Quadratic functions

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.

1003108307

Level: 
C
Choose the triple of points, such that the graph of any of the functions \( f(x)=ax^2+c \), where \( a\in\mathbb{R}\setminus{0} \), \( c\in\mathbb{R} \), does not pass through all three points.
\( [-2;5] \), \( [2;1] \), \( [0;3] \)
\( [-2;5] \), \( [2;5] \), \( [0;3] \)
\( [-2;5] \), \( [2;5] \), \( [0;7] \)
\( [-2;5] \), \( [0;0] \), \( [1;1] \)

1003124801

Level: 
C
Suppose we want to paint a cube so that there remains an unpainted stripe along all the edges on each face. The width of the stripe should be \( 1\,\mathrm{cm} \). The producer gives the paint consumption \( 100\,\mathrm{ml}/1\,\mathrm{m}^2 \). From the following functions choose the one that describes the dependence of the paint consumption \( V \) on the length of the cube edge \( a \). The paint consumption \( V \) is given in millilitres and the length of the cube edge \( a \) is given in meters.
\( V=\left(a-\frac1{50}\right)^2\cdot600 \)
\( V=\left(a-\frac1{50}\right)^2\cdot\frac3{50} \)
\( V=\left(a-\frac1{100}\right)^2\cdot600 \)
\( V=(a-2)^2\cdot100 \)

1003124802

Level: 
C
We want to plant flowers into rectangular flower bed with longer side by one meter longer than its shorter side. Each flower needs \( 1\,\mathrm{dm}^2 \) of free space. From the following functions, choose the one that describes the dependence of the number of planted flowers \( n \) on the length \( a \) of the shorter side of the flower bed. (Assume that the dimensions of the flower bed are given in whole meters.)
\( n=\left(a^2+a\right)\cdot100 \)
\( n=\left(a^2+a\right)\cdot\frac1{100} \)
\( n=(a+1)^2\cdot100 \)
\( n=\left(a^2+a\right) \)

1003124803

Level: 
C
The annulus shaped component parts are punched from sheet metal. Diameter of the circular hole is \( 25\,\% \) of the diameter of the whole component part. Choose the function that describes the dependence of the area (\( S \)) of material used to produce one component part on its outside diameter (\( d \)).
\( S=\frac{15}{64}\,\pi d^2 \)
\( S=\frac38\,\pi d^2 \)
\( S=\frac{15}{32}\,\pi d^2 \)
\( S=\frac{31}{64}\,\pi d^2 \)

1003124804

Level: 
C
In the centre of a square shaped square there is a water fountain. The fountain has a square ground plan with the side length \( 4.5\,\mathrm{m} \). The square should be paved with cobblestones of size \( 25\,\mathrm{cm} \times 25\,\mathrm{cm} \). Choose the function that describes the dependence of the number of cobblestones needed (\( n \)) on the length of the square (\( a \)) given in meters.
\( n=16a^2-324 \)
\( n=\frac{a^2}{625}-324 \)
\( n=16a^2-625 \)
\( n=\frac{a^2}{16}-324 \)

1003124805

Level: 
C
On a spool of mass \( 0.5\,\mathrm{kg} \) is winded an aluminium wire of length \( 100\,\mathrm{m} \). Choose the function that describes the dependence of a mass of the spool with the wire \( m \) (in kilograms) on a diameter of the wire \( d \) (in millimetres). Wire density is \( 2\,700\frac{kg}{m^3} \). \[ \] Hint: The density of an object is defined as the ratio of the mass and the volume of the object.
\( m=\frac{27\pi}{400} d^2+0.5 \)
\( m= 67 500\pi d^2+0.5 \)
\( m=\frac{27\pi}{400} d^2-0.5 \)
\( m=\frac{27\pi}{200} d^2+0.5 \)

1003124806

Level: 
C
We should fence the land in a shape of an equilateral triangle. Choose the function that describes the dependence of the fenced land area \( S \) (in square meters) on the length \( d \) (in meters) of the fence used.
\( S=\frac{\sqrt3}{36} d^2 \)
\( S=\frac{\sqrt3}{18} d^2 \)
\( S=\frac{\sqrt3}4 d^2 \)
\( S=\frac1{36} d^2 \)

1003148601

Level: 
C
Consider an object thrown upwards from the ground with the initial velocity of \( 30\frac{\mathrm{m}}{\mathrm{s}} \). The object moves upwards with decreasing vertical velocity until it stops. Then it starts moving vertically downwards. Find the greatest height above the ground the object does reach. \[ \] Note: The vertical distance \( y \) of a thrown object is described by the equation \( y=v_0t-\frac12gt^2 \), where \( v_0 \) is the initial velocity of the thrown object, \( g \) is gravitational acceleration (count with the rounded value \( 10\frac{\mathrm{m}}{\mathrm{s}^2}\)), and \( t \) is the time period of the object motion in seconds.
\( 45\,\mathrm{m} \)
\( 135\,\mathrm{m} \)
\( 360\,\mathrm{m} \)
\( 40\,\mathrm{m} \)

1003148602

Level: 
C
Consider an object thrown at an angle of \( 30^{\circ} \) above the horizontal with the initial velocity of \( 40\frac{\mathrm{m}}{\mathrm{s}} \). How long does it take for the object to reach its maximum height? \[ \] Note: The height \( y \) of an object thrown is described by the formula \( y=v_0t\sin\alpha-\frac12gt^2 \), where \( v_0 \) is the initial velocity, \( g \) is gravitational acceleration (count with the rounded value \( 10\frac{\mathrm{m}}{\mathrm{s}^2}\)), \( t \) is the time period of the object motion in seconds, and \( \alpha \) is the angle to the horizontal at which the object is thrown.
\( 2\,\mathrm{s} \)
\( 4\,\mathrm{s} \)
\( 8\,\mathrm{s} \)
\( 1\,\mathrm{s} \)