Algebraic expressions and formulae：練習問題

Simplify the algebraic expression:
\( 4(x - 3) - (2x + 5) \)

Express \( \frac{x+1}{x^2-4} + \frac{2}{x-2} \) as a single fraction in its simplest form.

Find the value of \( k \) such that \( 4x^2 + 20x + k \) is a perfect square.

Factorise the expression completely: \( 4ax + 12ay - bx - 3by \).

Factorise completely the expression \( 6ax - 2ay - 3bx + by \).

Given that \( a = 3 \) and \( b = -2 \), evaluate the expression \( 2a^2 - 5b \).

A sequence is given by \( 4, 7, 10, 13, \dots \). Find an algebraic expression for the \( n \)th term of this sequence.

Simplify the expression \( \frac{4a^2 - 9b^2}{2a^2 + ab - 3b^2} \) by factorising both the numerator and denominator.

Consider the algebraic expression \( P = 4x^2 - 25 + (2x - 5)(3x + 4) \).

(a) Factorise \( 4x^2 - 25 \) completely.
(b) Using your answer from part (a), factorise \( P \) completely.
(c) Hence, find the value of \( P \) when \( x = 10.5 \).

The formula for the total surface area, \( A \), of a solid cylinder with a hemispherical cap on one end is given by \( A = 2\pi r^2 + 2\pi rh + \pi r^2 \), where \( r \) is the radius and \( h \) is the height of the cylindrical part.

(a) Simplify the expression for \( A \).
(b) Rearrange the formula to make \( h \) the subject.
(c) Given that \( A = 120\pi \) and \( r = 4 \), find the value of \( h \).
(d) If the radius \( r \) is doubled while \( h \) remains constant, express the new surface area in terms of the original \( r \) and \( h \).