📘 Quiz

Solution: Step 1: Recall the property of square roots that √(a²) = |a|. So, √(x-1)² becomes |x-1|. And √(3-x)² becomes |3-x|. Step 2: Analyze the signs of the terms inside the absolute values based on the given inequality 1 < x < 2. For |x-1|: Since x > 1, the term (x-1) is positive. Therefore, |x-1| = x-1. For |3-x|: Since x < 2, it is also true that x < 3. Thus, the term (3-x) is positive. Therefore, |3-x| = 3-x. Step 3: Substitute the simplified absolute value expressions back into the original expression. (x - 1) + (3 - x). Step 4: Simplify the expression by combining like terms. x - 1 + 3 - x = (x - x) + (3 - 1) = 0 + 2 = 2.

Solution: Step 1: Understand the condition: 0 < x < 1. This means x is a positive fraction or decimal (e.g., 0.5, 1/2, 0.1, etc.). Step 2: When a number between 0 and 1 is raised to a positive power, its value decreases as the power increases. So, x² < x. Step 3: When a number between 0 and 1 is reciprocated (1/x), its value becomes greater than 1. For example, if x = 0.5, 1/x = 2. Step 4: Similarly, 1/x² will be even larger than 1/x because the denominator x² is smaller than x. Step 5: Let's test with an example, x = 0.5: - x = 0.5 - x² = (0.5)² = 0.25 - 2 = 2 (a constant) - 1/x² = 1 / (0.5)² = 1 / 0.25 = 4 Step 6: Comparing the values (0.5, 0.25, 2, 4), the greatest is 4. Step 7: In general, for 0 < x < 1, the order of increasing values is: x² < x < 1 < 1/x < 1/x². Also, the constant 2 falls between 1 and 1/x. Step 8: Therefore, 1/x² will always be the greatest among the given options when 0 < x < 1.

Solution: Step 1: Express a, b, and c in rationalized form: a = (√8 - √7) = 1/(√8 + √7) b = (√6 - √5) = 1/(√6 + √5) c = (√10 - 3) = 1/(√10 + 3) Step 2: Compare denominators: √10 + 3 > √8 + √7 > √6 + √5 Step 3: Since larger denominators result in smaller fractions: c < a < b Conclusion: c < a < b, which corresponds to z < x < y.

Solution: Step 1: We are given that `x` and `y` are positive real numbers and `xy = 8`. We need to find the minimum value of `2x + y`. Step 2: Apply the Arithmetic Mean - Geometric Mean (AM-GM) inequality, which states that for non-negative numbers `A` and `B`, `(A + B) / 2 \u2265 \u221a(AB)`. Equality holds when `A = B`. Step 3: Let `A = 2x` and `B = y`. Since `x` and `y` are positive, `2x` and `y` are also positive. So, `(2x + y) / 2 \u2265 \u221a(2x \u00d7 y)`. Step 4: Substitute the given condition `xy = 8` into the inequality: `(2x + y) / 2 \u2265 \u221a(2 \u00d7 8)` `(2x + y) / 2 \u2265 \u221a16` `(2x + y) / 2 \u2265 4`. Step 5: Multiply both sides by 2: `2x + y \u2265 8`. Step 6: The minimum value of `2x + y` is 8. Step 7: The minimum occurs when `A = B`, i.e., `2x = y`. Substitute `y = 2x` into `xy = 8`: `x(2x) = 8` `2x^2 = 8` `x^2 = 4` Since `x` is positive, `x = 2`. Step 8: Find `y`: `y = 2x = 2(2) = 4`. Step 9: Check the minimum value: `2x + y = 2(2) + 4 = 4 + 4 = 8`.

Solution: Step 1: Given a, b, c > 0 and a + b + c = 1. We need to find the least value of 1/a + 1/b + 1/c. Step 2: (Using AM-HM inequality) For positive numbers, the Arithmetic Mean (AM) is greater than or equal to the Harmonic Mean (HM). AM = (a + b + c) / 3 HM = 3 / (1/a + 1/b + 1/c) So, (a + b + c) / 3 ≥ 3 / (1/a + 1/b + 1/c). Step 3: Substitute a + b + c = 1: 1/3 ≥ 3 / (1/a + 1/b + 1/c). Step 4: Rearrange the inequality to find the minimum value of 1/a + 1/b + 1/c: 1/a + 1/b + 1/c ≥ 3 * 3 1/a + 1/b + 1/c ≥ 9. Step 5: The equality holds when a = b = c. Step 6: If a = b = c and a + b + c = 1, then 3a = 1, so a = 1/3. Step 7: In this case, 1/a = 1/(1/3) = 3. Step 8: So, 1/a + 1/b + 1/c = 3 + 3 + 3 = 9. Step 9: The least value of the expression is 9.

Solution: Step 1: Start with the given inequality: 12x - 66 ≤ 6. Step 2: Add 66 to both sides of the inequality: 12x ≤ 6 + 66. Step 3: Simplify the right side: 12x ≤ 72. Step 4: Divide both sides by 12. Since 12 is a positive number, the direction of the inequality sign remains unchanged. x ≤ 72 / 12. Step 5: Perform the division: x ≤ 6. Step 6: The solution to the inequality is x ≤ 6.

Solution: Step 1: Consider the two expressions to be compared: √(a+b) and √(a) + √(b). To compare them, square both expressions, as they are non-negative for natural numbers. Step 2: Square the first expression, √(a+b). (√(a+b))² = a + b Step 3: Square the second expression, √(a) + √(b). (√(a) + √(b))² = (√(a))² + (√(b))² + 2√(a)√(b) = a + b + 2√(ab) Step 4: Compare the results of the squared expressions. We are comparing (a + b) with (a + b + 2√(ab)). Step 5: Since 'a' and 'b' are distinct natural numbers, 'ab' will be a positive number. Therefore, 2√(ab) will be a positive value (> 0). Step 6: Conclude the inequality. Since (a + b + 2√(ab)) is greater than (a + b) by a positive amount (2√(ab)), it implies: (a + b) < (a + b + 2√(ab)) Therefore, √(a+b) < √(a) + √(b). Step 7: The true statement is √(a+b) < √(a) + √(b).

Solution: Step 1: Understand the given condition: 'p' is a positive fraction less than 1. This means 0 < p < 1. Step 2: Evaluate the option '2/p - p is a positive number'. Step 3: Since 0 < p < 1, it implies that its reciprocal, 1/p, must be greater than 1. (e.g., if p = 0.5, 1/p = 2). Step 4: Multiply both sides of 1/p > 1 by 2: 2/p > 2. Step 5: Now, consider the expression (2/p - p). Step 6: We know 2/p is a number greater than 2. Also, p is a positive number less than 1. Step 7: Subtracting a positive number less than 1 from a number greater than 2 will always result in a positive value. For example, if p=0.5, 2/0.5 - 0.5 = 4 - 0.5 = 3.5, which is positive. Step 8: Therefore, (2/p - p) is necessarily a positive number.

Solution: Step 1: Break the compound inequality into two separate inequalities: - Inequality 1: 4(4x + 5) > 2x - 1. - Inequality 2: 2x - 1 > 4x - 3. Step 2: Solve Inequality 1: - 16x + 20 > 2x - 1. - Subtract 2x from both sides: 14x + 20 > -1. - Subtract 20 from both sides: 14x > -21. - Divide by 14: x > -21/14, which simplifies to x > -3/2 or x > -1.5. Step 3: Solve Inequality 2: - 2x - 1 > 4x - 3. - Subtract 2x from both sides: -1 > 2x - 3. - Add 3 to both sides: 2 > 2x. - Divide by 2: 1 > x, which means x < 1. Step 4: Combine the solutions from both inequalities. We need a value of x that satisfies both x > -1.5 AND x < 1. Step 5: The combined inequality is -1.5 < x < 1. Step 6: Review the provided options to find an integer that falls within this range. The options are 2, 3, -2, 0. Step 7: The only integer among the options that is greater than -1.5 and less than 1 is 0. Step 8: Final result: x = 0.

Solution: Step 1: To maximize the product `abc` given the sum of squares `a^2 + b^2 + c^2 = 1`, we can use the AM-GM (Arithmetic Mean - Geometric Mean) inequality for non-negative terms. For non-negative x, y, z, (x + y + z) / 3 ≥ ∛(xyz). Step 2: Apply the AM-GM inequality to a^2, b^2, c^2 (since squares are non-negative). (a^2 + b^2 + c^2) / 3 ≥ ∛(a^2 × b^2 × c^2) (a^2 + b^2 + c^2) / 3 ≥ (abc)^(2/3) Step 3: Substitute the given condition a^2 + b^2 + c^2 = 1. 1 / 3 ≥ (abc)^(2/3) Step 4: To find the maximum value, equality holds when a^2 = b^2 = c^2. Since a^2 + b^2 + c^2 = 1, then 3a^2 = 1, which implies a^2 = 1/3. Taking the square root, a = ±1/√3. For `abc` to be maximum, `a`, `b`, `c` must all have the same sign. Let's assume a = b = c = 1/√3. Step 5: Calculate the maximum value of abc. abc = (1/√3) × (1/√3) × (1/√3) abc = 1 / (√3 × √3 × √3) abc = 1 / (3√3).

Solution: Step 1: Analyze the given condition: A and B are positive integers with B > A. Step 2: Consider the expression (A - B). Since B > A, (A - B) will always be a negative value. Step 3: Consider the expression (A + B). Since A and B are positive, (A + B) will always be positive. Step 4: Consider the expression (AB). Since A and B are positive, (AB) will always be positive. Step 5: Compare (A + B) and (AB) by testing different positive integer values for A and B where B > A. Step 6: Case 1: Let A = 1, B = 2. Step 7: A + B = 1 + 2 = 3. Step 8: AB = 1 × 2 = 2. Here, A + B > AB. Step 9: Case 2: Let A = 2, B = 3. Step 10: A + B = 2 + 3 = 5. Step 11: AB = 2 × 3 = 6. Here, AB > A + B. Step 12: Since the relationship between (A + B) and (AB) changes depending on the specific values of A and B, we cannot definitively say which expression will always have the highest value.

Solution: Step 1: Analyze the given equation: x - y = 8. This can be rewritten as y = x - 8 or x = y + 8. Step 2: Evaluate Statement I: 'Both x and y are positive.' Consider x = 5. Then y = 5 - 8 = -3. In this case, y is negative. Therefore, Statement I is not necessarily true. Step 3: Evaluate Statement II: 'If x is positive, y must be positive.' Using the example from Step 2, if x = 5 (positive), then y = -3 (negative). This contradicts the statement. Therefore, Statement II is not necessarily true. Step 4: Evaluate Statement III: 'If x is negative, y must be negative.' If x is a negative number (x < 0), then y = x - 8. Since we are subtracting 8 from an already negative number, y will become an even smaller (more negative) number. For example, if x = -1, then y = -1 - 8 = -9. If x = -10, then y = -10 - 8 = -18. In all cases where x is negative, y will also be negative. Step 5: Therefore, only Statement III must be true.

Solution: Step 1: Analyze each fraction given that x is a positive number. a) x/x = 1 (Since x is positive, it's not zero, so the fraction is 1). b) x/(x+1): For any positive x, the numerator (x) is smaller than the denominator (x+1). Therefore, this fraction is always less than 1 (x/(x+1) < 1). c) (x+1)/x: For any positive x, the numerator (x+1) is larger than the denominator (x). Therefore, this fraction is always greater than 1 ((x+1)/x > 1). d) (x+2)/(x+3): For any positive x, the numerator (x+2) is smaller than the denominator (x+3). Therefore, this fraction is always less than 1 ((x+2)/(x+3) < 1). Step 2: Compare the values. From Step 1, we observe: - x/x = 1 - x/(x+1) < 1 - (x+1)/x > 1 - (x+2)/(x+3) < 1 Step 3: Conclude the greatest value. The only fraction among the options that is greater than 1 is (x+1)/x. Therefore, (x+1)/x has the greatest value.

Solution: Step 1: Let P(x) = x³ - 7x² + 11x - 5. We need to find x such that P(x) ≥ 0. Step 2: First, find the roots of the polynomial to factor it. Test integer divisors of the constant term (-5), which are ±1, ±5. Check x = 1: P(1) = (1)³ - 7(1)² + 11(1) - 5 = 1 - 7 + 11 - 5 = 0. So, (x - 1) is a factor of P(x). Step 3: Divide P(x) by (x - 1) to find the remaining quadratic factor. Using polynomial division or synthetic division, we get: (x³ - 7x² + 11x - 5) / (x - 1) = x² - 6x + 5. Step 4: Factor the quadratic expression x² - 6x + 5: x² - 6x + 5 = (x - 1)(x - 5). Step 5: So, the fully factored form of P(x) is (x - 1)(x - 1)(x - 5) = (x - 1)²(x - 5). Step 6: We need to solve the inequality (x - 1)²(x - 5) ≥ 0. Step 7: Analyze the terms: The term (x - 1)² is always non-negative (≥ 0) for any real value of x. Step 8: For the product (x - 1)²(x - 5) to be ≥ 0, considering (x - 1)² is always ≥ 0, we require (x - 5) to be ≥ 0 (unless (x-1)² = 0). Case 1: If x - 5 ≥ 0, then x ≥ 5. In this range, (x - 1)²(x - 5) will be ≥ 0. Case 2: If x - 5 < 0, then x < 5. In this case, for the product to be ≥ 0, (x - 1)² must be 0. (x - 1)² = 0 ⇒ x - 1 = 0 ⇒ x = 1. Step 9: Combining both cases, the inequality is satisfied for x ≥ 5 or x = 1. Step 10: The question asks for the minimum value of x that satisfies this. Comparing the values x=1 and all values x ≥ 5, the minimum value is 1.

Solution: Step 1: Start with the equation a(a - 2) = 5a - 10 Step 2: Expand the left side: a^2 - 2a = 5a - 10 Step 3: Move all terms to one side: a^2 - 7a + 10 = 0 Step 4: Factorize the quadratic equation: (a - 2)(a - 5) = 0 Step 5: Solve for a: a = 2 or a = 5 Step 6: Check inequality conditions: a > 2 for the equation to hold true Step 7: Valid solution: 2 < a ≤ 5

Solution: Let's assume a=5 (since a>4) and b=-2 (since b<-1). Now, let's evaluate each option: Option A: 2a+b<0 => 2*5 + (-2) = 10 - 2 = 8, which is not less than 0. Option B: 4a<3b => 4*5 < 3*(-2) => 20 < -6, which is not true. Option C: a>4b => 5 > 4*(-2) => 5 > -8, which is true. Therefore, the correct option is C: a>4b.

Solution: Step 1: Simplify the algebraic expression (a + b)² - (a - b)². Recall the identities: (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b². Step 2: Substitute these into the given inequality: (a² + 2ab + b²) - (a² - 2ab + b²) > 29. Step 3: Distribute the negative sign and simplify: a² + 2ab + b² - a² + 2ab - b² > 29. 4ab > 29. Step 4: Since a and b are positive integers, 4ab must be an integer greater than 29. The smallest integer greater than 29 is 30. Step 5: So, we need 4ab ≥ 30. Divide by 4: ab ≥ 30/4 = 7.5. Step 6: Since a and b are integers, the smallest possible integer value for the product ab is 8. Step 7: We are given that a > b and both are positive integers. We need to find the smallest value of 'a' such that ab ≥ 8. Step 8: List pairs of positive integers (a, b) where a > b and ab ≥ 8: - If b = 1, then a × 1 ≥ 8 ⇒ a ≥ 8. Smallest a is 8. (Pair: 8, 1) - If b = 2, then a × 2 ≥ 8 ⇒ a ≥ 4. Since a > b, a > 2. Smallest a is 4. (Pair: 4, 2) - If b = 3, then a × 3 ≥ 8 ⇒ a ≥ 8/3 ⇒ a ≥ 2.66. Since a > b, a > 3. Smallest a is 3. But this is not valid, as a must be > b, so 3,3 is not allowed. Next smallest is 4. (Pair: 4, 3) gives ab=12. Step 9: Comparing the smallest values of 'a' from these valid pairs (8, 4), the overall smallest value of 'a' that satisfies the conditions is 4 (when b=2, then ab=8, which satisfies ab>=8, and a>b is 4>2).

Solution: Step 1: Let 'X' be the initial number chosen by Amitabh. Step 2: Track the value of the number at each stage: - Amitabh's turn 1 (gives to Sashi): 2X - Sashi's turn 1 (gives to Amitabh): 2X + 50 - Amitabh's turn 2 (gives to Sashi): 2 * (2X + 50) = 4X + 100 - Sashi's turn 2 (gives to Amitabh): (4X + 100) + 50 = 4X + 150 - Amitabh's turn 3 (gives to Sashi): 2 * (4X + 150) = 8X + 300 - Sashi's turn 3 (gives to Amitabh): (8X + 300) + 50 = 8X + 350 - Amitabh's turn 4 (gives to Sashi): 2 * (8X + 350) = 16X + 700 - Sashi's turn 4 (gives to Amitabh): (16X + 700) + 50 = 16X + 750 Step 3: Amitabh wins if his own result is not greater than 1000, but Sashi's *next* result (if Sashi plays) *is* greater than 1000. This means Sashi loses on his turn. Step 4: We need the smallest 'X' such that Sashi's final value (16X + 750) is greater than 1000, AND Amitabh's preceding value (16X + 700) is less than or equal to 1000. Step 5: Set up the inequality for Sashi to lose: 16X + 750 > 1000. 16X > 1000 - 750 16X > 250 X > 250 / 16 X > 15.625 Step 6: The smallest integer 'X' that satisfies X > 15.625 is X = 16. Step 7: Verify that Amitabh does not lose at his preceding turn with X=16: Amitabh's last value was 16X + 700 = 16(16) + 700 = 256 + 700 = 956. Since 956 ≤ 1000, Amitabh does not lose at this point. Step 8: Therefore, the smallest initial number 'X' for Amitabh to win is 16. Step 9: Calculate the sum of the digits of X: 1 + 6 = 7.

Solution: Step 1: Write the given condition for the arithmetic mean as an inequality: (3a + 4b) / 2 > 50 Step 2: Simplify the inequality by multiplying both sides by 2: 3a + 4b > 100. Step 3: Use the second given condition that 'a' is twice 'b': a = 2b. Step 4: Substitute 'a = 2b' into the inequality from Step 2: 3(2b) + 4b > 100 6b + 4b > 100 10b > 100. Step 5: Solve for 'b' by dividing by 10: b > 10. Step 6: Since a = 2b, multiply the inequality for b by 2: 2b > 2 * 10 a > 20. Step 7: We need the smallest possible integer value for 'a'. Since 'a' must be strictly greater than 20, the smallest integer value for 'a' is 21. The smallest possible integer value of 'a' is 21.