1003020402
Level:
A
The equation of an ellipse is given by \( x^2+4y^2-6x+32y+48=0 \). The length of its semi-minor axis is:
\( \frac52 \)
\( 5 \)
\( 25 \)
\( \frac{25}4 \)
\( \frac{15}2 \)
A
1003020401
Level:
A
The equation of an ellipse is given by \( 5x^2+9y^2-30x-18y+9=0 \). The length of its semi-major axis is:
\( 3 \)
\( 5 \)
\( 9 \)
\( \sqrt5 \)
\( 14 \)
1003020204
Level:
A
Assuming \( x\in\mathbb{Z} \), find the solution set of the equation.
\[ 3-6x+3\cdot\left\{x-\left[2-(x+1)\right]\right\}=0\]
\( \mathbb{Z} \)
\( \emptyset \)
\( \{1\} \)
\( \{0\} \)
1003020203
Level:
A
Find the solution set of the following equation.
\[ \frac{x+5}2-\frac{x+1}4=\frac{22+2x}8\]
\( \emptyset \)
\( \mathbb{R} \)
\( \{4\} \)
\( \{-4\} \)
1003020202
Level:
A
Assuming \( x\in\mathbb{N} \), find the solution set of the equation.
\[ 3x-\left[1+2\cdot(3x-2)\right]=9\]
\( \emptyset \)
\( \mathbb{N} \)
\( \{-2\} \)
\( \left\{\frac{14}9\right\} \)
1003020201
Level:
A
Find the solution set of the following equation.
\[ 2-\frac{2-x}3=\frac x2+\frac{8-x}6 \]
\( \mathbb{R} \)
\( \emptyset \)
\( \{0\} \)
\( \{4\} \)
1103030705
Level:
A
Let there be a triangle KLM and vectors \( \vec{a} \), \( \vec{c} \) in the coordinate system. Triangle \( KLM \) and vectors \( \vec{a} \), \( \vec{c} \) are
given in the coordinate system shown in the picture. Point T is the centroid of the triangle KLM. Express vector \( \vec{x} \), where \( \vec{x}=\overrightarrow{KT} \) as a linear combination of \( \vec{a} \) and \( \vec{c} \) and evaluate \( \left|\vec{x}\right| \).
\( \vec{x}=\frac13 \vec{a}+\frac13 \vec{c} \), \( \left|\vec{x}\right|=5 \)
\( \vec{x}=\frac23 \vec{a}+\frac23 \vec{c} \), \( \left|\vec{x}\right|=10 \)
\( \vec{x}=\frac12 \vec{a}+\frac12 \vec{c} \), \( \left|\vec{x}\right|=\frac{15}2 \)
\( \vec{x}=\frac14 \vec{a}+\frac14 \vec{c} \), \( \left|\vec{x}\right|=\frac{225}{12} \)
1103030704
Level:
A
We are given points \( A = [2,1] \), \( B = [4,-1] \), and \( T = [6,2] \), where point \( T \) is the centroid of triangle \( ABC \). Find the length of the median of triangle \( ABC \) to side \( AC \).
\( |t_b|=\frac{\sqrt{117}}2 \)
\( |t_b|=\frac{\sqrt{45}}2 \)
\( |t_b|=\frac{\sqrt{153}}2 \)
\( |t_b|=\sqrt{117} \)
1103030703
Level:
A
We are given points \( A = [2,1] \), \( B = [4,-1] \), and \( T = [6,2] \), where point \( T \) is the centroid of triangle \( ABC \). Find the coordinates of \( C \), which is the vertex of \( ABC \).
\( C = [12,6] \)
\( C = [8,4] \)
\( C = [9,6] \)
\( C = [8,5] \)
1103030702
Level:
A
We are given points \( A = [1,-1,2] \), \( B = [0,5,-3] \), \( S = [2,0,5] \). Point \( S \) is the centre of a parallelogram \( ABCD \). Evaluate the length of \( AD \).
\( |AD|=\sqrt{146} \)
\( |AD|=\sqrt{114} \)
\( |AD|=\sqrt{44} \)
\( |AD|=\sqrt{172} \)