JAC Board 2024 Class 12 Math Questions

Overview of the Video

• Solving the previous year's questions is beneficial for understanding the question pattern and securing approximately 10 marks in the exam.
• Solving the model sets of 2023 and 2024, helps secure an additional 5 marks.
• Up to 15 marks can be secured by doing so.
• Students should review Sharma Gas solutions for concept clarity.

Function Analysis and Types

• To check if the functions are one-one (injective) and onto (surjective).
• If the function is from real numbers to real numbers, use a graph.
• If not real to real (e.g., natural to natural, integer to integer), use an arrow diagram.

Example Function and Arrow Diagram Illustration

• Function defined as f(n) = 2n + 3, from natural numbers to natural numbers.
• Arrow diagrams are used due to the natural to natural mapping of the functions.
• f(1) = 5, f(2) = 7, f(3) = 9, indicating that each input maps to a unique output.
• The function is one-one since each element in the domain maps to a unique element in the co-domain.
• The function is not onto because some elements in the co-domain (e.g., 1, 2, 3, 4) are not mapped to by any element in the domain.
• The function is one-one but not onto.

Inverse Functions

• Given a function f(x), the inverse function is denoted as f⁻¹(x).
• If y = f(x), then x = f⁻¹(y).
• For a function f(x) = x², finding the inverse involves solving for x.
• In this case, x = ±√y, and thus f⁻¹(y) = ±√y.
• When the function is from Rationals to Rationals, both positive and negative solutions are valid.
• Example: f⁻¹(9) = ±3.
• If the domain/codomain is Natural number, you only use the +3 result.

Trigonometric Inverse

• sin⁻¹(-x) = -sin⁻¹(x)
• cosec⁻¹(-x) = -cosec⁻¹(x)
• tan⁻¹(-x) = -tan⁻¹(x)
• cos⁻¹(-x) = π - cos⁻¹(x)
• cot⁻¹(-x) = π - cot⁻¹(x)
• sec⁻¹(-x) = π - sec⁻¹(x)
• A worked example for solving cosec⁻¹(-√2) is provided.
• An example of how to solve cos⁻¹(-1/√2 ) is provided

Solving Matrix Equations

• In the scenario where two matrices are equal, their corresponding elements are also equal.
• Summing the variables x, y, and z requires solving for the individual values first, then addition.

Inverse Matrices

• If matrices A and B are inverses of each other, then AB = BA = I, where I is the identity matrix.

Determinant Calculations

• Solving a 3x3 determinant using expansion (plus, minus, plus)
• Remember that after that you need to do the following operation
• ad - bc

Cofactors

• When taking cofactors, add the position of aij.
• If the result is even, there is no sign change.
• If the result is odd, there is a sing change.
• Example of solving minor and cofactors for the 3x3

Integration

• Discusses the derivatives of exponential functions of the form a^x.
• d/dx (a^x) = a^x lna
• Integration formula for x^n is x^(n+1) / n+1
• When integrating an e^x, the original exponent in the expression does not change

Trigonometric Integration

• sin^2(x) and cos^2(x) are very common trig identities.
• The function you use as substitute will depend on what trigger function you are using (sin, cos, cot, tan)
• To solve one divided by a quadratic, recall the following integration formulas:
• One divided by x^2 +a^2 = 1/a *tan-1 (x/a)
• One divided by x^2 - a^2 = 1/2a * log (x-a)/(x+a)
• When integrating 1/x with limits a and b: Ln(a) - Ln(b) = Ln (a/b)

Integration Properties

• There's properties where you can set it to 0 when it's integrate from -a to +a
• But not from 0 to a

Deriving one divided by x

• If you want to take the derivative, all you do is add one to the current power
• Put the amount of derivative's original power on the top
• Do not forget a negative sign
• Example: 1/x^2
• d/dx 1/x^2 = -2/ x^3

Logarithmic Derivative

• When doing a logarithmic derivative, its just one divided by the original expression

Log x

• When doing the derivative of Log(x) = 1/x

d/dx f(g(x))

• To solve, the following steps
• d/dx f(g(x)) = f'(g(x)) * g'(x)

Differential Equations

• Variable Separable
• Homogeneous
• Always start with variable separable
• Remember that integration constant is always plus
• The integration constant should ALWAYS have Ln added IF there is a Ln variable on the left / right

Integrating Factor

• If you want to take the integrating factor of a differential equation, its just e to the power of p
• e^-1 * dx = e^-x

Vector Analysis

• Dot product rule
• Dot product by definition = Magnitude_A * Magnitude_B * Cos ( Angle )
• Cross Product by definition = Magnitude_A * Magnitude_B * Sin( Angle )
• Projection Formula A onto B = A dot B / Magnitude (B)

Vector Theory and Formulas

• The text offers a concise summary for direct formula application to solve vector problems.
• The problem involves finding theta given magnitudes of vectors a and b.
• The formula to remember is related to finding the cosine of the angle between vectors.
• Given |a x b| = 1 and |a| * |b| = 1 * 2 = 2, then cos(theta) = 1/2.
• Because cos(θ) = 1/2, θ can be found; the solution is option 3.
• θ = cos^(-1)(1/2), and in radians, it's π/3 (60 degrees).

Cross Product

• The text discusses solving cross product problems using a circle diagram with i, j, k.
• For cross products, create a circle with i, j, and k in that order.
• Moving counter-clockwise is positive, while clockwise is negative.
• i x j = k (positive, counter-clockwise)
• j x k = i (positive, counter-clockwise)
• k x i = j (positive, counter-clockwise)
• k x j = -i (negative, clockwise)
• k x j is provided in the question; its result is -i, making option B correct.

Direction Ratios

• The equation of a line is given in the form (x - x1)/a = (y - y1)/b = (z - z1)/c.
• The direction ratios are a, b, and c.
• In the given question, the direction ratios are 1, 1, and 1.
• If direction cosines were requested, you'd calculate a/√(a²+b²+c²), b/√(a²+b²+c²), and c/√(a²+b²+c²).

Equation of X-Axis

• The equation of the x-axis.
• In 2D, the equation of the x-axis is y = 0.
• In 3D, both y and z are zero on the x-axis.
• The equation for the x-axis in 3D is where y = 0 and z = 0.
• For the y-axis, x and z would be zero.

Mutual Perpendicularity

• Condition for perpendicular lines needs to be understood.
• For perpendicular lines, a1a2 + b1b2 + c1*c2 = 0.
• Given lines are mutually perpendicular, use this condition to find the value of k.
• Given two lines, (1, 1, k) and (k, 2, -2), apply the condition of perpendicularity.
• k + 2 - 2k = 0, which simplifies to -k + 2 = 0, so k = 2.
• Option D is the correct answer.

Distance of a Point from X-Axis.

• The question comes from class 11th material.
• Formula for distance between two points and conceptualizing points on the x-axis are key.
• The point is p(a, b, c) and we're finding its distance from the x-axis.
• On the x-axis, the point is (a, 0, 0).
• Applying the distance formula: √((a-a)² + (0-b)² + (0-c)²) = √(b² + c²).

Conditional Probability

• Will certainly be on the exam
• The formula for conditional probability is P(A|B) = P(A ∩ B) / P(B)
• Given P(A ∩ B) = 7/10 and P(B) = 17/20, calculate P(A|B).
• P(A|B) = (7/10) / (17/20) = (7/10) * (20/17) = 14/17, giving option A as correct.

Independent Events

• Focus on the conceptual definition of independent events.
• If A and B are independent, P(A ∩ B) = P(A) * P(B).
• Independent event questions may arise from exercise 13.2.
• Mutually exclusive and exhaustive events should be reviewed from initial theory.