ENEM Mathematics Flashcards

24 sample cards from the Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. What is the general term of an arithmetic progression (AP)?

   $a_{n} = a_{1} + (n-1)r$, where $r$ is the common difference.

2. Give the formula for the sum of the first $n$ terms of an AP.

   $S_{n} = \dfrac{(a_{1}+a_{n}) \cdot n}{2}$.

3. What is the general term of a geometric progression (GP)?

   $a_{n} = a_{1} \cdot q^{\,n-1}$, where $q$ is the common ratio.

4. Give the sum of an infinite convergent GP with $|q|<1$.

   $S = \dfrac{a_{1}}{1-q}$.

5. State the area of a triangle given base $b$ and height $h$, and Heron's formula.

   $A = \dfrac{b \cdot h}{2}$; Heron: $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \dfrac{a+b+c}{2}$.

6. State the Pythagorean theorem.

   In a right triangle, $a^{2} + b^{2} = c^{2}$, where $c$ is the hypotenuse and $a, b$ the legs.

7. Give the formulas for the circumference and area of a circle of radius $r$.

   Circumference $C = 2\pi r$; area $A = \pi r^{2}$.

8. What is the sum of the interior angles of a convex polygon with $n$ sides?

   $S = (n-2) \cdot 180^{\circ}$.

9. Give the volume formulas for a rectangular box (prism) and a cylinder.

   Box: $V = a \cdot b \cdot c$ (product of edges). Cylinder: $V = \pi r^{2} h$.

10. Give the volume formulas for a cone and a sphere.

    Cone: $V = \dfrac{1}{3}\pi r^{2} h$. Sphere: $V = \dfrac{4}{3}\pi r^{3}$.

11. State the volume of a pyramid.

    $V = \dfrac{1}{3} \cdot A_{\text{base}} \cdot h$, one-third of the base area times the height.

12. Give the distance formula between two points in the plane.

    $d = \sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}}$.

13. Give the midpoint formula for the segment joining $(x_1,y_1)$ and $(x_2,y_2)$.

    $M = \left(\dfrac{x_{1}+x_{2}}{2}, \dfrac{y_{1}+y_{2}}{2}\right)$.

14. What is the slope-intercept equation of a line and how is slope computed from two points?

    $y = mx + b$, where $m = \dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}$ is the slope.

15. Give the standard equation of a circle with center $(a,b)$ and radius $r$.

    $(x-a)^{2} + (y-b)^{2} = r^{2}$.

16. Define the sine, cosine, and tangent ratios in a right triangle.

    $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$.

17. State the fundamental trigonometric identity.

    $\sin^{2}\theta + \cos^{2}\theta = 1$.

18. State the Law of Sines and the Law of Cosines.

    Law of Sines: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$. Law of Cosines: $a^{2} = b^{2} + c^{2} - 2bc\cos A$.

19. Give the exact values of $\sin$, $\cos$, and $\tan$ for $30^{\circ}$.

    $\sin 30^{\circ} = \dfrac{1}{2}$, $\cos 30^{\circ} = \dfrac{\sqrt{3}}{2}$, $\tan 30^{\circ} = \dfrac{\sqrt{3}}{3}$.

20. How do the mean, median, and mode differ as measures of central tendency?

    Mean is the arithmetic average $\bar{x}=\dfrac{\sum x_i}{n}$; median is the middle value of ordered data; mode is the most frequently occurring value.

21. Define variance and standard deviation of a data set.

    Variance $\sigma^{2} = \dfrac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}$ (mean squared deviation); standard deviation $\sigma = \sqrt{\sigma^{2}}$.

22. State the classical (Laplace) definition of the probability of an event.

    $P(A) = \dfrac{\text{number of favorable outcomes}}{\text{number of possible outcomes}}$, for equally likely outcomes, with $0 \leq P(A) \leq 1$.

23. Give the addition rule for the probability of $A$ or $B$.

    $P(A \cup B) = P(A) + P(B) - P(A \cap B)$; for mutually exclusive events $P(A \cap B)=0$.

24. Give the formulas for the number of arrangements (permutations $P_n$) and combinations $C_{n,p}$.

    $P_{n} = n!$ orderings of $n$ elements; $C_{n,p} = \dbinom{n}{p} = \dfrac{n!}{p!\,(n-p)!}$ chooses $p$ from $n$ ignoring order.

What this deck covers

Mathematics flashcards FAQ