Night Study 10th Maths 1st - Tamil Nadu Board Mathematics

Key 5-Mark Questions in Class 10 Mathematics (TN Board)

Cartesian Product and Set Operations

The Cartesian product of sets and operations like union, intersection, and difference are foundational for set theory questions. Understanding how to convert between set-builder and roster forms, and performing operations stepwise, is essential for full marks.

• Cartesian Product ($A \times B$): Pair each element of $A$ with every element of $B$.
• Union ($A \cup B$): Combine all unique elements from both sets.
• Intersection ($A \cap B$): List only elements common to both sets.
• Difference ($A - B$): List elements in…

Key 5-Mark Questions in Class 10 Mathematics (TN Board)

Cartesian Product and Set Operations

The Cartesian product of sets and operations like union, intersection, and difference are foundational for set theory questions. Understanding how to convert between set-builder and roster forms, and performing operations stepwise, is essential for full marks.

• Cartesian Product ($A \times B$): Pair each element of $A$ with every element of $B$.
• Union ($A \cup B$): Combine all unique elements from both sets.
• Intersection ($A \cap B$): List only elements common to both sets.
• Difference ($A - B$): List elements in $A$ not in $B$.

Example Steps:

1. Write sets in roster form if given in set-builder form.
2. For $A \times (B \cup C)$, first find $B \cup C$, then pair each $A$ element with each in $B \cup C$.
3. For $(A \times B) \cup (A \times C)$, find both products, then take their union.
4. For $(A \times B) \cap (A \times C)$, find both products, then list common pairs.

Representation of Relations and Functions

Relations and functions can be represented in multiple forms: roster, arrow diagram, table, and graph. For exam questions, be able to convert between these forms and plot ordered pairs.

• Roster Form: List all ordered pairs explicitly.
• Arrow Diagram: Draw arrows from each $x$ in domain to corresponding $y$ in codomain.
• Table Form: List $x$ and $f(x)$ values in columns.
• Graph: Plot each $(x, y)$ pair on coordinate axes.

Example: For $x = 2y$ with $x \in \{2,3,4,5\}$, $y \in \{1,2,3,4\}$:

• Substitute values to find valid pairs: $(2,1)$ and $(4,2)$.
• Arrow diagram: Draw arrows from 2 to 1, 4 to 2.
• Table: $x$ in one column, $y$ in another.
• Graph: Plot points $(2,1)$ and $(4,2)$.

Piecewise Functions and Number Line Interpretation

Piecewise functions assign different expressions to different intervals. Use the number line to determine which expression to use for a given input.

• For $f(x)$ defined as:
  • $6x+1$ for $-5 \leq x < 2$
  • $5x^2-1$ for $2 \leq x < 6$
  • $3x-4$ for $6 \leq x \leq 9$
• To find $f(-3) + f(2)$:
  • $f(-3) = 6(-3) + 1 = -17$
  • $f(2) = 5(2^2) - 1 = 19$
  • Sum: $-17 + 19 = 2$

Key Steps:

1. Draw the number line and mark intervals.
2. Identify which interval the input falls into.
3. Substitute and compute.

Composition of Functions

Composition involves applying one function to the result of another. For $f \circ g \circ h$, substitute stepwise from innermost to outermost.

• If $f(x) = x-4$, $g(x) = x^2$, $h(x) = 3x-5$:
  • $f(g(x)) = (x^2) - 4$
  • $f(g(h(x))) = (3x-5)^2 - 4$
  • $g(h(x)) = (3x-5)^2$
  • $f(g(h(x))) = (3x-5)^2 - 4$

Proof Structure:

• Compute $f(g(h(x)))$ and $f((g \circ h)(x))$ separately.
• Show both simplify to the same expression.

Euclid’s Division Algorithm and HCF

Euclid’s Division Algorithm is used to find the Highest Common Factor (HCF) of two or more numbers by repeated division.

• For numbers $a$ and $b$ ($a > b$): $a = bq + r$, $0 \leq r < b$.
• Repeat with $b$ and $r$ until remainder is $0$.
• The last non-zero remainder is the HCF.

Example:

• $504 = 396 \times 1 + 108$
• $396 = 108 \times 3 + 72$
• $108 = 72 \times 1 + 36$
• $72 = 36 \times 2 + 0$
• HCF is $36$.

For three numbers, find HCF of first two, then with the third.

Fundamental Theorem of Arithmetic and Prime Factorization

Every integer greater than 1 can be expressed as a product of primes in ascending order.

• Example: $13400 = 2^3 \times 3^4 \times 5^2 \times 7^1$
• List all prime factors with their exponents.

Table Example:

Greatest 6-Digit Number Divisible by Given Numbers

To find the greatest 6-digit number exactly divisible by $24$, $15$, and $36$:

• Find LCM of $24$, $15$, $36$:
  • $24 = 2^3 \times 3$
  • $15 = 3 \times 5$
  • $36 = 2^2 \times 3^2$
  • LCM: $2^3 \times 3^2 \times 5 = 360$
• Divide $999999$ by $360$, find remainder.
• Subtract remainder from $999999$ to get the required number.

Arithmetic Progression (AP): nth Term and Sum

The $n$th term of an AP: $a_n = a + (n-1)d$.

• For $l$th term $= x$, $m$th term $= y$, $n$th term $= z$:
  • $x = a + (l-1)d$
  • $y = a + (m-1)d$
  • $z = a + (n-1)d$
• For sum of first $n$ terms: $S_n = \frac{n}{2}(2a + (n-1)d)$ or $S_n = \frac{n}{2}(a + l)$.

Special Result: For four consecutive terms $a-3d, a-d, a+d, a+3d$:

• Their sum: $4a = 28 \implies a = 7$
• Sum of squares: $a^2 + 5d^2 = 69 \implies d^2 = 4, d = \pm2$
• The four terms: $1, 5, 9, 13$ or $13, 9, 5, 1$

Application Problem: House Number Puzzle

Given house numbers from $1$ to $49$, find the house number $x$ such that the sum of numbers before $x$ equals the sum after $x$.

• Sum before $x$: $S_{x-1} = \frac{(x-1)x}{2}$
• Sum after $x$: $S_{49} - S_x = \frac{49 \times 50}{2} - \frac{x(x+1)}{2}$
• Equate and solve: $x^2 = 1225 \implies x = 35$

Geometric Progression (GP): nth Term and Application

The $n$th term of a GP: $a_n = ar^{n-1}$.

• If $9$th term $= 32805$, $6$th term $= 1215$:
  • $ar^8 = 32805$, $ar^5 = 1215$
  • Divide: $r^3 = 27 \implies r = 3$
  • Find $a$: $a \times 3^5 = 1215 \implies a = 5$
  • $12$th term: $a \times r^{11} = 5 \times 3^{11}$

Special Series and Summation

For series like $5 + 55 + 555 + \ldots$ up to $n$ terms:

• Express each term as $5 \times (10^k - 1)/9$ for $k = 1$ to $n$.
• Sum: $S_n = \frac{5}{9}[(10 + 100 + \ldots + 10^n) - n]$
• Use GP sum formula for $10 + 100 + \ldots + 10^n$:$$S = \frac{5}{9} \left( \frac{10(10^n - 1)}{9} - n \right )$$

Simultaneous Linear Equations in Three Variables

To solve three equations in $x, y, z$:

• Write all equations in standard form.
• Use elimination or substitution to reduce to two variables, then one.
• Example:
  • $2x - 3y + 20 = 0$
  • $3y - 4z + 10 = 0$
  • $3z + y = 105$
• Solve stepwise to find $z = 25$, $y = 30$, $x = 35$.

Exam-Focused Takeaways

• Practice converting between set forms and performing set operations stepwise.
• Master all representations of relations and functions.
• For piecewise functions, always check the interval before substituting.
• In AP/GP, memorize and apply the nth term and sum formulas as given.
• For HCF, strictly follow Euclid’s algorithm steps.
• For prime factorization, write all exponents and primes in ascending order.
• For word problems, translate statements into equations before solving.
• For simultaneous equations, elimination and substitution methods are both valid—choose based on the question structure.
• Always write final answers with clear justification and all steps for full marks.