Algebra and functions：練習問題

Simplify the expression \( \frac{(x^2 y^3)^4}{x^5 y^2} \) using the laws of indices.

The curve \( C \) has the equation \( y = x^2 - 4x + 7 \). By completing the square, find the coordinates of the turning point of the curve.

Find the discriminant of the quadratic function \( f(x) = 3x^2 - 5x + 2 \) and determine the nature of its roots.

Solve the following simultaneous equations:
\( y = x + 2 \)
\( x^2 + y^2 = 10 \)

Given that \( x = 1 \) is a root of the cubic equation \( f(x) = x^3 + kx^2 - 4x - 12 = 0 \), find the value of the constant \( k \).

Find the value of the constant \( k \) for which the quadratic equation \( 3x^2 - 6x + k = 0 \) has exactly one real root.

(a) Express \(\frac{12}{3 - \sqrt{3}}\) in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers.
(b) Given that \(x = \frac{12}{3 - \sqrt{3}}\), show that the value of \(x^2 - 12x\) is an integer and state its value.

A curve has the equation \( y = f(x) \), where \( f(x) = x^2 - (k+3)x + (3k+1) \) and \( k \) is a constant.
(a) Given that the equation \( f(x) = 0 \) has equal roots, find the possible values of \( k \).
(b) For one of these values of \( k \), the curve has a minimum point at \( (x, y) \). Use the method of completing the square to find the coordinates of this minimum point.
(c) Sketch the graph of \( y = f(x) \) for the larger value of \( k \) found in part (a), showing clearly the coordinates of any points where the curve touches or crosses the coordinate axes.