1003107203
Level:
C
Insert such $3$ numbers between the roots of the equation $x^2-10x-119=0$, so that together with the roots of the equation they form $5$ consecutive arithmetic sequence terms. What is the central term?
$5$
$17$
$6$
$-1$
$-7$
C
1003107202
Level:
C
The second term of an arithmetic sequence is $-21$, the fifth term is $21$. How many of the first consecutive terms do we have to add up to get a sum of more than $50$?
$8$
$7$
$6$
$9$
$5$
1003107201
Level:
C
The sum of the first ten terms of an arithmetic sequence with odd subscripts is $190$, the sum of the first ten terms with even subscripts is $230$. Find the first term.
$-17$
$-34$
$5$
$4$
$10$
1003124304
Level:
C
Given a function \( f(x)=ax^4+bx \), find real numbers \( a \) and \( b \), such that \( \int\limits_0^1f(x)\,\mathrm{d}x=27 \) and \( \int\limits_{-1}^0f(x)\,\mathrm{d}x=57 \).
\( a=210 \), \( b=-30 \)
\( a=210 \), \( b=30 \)
\( a=75 \), \( b=60 \)
\( a=30 \), \( b=210 \)
1003124308
Level:
C
Which of the values of a real number \( a\in(\pi;2\pi) \) makes the equality \( \int\limits_{\pi}^{a+\pi} \sin2x\,\mathrm{d}x=1 \) true?
\( a=\frac32\pi \)
\( a=\frac34\pi \)
\( a=\frac12\pi \)
\( a=\frac23\pi \)
1003124307
Level:
C
Which of the positive values of a real number \(a\) makes the equality \(\int\limits_a^7 \frac{6x-5}{3x^2-5x}\,\mathrm{d}x=\ln 56 \) true?
\( a=2 \)
\( a=1 \)
\( a=3 \)
\( a=5 \)
1003124306
Level:
C
Which of the positive values of a real number \(a\) makes the equality \( \int\limits_1^a \frac3{3x+5}\,\mathrm{d}x=\ln\frac74 \) true?
\( a = 3 \)
\( a = 2 \)
\( a = 4 \)
\( a = 5 \)
1003124305
Level:
C
Given a function \( f(x)=ax^6+bx^3+cx+8 \), find real numbers \( a \), \( b \) and \( c \), such that \( \int\limits_0^1f(x)\,\mathrm{d}x=\frac{35}4 \), \( f'(0)=2 \) and \( f'(1)=180 \).
\( a=7 \), \( b=-5 \), \( c=2 \)
\( a=7 \), \( b=5 \), \( c=2 \)
\( a=-7 \), \( b=-5 \), \( c=2 \)
\( a=-7 \), \( b=5 \), \( c=-2 \)
1003124303
Level:
C
Which of the given values of real numbers \( a \), \( b\in\left(0;\frac{\pi}2\right) \), such as \( a < b \), makes the equality \( \int\limits_a^b \cos x\,\mathrm{d}x=2\cos\frac{\pi}4\cdot\sin\frac{\pi}{12} \) true?
\( a=\frac{\pi}6 \), \( b=\frac{\pi}3 \)
\( a=\frac{\pi}3 \), \( b=\frac{\pi}6 \)
\( a=\frac{\pi}3 \), \( b=\frac{\pi}4 \)
\( a=\frac{\pi}4 \), \( b=\frac{\pi}3 \)
1003124302
Level:
C
Which of the given values of a real number \( a\in\left(\frac{\pi}2;\pi\right) \) makes the equality \( \int\limits_a^{2a} (3\sin x-4x)\,\mathrm{d}x=-6a^2 \) true?
\( a =\frac23\pi \)
\( a=\frac56\pi \)
\( a=\frac34\pi \)
\( a=\frac78\pi \)