9000070501
Level:
B
Consider the geometric sequence with \(a_{2} = 50\)
and \(a_{3} = 25\).
Find the sum of the first four terms.
\(187.5\)
\(93.75\)
\(250\)
\(375\)
\(500\)
B
9000070808
Level:
B
Differentiate the following function.
\[
f(x)= \frac{x}
{x + 1}
\]
\(f'(x) = \frac{1}
{(x+1)^{2}} ;\ x\in \mathbb{R}\setminus \{ - 1\}\)
\(f'(x) = - \frac{1}
{(x+1)^{2}} ;\ x\in \mathbb{R}\setminus \{ - 1\}\)
\(f'(x) = \frac{x}
{(x+1)^{2}} ;\ x\in \mathbb{R}\setminus \{ - 1\}\)
\(f'(x) = - \frac{x}
{(x+1)^{2}} ;\ x\in \mathbb{R}\setminus \{ - 1\}\)
9000070809
Level:
B
Differentiate the following function.
\[
f(x)= 3x^{2}\sin x
\]
\(f'(x) = 6x\sin x + 3x^{2}\cos x;\ x\in \mathbb{R}\)
\(f'(x) = 6x\cos x;\ x\in \mathbb{R}\)
\(f'(x) = 3x^{2}\sin x\cos x;\ x\in \mathbb{R}\)
\(f'(x) = -3x^{2}\sin x\cos x;\ x\in \mathbb{R}\)
9000070502
Level:
B
Consider the geometric sequence with the first term
\(a_{1} = 243\) and quotient
\(q = \frac{1}
{3}\). Find
\(n\) such that the
sum of the first \(n\)
terms equals \(363\).
\(n = 5\)
\(n = 2\)
\(n = 3\)
\(n = 4\)
\(n = 6\)
9000065906
Level:
B
Evaluate the following integral on the interval \((-3;+\infty)\).
\[
\int \frac{x^{2} - 9}
{x + 3} \, \text{d}x
\]
\(\frac{1}
{2}x^{2} - 3x + c,\ c\in \mathbb{R}\)
\(\frac{1}
{3}x^{3} - 9x +\ln |x + 3| + c,\ c\in \mathbb{R}\)
\(2x - x^{-2} + c,\ c\in \mathbb{R}\)
\(\frac{1}
{2}x^{2} + 3x + c,\ c\in \mathbb{R}\)
9000069907
Level:
B
Find the quadratic equation with real valued coefficients and one of the solutions
\(x_{1} = -5 + \mathrm{i}\).
\(x^{2} + 10x + 26 = 0\)
\(x^{2} - 10x + 26 = 0\)
\(x^{2} - 10x - 24 = 0\)
\(x^{2} + 10x + 24 = 0\)
9000065907
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int \frac{x^{4} - 1}
{x^{2} + 1}\, \text{d}x
\]
\(\frac{1}
{3}x^{3} - x + c,\ c\in \mathbb{R}\)
\(\frac{1}
{3}x^{3} + x + c,\ c\in \mathbb{R}\)
\(\frac{1}
{5}x^{5} - x +\ln |x^{2} - 1| + c,\ c\in \mathbb{R}\)
\(3x^{2} -\ln |x^{2} - 1| + c,\ c\in \mathbb{R}\)
9000069908
Level:
B
One of the solutions of the quadratic equation
\[
2x^{2} + px + 5 = 0
\]
with a real parameter \(p\)
is
\[
x_1 = -1 + \frac{\sqrt{6}}
{2} \mathrm{i}.
\]
Find the value of \(p\).
\(4\)
\(- 4\)
\(8\)
\(- 8\)
9000066003
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int x\cos x\, \mathrm{d}x
\]
\(x\sin x +\cos x + c,\ c\in \mathbb{R}\)
\(- x\cos x +\sin x + c,\ c\in \mathbb{R}\)
\(x\cos x -\sin x + c,\ c\in \mathbb{R}\)
\(x\sin x -\cos x + c,\ c\in \mathbb{R}\)
9000069909
Level:
B
One of the solutions of the quadratic equation
\[
9x^{2} - 6x + p = 0
\]
with a real parameter \(p\)
is
\[
x_1=\frac{1}
{3} + \mathrm{i}.
\]
Find the value of \(p\).
\(10\)
\(- 10\)
\(3\)
\(- 1\)