Find the coordinates of the centre of the circle \(4x^{2} + 4y^{2} - 5x + 3y - 2 = 0\).
\((\frac{-5}{4}, \frac{3}{4})\)
\((\frac{3}{8}, -\frac{5}{8})\)
\((\frac{5}{8}, -\frac{3}{8})\)
\((\frac{5}{4}, -\frac{3}{4})\)
Correct answer is C
Equation : \((x - a)^{2} + (y - b)^{2} = r^{2}\)
Expanding : \(x^{2} + y^{2} - 2ax - 2by + a^{2} + b^{2} = r^{2}\)
Given, \(4x^{2} + 4y^{2} - 5x + 3y - 2 = 0\)
Divide through by 4 to make the coefficient of \(x^{2}\) and \(y^{2}\) to be 1.
\(x^{2} + y^{2} - \frac{5}{4}x + \frac{3}{4}y - \frac{1}{2} = 0\)
Comparing, \(2a = \frac{5}{4} \implies a = \frac{5}{8}\)
\(2b = -\frac{3}{4} \implies b = -\frac{3}{8}\)
\((a, b) = (\frac{5}{8}, -\frac{5}{8})\)
-8
-5
4
-3
Correct answer is D
\(\begin{pmatrix} 1 & -3 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} -6 \\ P \end{pmatrix} = \begin{pmatrix} 3 \\ -26 \end{pmatrix}\)
\(\implies (1 \times -6) + (-3 \times P) = 3\)
\(-6 - 3P = 3 \implies -3P = 9\)
\(P = -3\)
12
16
6
4
Correct answer is D
\(F = ma\)
\(32 = 8m \implies m = 4 kg\)
Find the least value of the function \(f(x) = 3x^{2} + 18x + 32\)
5
4
-3
-2
Correct answer is A
\(f(x) = 3x^{2} + 18x + 32\)
\(\frac{\mathrm d y}{\mathrm d x} = 6x + 18 = 0\)
\(6x = -18 \implies x = -3\)
\(f(-3) = 3(-3^{2}) + 18(-3) + 32 = 27 - 54 + 32 = 5\)
Find the coordinates of the point on the curve \(y = x^{2} + 4x - 2\), where the gradient is zero.
(-2, 10)
(-2, 2)
(-2, -2)
(-2, -6)
Correct answer is D
\(y = x^{2} + 4x - 2\)
\(\frac{\mathrm d y}{\mathrm d x} = 2x + 4 = 0\)
\(2x = -4 \implies x = -2\)
\(y(-2) = (-2^{2}) + 4(-2) - 2 = 4 - 8 - 2 = -6\)
\(\therefore (x, y) = (-2, -6)\)