Find the equation of the tangent to the curve \(y = 4x^{2} - 12x + 7\) at point (2, -1).
y + 4x - 9 = 0
y - 4x - 9 = 0
y - 4x + 9 = 0
y + 4x + 9 = 0
Correct answer is C
\(y = 4x^{2} - 12x + 7\)
\(\frac{\mathrm d y}{\mathrm d x} = 8x - 12\)
At x = 2, y = 8(2) - 12 = 4
Equation of the tangent to the curve: \(y - (-1) = 4(x - 2)\)
\(y + 1 = 4x - 8 \implies y - 4x + 1 + 8 = y - 4x + 9 = 0\)
Find the axis of symmetry of the curve \(y = x^{2} - 4x - 12\).
x = -2
y = -2
x = 2
y = 2
Correct answer is C
The vertical line \(x = \frac{-b}{2a}\) is the axis of symmetry of the curve.
\(y = x^{2} - 4x - 12\)
\(\text{Axis of symmetry} = x = \frac{-(-4)}{2(1)} = \frac{4}{2} = 2\)
The third of geometric progression (G.P) is 10 and the sixth term is 80. Find the common ratio.
2
3
4
8
Correct answer is A
\(T_{n} = ar^{n - 1}\) ( Geometric Progression)
\(T_{3} = ar^{3 - 1} = ar^{2} = 10 .... (1)\)
\(T_{6} = ar^{6 - 1} = ar^{5} = 80 .....(2)\)
Divide (2) by (1)
\(r^{5 - 2} = r^{3} = 8 \)
\(r = \sqrt[3]{8} = 2\)
{-1, 5, 13}
{5, 13, 49}
{1, 2, 3, 6}
{-1, 5, 13, 49}
Correct answer is D
\(P = {x : \text{x is a factor of 6}} \implies P = {1, 2, 3, 6}\)
\(g(x) = x^{2} + 3x - 5\)
\(g(1) = 1^{2} + 3(1) - 5 = 1 + 3 - 5 = -1\)
\(g(2) = 2^{2} + 3(2) - 5 = 4 + 6 - 5 = 5\)
\(g(3) = 3^{2} + 3(3) - 5 = 9 + 9 - 5 = 13\)
\(g(6) = 6^{2} + 3(6) - 5 = 36 + 18 - 5 = 49\)
\(\therefore Range(g(x)) = {-1, 5, 13, 49}\)
If \(\begin{vmatrix} 3 & x \\ 2 & x - 2 \end{vmatrix} = -2\), find the value of x.
-8
4
-4
8
Correct answer is A
\(\begin{vmatrix} 3 & x \\ 2 & x - 2 \end{vmatrix} = 3(x - 2) - 2x = -2\)
\(3x - 6 - 2x = -2 \implies x - 6 = -2\)
\(x = -2 + 6 = 4\)