ICSE CLASS X MATHEMATICS
Time Allowed: 1.5 Hours | Maximum Marks: 40
Instructions: This paper is divided into two sections. Section A (20 Marks) is compulsory. Section B (20 Marks) contains three questions; answer any two.
SECTION A (20 Marks)
(Attempt all questions from this section)
Question 1: Multiple Choice Questions (1 × 5 = 5 Marks)
i. A manufacturing company in Maharashtra sells an automated machine to a dealer in the same state for ₹50,000. The dealer sells it to a consumer in Goa at a profit of ₹10,000. If the GST rate is 18%, the Integrated GST (IGST) paid by the dealer to the government is:
(a) ₹1,800
(b) ₹10,800
(c) ₹0
(d) ₹9,000
ii. Rohan opens a Recurring Deposit (RD) account in a bank for 2 years. If the bank increases its interest rate midway through the tenure, which of the following statements is true regarding his maturity value?
(a) Only the interest on the remaining monthly installments will change.
(b) The interest on all 24 installments will be recalculated from month one.
(c) The maturity value remains completely unaffected as RDs have fixed returns.
(d) The total principal deposited changes.
iii. Given the linear inequation $x \in W$ (Whole Numbers), the solution set for $-3 < 2x - 1 \le 5$ contains how many elements?
(a) 3
(b) 4
(c) 5
(d) Infinite
iv. If $A$ is a matrix of order $2 \times 3$ and $B$ is a matrix such that both $AB$ and $BA$ are defined, then the order of matrix $B$ must be:
(a) $2 \times 3$
(b) $3 \times 2$
(c) $3 \times 3$
(d) $2 \times 2$
v. Two similar triangles, $\Delta ABC$ and $\Delta PQR$, have their corresponding heights in the ratio $4 : 9$. The ratio of the area of $\Delta ABC$ to the area of $\Delta PQR$ is:
(a) $2 : 3$
(b) $4 : 9$
(c) $16 : 81$
(d) $81 : 16$
Question 2 (7 Marks)
(a) A shopkeeper buys a smart television from a wholesaler for ₹40,000. He marks up the price by 25%. A customer buys this TV from the shopkeeper and receives a 10% discount on the marked price. If the intra-state rate of GST is 28%, find:
The total GST paid by the shopkeeper to the State Government (SGST).
The final amount (inclusive of tax) paid by the consumer. [4]
(b) Solve the following inequation and represent the solution set on a real number line: [3]
$$-2 \frac{2}{3} \le x + \frac{1}{3} < 3 \frac{1}{3}, \quad x \in R \text{ (Real Numbers)}$$
Question 3 (8 Marks)
(a) Mrs. Kapoor opened a Recurring Deposit account in State Bank of India for a period of 3 years. If the bank pays interest at the rate of 8% per annum and she receives ₹23,610 as interest at the time of maturity, find:
The monthly installment amount ($P$).
The total maturity value of the account. [4]
(b) Find the values of $x$ and $y$ if they satisfy the following matrix equation: [4]
$$\begin{pmatrix} 2 & x \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \end{pmatrix} + \begin{pmatrix} y \\ -2 \end{pmatrix} = \begin{pmatrix} 14 \\ 2 \end{pmatrix}$$
SECTION B (20 Marks)
(Attempt any two questions from this section)
Question 4 (10 Marks)
(a) Case Study Based (GST & Banking):
An IT consultant, Ananya, manages her corporate expenses through a business bank account. She purchases office furniture listed at ₹1,00,000 from a local vendor. She is offered two choices:
Option A: A direct discount of 20%, with GST charged at 18% on the discounted price.
Option B: No discount, but the vendor offers to absorb 5% of the total 18% GST (effectively charging only 13% GST).
Mathematically analyze which option saves Ananya more money and by how much.
If she deposits her savings from the better option into a 1-year Recurring Deposit paying 6% simple interest per annum with a ₹500 monthly installment, will her total savings cover the interest generated? Justify. [5]
(b) In the given figure, $DE \parallel BC$. If $AD = x$, $DB = x-2$, $AE = x+2$, and $EC = x-1$:
Set up an algebraic equation to find the value of $x$.
Find the ratio of $\text{Area}(\Delta ADE) : \text{Area}(\Delta ABC)$. [5]
Question 5 (10 Marks)
(a) Given the matrices $A = \begin{pmatrix} 2 & -1 \\ 2 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} -3 & 2 \\ 4 & 0 \end{pmatrix}$. Find a matrix $C$ such that $A^2 + C = 3B$. [4]
(b) A person has a choice of investing in two different domain-linked accounts. Account X is a linear system represented by the inequation group:
$$3x - 2 < 7 \quad \text{and} \quad 4x + 1 \ge -7, \quad x \in I \text{ (Integers)}$$Account Y is defined by the solution set of:
$$-1 \le 2x - 3 < 5, \quad x \in W \text{ (Whole Numbers)}$$Find the intersection of the solution sets of Account X and Account Y. List all common integral values. [6]
Question 6 (10 Marks)
(a) High-Level Similarity Application:
A vertical stick of length 1.2 meters casts a shadow 80 cm long on the ground. At the same time, a multi-story building nearby casts a shadow 40 meters long.
Using the concept of similar triangles, calculate the height of the building.
If a drone is hovering exactly midway between the top of the stick and the top of the building horizontally, what is its vertical height from the ground? [5]
(b) Let $M$ be a $2 \times 2$ matrix. It is given that:
$$M \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} \quad \text{and} \quad M \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$$Determine the matrix $M$.
Evaluate $M^2$. [5]