Key Topics in ICSE Class 10 - Class 10 ICSE Mathematics

Key Topics in ICSE Class 10 Mathematics

This summary covers high-yield concepts, formulas, and problem-solving strategies for ICSE Class 10 Mathematics, focusing on chapters and question types repeatedly emphasized in exams. The content is structured to maximize recall and exam performance, with explicit attention to GST, Banking, Shares & Dividends, Proportion, Matrices, AP/GP, Linear Inequations, and Quadratic Equations.

Goods and Services Tax (GST)

GST is a tax applied as a percentage of the selling or marked price of an article. For intrastate transactions, GST is split equally between SGST (State GST) and CGST (Central GST). The total GST…

Key Topics in ICSE Class 10 Mathematics

This summary covers high-yield concepts, formulas, and problem-solving strategies for ICSE Class 10 Mathematics, focusing on chapters and question types repeatedly emphasized in exams. The content is structured to maximize recall and exam performance, with explicit attention to GST, Banking, Shares & Dividends, Proportion, Matrices, AP/GP, Linear Inequations, and Quadratic Equations.

Goods and Services Tax (GST)

GST is a tax applied as a percentage of the selling or marked price of an article. For intrastate transactions, GST is split equally between SGST (State GST) and CGST (Central GST). The total GST is calculated directly as a percentage of the selling price unless otherwise specified.

• GST Calculation: If GST is 12% and the total GST paid is ₹600, then $0.12 \times \text{Selling Price} = 600$.
• Finding Selling Price: $\text{Selling Price} = \frac{600 \times 100}{12} = ₹5000$.
• Discount Calculation: If a new marked price $x$ is set so that after GST, the total payable is a specific amount, set up the equation: $x + \text{GST\% of } x = \text{Total Paid}$, solve for $x$, then calculate discount as $(\text{Original Marked Price} - x)/\text{Original Marked Price} \times 100$.
• GST Rate Determination: If GST paid and marked price are given, let rate be $x\%$, then $x\%$ of marked price $=$ GST paid.

Banking and Recurring Deposit Accounts

Recurring Deposit (RD) accounts involve monthly deposits for a fixed period, with interest calculated on the sum deposited.

• Principal (P): Amount deposited per month.
• Time (n): Always taken in months.
• Maturity Value (MV): $P \times n + \text{Interest}$
• Interest Formula: $I = \frac{P \times n(n+1)}{2 \times 12} \times \frac{R}{100}$
• Interest from Maturity Value: $I = \text{Maturity Value} - (P \times n)$
• Rate of Interest Calculation: Rearranged from the interest formula if $I$, $P$, and $n$ are known.

Shares and Dividends

Shares have a nominal (face) value and a market value. Dividends are paid as a percentage of the nominal value, not the market value.

• Dividend Formula: $\text{Dividend} = \text{Number of Shares} \times \text{Nominal Value} \times \frac{\text{Rate of Dividend}}{100}$
• Total Dividend is always calculated on the nominal value, not the market value.
• Return vs Dividend: Dividend is income received; return is income as a percentage of investment.
• Return Formula: $\text{Return} = \frac{\text{Dividend}}{\text{Investment}} \times 100\%$
• Market Value with Return: If return and dividend rate are given, $\text{Market Value} = \frac{\text{Nominal Value} \times \text{Dividend Rate}}{\text{Return Rate}}$

Proportion, Ratio, and Continued Proportion

Proportion problems often use compounding and dividing properties.

• Continued Proportion: If $a$, $b$, $c$ are in continued proportion, then $a/b = b/c$ or $b^2 = ac$.
• Compounded and Divided Proportion: If $a/b = c/d$, then $(a+b)/(a-b) = (c+d)/(c-d)$.
• Mean Proportion: For $a$, $b$, $c$ in continued proportion, $b$ is the mean proportion between $a$ and $c$.
• Third Proportional: If $a$, $b$, $c$ are in proportion, $c$ is the third proportional to $a$ and $b$.

Matrices

Matrix multiplication is only possible if the number of columns in the first matrix equals the number of rows in the second. The order of the resulting matrix is (rows of first) × (columns of second).

• Null Matrix: All elements are zero.
• Order Determination: For $A$ of order $2 \times 2$ and $B$ of order $2 \times 1$, $A \times B$ is $2 \times 1$.
• Solving for Variables: Set up equations by equating the product to the given matrix and solve for unknowns.
• Identity Matrix ($I$): Diagonal elements are 1, others are 0.

Arithmetic Progression (AP) and Geometric Progression (GP)

AP and GP are sequences with constant differences or ratios, respectively.

• AP nth Term: $a_n = a + (n-1)d$
• AP Sum of n Terms: $S_n = \frac{n}{2}[2a + (n-1)d]$
• Finding Terms Divisible by Numbers: Use LCM to determine the sequence.
• GP nth Term: $a_n = ar^{n-1}$
• GP Sum (finite, $r > 1$): $S_n = a \frac{r^n - 1}{r - 1}$
• Finding Common Ratio: Divide two consecutive terms.
• Solving for Terms: Use given sums or terms to set up equations and solve for $a$ and $r$.

Linear Inequations

Linear inequations are solved by isolating $x$ and applying rules for inequalities.

• Solving Steps:
  1. Isolate $x$ by moving terms.
  2. When multiplying/dividing by a negative, reverse the inequality sign.
  3. Express solution as an interval.
• Number Line Representation: Use open/closed circles for strict/weak inequalities and shade the solution region.

Quadratic Equations and Factor Theorem

Quadratic equations can be solved by factorization, quadratic formula, or by using the discriminant.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Discriminant ($D$): $D = b^2 - 4ac$
  • If $D \geq 0$, real roots exist.
• Factor Theorem: If $x + a$ is a factor of a polynomial, substituting $x = -a$ gives zero.
• Remainder Theorem: Substituting $x = -a$ into the polynomial gives the remainder when divided by $x + a$.
• Finding Value to Add/Subtract for Divisibility: Substitute the required $x$ value, set the result plus $k$ to zero, and solve for $k$.

Assertion and Reasoning in MCQs

Assertion and reasoning questions require checking the truth of both statements and whether the reason correctly explains the assertion.

• Both True and Reason is Correct Explanation: Mark accordingly if both statements are true and the reason explains the assertion.
• Interest in RD: Interest earned equals maturity value minus total amount deposited.

Common Pitfalls and Exam Tips

Certain chapters like GST, Banking, Linear Inequations, Matrices, and Shares & Dividends are high-yield and should not be skipped. Always check for:

• Whether the question asks for amount, rate, or value.
• Whether to use nominal or market value.
• Whether to use AP or GP formulas based on the sequence.
• Proper handling of negative signs in inequalities.
• Correct application of compounding/dividing properties in proportion problems.
• For matrices, ensure compatibility before attempting multiplication.

Key formulas and methods are repeatedly tested; practice their direct application and avoid overcomplicating with unnecessary steps.