1003086002
Level:
C
The equation \( (\sin x + \cos x)^2 = 1.5 \) has two solutions on the interval \( \left(0^{\circ}; 90^{\circ}\right) \). The larger one is:
\( 75^{\circ} \)
\( 15^{\circ} \)
\( 30^{\circ} \)
\( 65^{\circ} \)
C
1003086001
Level:
C
The arithmetic mean of all \( x\in\left[0^{\circ};360^{\circ}\right] \) satisfying the equation \( \sin 2x = \sin x \) is:
\( 180^{\circ} \)
\( 240^{\circ} \)
\( 360^{\circ} \)
\( 270^{\circ} \)
1003032508
Level:
C
Find the term that does not contain \( x \) and \( y \) (constant term) in the expansion of \( (x+y)^4(\frac1x+\frac1y)^4 \).
\( 70 \)
\( 2 \)
\( 36 \)
\( 0 \)
1003032507
Level:
C
Express \( v_1 \) from the formula \( v=\frac{v_1v_2(d_1+d_2 )}{d_1v_2+d_2v_1} \).
\( v_1=\frac{vd_1 v_2}{v_2d_1+v_2d_2-vd_2} \)
\( v_1=\frac{vv_2 d_1}{vd_2-v_2d_1-v_2d_2} \)
\( v_1=\frac{vv_2(d_1+d_2)}{d_1v_2+d_2 v} \)
\( v_1=\frac{v(d_1v_2+d_2 v_1 )}{v_2 (d_1+d_2 )} \)
1003032506
Level:
C
In a parallel circuit the total resistance \( R \) of three components with the resistances \( R_1 \), \( R_2 \), \( R_3 \) is given by the formula \( \frac1R=\frac1{R_1}+\frac1{R_2}+\frac1{R_3} \). Express \( R_1 \) from this formula.
\( R_1=\frac{RR_2R_3}{R_2R_3-R(R_2+R_3)} \)
\( R_1=\frac{R-R_2-R_3}{RR_2R_3} \)
\( R_1=R-\frac{R_2R_3}{R_2+R_3} \)
\( R_1=\frac{R_2R_3-R(R_2+R_3)}{RR_2R_3} \)
1003032505
Level:
C
Factoring the polynomial \( x^4+4 \) you get:
\( \left(x^2-2x+2\right)\left(x^2+2x+2\right) \)
\( \left(x^2+2\right)\left(x^2-(-2)\right) \)
\( \left( x+\sqrt2 \right)^4 \)
\( \left(x^2-\sqrt2x+2\right)\left(x^2+\sqrt2x+2\right) \)
1003032503
Level:
C
The polynomial \( 3x^5+px^3-(p-1)x^2+5x-9 \) is divisible by the binomial \( x^2-1 \) if \( p \) is equal to:
\( -8 \)
\( -16 \)
\( 4 \)
\( 6 \)
1003032502
Level:
C
Let the polynomial \( (x-2)^5-(x+2)^5 \) be expressed in the form \( a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \). Give the sum of \( a_5+a_4+a_3+a_2+a_1 \).
\( -180 \)
\( -244 \)
\( -242 \)
\( -212 \)
1003086108
Level:
C
The solution set of the equation \( 1 + \sin x \cdot \cos 2x = \sin x + \cos 2x \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}2+2k\pi;k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}2+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{2k\pi\right\} \)
1003086107
Level:
C
The solution set of the equation \( 2\mathrm{tg}^2x + 4\cos^2x = 7 \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}3+k\pi;\frac{2\pi}3+k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}3+2k\pi;\frac{2\pi}3+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}3+k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{2\pi}3+k\pi\right\} \)