What is the coordinate of the centre of the circle \(5x^{2} + 5y^{2} - 15x + 25y - 3 = 0\)?
\((\frac{15}{2}, -\frac{25}{2})\)
\((\frac{3}{2}, -\frac{5}{2})\)
\((-\frac{3}{2}, \frac{5}{2})\)
\((-\frac{15}{2}, \frac{25}{2})\)
Correct answer is B
Equation for a circle: \((x - a)^{2} + (y - b)^{2} = r^{2}\)
Expanding, we have:
\(x^{2} - 2ax + a^{2} + y^{2} - 2by + b^{2} = r^{2}\)
Given: \(5x^{2} + 5y^{2} - 15x + 25y - 3 = 0\)
Divide through by 5,
\(= x^{2} + y^{2} - 3x + 5y - \frac{3}{5} = 0\)
Comparing, we have
\(- 2a = -3; a = \frac{3}{2}\)
\(-2b = 5; b = -\frac{5}{2}\)
Which of the following quadratic curves will not intersect with the x- axis?
\(y = 2 - 4x - x^{2}\)
\(y = x^{2} - 5x -1\)
\(y = 2x^{2} - x - 1\)
\(y = 3x^{2} - 2x + 4\)
Correct answer is D
The criterion for the quadratic curve to intersect the x- axis is \(b^{2} > 4ac\).
If \(2\log_{4} 2 = x + 1\), find the value of x.
-2
-1
0
1
Correct answer is C
\(2\log_{4} 2 = x + 1\)
\(\log_{4} 2^{2} = \log_{4} 4 = 1\)
\(x + 1 = 1 \implies x = 0\)
8
6
-6
-8
Correct answer is B
Remainder for f(2) = 20.
\(f(2) = 2(2^{3}) + 2^{2} - 3(2) + p = 20\)
\(16 + 4 - 6 + p = 20\)
\(14 + p = 20\)
\(p = 6\)
If \(f(x) = 2x^{2} - 3x - 1\), find the value of x for which f(x) is minimum.
\(\frac{3}{2}\)
\(\frac{4}{3}\)
\(\frac{3}{4}\)
\(\frac{2}{3}\)
Correct answer is C
\(y = 2x^{2} - 3x - 1\)
\(\frac{\mathrm d y}{\mathrm d x} = 4x - 3 = 0\) (At turning point)
\(4x = 3 \implies x = \frac{3}{4}\)