TMUA Pure Mathematics 5

Question 1

What is the derivative of y = e^(2x) · sin(x)?

Accepted Answer

e^(2x)(2sin(x) + cos(x))

Answer

By the product rule: dy/dx = 2e^(2x)sin(x) + e^(2x)cos(x) = e^(2x)(2sin(x) + cos(x)).

Question 2

Find the equation of the circle with centre (3, −2) and radius 5.

Accepted Answer

(x−3)² + (y+2)² = 25

Answer

Standard form (x − h)² + (y − k)² = r² with (h,k) = (3,−2) and r = 5 gives (x−3)² + (y+2)² = 25.

Question 3

Rationalise the denominator of 1/(3 − √2).

Accepted Answer

(3 + √2)/7

Answer

Multiply numerator and denominator by (3 + √2): (3 + √2)/(9 − 2) = (3 + √2)/7.

Question 4

If log₃(x) = 4, what is x?

Accepted Answer

81

Answer

log₃(x) = 4 means x = 3⁴ = 81.

Question 5

A function satisfies f(x) = f(−x) for all x. What type of symmetry does its graph have?

Accepted Answer

Symmetry about the y-axis

Answer

f(x) = f(−x) defines an even function, which is symmetric about the y-axis.

(TMUA) Test of Mathematics for University Admission Practice Test