Explanation: For independent events, the occurrence of one does not affect the probability of the other. The probability of both happening together (Intersection) is the product of their individual probabilities. This is the Multiplication Rule of Probability. If events are dependent, conditional probability is used.
42
What is the value of i² (iota squared)?
✅ Correct Answer:-1
Explanation: In complex numbers, the imaginary unit 'i' is defined as the square root of -1 (i = √-1). Therefore, squaring both sides gives i² = -1. This property allows us to solve equations like x² + 1 = 0, which have no solutions in the real number system.
43
Which vector has a magnitude of 1?
✅ Correct Answer:Unit Vector
Explanation: A Unit Vector is defined as a vector that has a magnitude (length) of exactly 1. It is used to specify a direction without describing length. It is usually denoted by a 'hat' symbol (e.g., â). Any non-zero vector can be converted into a unit vector by dividing it by its own magnitude.
44
Find the 10th term of the sequence 2, 4, 6, 8...
✅ Correct Answer:20
Explanation: This is an Arithmetic Progression (AP) where the first term a=2 and the common difference d=2. The formula for the nth term is an = a + (n-1)d. Putting n=10: a10 = 2 + (10-1)2 = 2 + 18 = 20. Basically, it's just the table of 2.
45
The transpose of a Row Matrix is a:
✅ Correct Answer:Column Matrix
Explanation: Transposing a matrix means switching its rows and columns. A Row Matrix has dimensions 1 × n. When transposed, it becomes n × 1, which is a Column Matrix. For example, the transpose of [1 2 3] is a vertical column with entries 1, 2, 3.
46
The sum of an infinite Geometric Series exists only if the common ratio 'r' is:
✅ Correct Answer:|r| < 1
Explanation: An infinite Geometric Series converges (has a finite sum) only if the absolute value of the common ratio |r| is strictly less than 1 (-1 < r < 1). The formula for the sum is S = a / (1 - r). If |r| ≥ 1, the terms keep growing or staying constant, so the sum goes to infinity (diverges).
47
Two matrices A and B can be multiplied only if:
✅ Correct Answer:Columns of A = Rows of B
Explanation: Matrix multiplication AB is possible only if the number of Columns in the first matrix (A) is equal to the number of Rows in the second matrix (B). If A is of order m×n and B is of order n×p, the resulting matrix will have order m×p. Otherwise, the product is undefined.
48
Integration is the reverse process of:
✅ Correct Answer:Differentiation
Explanation: Integration is often called 'anti-differentiation' because it reverses the process of finding a derivative. If differentiation gives you the rate of change (slope), integration gives you the accumulation of quantities (area under the curve). The Fundamental Theorem of Calculus links these two operations.
49
The value of Limit (sin x) / x as x approaches 0 is:
✅ Correct Answer:1
Explanation: This is a standard limit in calculus: lim (x→0) (sin x / x) = 1. Even though direct substitution results in 0/0 (indeterminate form), applying L'Hôpital's Rule or geometric proofs shows the value approaches 1. This limit is crucial for finding derivatives of trigonometric functions.
50
What is the derivative of Sin(x)?
✅ Correct Answer:Cos(x)
Explanation: In Calculus, the derivative measures the rate of change of a function. The derivative of the sine function, d/dx(sin x), is Cos(x). This is a fundamental rule in differentiation. Conversely, the derivative of Cos(x) is -Sin(x). These rules are essential for solving problems involving rates of change and motion.
51
The intersection of two sets A and B is empty. They are called:
✅ Correct Answer:Disjoint sets
Explanation: Two sets are called Disjoint if they have no elements in common. Their intersection is the empty set (null set). For example, if A={1,2} and B={3,4}, A ∩ B = ∅. They do not overlap in a Venn diagram.
52
What is the derivative of x²?
✅ Correct Answer:2x
Explanation: The power rule for differentiation states that if f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. Here n=2. So, the derivative is 2x²⁻¹ = 2x¹. The derivative represents the slope of the tangent to the curve at any point x.
53
Factorize: x² - 4
✅ Correct Answer:(x-2)(x+2)
Explanation: This is a difference of two squares: a² - b². The formula is (a-b)(a+b). Here, x² - 2². So the factors are (x-2)(x+2). Expanding this back: x(x) + 2x - 2x - 4 = x² - 4. This confirms the factors are correct.
54
Which is greater: 1/3 or 0.3?
✅ Correct Answer:1/3
Explanation: Convert 1/3 to decimal. 1 ÷ 3 = 0.3333... (recurring). Compare this to 0.3 (which is 0.3000...). Since 0.333... > 0.300, 1/3 is greater than 0.3. The difference is small (0.033...) but mathematically significant.
55
If a = 3 and b = -2, find a² + b².
✅ Correct Answer:13
Explanation: Substitute the values: a² = 3² = 9. b² = (-2)² = 4 (negative becomes positive when squared). Now add them: 9 + 4 = 13. A common error is calculating -2² as -4, which is incorrect; (-2)×(-2) is +4.
56
The sum of exterior angles of any polygon is:
✅ Correct Answer:360°
Explanation: Regardless of the number of sides, the sum of the exterior angles of any convex polygon (taken one at each vertex) is always 360 degrees. This property is very useful for finding the number of sides of a regular polygon given an exterior angle.
57
Convert 25% to a decimal.
✅ Correct Answer:0.25
Explanation: To convert a percentage to a decimal, divide by 100 or move the decimal point two places to the left. 25 / 100 = 0.25. 'Percent' literally means 'per 100'. So 25% is 25 parts out of 100.
58
Find the distance between points (0,0) and (3,4).
✅ Correct Answer:5
Explanation: Use the distance formula: d = √((x2-x1)² + (y2-y1)²). Here, (x1,y1)=(0,0) and (x2,y2)=(3,4). d = √((3-0)² + (4-0)²) = √(3² + 4²) = √(9 + 16) = √25 = 5. This is a 3-4-5 Pythagorean triangle.
59
Which represents the Identity Matrix?
✅ Correct Answer:[1 0; 0 1]
Explanation: An Identity Matrix (denoted I) is a square matrix with 1s on the main diagonal (top-left to bottom-right) and 0s everywhere else. Multiplying any matrix by the Identity Matrix results in the original matrix unchanged, similar to multiplying a number by 1.
60
If x/2 = 4, then x = ?
✅ Correct Answer:8
Explanation: To isolate x, multiply both sides by 2 (the inverse of dividing by 2). x = 4 × 2. Therefore, x = 8. Checking the answer: 8 / 2 = 4, which is correct. This is a one-step linear equation.