. Front elevation (view from front): a 3-column by 2-row grid where the top row has only the right cell filled, and the bottom row has all three cells filled. Plan view (view from above): an L-shaped 3-column by 2-row grid where the top row has 2 cells filled (left and middle, but not right), and the bottom row has all 3 cells filled.

Aexactly 6
Beither 6 or 7
Cexactly 7
Deither 7 or 8
Eexactly 8
Feither 8 or 9
Gexactly 9

Each interior angle of a regular polygon with

n

sides is

\frac{3}{4}

of each interior angle of a second regular polygon with

m

sides.

How many pairs of positive integers

n

and

m

are there for which this statement is true?

Anone
B1
C2
D3
E4
F5
G6
Hinfinitely many

End of paper.

← 2016 · Paper 1 All TMUA sittings 2017 · Paper 1 →

E

\frac{163}{15}

F

18

B

\frac{9}{2} x^{\frac{5}{4}} + \frac{5}{2} x^{- \frac{3}{4}}

C

8 x^{\frac{9}{4}} + \frac{40}{3} x^{- \frac{1}{4}}

D

\frac{8}{13} x^{\frac{13}{4}} + 8 x^{\frac{5}{4}}

C

(\frac{1 0 ^{20} - 1}{9}) x^{10}

D

(\frac{1 0 ^{21} - 1}{9}) x^{10}

E

(\frac{( 10 x ) ^{20} - 1}{10 x - 1}) x^{10}

F

(\frac{( 10 x ) ^{21} - 1}{10 x - 1}) x^{10}

G

x^{10} + x^{9} + x^{8} + \dots + x + 1

H

x^{10} + 10 x^{9} + (10 \times 9) x^{8} + \dots + (10 \times 9 \times \dots \times 2) x + (10 \times 9 \times \dots \times 2 \times 1)

C

p = 1, q = 4, r = 3

and

s = 2

D

p = 2, q = 4, r = 3

and

s = 1

E

p = 4, q = 3, r = 2

and

s = 1

C

f (x) = x + x^{3}

D

f (x) = x - x^{3}

E

f (x) = x^{2} + x^{4}

F

f (x) = x^{2} - x^{4}