3142 AI Curriculum Vault • 20-Marks Model Question Papers

ICSE & CBSE AI Sample Question Papers

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CISCE / ICSE BOARD SPECIMEN PATTERN • 3142 AI GENERATED

ICSE Class 10 Mathematics — Unit Test Paper

Commercial Mathematics & Algebra • Time Allowed: 45 Minutes • Maximum Marks: 20

SECTION A (Compulsory — 6 Marks) 3 Questions x 2 Marks

Q1. Goods and Services Tax (GST): [2 Marks]

A shopkeeper in Kolkata buys an article with marked price \(\text{₹} 8,000\) at a discount of \(10\%\) from a wholesaler in Kolkata. If the rate of GST is \(18\%\), find:
(i) The amount of CGST and SGST paid by the shopkeeper.
(ii) The total price paid by the shopkeeper inclusive of tax.

Q2. Recurring Deposit Scheme: [2 Marks]

Rohan deposited \(\text{₹} 600\) per month in a bank for \(2\) years in a Recurring Deposit account. If the rate of interest is \(10\%\) per annum, find the maturity value of the account.

Q3. Linear Inequations: [2 Marks]

Solve the following inequation and represent the solution set on the real number line: \[ -3 \le 2x - 1 < 5, \quad x \in \mathbb{R} \]

SECTION B (Structured Problems — 14 Marks) Questions Q4 (4M), Q5 (5M), Q6 (5M)

Q4. Matrix Algebra & Multiplicative Identity: [4 Marks]

Given matrices \(A = \begin{bmatrix} 2 & -1 \\ 1 & 3 \end{bmatrix}\) and \(B = \begin{bmatrix} 4 & 1 \\ 0 & 2 \end{bmatrix}\). Find matrix \(X\) such that: \[ 2A + X = 3B \] Hence, verify whether \(X\) has an inverse.

Q5. Quadratic Equation (Word Problem): [5 Marks]

A fast train takes \(3\) hours less than a slow train for a journey of \(600\text{ km}\). If the speed of the slow train is \(10\text{ km/h}\) less than that of the fast train, find the speeds of the two trains.

Q6. Remainder & Factor Theorem: [5 Marks]

Using remainder theorem, factorize completely the polynomial: \[ f(x) = 2x^3 + x^2 - 13x + 6 \]

CBSE STANDARD PATTERN • 3142 AI GENERATED

CBSE Class 10 Mathematics — Periodic Assessment Paper

Real Numbers, Polynomials & Linear Equations • Time Allowed: 45 Minutes • Maximum Marks: 20

SECTION A: Objective & Short Answer (8 Marks) 4 Questions x 2 Marks

Q1. Fundamental Theorem of Arithmetic: [2 Marks]

Given that \(\text{HCF}(306, 657) = 9\), find \(\text{LCM}(306, 657)\) using prime factorization relation.

Q2. Polynomial Zeroes & Coefficients: [2 Marks]

Find a quadratic polynomial whose zeroes are \(\alpha = 3 + \sqrt{2}\) and \(\beta = 3 - \sqrt{2}\).

Q3. Consistency of Linear Equations: [2 Marks]

For what value of \(k\) will the following system of linear equations have infinitely many solutions? \[ 2x + 3y = 7, \quad (k-1)x + (k+2)y = 3k \]

Q4. Prove Irrationality: [2 Marks]

Prove by contradiction that \(\sqrt{5}\) is an irrational number.

SECTION B: Structured & Case-Based (12 Marks) Questions Q5 (6M), Q6 (6M)

Q5. Upstream & Downstream Motion: [6 Marks]

A motor boat whose speed is \(18\text{ km/h}\) in still water takes \(1\) hour more to go \(24\text{ km}\) upstream than to return downstream to the same spot. Find the speed of the stream.

Q6. Applied Geometry & Quadratic Roots: [6 Marks]

The diagonal of a rectangular field is \(60\text{ meters}\) more than the shorter side. If the longer side is \(30\text{ meters}\) more than the shorter side, find the sides of the field.

ISC & CBSE CLASS 12 MASTER PATTERN • 3142 AI GENERATED

Class 12 Mathematics — Calculus Mastery Paper

Derivatives, Integrals & Optimization • Time Allowed: 45 Minutes • Maximum Marks: 20

SECTION A: Core Proofs & Integrals (8 Marks) 2 Questions x 4 Marks

Q1. Indefinite Integration by Partial Fractions: [4 Marks]

Evaluate the integral: \[ \int \frac{x^2 + 1}{(x-1)(x-2)(x-3)} \, dx \]

Q2. Definite Integral Property: [4 Marks]

Evaluate using properties of definite integrals: \[ I = \int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \]

SECTION B: Maxima & Minima Optimization (12 Marks) Questions Q3 (6M), Q4 (6M)

Q3. Cylinder Inscribed in a Sphere: [6 Marks]

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius \(R\) is \(\frac{2R}{\sqrt{3}}\). Also find the maximum volume.

Q4. Linear Differential Equation: [6 Marks]

Find the particular solution of the differential equation: \[ \frac{dy}{dx} + y \cot x = 2x + x^2 \cot x, \quad \text{given } y = 0 \text{ when } x = \frac{\pi}{2} \]

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ICSE & CBSE AI Sample Question Papers

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ICSE Class 10 Mathematics — Unit Test Paper

CBSE Class 10 Mathematics — Periodic Assessment Paper

Class 12 Mathematics — Calculus Mastery Paper