C

1003047602

Level: 
C
Choose the step to take first to efficiently evaluate the limit of the sequence \( \left(n-\sqrt{n^2-1} \right)_{n=1}^{\infty} \).
We expand with the expression \( n+\sqrt{n^2-1} \).
We expand with the expression \( n-\sqrt{n^2-1} \).
We expand with \( n \).
We multiply by the expression \( n+\sqrt{n^2-1} \).
We multiply by the expression \( n-\sqrt{n^2-1} \).
We substitute \( n=\infty \).

1003083110

Level: 
C
The graphs of the quadratic functions \( f \) and \( g \) have not the same vertex and \( f(x)=ax^2+bx+c \), where \( a \), \( b \), \( c \) are nonzero real numbers. Find \( g(x) \) such that the graph of \( g \) is the reflection of the graph of \( f \) about \( y \)-axis.
\( g(x)=ax^2-bx+c \), i.e. the equation of \( f \) and \( g \) differ in the sign of the coefficient at the linear term only
\( g(x)=-ax^2+bx+c \), i.e the equation of \( f \) and \( g \) differ in the sign of the coefficient at the quadratic term only
\( g(x)=ax^2+bx-c \), i.e. the equation of \( f \) and \( g \) differ in the sign of the coefficient at the absolute term only
\( g(x)=-ax^2-bx-c \), i.e. \( g(x)=-f(x) \)
None of the statements above is true.

1003083108

Level: 
C
The parabolas of the functions \( f \) and \( g \) have the same vertex \( V \) and \( f(x)=ax^2+c \), where \( a \) and \( c \) are nonzero real numbers. Find \( g(x) \) such that the graphs of \( f \) and \( g \) are symmetric about the vertex \( V \) and that \( y \)-axis is their line of symmetry.
\( g(x)=-ax^2+c\), i.e. the equations of \( f \) and \( g \) differ in the sign of the coefficient at the quadratic term only
\( g(x)=ax^2-c\), i.e. the equations of \( f \) and \( g \) differ in the sign of the coefficient at the linear term only
\( g(x)=-ax^2-c \), i.e. \( g(x)=-f(x) \)
None of the statements above is true.

1003109404

Level: 
C
One of the roots of the quadratic equation \( x^2 + px + 1 - 3\,\mathrm{i} = 0 \) with a complex parameter \( p \) is \( x_1 = -\mathrm{i} \). Choose the equivalent form of the given equation.
\( (x + \mathrm{i})(x -3 - \mathrm{i}) = 0 \)
\( (x + \mathrm{i})(x - 3 +\mathrm{i}) = 0 \)
\( (x -\mathrm{i})(x- 3-\mathrm{i}) = 0 \)
\( (x +\mathrm{i})(x + 3 + \mathrm{i}) = 0 \)
\( (x -\mathrm{i})(x- 3 + \mathrm{i}) = 0 \)
\( (x -\mathrm{i})(x + 3 +\mathrm{i}) = 0 \)

1003109403

Level: 
C
One of the following equations has the solutions \( x_1=\frac12-\mathrm{i} \) and \( x_2=-\frac12+2\,\mathrm{i} \). Find this equation.
\( 4x^2-4\,\mathrm{i}\,x+7+6\,\mathrm{i}=0 \)
\( 4x^2-4\,\mathrm{i}\,x-9+3\,\mathrm{i}=0 \)
\( 4x^2+4\,\mathrm{i}\,x+7+6\,\mathrm{i}\,=0 \)
\( 4x^2+4\,\mathrm{i}\,x-9+3\,\mathrm{i}=0 \)
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