NTS NAT-ICS Quantitative Reasoning Flashcards

24 sample cards from the Quantitative Reasoning 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 slope-intercept form of a straight line, and what do its parts represent?

   The slope-intercept form is y = mx + c, where m is the slope (gradient = rise/run) and c is the y-intercept (the value of y where the line crosses the y-axis). A positive m rises left-to-right; a negative m falls.

2. What are the main methods for solving a system of two linear equations in two variables?

   The three standard methods are: (1) Substitution — solve one equation for a variable and substitute into the other; (2) Elimination — add or subtract the equations to cancel one variable; (3) Graphical — plot both lines and read the intersection point as the solution.

3. How do the slopes of parallel and perpendicular lines compare?

   Parallel lines have equal slopes (m1 = m2). Perpendicular lines have slopes that are negative reciprocals, so their product is −1 (m1 × m2 = −1). A horizontal line (slope 0) is perpendicular to a vertical line (undefined slope).

4. What is the general form of a quadratic equation, and what is the quadratic formula?

   The general form is ax² + bx + c = 0 (a ≠ 0). The quadratic formula gives its roots: x = [−b ± √(b² − 4ac)] ÷ (2a). The two solutions come from the ± sign.

5. What is the discriminant of a quadratic equation, and what does its value tell you about the roots?

   The discriminant is D = b² − 4ac. If D > 0 there are two distinct real roots; if D = 0 there is one real (repeated) root; if D < 0 there are no real roots (the roots are complex/imaginary).

6. What are the relationships between the roots and coefficients of ax² + bx + c = 0?

   If the roots are α and β, then the sum of the roots α + β = −b/a, and the product of the roots α × β = c/a. These let you analyze roots without solving the equation.

7. What does it mean to solve a quadratic equation by factoring?

   Express the quadratic as a product of two binomials equal to zero, e.g. (x − p)(x − q) = 0. By the zero-product property, the solutions are x = p and x = q. This works cleanly only when the quadratic factors into rational terms.

8. State the basic product and quotient laws of exponents.

   Product law: a^m × a^n = a^(m+n) (same base, add exponents). Quotient law: a^m ÷ a^n = a^(m−n) (same base, subtract exponents). Power of a power: (a^m)^n = a^(mn).

9. What do a zero exponent and a negative exponent mean?

   Any nonzero base raised to the power 0 equals 1 (a^0 = 1). A negative exponent denotes a reciprocal: a^(−n) = 1/a^n. For example 5^0 = 1 and 2^(−3) = 1/8.

10. How is a fractional (rational) exponent interpreted in terms of radicals?

    A fractional exponent represents a root: a^(1/n) = the n-th root of a, and a^(m/n) = the n-th root of a^m = (n-th root of a)^m. For example, 8^(2/3) = (cube root of 8)² = 2² = 4.

11. What is the rule for multiplying and dividing radicals with the same index?

    For the same index, √a × √b = √(ab) and √a ÷ √b = √(a/b). To simplify a radical, factor out perfect squares (or n-th powers), e.g. √50 = √(25×2) = 5√2.

12. What is the definition of a logarithm, and how is it related to exponents?

    A logarithm is the inverse of exponentiation: log_b(x) = y means b^y = x (b > 0, b ≠ 1). For example, log_2(8) = 3 because 2³ = 8. The base b is the number being raised to a power.

13. State the product, quotient, and power laws of logarithms.

    Product: log_b(MN) = log_b(M) + log_b(N). Quotient: log_b(M/N) = log_b(M) − log_b(N). Power: log_b(M^p) = p · log_b(M). These convert multiplication/division/powers into addition/subtraction/multiplication.

14. What is the difference between a common logarithm and a natural logarithm?

    A common logarithm has base 10 and is written log x (so log 1000 = 3). A natural logarithm has base e (≈ 2.718) and is written ln x. Both follow the same logarithm laws; only the base differs.

15. What are log_b(1) and log_b(b), and why?

    log_b(1) = 0 because any base raised to the power 0 equals 1 (b^0 = 1). log_b(b) = 1 because the base raised to the power 1 equals itself (b^1 = b). These hold for any valid base b > 0, b ≠ 1.

16. What is the change-of-base formula for logarithms?

    log_b(x) = log_c(x) ÷ log_c(b), for any new base c > 0, c ≠ 1. This lets you compute a logarithm in any base using base-10 or natural logs, e.g. log_2(7) = log(7)/log(2).

17. What is the difference between a monomial, a binomial, and a trinomial?

    These are classifications of polynomials by number of terms. A monomial has one term (e.g. 5x²). A binomial has two terms (e.g. x + 3). A trinomial has three terms (e.g. x² + 5x + 6).

18. State the algebraic identities for (a + b)², (a − b)², and (a + b)(a − b).

    (a + b)² = a² + 2ab + b². (a − b)² = a² − 2ab + b². (a + b)(a − b) = a² − b² (the difference of two squares). These are core identities used to expand and factor expressions.

19. How do you factor a quadratic trinomial of the form x² + bx + c?

    Find two numbers whose product is c and whose sum is b. Then write the trinomial as (x + p)(x + q), where p and q are those numbers. For example, x² + 5x + 6 = (x + 2)(x + 3) because 2×3 = 6 and 2 + 3 = 5.

20. What is the difference between an algebraic expression and an algebraic equation?

    An expression is a combination of variables, numbers, and operations with no equals sign (e.g. 3x + 5); it can be simplified or evaluated but not solved. An equation states that two expressions are equal using an equals sign (e.g. 3x + 5 = 11) and can be solved for the variable.

21. What is the difference between complementary and supplementary angles?

    Two angles are complementary if their measures add up to 90° (a right angle). Two angles are supplementary if their measures add up to 180° (a straight angle). For example, 30° and 60° are complementary; 110° and 70° are supplementary.

22. When two parallel lines are cut by a transversal, what is the relationship between corresponding angles and co-interior (allied) angles?

    Corresponding angles are equal. Alternate (interior and exterior) angles are also equal. Co-interior (allied/same-side interior) angles are supplementary, adding up to 180°. Vertically opposite angles formed at any intersection are always equal.

23. State the angle sum property of a triangle and how to find the sum of interior angles of any polygon.

    The interior angles of any triangle sum to 180°. For a polygon with n sides, the interior angles sum to (n − 2) × 180°. The exterior angles of any convex polygon always sum to 360°.

24. How are triangles classified by their sides and by their angles?

    By sides: equilateral (all three sides equal), isosceles (two sides equal), and scalene (no sides equal). By angles: acute (all angles less than 90°), right (one angle equals 90°), and obtuse (one angle greater than 90°).

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