January 25, 2025 by akhilendra
Ace Your Board Exams: Class 12 Math Sample Questions for Practice
Class 12 Sample Questions
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Class 12 Maths Practice Sample questions
Here are 50 original, high-quality questions for the Class 12 CBSE Maths exam, distributed as specified, with a focus on complex concepts and logical progression:
Algebra (10 Questions)
Matrices: Given matrices A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], find a matrix X such that 2A + 3X = B.
Matrices: If A = [[cosθ, sinθ], [-sinθ, cosθ]], verify that A' * A = I, where A' is the transpose of A and I is the identity matrix.
Determinants: Without expanding, prove that | a b c | |a+2x b+2y c+2z | = 0
| x y z |
|u v w|Determinants: If A is a square matrix of order 3 and |A| = 5, find the value of |adj(A)|.
Determinants: Using properties of determinants, show that | a b c | = (a-b)(b-c)(c-a)(a+b+c)
| a² b² c²|
| b+c c+a a+b |Inverse: Using elementary row operations, find the inverse of the matrix A = [[2, 1], [5, 3]].
Inverse: If A = [[2, -1, 1], [-1, 2, -1], [1, -1, 2]], verify that A³ - 6A² + 9A - 4I = 0, and hence find A⁻¹.
System of Equations: Solve the following system of linear equations using the matrix method:
2x + y - z = 1
x - 2y + z = 4
3x + y + 2z = 7Matrices: For the given matrix A = [ [1, -2, 3], [0, 4, -1], [2, 1, 5]], find a matrix X such that AX = I where I is an identity matrix.
Matrices: Determine the values of ‘a’ and ‘b’ such that the matrix A = [ [a, 2, 1], [b, a, 2], [1, -1, a] ] is symmetric.
Calculus (10 Questions)
Continuity and Differentiability: Examine the continuity and differentiability of the function f(x) = |x - 2| + |x + 2| at x = 2 and x = -2.
Application of Derivatives: Find the equation of the tangent to the curve y = x³ - 2x² + x - 1 at the point where x = 2.
Application of Derivatives: A particle moves along the curve 6y = x³ + 2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.
Maxima and Minima: Find the absolute maximum and minimum values of the function f(x) = 2x³ - 15x² + 36x + 1 in the interval [1, 5].
Integrals: Evaluate ∫(x² + 1) / (x⁴ + 1) dx.
Integrals: Evaluate ∫₀^(π/2) sin²x cos²x dx.
Definite Integrals: Evaluate ∫₂⁴ (x-2) / (x-2 + √x-2) dx
Differential Equations: Solve the differential equation: x dy/dx + y = x log x.
Differential Equations: Form the differential equation of the family of circles touching the y-axis at the origin.
Application of Integrals: Find the area bounded by the curves y = x^2 and y = |x|.
Vectors and 3D Geometry (8 Questions)
Vectors: If vectors a = 2i + j - k and b = i - 2j + 3k, find a unit vector perpendicular to both a and b.
Vectors: Find the projection of the vector a = 2i - j + k on the vector b = i + 2j + 2k.
3D Geometry: Find the angle between the line (x-2)/2 = (y-1)/-3 = (z+2)/1 and the plane x+2y-z=5.
3D Geometry: Find the equation of the plane passing through the points (1, 2, 3), (2, -1, 1) and (1, 1, -2).
3D Geometry: Find the distance of the point (2, 3, -1) from the plane x + 2y - z = 4.
Vectors: Find the area of the parallelogram whose adjacent sides are given by the vectors a = 2i-j+3k and b = i+3j-k.
3D Geometry: Find the shortest distance between the lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-2)/3 = (y-4)/4 = (z-5)/5.
Vectors: If a,b,c are three vectors such that a+b+c = 0 and |a| = 3, |b|=4, |c|= 5, find the value of a.b + b.c + c.a.
Linear Programming (6 Questions)
Linear Programming: Maximize Z = 5x + 3y subject to the constraints:
x + 2y ≤ 10
2x + y ≤ 8
x - y ≤ 2
x, y ≥ 0Linear Programming: Minimize Z = 2x + 3y subject to:
x + y ≥ 6
x + 2y ≥ 8
2x + y ≥ 10
x, y ≥ 0Linear Programming: A furniture dealer deals in only two items - tables and chairs. He has Rs 50,000 to invest and has storage space of at most 60 pieces. A table costs Rs 2,500 and a chair costs Rs 500. He estimates that from the sale of one table, he can make a profit of Rs 250 and from the sale of one chair, a profit of Rs 75. Assuming that he can sell all the items which he buys, how should he invest his money in order that he may maximize his profit?
Linear Programming: A factory manufactures two types of screws, A and B. Each screw of type A requires 4 minutes on the automatic machine and 6 minutes on the hand machine. Each screw of type B requires 6 minutes on the automatic machine and 3 minutes on the hand machine. It is possible to run the automatic machine for at most 4 hours and the hand machine for at most 6 hours on any day. If the profit on a screw of type A is Rs 7 and on a screw of type B is Rs 10, formulate the above problem as a linear programming problem and find the number of each type of screws the company must manufacture in order to maximize the profit.
Linear Programming: Determine the graphical solution for the following LPP.
Maximize z = x + y
Subject to x – y ≤ 10,
x + y ≤ 20,
x ≥ 0, y ≥ 0Linear Programming: Determine the graphical solution for the following LPP.
Minimize z = 2x + y
Subject to 5x + 10y ≤ 50,
x + y ≥ 1,
y ≤ 4
x ≥ 0, y ≥ 0Probability (6 Questions)
Conditional Probability: A die is thrown twice, and the sum of the numbers appearing is observed to be 6. What is the probability that the number 4 has appeared at least once?
Bayes' Theorem: A factory has three machines A, B, and C that produce 20%, 30%, and 50% of the items, respectively. The defect rates for these machines are 2%, 3%, and 5%, respectively. If a randomly selected item is found to be defective, what is the probability that it was produced by machine C?
Random Variables: A random variable X has the following probability distribution:
X 0 1 2 3 4 5 6 P(X) 0.1 k 2k 2k 3k k² 2k² Find (i) k, (ii) P(X < 3), (iii) P(2 < X ≤ 5)
Probability Distribution: Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the number of aces obtained. Find the probability distribution of X and hence the mean and the variance.
Conditional Probability: A bag contains 4 red and 3 black balls. A second bag contains 2 red and 3 black balls. One bag is selected at random and a ball is drawn from it. If the ball is found to be red, find the probability that the ball was drawn from the first bag.
Bayes' Theorem: A test is being conducted in a college. The probability of a student passing the test is 3/5, given that they study. The probability of passing is 1/5 given that they don't study. It is found that 2/3rd of the students study. A student is picked at random. They pass the test. Find the probability that they studied.
Relations and Functions (5 Questions)
Relations: Let R be a relation on the set of natural numbers N, defined as R = {(a, b) : a divides b}. Determine if R is reflexive, symmetric, or transitive. Is R an equivalence relation?
Functions: Show that the function f : R -> R defined by f(x) = 3x - 5 is one-one and onto. Also, find its inverse.
Functions: Consider the functions f : R → R and g : R → R defined as f(x) = 2x + 3 and g(x) = x² + 1. Find (g o f)(x) and (f o g)(x).
Relations: Check if the relation R defined in the set A = {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric, and transitive.
Functions: Let f : R→ R be given by f(x) = |x|. Check if the function is one-one and onto.
Applications of Derivatives (5 Questions)
Increasing and Decreasing Functions: Find the intervals in which the function f(x) = x^4 - 8x^3 + 18x^2 - 16x is strictly increasing or strictly decreasing.
Tangents and Normals: Find the equation of the normal to the curve y = x^3 - 3x^2 + 2x + 1 at the point (2, 1).
Approximation Using Derivatives: Use differentials to find the approximate value of (3.02)².
Maxima and Minima: A rectangle is inscribed in a circle of radius 10 cm. Find the maximum area of the rectangle.
Rate of Change: A ladder 10 m long leans against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2 m/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 m from the wall?
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