B | math4u.vsb.cz

9000104310

Level:

Assuming \(a\in \left (0,1\right )\), solve the following inequality. \[ 2a\left (1 - a\right )x > 3 \]

\(\left ( \frac{3} {2a\left (1-a\right )},\infty \right )\)

\(\left (- \frac{3} {2a\left (1-a\right )},\infty \right )\)

\(\left (- \frac{3} {2a\left (1-a\right )}, \frac{3} {2a\left (1-a\right )}\right )\)

\(\left (-\infty , \frac{3} {2a\left (1-a\right )}\right )\)

9000101805

Level:

Find the vector \(\vec{v}\) such that the length of this vector is \(5\) and \(\vec{v}\) is perpendicular to the vector \(\vec{u} = (-1,0.75)\).

\(\vec{v} = (3,4)\)

\(\vec{v} = (3,-4)\)

\(\vec{v} = (4,-3)\)

\(\vec{v} = (5,0)\)

9000101705

Level:

Factor the following polynomial expression. \[ 16a^{2}b^{2} - 4a^{2}c^{2} - 16b^{2}d^{2} + 4c^{2}d^{2} \]

\(4\left (a - d\right )\left (a + d\right )\left (2b + c\right )\left (2b - c\right )\)

\(4\left (a + b\right )^{2}\left (2b + c\right )^{2}\)

\(4\left (a - b\right )\left (a + b\right )\left (2b + c\right )\left (2b - c\right )\)

\(4\left (a - c\right )\left (a + c\right )\left (2b + d\right )\left (2b - d\right )\)

9000101105

Level:

Find the distance between two parallel planes \(\alpha \) and \(\beta \). \[ \begin{aligned} \alpha& \colon x - 1.5y - 3z - 1 = 0,\\ \beta &\colon 2x - 3y - 6z + 3 = 0 \end{aligned}\]

\(\frac{5} {7}\)

\(\frac{1} {7}\)

\(\frac{2} {49}\)

\(\frac{2} {41}\)

9000100706

Level:

Given vectors \(\vec{a} = (-1,2,-3)\), \(\vec{b} = (0,1,-1)\), find the vector \(\vec{c}\) perpendicular to both of them.

\(\vec{c} = (-1,1,1)\)

\(\vec{c} = (-3,0,1)\)

\(\vec{c} = (2,4,2)\)

\(\vec{c} = (-1,-1,1)\)

9000100707

Level:

Consider points \(A = [-2,-1]\), \(B = [1,y]\), \(C = [3,-4]\). Find the coordinate \(y\) which ensures that the vectors \(\overrightarrow{AB } \) and \(\overrightarrow{AC } \) are perpendicular.

\(y = 4\)

\(y = -4\)

\(y = 0.8\)

\(y = -0.8\)

9000100708

Level:

Given points \(A = [-2,-1]\), \(B = [x,-3]\), \(C = [4,-4]\), find the coordinate \(x\) which ensures that the vectors \(\overrightarrow{AB } \) and \(\overrightarrow{AC } \) are parallel.

\(x = 2\)

\(x = 7\)

\(x = -3\)

\(x = 3\)

9000100709

Level:

Find the value of the parameter \(z\) so that the vector \(\vec{w} = (8,2,z)\) is perpendicular to the vectors \(\vec{a} = (1,2,-3)\) and \(\vec{b} = (-1,2,1)\).

\(z = 4\)

\(z = -12\)

\(z = 2\)

\(z = -4\)

9000101605

Level:

Expand \(\left (4x^{2}y + 2xy^{2}\right )^{3}\).

\(64x^{6}y^{3} + 96x^{5}y^{4} + 48x^{4}y^{5} + 8x^{3}y^{6}\)

\(16x^{2}y^{3} + 24x^{3}y^{3} + 8x^{3}y^{6}\)

\(64x^{6}y^{3} + 96x^{3}y^{3} + 96x^{4}y^{5} + 8x^{3}y^{6}\)

\(64x^{6}y^{3} + 8x^{3}y^{6}\)

Among vectors \(\vec{u} = \left (- \frac{2} {\sqrt{2}},2\sqrt{2}\right )\), \(\vec{v} = (-5,10)\), \(\vec{w} = (2.5,-5)\), \(\vec{r} = (-3.5,6)\) find the vector which is not parallel to the vector \(\vec{a} = (1,-2)\).

\(\vec{r}\)

\(\vec{w}\)

\(\vec{v}\)

\(\vec{u}\)

B

9000104310

9000101805

9000101705

9000101105

9000100706

9000100707

9000100708

9000100709

9000101605

9000101802

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