1003027705
Level:
A
How much smaller is \( \int\limits_0^{1} x^{15}\,\mathrm{d}x \) than \( \int\limits_0^{1} x^{5}\,\mathrm{d}x \)?
\( \frac5{48} \)
\( \frac{11}{48} \)
\( \frac5{32} \)
\( \frac{11}{32} \)
A
1003027704
Level:
A
Compare two definite integrals \( I_1=\int\limits_{-1}^1x^8\,\mathrm{d}x \) and \( I_2=\int\limits_{-1}^1x^2\,\mathrm{d}x \).
\( I_2 \) is bigger than \( I_1 \) by \( \frac49 \).
\( I_1 \) is bigger than \( I_2 \) by \( \frac49 \).
\( I_2 \) is bigger than \( I_1 \) by \( \frac29 \).
\( I_1 \) is bigger than \( I_2 \) by \( \frac29 \).
1003027703
Level:
A
Compare two definite integrals \( I_1=\int\limits_0^{\frac{\pi}2}\cos x\,\mathrm{d}x \) and \( I_2=\int\limits_0^{2\pi}2\cos x\,\mathrm{d}x \).
\( I_1 \) is bigger than \( I_2 \) by \( 1 \).
\( I_1 \) is smaller than \( I_2 \) by \( 1 \).
\( I_1 \) is equal to \( I_2 \).
\( I_1 \) is smaller than \( I_2 \) by \( 2 \).
1003027702
Level:
A
How much smaller is \( \int\limits_{-\frac{\pi}2}^{\frac{\pi}2}\sin x\,\mathrm{d}x \) than \( \int\limits_{0}^{\pi}(\cos x + 1)\,\mathrm{d}x \)?
\( \pi \)
\( 1-\pi \)
\( \pi-1 \)
\( 2\pi \)
1003027701
Level:
A
How much bigger is \( \int\limits_0^{\pi}(x+\sin x)\,\mathrm{d}x \) than \( \int\limits_1^{\pi}x\,\mathrm{d}x \)?
\( \frac52 \)
\( 0.5 \)
\( \pi-1 \)
\( \frac32 \)
1103040107
Level:
A
The picture below shows an ellipse drawn in the rectangular coordinate system. What is the semi-minor axis of this ellipse?
The segment \( SM \)
The segment \( SL \)
The line \( MN \)
The line \( EF \)
1103040106
Level:
A
A circle in the rectangular coordinate system is shown in the picture. What is the standard form equation of this circle?
\( x^2+(y-3)^2=4 \)
\( x^2+(y+3)^2=4 \)
\( x^2+(y-3)^2=2 \)
\( x^2+(y+3)^2=2 \)
1103040103
Level:
A
The diagram of an ellipse in the rectangular coordinate system is shown in the picture.What is the semi-major axis of this ellipse?
The segment \( SN \)
The segment \( SK \)
The line \( EF \)
The line \( KL \)
1103040102
Level:
A
The diagram of an ellipse in the rectangular coordinate system is shown in the picture. Find the standard form equation of this ellipse:
\( \frac{(x-3)^2}4+\frac{(y-3)^2}9=1 \)
\( \frac{(x-3)^2}9+\frac{(y-3)^2}4=1 \)
\( \frac{(x+3)^2}4+\frac{(y+3)^2}9=1 \)
\( \frac{(x+3)^2}9+\frac{(y+3)^2}4=1 \)
1103040101
Level:
A
The equation of an ellipse is given by \( \frac{(x-2)^2}4+y^2=1 \).
Which of the pictures below shows the given ellipse in the rectangular coordinate system?
