Mock Test Review: quick revision, 19 bites
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Rational Number Closure
Division breaks closure for rationals.
Rational numbers are **closed** under addition, subtraction, and multiplication, meaning the result is always a rational number. However, they are **not closed under division** because division by zero is undefined, which is not a rational number.
Takeaway. Rational numbers are closed under +, -, × but not ÷ (due to zero).
Multiplicative Inverse
Zero has no multiplicative inverse.
The **multiplicative inverse** of a number $x$ is $\frac{1}{x}$, such that $x \times \frac{1}{x} = 1$. Every non-zero rational number has a multiplicative inverse.
Takeaway. Zero is the only rational number without a multiplicative inverse.
Non-Associative Operations
Subtraction and division are not associative.
An operation is **associative** if $(a \circ b) \circ c = a \circ (b \circ c)$. Addition and multiplication of rationals are associative, but **subtraction and division are not**.
Takeaway. For rationals, + and × are associative; − and ÷ are not.
GP nth Term Formula
GP grows by common ratio.
In a **geometric progression (GP)**, each term is found by multiplying the previous term by a fixed **common ratio** $r$. The $n$th term is given by $a_n = a_1 \cdot r^{n-1}$.
Takeaway. GP nth term = first term × (common ratio)^(n-1).
Difference of Squares
a² − b² = (a+b)(a−b).
The **difference of squares** identity states $a^2 - b^2 = (a+b)(a-b)$. This is useful for quick mental math with large numbers.
Takeaway. Use (a+b)(a−b) to quickly compute differences of squares.
Parallelogram Diagonal Midpoint
Diagonals bisect each other.
In a parallelogram, the **diagonals bisect each other**, meaning they intersect at their midpoints. The intersection point is the midpoint of both diagonals.
Takeaway. Parallelogram diagonals intersect at their common midpoint.
Midpoint Application
Find relation between a and b.
Given A(-5, 6) and C(9, 1) as opposite vertices of a parallelogram, the intersection P(a, b) is the midpoint of AC.
Takeaway. Midpoint of diagonal gives intersection point coordinates.
AP nth Term
Find term 130 more than 31st.
In an **arithmetic progression (AP)**, the $n$th term is $a_n = a + (n-1)d$. To find which term is $k$ more than the 31st term, set up the equation.
Takeaway. Set up equation aₙ = a₃₁ + 130 to solve for n.
Recurring Decimal to Fraction
Convert 1.272727... to fraction.
To convert a **recurring decimal** to a fraction, let $x$ equal the decimal, multiply by a power of 10 to shift the repeating part, then subtract.
Takeaway. Subtract shifted decimals to eliminate the repeating part.
Additive Identity
Zero is the additive identity.
**Zero (0)** is the **identity for addition** of rational numbers because adding 0 to any rational number leaves it unchanged: $x + 0 = x$.
Takeaway. 0 is the additive identity; 1 is the multiplicative identity.
GP Sum and Product
Find three numbers in GP.
For three numbers in GP, let them be $\frac{a}{r}, a, ar$. Their product is $a^3$ and their sum is given.
Takeaway. Represent GP terms as a/r, a, ar to solve sum and product.
Probability Complement
P(E) + P(not E) = 1.
The **complement rule** states that the probability of an event happening plus the probability of it not happening equals 1: $P(E) + P(\text{not } E) = 1$.
Takeaway. Probability of complement = 1 − probability of event.
Distance Formula
Find x given distance = 10.
The **distance** between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula.
Takeaway. Square both sides to solve distance problems.
Rectangle in III Quadrant
All coordinates are negative.
In the **III quadrant**, both x and y coordinates are negative. A rectangle with length $p$ on X-axis and breadth $q$ on Y-axis placed here has all vertices with negative coordinates.
Takeaway. III quadrant points have both coordinates negative.
Chord Length Formula
Use perpendicular distance from center.
The **length of a chord** in a circle can be found using the radius and the perpendicular distance from the center to the chord.
Takeaway. Chord length = 2√(r² − d²).
Polynomial Degrees
Degree = highest power of variable.
The **degree of a polynomial** is the highest power of the variable in the polynomial. A constant has degree 0.
Takeaway. Degree is the highest exponent; constants have degree 0.
Polynomial Multiplication
Use distributive property.
Multiply polynomials using the **distributive property** (FOIL method for binomials).
Takeaway. Multiply each term in first bracket by each in second.
Brahmagupta's Sign Rules
Sign rules for integer operations.
**Brahmagupta's laws** define operations with negative numbers: product of same signs is positive, different signs is negative; subtracting a negative is addition.
Takeaway. Same signs multiply to positive; different signs to negative.
AP Middle Terms
37 terms → middle is 19th.
For an AP with 37 terms, the **three middle terms** are the 18th, 19th, and 20th terms. Their sum and the sum of last three terms can be used to find the AP.
Takeaway. Middle terms of odd-count AP are centered around the middle term.
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