Correct answer: 1/2. √2 − 1/√2 = (2 − 1)/√2 = 1/√2, and (1/√2)² = 1/2.
Correct answer: 90. For an odd power n, aⁿ + bⁿ is divisible by a + b. Here 47 + 43 = 90, so the sum is divisible by 90.
Correct answer: 285. A 30-day month beginning on Sunday has 5 Sundays and 25 other days. Total = 5 × 510 + 25 × 240 = 8550, and 8550 ÷ 30 = 285.
Correct answer: 269. After 3 minutes: 428 − 3 × 23 = 359 °C. The next 6 minutes (to minute 9) fall by 6 × 15 = 90 °C, giving 269 °C.
Correct answer: 43.75%. New extension = 0.75d, so new energy = (0.75)² E = 0.5625E. The decrease is 1 − 0.5625 = 0.4375, i.e. 43.75%.
Correct answer: 0.06. 0.1 × 0.6 = 0.06, since 1 × 6 = 6 and there are two decimal places in total.
The correct answer is Same. In geometry, the distance between two parallel lines remains same at all points along their length. Parallel lines are defined as lines in a plane that never meet and are always the same distance apart. This distance is constant and does not decrease or increase. Therefore, the correct answer is Same.
The correct answer is 17/12. To add fractions 2/3 and 3/4, we need a common denominator. The least common denominator (LCD) of 3 and 4 is 12. Convert each fraction: 2/3 = 8/12, 3/4 = 9/12. Adding them: 8/12 + 9/12 = 17/12. 5/7 is the sum of numerators only, 5/12 is incorrect, and 7/12 is also incorrect. Therefore, 17/12 is the correct answer.
The correct answer is 9cm. The area of a triangle is given by the formula: Area = (1/2) × base × height. We are given Area = 126 cm² and base = 28 cm. So: 126 = (1/2) × 28 × height = 14 × height. Therefore, height = 126 ÷ 14 = 9 cm. 7cm, 8cm, and 10cm are incorrect values that do not satisfy the formula. Therefore, 9cm is the correct answer.
The correct answer is Rs. 200. This is a direct proportion problem solved using the unitary method.
Hence, 8 notebooks will cost Rs. 200.
The correct answer is 4/5. When adding fractions with the same denominator, you simply add the numerators and keep the denominator unchanged.
Thus, 3/5 + 1/5 = 4/5.
The correct answer is 200 km. Distance is calculated using the formula: Distance = Speed × Time.
Therefore, the vehicle covers 200 km.
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The correct answer is 28. To find the unknown number, we translate the percentage statement into an algebraic equation and solve for the variable.
Hence, the number is 28.
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The correct answer is A ∩ B = φ. This notation states that the intersection of events A and B is the empty set, which perfectly describes mutual exclusivity.
Therefore, the correct relation for mutually exclusive events A and B is A ∩ B = φ.
The correct answer is 720. The factorial notation (!) means multiplying all positive integers from the number down to 1. The expression 10! / 7! can be simplified by expanding the factorials and cancelling common terms.
Thus, 10! / 7! equals 720.
The correct answer is 300. We are told that 45% of the participants are boys, and the number of boys is 135. Using this information, we can find the total number of students by setting up a simple percentage equation.
Hence, the total number of students participating is 300.
The correct answer is -1. This is a basic linear equation. We will isolate the variable x step by step using algebraic operations.
Therefore, the value of x is -1.
The correct answer is 25%. Percentage increase is calculated by finding the absolute increase (new value minus original value), dividing it by the original value, and then multiplying by 100.
Thus, the increasing percentage is 25%.
The correct answer is 50p² + 72q². This algebraic simplification uses the identities for the square of a binomial: (a - b)² = a² - 2ab + b² and (a + b)² = a² + 2ab + b². When added, the middle terms cancel beautifully.
Thus, the simplified expression is 50p² + 72q².
The correct answer is 99. The square root of 9801 is the number which, when multiplied by itself, gives 9801. By calculation or recall, we know that 99 × 99 = 9801, so √9801 = 99.
All other options produce a square greater than 9,801, while 99² exactly equals the given value. Therefore, the only correct choice is 99 (Option 1).
The correct answer is 5. A ratio expresses the proportional relationship between parts of a whole. In the given ratio 1 : x : 5, the three parts are represented by 1, x, and 5 respectively. Since the problem states that their total sum equals 11, we can directly translate this into a simple algebraic equation.
Thus, the missing term in the ratio is 5, making the complete ratio 1 : 5 : 5. A quick verification confirms our solution: 1 + 5 + 5 = 11, which perfectly matches the given sum.
Hence, the only value that satisfies the condition is x = 5, which corresponds to Option 1.
Correct Answer: 7
To evaluate the expression 16/4 + 9 − 6, we apply the order of operations (BODMAS/PEMDAS). Division is performed before addition and subtraction. First, divide 16 by 4 to obtain 4. The expression then becomes 4 + 9 − 6. Next, perform addition: 4 + 9 = 13. Finally, subtract 6 from 13 to get 7. Following the correct order of operations ensures accurate results in arithmetic calculations. Questions of this type are commonly asked in competitive examinations to test understanding of mathematical rules and calculation speed.
Correct Answer: 5 days
Work problems are based on the principle that the amount of work remains constant. Total work can be calculated by multiplying the number of workers by the number of days. Here, 5 men × 6 days = 30 man-days. If the same work is to be completed by 6 men, the required time is 30 ÷ 6 = 5 days. This demonstrates an inverse relationship between the number of workers and the time taken to complete a fixed amount of work. As the number of workers increases, the number of days decreases proportionally, assuming every worker performs at the same rate and works for the same duration each day.
To simplify the expression x(x²)/x³, first multiply the terms in the numerator. Using the law of exponents, x × x² = x³. The expression therefore becomes x³/x³. According to the exponent rule am/an = am−n (where a ≠ 0), we subtract the exponents: x3−3 = x⁰. Since any non-zero number raised to the power of zero equals 1, the simplified value of the expression is 1.
Exponent laws are fundamental in algebra and are widely used to simplify mathematical expressions. Some important rules include: am × an = am+n, am/an = am−n, and a⁰ = 1 for a ≠ 0. These principles are frequently tested in school mathematics as well as competitive examinations because they provide a quick and systematic method for solving algebraic problems involving powers and exponents.
Why the other options are incorrect:
The correct answer is C: x² - 2x + √2. A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. This means that variables cannot be in the denominator, cannot have fractional exponents (like square roots of variables), and cannot have negative exponents.
Let's look at why the other options are not polynomials: