The Officials Mth101 CBT Practice

Explanation
The correct answer is {6,8}. This is because the union of sets A, B, and C includes all the elements that are in any of the sets. In this case, the elements 6 and 8 are not present in any of the sets A, B, or C, so they are included in the union. Therefore, the answer is {6,8}.

Explanation
The correct answer is 4 because the power set of a set with n elements has 2^n elements. In this case, A has 4 elements, so the power set of A will have 2^4 = 16 elements.

Explanation
The correct answer is 1 because when simplifying a degree, it remains the same. Therefore, a degree symbol does not affect the value of the number.

Explanation
To solve the given exponential equation, we can rewrite it as a quadratic equation by substituting 2^x with a variable, let's say y. Therefore, the equation becomes y^2 - 6y + 8 = 0. Factoring this quadratic equation, we get (y-2)(y-4) = 0. Setting each factor equal to zero, we find that y = 2 or y = 4. Substituting back 2^x for y, we get 2^x = 2 or 2^x = 4. Solving these equations, we find x = 1 or x = 2. Therefore, the correct answer is X = 1 or 2.

Explanation
The inequality |x-3a| > |x-a| can be simplified by considering the different cases for x. When x < a, the inequality becomes -(x-3a) > -(x-a), which simplifies to 3a > a. When a < x < 2a, the inequality becomes x-3a > -(x-a), which simplifies to 4a > 2a. When x > 2a, the inequality becomes x-3a > x-a, which simplifies to 0 > 2a. However, when x < 2a, the inequality holds true. Therefore, the answer is x < 2a.

Explanation
The correct answer is 12 - 5n. This can be determined by observing the pattern in the sequence. Each term is obtained by subtracting 5 from the previous term. The first term is 7, so the sequence can be represented as 7, 7 - 5, 7 - 2(5), 7 - 3(5), and so on. Simplifying this expression gives us 12 - 5n, which represents the nth term of the sequence.

Explanation
The common difference between successive terms in an arithmetic sequence is found by subtracting the first term from the second term. In this case, the first term is 5 and the 32nd term is -119. Subtracting these two terms gives us a difference of -124. Therefore, the common difference between the successive terms is -124.

Explanation
The given series consists of numbers that are multiples of 5, starting from 5 and increasing by 5 each time. To find the sum of the series, we can use the formula for the sum of an arithmetic series: Sn = (n/2)(a + l), where Sn is the sum, n is the number of terms, a is the first term, and l is the last term. In this case, n = 8 (number of terms), a = 5 (first term), and l = 40 (last term). Plugging these values into the formula, we get Sn = (8/2)(5 + 40) = 4 * 45 = 180. Therefore, the correct answer is 180.

Explanation
The sum of the 3rd term and the 7th term of the sequence is 6, and their product is 8. This means that the 3rd term and the 7th term are the roots of a quadratic equation. By solving the equation x^2 - 6x + 8 = 0, we find that the roots are x = 2 and x = 4. Therefore, the sequence is an arithmetic sequence with a common difference of 2. The sum of the first 16 terms of an arithmetic sequence can be found using the formula n/2 * (2a + (n-1)d), where n is the number of terms, a is the first term, and d is the common difference. Plugging in the values, we get 16/2 * (2*26 + (16-1)*2) = 8 * (52 + 30) = 8 * 82 = 656. Therefore, the sum of the first 16 terms of the sequence is 656.

Explanation
To find the sum of all the integers between 100 and 400 that are divisible by 7, we need to first determine the first and last number in this range that are divisible by 7. The first number divisible by 7 is 105, and the last number divisible by 7 is 399. We can then use the formula for the sum of an arithmetic series: sum = (n/2)(first term + last term), where n is the number of terms. In this case, n = (399-105)/7 + 1 = 42. Plugging in the values, we get sum = (42/2)(105+399) = 21(504) = 10584. Therefore, the correct answer is 10836.

Explanation
The given sequence is a geometric sequence where each term is obtained by dividing the previous term by 2. Starting with 8, we divide by 2 successively to get 4, 2, and so on. To find the eighth term, we divide the initial term 8 by 2 seven times. This can be written as 8/(2^7) which simplifies to 1/16. Therefore, the eighth term of the sequence is 1/16.

Explanation
The common ratio of a geometric progression (GP) can be found by dividing any term in the progression by its preceding term. In this case, we can divide the 6th term (-32) by the 3rd term (108) to find the common ratio. -32/108 simplifies to -2/3, so the common ratio of this GP is -2/3.

Explanation
The correct answer is 0.5 or -4. To solve the equation 2x^2 + 7x - 4 = 0, we can use the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula x = (-b ± √(b^2 - 4ac)) / (2a). In this case, a = 2, b = 7, and c = -4. Plugging these values into the formula, we get x = (-7 ± √(7^2 - 4(2)(-4))) / (2(2)). Simplifying further, we get x = (-7 ± √(49 + 32)) / 4, which becomes x = (-7 ± √81) / 4. Taking the square root of 81 gives us x = (-7 ± 9) / 4. This gives us two possible solutions: x = (-7 + 9) / 4 = 2/4 = 0.5, and x = (-7 - 9) / 4 = -16/4 = -4. Therefore, the correct answer is 0.5 or -4.

Explanation
To find the minimum value of the quadratic equation y = 3x^2 - 2x + 1, we can use the formula for the x-coordinate of the vertex, which is given by -b/2a. In this equation, a = 3 and b = -2. Plugging these values into the formula, we get x = -(-2)/(2*3) = 2/6 = 1/3. To find the corresponding y-coordinate, we substitute this value of x back into the equation: y = 3(1/3)^2 - 2(1/3) + 1 = 3/9 - 2/3 + 1 = 1/3 - 2/3 + 1 = 0. Therefore, the minimum value of y is 2/3.

Explanation
The sum of the roots of a quadratic equation can be found by dividing the coefficient of the linear term by the coefficient of the quadratic term, with the opposite sign. In this case, the coefficient of the linear term is -1 and the coefficient of the quadratic term is 3. Therefore, the sum of the roots is -(-1/3) which simplifies to 1/3.

Explanation
Pascal's triangle is used to solve problems in Binomial expansion. It is a triangular array of numbers in which each number is the sum of the two numbers directly above it. The triangle is used to find the coefficients of the terms in the expansion of a binomial raised to a power. Each row of the triangle represents the coefficients of the terms in the expansion of (a + b)^n, where n is the row number. Pascal's triangle provides a systematic way to determine these coefficients without having to perform lengthy calculations.

Explanation
The word "SUCCESSION" has 10 letters, including 3 "S"s, 2 "C"s, and 2 "E"s. To find the number of ways it can be arranged, we can use the formula for permutations of objects with repetition. The formula is n! / (n1! * n2! * n3! * ...), where n is the total number of objects and n1, n2, n3, ... are the frequencies of each repeated object. In this case, the formula becomes 10! / (3! * 2! * 2!). Calculating this gives us 302400 ways.

Explanation
There are two possible scenarios for two people sitting next to each other in a round table. First, we can consider them as a single entity, which means treating them as one seat. In this case, there are 6 ways to arrange the group of two people and 4! ways to arrange the remaining 4 people. Second, we can consider them as separate entities sitting next to each other. In this case, there are 6 ways to choose the first person, and 5 ways to choose the second person. For each choice, there are 4! ways to arrange the remaining 4 people. Adding up both scenarios, we get a total of 30 ways.

Explanation
The number of ways five men can be chosen out of seven for the job can be calculated using the combination formula. The formula for combination is nCr = n! / (r!(n-r)!), where n is the total number of options and r is the number of options to be chosen. In this case, n = 7 and r = 5. Plugging these values into the formula, we get 7C5 = 7! / (5!(7-5)!) = 7! / (5!2!) = (7x6x5x4x3x2x1) / ((5x4x3x2x1)(2x1)) = 21. Therefore, there are 21 ways to choose five men out of seven for the job.

Explanation
Plank's method is not a valid rule/method for solving simultaneous equations in matrices. Plank's method is used for finding the determinant of a matrix, not for solving systems of equations. The other three methods mentioned - inverse method, Gauss elimination method, and Crammer's rule - are commonly used techniques for solving simultaneous equations in matrices.

Explanation
To simplify the given expression, we need to combine the real parts and the imaginary parts separately. The real parts are 6 and 7, so when we add them together, we get 13. The imaginary parts are 3i and -i, so when we add them together, we get 2i. Therefore, the simplified expression is 13 + 2i.

Explanation
The given expression involves adding and subtracting complex numbers. To simplify, we can combine the real parts and the imaginary parts separately. The real parts are (7 + 3 - 5) = 5, and the imaginary parts are (2i - 4i - i) = -3i. Therefore, the simplified form of the expression is 5 - 3i.

Explanation
To find the product of (3 + i) and (4 - 2i), we use the distributive property. Multiplying the real parts gives us 3 * 4 = 12, and multiplying the imaginary parts gives us i * (-2i) = -2i^2 = -2 * (-1) = 2. Therefore, the product is 12 + 2i. Simplifying further, we get 14 + 2i. However, since the question asks for the expression in the form of a + ib, we write it as 14 - 2i.

Explanation
To convert degrees to radians, we need to multiply the number of degrees by π/180. In this case, we have 100 degrees. Multiplying 100 by π/180 gives us 100π/180, which simplifies to 5π/9. Therefore, 100 degrees is equal to 5π/9 radians.

Explanation
To express 2pi/5 in degrees, we need to convert the given radian measure to degrees. Since there are 2pi radians in a full circle (360 degrees), we can set up a proportion: 2pi/5 = 360/x, where x represents the number of degrees. Cross multiplying, we get 2pi * x = 5 * 360. Simplifying, we have 2pi * x = 1800. Dividing both sides by 2pi, we find x = 1800/(2pi). Evaluating this expression using a calculator, we get x ≈ 114.59 degrees. Since 72 degrees is the closest option to 114.59 degrees, it is the correct answer.