Chapter 1: Relations and Functions
Topic: Relations
Subtopic: Reflexive, Symmetric & Transitive
Q. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
Q. Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.
A)
R is reflexive and symmetric but not transitive
B)
R is reflexive and transitive but not symmetric
✓ Correct
C)
R is symmetric and transitive but not reflexive
D)
R is an equivalence relation
💡 Hint:Check reflexivity (a,a for all a), symmetry (a,b => b,a), and transitivity (a,b & b,c => a,c).
✅
Solution:For every a in {1,2,3,4}, (a,a) exists in R, so it is reflexive. (1,2) is in R but (2,1) is not, so it is not symmetric. All conditions for transitivity are met.
Chapter 2: Inverse Trigonometric Functions
Topic: Principal Value Branch
Q. The principal value of sin⁻¹(1/2) is:
A)
π/6
✓ Correct
B)
π/3
C)
π/4
D)
π/2
💡 Hint:Recall the value of sin for which the result is 1/2 in the principal range [-π/2, π/2].
✅
Solution:sin(π/6) = 1/2. Since π/6 lies in the range [-π/2, π/2], the value is π/6.
Chapter 3: Matrices
Topic: Matrix Operations
Q. If A is a square matrix such that A² = A, then (I + A)³ - 7A is equal to:
A)
A
B)
I - A
C)
I
✓ Correct
D)
3A
💡 Hint:Expand (I+A)³ and use the property A² = A repeatedly.
✅
Solution:(I+A)³ = I³ + 3I²A + 3IA² + A³ = I + 3A + 3A + A = I + 7A. Thus, (I+7A) - 7A = I.
Topic: Symmetric Matrices
Q. If A and B are symmetric matrices of the same order, then show that AB – BA is a skew symmetric matrix.
Chapter 4: Determinants
Topic: Adjoint & Inverse
Q. If A is an invertible matrix of order 2, then det(A⁻¹) is equal to:
A)
det(A)
B)
1/det(A)
✓ Correct
C)
1
D)
0
💡 Hint:Use the property det(A * A⁻¹) = det(I).
✅
Solution:det(A * A⁻¹) = det(A) * det(A⁻¹). Since det(I) = 1, we have det(A) * det(A⁻¹) = 1 => det(A⁻¹) = 1/det(A).
Chapter 5: Continuity and Differentiability
Topic: Continuity
Q. The function f(x) = |x| is:
A)
Continuous and differentiable everywhere
B)
Continuous everywhere but not differentiable at x=0
✓ Correct
C)
Neither continuous nor differentiable at x=0
D)
Differentiable at x=0 but not continuous
💡 Hint:Consider the limit of f(x) and the derivative definition at x=0.
✅
Solution:The graph has no breaks, but it has a sharp corner at x=0, making it non-differentiable there.
Topic: Chain Rule
Q. Find the derivative of sin(cos(x^2)) with respect to x.
Chapter 7: Integrals
Topic: By Parts
Q. Evaluate the integral of x * e^x dx using the integration by parts method.
Chapter 10: Vector Algebra
Topic: Dot Product
Subtopic: Angle Between Vectors
Q. Find the angle between two vectors a and b with magnitudes sqrt(3) and 2 respectively, having a.b = sqrt(6).
Chapter 13: Probability
Topic: Bayes Theorem
Subtopic: Conditional Probability Applications
Q. A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag using Bayes Theorem.