1003076603
Level:
B
How many points of intersection has the graph of the function \( f(x)=- 2\sin2x \) with the \( x \)-axis on the interval \( [-2\pi; 2\pi] \)?
\( 9 \)
\( 8 \)
\( 10 \)
\( 11 \)
B
1003076602
Level:
B
How many \( x \)-intercepts has the graph of the function \( f(x)=\sin 3x \) on the interval \( [-\pi; 3\pi] \)?
\( 13 \)
\( 10 \)
\( 14 \)
\( 8 \)
1003076601
Level:
B
How many \( x \)-intercepts has the graph of the function \( f(x)=\cos 2x \) on the interval \( [-\pi; 2\pi] \)?
\( 6 \)
\( 4 \)
\( 5 \)
\( 3 \)
1003107309
Level:
B
We are given the sequence \( \left(\log2^n \right)^{\infty}_{n=1} \).
Find the recursive formula of such sequence.
\( a_1=\log 2\,;\ a_{n+1}=a_n+\log 2,\ n\in\mathbb{N} \)
\( a_1=\log 2\,;\ a_{n+1}=a_n\cdot\log 2,\ n\in\mathbb{N} \)
\( a_1=\log 2\,;\ a_{n+1}=a_n-\log 2,\ n\in\mathbb{N} \)
\( a_1=\log 2\,;\ a_{n+1}=a_n+\log 2^n,\ n\in\mathbb{N} \)
1003107307
Level:
B
We are given a sequence \( \left( a_n \right)^{\infty}_{n=1} \) defined recursively by: \( a_1=1,\ a_2=5\) and \(\ a_{n+2}=a_{n+1}-a_n+d\), where \(\ n\in\mathbb{N} \).
Find the value of an unknown constant \( d\in\mathbb{R} \) and of the term \( a_5 \) if \( a_3 = 10 \).
\( d=6,\ a_5=7 \)
\( d=6,\ a_5=6 \)
\( d=7,\ a_5=6 \)
\( d=7,\ a_5=7 \)
1003107306
Level:
B
We are given a sequence \( \left( a_n \right)^{\infty}_{n=1} \) defined recursively by: \( a_1=1\,;\ a_{n+1}=2a_n,\ n\in\mathbb{N} \).
Find the \( n \)th term of this sequence.
\( a_n=2^{n-1},\ n\in\mathbb{N} \)
\( a_n=2^n,\ n\in\mathbb{N} \)
\( a_n=2^{n+1},\ n\in\mathbb{N} \)
\( a_n=2^n-1,\ n\in\mathbb{N} \)
1003107305
Level:
B
We are given a sequence \( \left( a_n \right)^{\infty}_{n=1} \) defined recursively by: \( a_1=5\,;\ a_{n+1}=a_n+4,\ n\in\mathbb{N} \).
Find the \( n \)th term of this sequence.
\( a_n=4n+1,\ n\in\mathbb{N} \)
\( a_n=4n-1,\ n\in\mathbb{N} \)
\( a_n=4n,\ n\in\mathbb{N} \)
\( a_n=5n,\ n\in\mathbb{N} \)
1003107304
Level:
B
We are given a sequence \( \left( a_n \right)^{\infty}_{n=1} \) defined recursively by: \(a_1=0\,;\ a_{n+1}=2-a_n,\ n\in\mathbb{N} \).
Find the \( n \)th term of this sequence.
\( a_n=1+(-1)^n,\ n\in\mathbb{N} \)
\( a_n=1+(-1)^{n+1},\ n\in\mathbb{N} \)
\( a_n=1+(-1)^{n-1},\ n\in\mathbb{N} \)
\( a_n=1-1^n,\ n\in\mathbb{N} \)
1003107303
Level:
B
We are given the sequence \( \left( 2^{2-n} \right)^{\infty}_{n=1} \). Find the recursive formula of such sequence.
\( a_1=2\,;\ a_{n+1}=a_n\cdot\frac12,\ n\in\mathbb{N} \)
\( a_1=2\,;\ a_{n+1}=a_n\cdot2,\ n\in\mathbb{N} \)
\( a_1=2\,;\ a_{n+1}=a_n\cdot2^n,\ n\in\mathbb{N} \)
\( a_1=2\,;\ a_{n+1}=a_n\cdot\frac1{2^n},\ n\in\mathbb{N} \)
1003107302
Level:
B
We are given the sequence \( \left( \frac{\sqrt2}4\left( \sqrt2 -1 \right)^n \right)^{\infty}_{n=1} \). Find the recursive formula of such sequence.
\( a_1=\frac{\sqrt2}4\left(\sqrt2-1\right);\ a_{n+1}=a_n\left(\sqrt2-1\right),\ n\in\mathbb{N} \)
\( a_1=\frac{\sqrt2}4;\ a_{n+1}=a_n\left(\sqrt2-1\right),\ n\in\mathbb{N} \)
\( a_1=\sqrt2-1\,;\ a_{n+1}=a_n\left(\sqrt2-1\right),\ n\in\mathbb{N} \)
\( a_1=\frac{\sqrt2}4\left(\sqrt2-1\right);\ a_{n+1}=a_n\left(\sqrt2-1\right)^2,\ n\in\mathbb{N} \)