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)
\(4x = 3 \implies x = \frac{3}{4}\)
n + 1
2n + 1
3n + 1
4n + 1
Correct answer is B
\(S_{n} = \frac{n}{2}(2a + (n - 1) d = n^{2} + 2n\)
\(n(2a + (n - 1) d = 2n^{2} + 4n\)
\(2an + n^{2}d - nd = 2n^{2} + 4n\)
\(n^{2}d = 2n^{2}\)
\(d = 2\)
\((2a - d) n = 4n\)
\(2a - d = 4 \implies 2a = 4 + d = 4 + 2 = 6\)
\(a = 3\)
\(T_{n} = a + (n - 1)d\)
= \(3 + (n - 1)2 = 3 + 2n - 2 = 2n + 1\)