Q: If $0 < a < 1$, then the value of $a + \frac{1}{a}$ is

Concept & Logic: This problem relies on the foundational arithmetic inequality regarding a positive number and its reciprocal. For any positive real number a, the sum of the number and its reciprocal is always greater than or equal to 2. a + (1/a) \geq 2 Equality holds strictly when a = 1. Step-by-Step Solution: Given the condition: 0 < a < 1 This means a is a positive fraction strictly less than 1. According to the Arithmetic Mean-Geometric Mean (AM-GM) inequality for positive numbers: (a + \frac{1/a)}{2} \geq \sqrt{a \cdot (1/a)} (a + \frac{1/a)}{2} \geq 1 a + (1/a) \geq 2 Since the equality a + (1/a) = 2 only occurs when a = 1, and we are given that a < 1, the sum must be strictly greater than 2. Exam Strategy & Shortcut: Value Assumption: Pick a simple fraction between 0 and 1. Let a = 0.5 (or (1/2)). Then, (1/a) = 2. Sum = 0.5 + 2 = 2.5. 2.5 is clearly greater than 2. Common Pitfall: Students might confuse the bounds and think that because a is small (less than 1), the overall sum will also be small. They forget that as a gets smaller, its reciprocal (1/a) gets exponentially larger, driving the sum well above 2. Final Answer: Therefore, the correct answer is greater than 2.

Q: If $\frac{a}{x} + \frac{y}{b} = 1$ and $\frac{b}{y} + \frac{z}{c} = 1$, then $\frac{x}{a} + \frac{c}{z}$ will be equal to:

Concept & Strategy: This is a cyclic equation problem. The key is to isolate the inverse of the required terms ((x/a) and (c/z)) from the given equations and express them in terms of the common linking variables (y and b). Step-by-Step Solution: Given Equations: 1) (a/x) + (y/b) = 1 2) (b/y) + (z/c) = 1 We need to find the value of: (x/a) + (c/z) From equation (1), isolate (a/x): (a/x) = 1 - (y/b) (a/x) = (b - y/b) Taking the reciprocal to find (x/a): (x/a) = (b/b - y) From equation (2), isolate (z/c): (z/c) = 1 - (b/y) (z/c) = (y - b/y) Taking the reciprocal to find (c/z): (c/z) = (y/y - b) Now, substitute these into the expression we need to evaluate: (x/a) + (c/z) = (b/b - y) + (y/y - b) To make the denominators the same, rewrite (y/y - b) as -(y/b - y): = (b/b - y) - (y/b - y) = (b - y/b - y) = 1 Exam Strategy & Shortcut: Value Putting Method: Assign values to variables that satisfy the given equations. Let (y/b) = 2. This means (b/y) = 0.5. From Eq 1: (a/x) + 2 = 1 \implies (a/x) = -1 \implies (x/a) = -1. From Eq 2: 0.5 + (z/c) = 1 \implies (z/c) = 0.5 \implies (c/z) = 2. Now find (x/a) + (c/z): -1 + 2 = 1. Common Pitfall: A major trap is attempting to solve for individual variables (a, b, c, x, y, z). Because there are 6 variables and only 2 equations, finding individual values is impossible. Always treat grouped fractions as single units. Final Answer: Therefore, the correct answer is 1.

Q: If $a$, $b$, $c$ are integers; $a^2 + b^2 = 45$ and $b^2 + c^2 = 40$, then the values of $a$, $b$ and $c$ respectively are

Concept & Strategy: This problem involves solving a system of nonlinear Diophantine equations (equations where the solutions must be integers). We can use algebraic manipulation or simply check the given options to find a valid integer set. Step-by-Step Solution: Given equations: 1) a^2 + b^2 = 45 2) b^2 + c^2 = 40 Subtract equation (2) from equation (1): (a^2 + b^2) - (b^2 + c^2) = 45 - 40 a^2 - c^2 = 5 Using the difference of squares formula: (a - c)(a + c) = 5 Since a and c are integers, their sum and difference must also be integers. The only integer factors of 5 are (1, 5) and (-1, -5). Assuming positive integers for a moment: a - c = 1 a + c = 5 Adding both equations: 2a = 6 \implies a = 3. Subtracting: 2c = 4 \implies c = 2. Now, substitute a = 3 back into equation (1) to find b: 3^2 + b^2 = 45 9 + b^2 = 45 b^2 = 36 \implies b = 6 (or -6). So the valid positive integer triplet is a = 3, b = 6, c = 2. Exam Strategy & Shortcut: Option Verification: Instead of solving mathematically, directly plug the options into the original equations. Check Option (a): a=2, b=6, c=3 a^2 + b^2 = 4 + 36 = 40 (But it must be 45). Incorrect. Check Option (b): a=3, b=2, c=6 a^2 + b^2 = 9 + 4 = 13 (But it must be 45). Incorrect. Check Option (c): a=5, b=4, c=3 a^2 + b^2 = 25 + 16 = 41 (But it must be 45). Incorrect. Since none of the options perfectly match the valid set (3, 6, 2), the answer is clearly "None of these". Common Pitfall: Students might calculate the correct values (a=3, b=6, c=2) and hastily select an option that contains these numbers in the wrong order, such as picking (b) just because it has a 3, a 2, and a 6. Pay strict attention to the word "respectively". Final Answer: Therefore, the correct answer is None of these.

Q: If $x + y = 15$ and $xy = 56$, then what is the value of $x^2 + y^2$?

Concept & Formula: This problem utilizes standard algebraic identities. The target expression x^2 + y^2 can be directly derived from the square of the sum of the variables. (x + y)^2 = x^2 + y^2 + 2xy Step-by-Step Solution: Given: x + y = 15 xy = 56 We need to find x^2 + y^2. Using the algebraic identity: (x + y)^2 = x^2 + y^2 + 2xy Rearrange the formula to solve for x^2 + y^2: x^2 + y^2 = (x + y)^2 - 2xy Substitute the given values into the equation: x^2 + y^2 = (15)^2 - 2(56) x^2 + y^2 = 225 - 112 x^2 + y^2 = 113 Exam Strategy & Shortcut: Number Guessing (Factorization): For problems with small, neat integers, try to guess the numbers logically. We need two numbers that multiply to 56 and add to 15. Think of the factors of 56: 1 × 56, 2 × 28, 4 × 14, 7 × 8. Notice that 7 + 8 = 15. So, the numbers are 7 and 8. Now, calculate x^2 + y^2 = 7^2 + 8^2 = 49 + 64 = 113. This bypasses algebraic formulas entirely! Common Pitfall: A common error is confusing the identity and calculating (x+y)^2 + 2xy instead of subtracting 2xy, leading to 225 + 112 = 337, which usually throws students off when it isn't an option. Final Answer: Therefore, the correct answer is 113.

Q: If $\frac{x}{y} = \frac{a + 2}{a - 2}$, then $\frac{x^2 - y^2}{x^2 + y^2}$ is equal to

Concept & Logic: This problem heavily relies on the mathematical property of proportions known as Componendo and Dividendo. If (p/q) = (r/s), then (p + q/p - q) = (r + s/r - s) Step-by-Step Solution: Given: (x/y) = (a + 2/a - 2) First, square both sides to match the terms in the target expression (x^2 and y^2): (x^2/y^2) = ((a + 2)^2/(a - 2)^2) Expand the terms on the right side using (a \pm b)^2 = a^2 \pm 2ab + b^2: (x^2/y^2) = (a^2 + 4a + 4/a^2 - 4a + 4) Now, apply the rule of Componendo and Dividendo (in the subtract/add format: \frac{Num - Den}{Num + Den}): (x^2 - y^2/x^2 + y^2) = ((a^2 + 4a + 4) - (a^2 - 4a + 4)/(a^2 + 4a + 4) + (a^2 - 4a + 4)) Simplify the numerator (subtracting the terms): a^2 + 4a + 4 - a^2 + 4a - 4 = 8a Simplify the denominator (adding the terms): a^2 + 4a + 4 + a^2 - 4a + 4 = 2a^2 + 8 = 2(a^2 + 4) Combine them back into the fraction: (8a/2(a^2 + 4)) Divide numerator and denominator by 2 to simplify: = (4a/a^2 + 4) Exam Strategy & Shortcut: Value Putting: Assign a simple number to a. Let a = 3. Then (x/y) = (3 + 2/3 - 2) = (5/1). So x = 5, y = 1. The required expression is (x^2 - y^2/x^2 + y^2) = (25 - 1/25 + 1) = (24/26) = (12/13). Now, plug a = 3 into the options to see which one gives (12/13): (a) (8(3)/9 + 4) = (24/13) (Incorrect) (b) (4(3)/9 - 4) = (12/5) (Incorrect) (d) (4(3)/9 + 4) = (12/13) (Correct!) Common Pitfall: Forgetting to square the right-hand side expressions before applying Componendo and Dividendo. Applying the rule immediately to x/y instead of x^2/y^2 will result in completely wrong algebraic terms. Final Answer: Therefore, the correct answer is (4a/a^2 + 4).

Q: If $3x + 7 = x^2 + P = 7x + 5$, what is the value of $P$?

Concept & Strategy: This problem involves a continuous equation (an equality with three parts). The strategy is to split it into two manageable equations. First, solve for x using the two linear parts, and then substitute that value into the quadratic part to solve for P. Step-by-Step Solution: Given the continuous equality: 3x + 7 = x^2 + P = 7x + 5 Step 1: Solve for x Take the first and third parts of the equation, as they only contain one variable (x): 3x + 7 = 7x + 5 Rearrange to isolate x: 7 - 5 = 7x - 3x 2 = 4x x = (2/4) = (1/2) Step 2: Solve for P Now, use the second and third parts (or first and second) and substitute x = 0.5: x^2 + P = 7x + 5 Substitute x = (1/2): ((1/2))^2 + P = 7((1/2)) + 5 (1/4) + P = 3.5 + 5 0.25 + P = 8.5 Isolate P: P = 8.5 - 0.25 P = 8.25 Convert the decimal back to a mixed fraction to match the options: 8.25 = 8 (1/4) Exam Strategy & Shortcut: Direct Substitution: You can quickly solve 3x + 7 = 7x + 5 mentally. The difference in x is 4, the difference in constants is 2. So x = 2/4 = 0.5. Plug 0.5 into the easiest side: 3(0.5) + 7 = 8.5. Now you know x^2 + P = 8.5. Since x^2 = 0.25, P must be 8.5 - 0.25 = 8.25. This avoids writing down multiple lines of algebra. Common Pitfall: Students often get intimidated by the double equals sign and try to set up complex quadratic formulas. Always look for the easiest path first—in this case, equating the linear ends completely bypasses the need for quadratic factoring. Final Answer: Therefore, the correct answer is 8 (1/4).

Q: The expression $\frac{x + y}{x - y} \div \frac{(x + y)^2}{(x^2 - y^2)}$ is equal to

Concept & Formula: This problem tests the simplification of rational expressions involving algebraic identities and fraction division. The key rule to remember is how to divide fractions (multiply by the reciprocal) and the difference of squares factorization. A^2 - B^2 = (A - B)(A + B) Step-by-Step Solution: Given expression: (x + y/x - y) \div ((x + y)^2/x^2 - y^2) Step 1: Convert division to multiplication To divide by a fraction, we multiply by its reciprocal (flip the second fraction). = (x + y/x - y) × (x^2 - y^2/(x + y)^2) Step 2: Factor the expressions Use the difference of squares identity on the numerator of the second fraction: x^2 - y^2 = (x - y)(x + y). Expand the denominator for clarity: (x + y)^2 = (x + y)(x + y). Substitute these factored forms back into the expression: = (x + y/x - y) × ((x - y)(x + y)/(x + y)(x + y)) Step 3: Cancel common terms Now, look for identical binomials in the numerators and denominators to cancel out. - The (x - y) in the denominator cancels the (x - y) in the numerator. - The two (x + y) terms in the numerator completely cancel the two (x + y) terms in the denominator. = (1/1) × (1/1) = 1 Exam Strategy & Shortcut: Value Putting (Zero Assumption): To solve this in 5 seconds, pick the easiest possible numbers that don't result in division by zero. Let x = 2 and y = 1. First part: (2+1/2-1) = (3/1) = 3. Second part: ((2+1)^2/2^2 - 1^2) = (3^2/4-1) = (9/3) = 3. Expression becomes: 3 \div 3 = 1. This completely avoids algebra and is foolproof. Common Pitfall: A very common mistake is confusing the \div symbol with a + sign due to poor printing quality on exam papers. Always double-check the operator before starting long algebraic simplifications. Additionally, forgetting to flip the second fraction when converting from division to multiplication is a frequent arithmetic error. Final Answer: Therefore, the correct answer is 1.

Q: $\frac{x}{x - 1} + \frac{1}{x + 1} + \frac{2x}{1 - x^2} =$ ?

Concept & Formula: This problem requires simplifying rational expressions by finding a common denominator. The key is recognizing the Difference of Squares identity in the third term's denominator. A^2 - B^2 = (A - B)(A + B) Step-by-Step Solution: Given expression: (x/x - 1) + (1/x + 1) + (2x/1 - x^2) Step 1: Standardize the denominators Notice the denominator of the third term is 1 - x^2. We can rewrite this to match the other denominators by factoring out a negative sign: 1 - x^2 = -(x^2 - 1) = -(x - 1)(x + 1) Rewrite the original expression with this substitution: (x/x - 1) + (1/x + 1) - (2x/(x - 1)(x + 1)) Step 2: Find a common denominator The least common multiple for the denominators is (x - 1)(x + 1). Adjust the first two fractions so they all share this common denominator: - Multiply the first term by (x + 1/x + 1) - Multiply the second term by (x - 1/x - 1) = (x(x + 1)/(x - 1)(x + 1)) + (1(x - 1)/(x - 1)(x + 1)) - (2x/(x - 1)(x + 1)) Step 3: Combine and simplify Combine all numerators over the common denominator: = (x^2 + x + x - 1 - 2x/(x - 1)(x + 1)) Group the like terms in the numerator: = (x^2 + 2x - 1 - 2x/(x - 1)(x + 1)) = (x^2 - 1/(x - 1)(x + 1)) Since x^2 - 1 factors to (x - 1)(x + 1): = (x^2 - 1/x^2 - 1) = 1 Exam Strategy & Shortcut: Value Assumption: To solve this rapidly during an exam, assign a small integer value to x. Let's pick x = 2 (avoid x = 1 or x = -1 to prevent dividing by zero). Substitute x = 2 into the expression: = (2/2 - 1) + (1/2 + 1) + (2(2)/1 - 2^2) = (2/1) + (1/3) + (4/1 - 4) = 2 + (1/3) - (4/3) = 2 - (3/3) = 2 - 1 = 1. Common Pitfall: Failing to flip the sign on the third fraction when converting 1 - x^2 to x^2 - 1. If you forget to change the + to a -, the numerator becomes x^2 + 4x - 1, which does not simplify, causing students to get stuck and waste time. Final Answer: Therefore, the correct answer is 1.

Question 1

If $\frac{a}{b + c} = \frac{b}{c + a} = \frac{c}{a + b} = K$, then the value of $K$ is

Accepted Answer

Concept & Formula: This problem requires evaluating a continuous proportion under different conditions. Like the previous problem, it utilizes the property of equal fractions. However, you must account for the mathematical condition where the sum of the variables could be zero, which splits the problem into two distinct cases. If (p/q) = (r/s) = (t/u) = K, then K = (p+r+t/q+s+u) (provided q+s+u 
eq 0) Step-by-Step Solution: • Given: • (a/b + c) = (b/c + a) = (c/a + b) = K • Calculation: 1. Case 1: Assume (a + b + c) 
eq 0 Apply the sum property of equal fractions. K = \frac{Sum of numerators}{Sum of denominators} K = (a + b + c/(b + c) + (c + a) + (a + b)) K = (a + b + c/2a + 2b + 2c) Factor the denominator: K = (a + b + c/2(a + b + c)). Cancel the common term: K = (1/2). 2. Case 2: Assume (a + b + c) = 0 If the sum is zero, we cannot use the theorem above as it would cause division by zero. Instead, rearrange the equation: b + c = -a. Substitute this into the very first given fraction: K = (a/b + c). K = (a/-a). K = -1. 3. Since both cases are mathematically valid depending on the values of a, b, c, K can be either (1/2) or -1. Exam Strategy & Shortcut: Test the extreme cases using assumed numbers. Case A (Symmetry): Let a = b = c = 1. K = (1/1+1) = (1/2). This eliminates options (c). Case B (Zero Sum): Let a = 2, b = -1, c = -1. (Notice 2 - 1 - 1 = 0). Check the fractions: (2/-1-1) = (2/-2) = -1 (-1/-1+2) = (-1/1) = -1 (-1/2-1) = (-1/1) = -1 So K also equals -1. Therefore, the answer must contain both values. Common Pitfall: The most common trap is only calculating Case 1 using the sum theorem and immediately choosing (1/2) (Option D). In higher-level exams, always check if the denominator sum can equal zero. If no constraint (like "a,b,c are positive integers") is given, you must evaluate the zero-sum scenario. Final Answer: Therefore, the correct answer is (1/2) or -1.

Question 2

If $a - 8 = b$, then determine the value of $|a - b| - |b - a|$.

Accepted Answer

Concept & Formula: This problem tests the fundamental property of absolute values related to distance. The absolute value |x - y| represents the distance between point x and point y on a number line. Because distance is scalar and directionless, the distance from x to y is exactly the same as the distance from y to x. Absolute Value Identity: |x - y| = |y - x| for any real numbers x, y Step-by-Step Solution: • Given: • Equation: a - 8 = b • Target expression: |a - b| - |b - a| • Calculation: 1. Rearrange the given equation to find the value of a - b. a - 8 = b \implies a - b = 8. 2. Rearrange the equation to find the value of b - a. Multiply the equation a - b = 8 by -1. -1(a - b) = -1(8) \implies b - a = -8. 3. Substitute these values into the target expression: |a - b| - |b - a|. |8| - |-8|. 4. Evaluate the absolute values. The absolute value of a number is its non-negative distance from zero. |8| = 8 |-8| = 8 5. Subtract the results: 8 - 8 = 0. Exam Strategy & Shortcut: You do not need to calculate the actual values of a - b or b - a. By definition, |X| = |-X|. Since (a - b) is simply the negative of (b - a), their absolute values will always be identical regardless of what numbers a and b are. Subtracting two identical values always yields 0. You can ignore the equation a - 8 = b entirely; it is just a distractor. Common Pitfall: Students often get confused by the negative signs inside the absolute value brackets and mistakenly calculate |8| - (-8) = 16. Remember that absolute value operations must be fully resolved into positive numbers *before* applying the subtraction sign that sits between them. Final Answer: Therefore, the correct answer is 0.

Question 3

If $0 < a < 1$, then the value of $a + \frac{1}{a}$ is

Accepted Answer

Concept & Logic: This problem relies on the foundational arithmetic inequality regarding a positive number and its reciprocal. For any positive real number a, the sum of the number and its reciprocal is always greater than or equal to 2. a + (1/a) \geq 2 Equality holds strictly when a = 1. Step-by-Step Solution: Given the condition: 0 < a < 1 This means a is a positive fraction strictly less than 1. According to the Arithmetic Mean-Geometric Mean (AM-GM) inequality for positive numbers: (a + \frac{1/a)}{2} \geq \sqrt{a \cdot (1/a)} (a + \frac{1/a)}{2} \geq 1 a + (1/a) \geq 2 Since the equality a + (1/a) = 2 only occurs when a = 1, and we are given that a < 1, the sum must be strictly greater than 2. Exam Strategy & Shortcut: Value Assumption: Pick a simple fraction between 0 and 1. Let a = 0.5 (or (1/2)). Then, (1/a) = 2. Sum = 0.5 + 2 = 2.5. 2.5 is clearly greater than 2. Common Pitfall: Students might confuse the bounds and think that because a is small (less than 1), the overall sum will also be small. They forget that as a gets smaller, its reciprocal (1/a) gets exponentially larger, driving the sum well above 2. Final Answer: Therefore, the correct answer is greater than 2.

Question 4

If $\frac{a}{x} + \frac{y}{b} = 1$ and $\frac{b}{y} + \frac{z}{c} = 1$, then $\frac{x}{a} + \frac{c}{z}$ will be equal to:

Accepted Answer

Concept & Strategy: This is a cyclic equation problem. The key is to isolate the inverse of the required terms ((x/a) and (c/z)) from the given equations and express them in terms of the common linking variables (y and b). Step-by-Step Solution: Given Equations: 1) (a/x) + (y/b) = 1 2) (b/y) + (z/c) = 1 We need to find the value of: (x/a) + (c/z) From equation (1), isolate (a/x): (a/x) = 1 - (y/b) (a/x) = (b - y/b) Taking the reciprocal to find (x/a): (x/a) = (b/b - y) From equation (2), isolate (z/c): (z/c) = 1 - (b/y) (z/c) = (y - b/y) Taking the reciprocal to find (c/z): (c/z) = (y/y - b) Now, substitute these into the expression we need to evaluate: (x/a) + (c/z) = (b/b - y) + (y/y - b) To make the denominators the same, rewrite (y/y - b) as -(y/b - y): = (b/b - y) - (y/b - y) = (b - y/b - y) = 1 Exam Strategy & Shortcut: Value Putting Method: Assign values to variables that satisfy the given equations. Let (y/b) = 2. This means (b/y) = 0.5. From Eq 1: (a/x) + 2 = 1 \implies (a/x) = -1 \implies (x/a) = -1. From Eq 2: 0.5 + (z/c) = 1 \implies (z/c) = 0.5 \implies (c/z) = 2. Now find (x/a) + (c/z): -1 + 2 = 1. Common Pitfall: A major trap is attempting to solve for individual variables (a, b, c, x, y, z). Because there are 6 variables and only 2 equations, finding individual values is impossible. Always treat grouped fractions as single units. Final Answer: Therefore, the correct answer is 1.

Question 5

If $a$, $b$, $c$ are integers; $a^2 + b^2 = 45$ and $b^2 + c^2 = 40$, then the values of $a$, $b$ and $c$ respectively are

Accepted Answer

Concept & Strategy: This problem involves solving a system of nonlinear Diophantine equations (equations where the solutions must be integers). We can use algebraic manipulation or simply check the given options to find a valid integer set. Step-by-Step Solution: Given equations: 1) a^2 + b^2 = 45 2) b^2 + c^2 = 40 Subtract equation (2) from equation (1): (a^2 + b^2) - (b^2 + c^2) = 45 - 40 a^2 - c^2 = 5 Using the difference of squares formula: (a - c)(a + c) = 5 Since a and c are integers, their sum and difference must also be integers. The only integer factors of 5 are (1, 5) and (-1, -5). Assuming positive integers for a moment: a - c = 1 a + c = 5 Adding both equations: 2a = 6 \implies a = 3. Subtracting: 2c = 4 \implies c = 2. Now, substitute a = 3 back into equation (1) to find b: 3^2 + b^2 = 45 9 + b^2 = 45 b^2 = 36 \implies b = 6 (or -6). So the valid positive integer triplet is a = 3, b = 6, c = 2. Exam Strategy & Shortcut: Option Verification: Instead of solving mathematically, directly plug the options into the original equations. Check Option (a): a=2, b=6, c=3 a^2 + b^2 = 4 + 36 = 40 (But it must be 45). Incorrect. Check Option (b): a=3, b=2, c=6 a^2 + b^2 = 9 + 4 = 13 (But it must be 45). Incorrect. Check Option (c): a=5, b=4, c=3 a^2 + b^2 = 25 + 16 = 41 (But it must be 45). Incorrect. Since none of the options perfectly match the valid set (3, 6, 2), the answer is clearly "None of these". Common Pitfall: Students might calculate the correct values (a=3, b=6, c=2) and hastily select an option that contains these numbers in the wrong order, such as picking (b) just because it has a 3, a 2, and a 6. Pay strict attention to the word "respectively". Final Answer: Therefore, the correct answer is None of these.

Question 6

If $x + y = 15$ and $xy = 56$, then what is the value of $x^2 + y^2$?

Accepted Answer

Concept & Formula: This problem utilizes standard algebraic identities. The target expression x^2 + y^2 can be directly derived from the square of the sum of the variables. (x + y)^2 = x^2 + y^2 + 2xy Step-by-Step Solution: Given: x + y = 15 xy = 56 We need to find x^2 + y^2. Using the algebraic identity: (x + y)^2 = x^2 + y^2 + 2xy Rearrange the formula to solve for x^2 + y^2: x^2 + y^2 = (x + y)^2 - 2xy Substitute the given values into the equation: x^2 + y^2 = (15)^2 - 2(56) x^2 + y^2 = 225 - 112 x^2 + y^2 = 113 Exam Strategy & Shortcut: Number Guessing (Factorization): For problems with small, neat integers, try to guess the numbers logically. We need two numbers that multiply to 56 and add to 15. Think of the factors of 56: 1 × 56, 2 × 28, 4 × 14, 7 × 8. Notice that 7 + 8 = 15. So, the numbers are 7 and 8. Now, calculate x^2 + y^2 = 7^2 + 8^2 = 49 + 64 = 113. This bypasses algebraic formulas entirely! Common Pitfall: A common error is confusing the identity and calculating (x+y)^2 + 2xy instead of subtracting 2xy, leading to 225 + 112 = 337, which usually throws students off when it isn't an option. Final Answer: Therefore, the correct answer is 113.

Question 7

If $\frac{x}{y} = \frac{a + 2}{a - 2}$, then $\frac{x^2 - y^2}{x^2 + y^2}$ is equal to

Accepted Answer

Concept & Logic: This problem heavily relies on the mathematical property of proportions known as Componendo and Dividendo. If (p/q) = (r/s), then (p + q/p - q) = (r + s/r - s) Step-by-Step Solution: Given: (x/y) = (a + 2/a - 2) First, square both sides to match the terms in the target expression (x^2 and y^2): (x^2/y^2) = ((a + 2)^2/(a - 2)^2) Expand the terms on the right side using (a \pm b)^2 = a^2 \pm 2ab + b^2: (x^2/y^2) = (a^2 + 4a + 4/a^2 - 4a + 4) Now, apply the rule of Componendo and Dividendo (in the subtract/add format: \frac{Num - Den}{Num + Den}): (x^2 - y^2/x^2 + y^2) = ((a^2 + 4a + 4) - (a^2 - 4a + 4)/(a^2 + 4a + 4) + (a^2 - 4a + 4)) Simplify the numerator (subtracting the terms): a^2 + 4a + 4 - a^2 + 4a - 4 = 8a Simplify the denominator (adding the terms): a^2 + 4a + 4 + a^2 - 4a + 4 = 2a^2 + 8 = 2(a^2 + 4) Combine them back into the fraction: (8a/2(a^2 + 4)) Divide numerator and denominator by 2 to simplify: = (4a/a^2 + 4) Exam Strategy & Shortcut: Value Putting: Assign a simple number to a. Let a = 3. Then (x/y) = (3 + 2/3 - 2) = (5/1). So x = 5, y = 1. The required expression is (x^2 - y^2/x^2 + y^2) = (25 - 1/25 + 1) = (24/26) = (12/13). Now, plug a = 3 into the options to see which one gives (12/13): (a) (8(3)/9 + 4) = (24/13) (Incorrect) (b) (4(3)/9 - 4) = (12/5) (Incorrect) (d) (4(3)/9 + 4) = (12/13) (Correct!) Common Pitfall: Forgetting to square the right-hand side expressions before applying Componendo and Dividendo. Applying the rule immediately to x/y instead of x^2/y^2 will result in completely wrong algebraic terms. Final Answer: Therefore, the correct answer is (4a/a^2 + 4).

Question 8

If $3x + 7 = x^2 + P = 7x + 5$, what is the value of $P$?

Accepted Answer

Concept & Strategy: This problem involves a continuous equation (an equality with three parts). The strategy is to split it into two manageable equations. First, solve for x using the two linear parts, and then substitute that value into the quadratic part to solve for P. Step-by-Step Solution: Given the continuous equality: 3x + 7 = x^2 + P = 7x + 5 Step 1: Solve for x Take the first and third parts of the equation, as they only contain one variable (x): 3x + 7 = 7x + 5 Rearrange to isolate x: 7 - 5 = 7x - 3x 2 = 4x x = (2/4) = (1/2) Step 2: Solve for P Now, use the second and third parts (or first and second) and substitute x = 0.5: x^2 + P = 7x + 5 Substitute x = (1/2): ((1/2))^2 + P = 7((1/2)) + 5 (1/4) + P = 3.5 + 5 0.25 + P = 8.5 Isolate P: P = 8.5 - 0.25 P = 8.25 Convert the decimal back to a mixed fraction to match the options: 8.25 = 8 (1/4) Exam Strategy & Shortcut: Direct Substitution: You can quickly solve 3x + 7 = 7x + 5 mentally. The difference in x is 4, the difference in constants is 2. So x = 2/4 = 0.5. Plug 0.5 into the easiest side: 3(0.5) + 7 = 8.5. Now you know x^2 + P = 8.5. Since x^2 = 0.25, P must be 8.5 - 0.25 = 8.25. This avoids writing down multiple lines of algebra. Common Pitfall: Students often get intimidated by the double equals sign and try to set up complex quadratic formulas. Always look for the easiest path first—in this case, equating the linear ends completely bypasses the need for quadratic factoring. Final Answer: Therefore, the correct answer is 8 (1/4).

Question 9

If $(a + b + 2c + 3d)(a - b - 2c + 3d) = (a - b + 2c - 3d)(a + b - 2c - 3d)$, then $2bc$ is equal to

Accepted Answer

Concept & Strategy: This problem tests your ability to recognize algebraic structures within large polynomials. By strategically grouping terms, we can transform these complex four-term expressions into simple Difference of Squares formats. (X + Y)(X - Y) = X^2 - Y^2 Step-by-Step Solution: Given equation: (a + b + 2c + 3d)(a - b - 2c + 3d) = (a - b + 2c - 3d)(a + b - 2c - 3d) Step 1: Group the terms on the Left Hand Side (LHS) Notice that a and 3d keep their positive signs in both brackets, while b and 2c change signs. Let's group them: LHS = [(a + 3d) + (b + 2c)] \cdot [(a + 3d) - (b + 2c)] Using (X+Y)(X-Y) = X^2 - Y^2, this becomes: LHS = (a + 3d)^2 - (b + 2c)^2 Step 2: Group the terms on the Right Hand Side (RHS) Here, a keeps its positive sign, and -3d keeps its negative sign. The terms b and 2c alternate signs. Group them carefully: RHS = [(a - 3d) - (b - 2c)] \cdot [(a - 3d) + (b - 2c)] Applying the difference of squares again: RHS = (a - 3d)^2 - (b - 2c)^2 Step 3: Equate and simplify Equating the simplified LHS and RHS: (a + 3d)^2 - (b + 2c)^2 = (a - 3d)^2 - (b - 2c)^2 Rearrange to put similar binomials on the same sides: (a + 3d)^2 - (a - 3d)^2 = (b + 2c)^2 - (b - 2c)^2 We use the algebraic identity: (X + Y)^2 - (X - Y)^2 = 4XY. Apply this to both sides: Left side: 4(a)(3d) = 12ad Right side: 4(b)(2c) = 8bc So, 12ad = 8bc. Divide by 4: 3ad = 2bc The question asks for the value of 2bc. Exam Strategy & Shortcut: Sign Grouping Method: In massive expansion problems, immediately look for which variables flip signs. On the left: (b+2c) flips to -(b+2c). On the right: (b-2c) flips to -(b-2c). This immediately tells you the structure is squares of (a \pm 3d) minus squares of (b \pm 2c). Jumping straight to (a+3d)^2 - (a-3d)^2 = (b+2c)^2 - (b-2c)^2 saves enormous amounts of time. Common Pitfall: Attempting to multiply out the brackets term-by-term. Expanding a 4x4 polynomial creates 16 terms on the left and 16 terms on the right. While they will eventually cancel out, it is guaranteed to consume 5+ minutes and almost always results in a sign error. Final Answer: Therefore, the correct answer is 3ad.

Question 10

The expression $\frac{x + y}{x - y} \div \frac{(x + y)^2}{(x^2 - y^2)}$ is equal to

Accepted Answer

Concept & Formula: This problem tests the simplification of rational expressions involving algebraic identities and fraction division. The key rule to remember is how to divide fractions (multiply by the reciprocal) and the difference of squares factorization. A^2 - B^2 = (A - B)(A + B) Step-by-Step Solution: Given expression: (x + y/x - y) \div ((x + y)^2/x^2 - y^2) Step 1: Convert division to multiplication To divide by a fraction, we multiply by its reciprocal (flip the second fraction). = (x + y/x - y) × (x^2 - y^2/(x + y)^2) Step 2: Factor the expressions Use the difference of squares identity on the numerator of the second fraction: x^2 - y^2 = (x - y)(x + y). Expand the denominator for clarity: (x + y)^2 = (x + y)(x + y). Substitute these factored forms back into the expression: = (x + y/x - y) × ((x - y)(x + y)/(x + y)(x + y)) Step 3: Cancel common terms Now, look for identical binomials in the numerators and denominators to cancel out. - The (x - y) in the denominator cancels the (x - y) in the numerator. - The two (x + y) terms in the numerator completely cancel the two (x + y) terms in the denominator. = (1/1) × (1/1) = 1 Exam Strategy & Shortcut: Value Putting (Zero Assumption): To solve this in 5 seconds, pick the easiest possible numbers that don't result in division by zero. Let x = 2 and y = 1. First part: (2+1/2-1) = (3/1) = 3. Second part: ((2+1)^2/2^2 - 1^2) = (3^2/4-1) = (9/3) = 3. Expression becomes: 3 \div 3 = 1. This completely avoids algebra and is foolproof. Common Pitfall: A very common mistake is confusing the \div symbol with a + sign due to poor printing quality on exam papers. Always double-check the operator before starting long algebraic simplifications. Additionally, forgetting to flip the second fraction when converting from division to multiplication is a frequent arithmetic error. Final Answer: Therefore, the correct answer is 1.

Question 11

If $x + y + z = 0$, then $x^2 + xy + y^2$ equals

Accepted Answer

Concept & Strategy: When given the cyclic condition x + y + z = 0, expressions of the form x^2 + xy + y^2 exhibit a unique mathematical symmetry. By expressing one variable in terms of the other two, we can seamlessly convert the expression into terms of y and z. Step-by-Step Solution: Given: x + y + z = 0 We need to find the equivalent of: x^2 + xy + y^2 Method: Algebraic Substitution From the given condition, isolate x: x = -(y + z) Now, substitute this expression for x into our target formula: = [-(y + z)]^2 + [-(y + z)]y + y^2 Expand the terms: The square of a negative is positive, so [-(y + z)]^2 = (y + z)^2 = y^2 + 2yz + z^2. Distribute the -y in the middle term: -y(y + z) = -y^2 - yz. Combine everything together: = (y^2 + 2yz + z^2) - y^2 - yz + y^2 Combine like terms: - The y^2 terms: y^2 - y^2 + y^2 = y^2 - The yz terms: 2yz - yz = yz - The z^2 terms: + z^2 Result: = y^2 + yz + z^2 *(Note: Due to the symmetrical nature of x+y+z=0, the expression x^2+xy+y^2 is also mathematically equal to z^2+zx+x^2, which is option D. However, direct consecutive cyclic substitution conventionally yields option A first.)* Exam Strategy & Shortcut: Value Verification: Pick three numbers that sum to zero. Let x = 1, y = 2, z = -3. Calculate the target expression x^2 + xy + y^2: = (1)^2 + (1)(2) + (2)^2 = 1 + 2 + 4 = 7. Now, plug these values into the options to find which equals 7: (a) y^2 + yz + z^2 = (2)^2 + (2)(-3) + (-3)^2 = 4 - 6 + 9 = 7. (Matches) (b) y^2 - yz + z^2 = 4 - (-6) + 9 = 19. (Incorrect) (c) z^2 - zx + x^2 = 9 - (-3)(1) + 1 = 13. (Incorrect) (d) z^2 + zx + x^2 = 9 + (-3)(1) + 1 = 7. (Also matches) As verified algebraically, A and D are identical in value. Usually, A is the intended direct cyclic derivation. Common Pitfall: Students often try to blindly square the equation (x + y + z)^2 = 0 to get x^2 + y^2 + z^2 + 2xy + 2yz + 2zx = 0 and then get stuck trying to manipulate this massive string into x^2 + xy + y^2. Targeted substitution is much cleaner. Final Answer: Therefore, the correct answer is y^2 + yz + z^2.

Question 12

$\frac{x}{x - 1} + \frac{1}{x + 1} + \frac{2x}{1 - x^2} =$ ?

Accepted Answer

Concept & Formula: This problem requires simplifying rational expressions by finding a common denominator. The key is recognizing the Difference of Squares identity in the third term's denominator. A^2 - B^2 = (A - B)(A + B) Step-by-Step Solution: Given expression: (x/x - 1) + (1/x + 1) + (2x/1 - x^2) Step 1: Standardize the denominators Notice the denominator of the third term is 1 - x^2. We can rewrite this to match the other denominators by factoring out a negative sign: 1 - x^2 = -(x^2 - 1) = -(x - 1)(x + 1) Rewrite the original expression with this substitution: (x/x - 1) + (1/x + 1) - (2x/(x - 1)(x + 1)) Step 2: Find a common denominator The least common multiple for the denominators is (x - 1)(x + 1). Adjust the first two fractions so they all share this common denominator: - Multiply the first term by (x + 1/x + 1) - Multiply the second term by (x - 1/x - 1) = (x(x + 1)/(x - 1)(x + 1)) + (1(x - 1)/(x - 1)(x + 1)) - (2x/(x - 1)(x + 1)) Step 3: Combine and simplify Combine all numerators over the common denominator: = (x^2 + x + x - 1 - 2x/(x - 1)(x + 1)) Group the like terms in the numerator: = (x^2 + 2x - 1 - 2x/(x - 1)(x + 1)) = (x^2 - 1/(x - 1)(x + 1)) Since x^2 - 1 factors to (x - 1)(x + 1): = (x^2 - 1/x^2 - 1) = 1 Exam Strategy & Shortcut: Value Assumption: To solve this rapidly during an exam, assign a small integer value to x. Let's pick x = 2 (avoid x = 1 or x = -1 to prevent dividing by zero). Substitute x = 2 into the expression: = (2/2 - 1) + (1/2 + 1) + (2(2)/1 - 2^2) = (2/1) + (1/3) + (4/1 - 4) = 2 + (1/3) - (4/3) = 2 - (3/3) = 2 - 1 = 1. Common Pitfall: Failing to flip the sign on the third fraction when converting 1 - x^2 to x^2 - 1. If you forget to change the + to a -, the numerator becomes x^2 + 4x - 1, which does not simplify, causing students to get stuck and waste time. Final Answer: Therefore, the correct answer is 1.

Question 13

The cost of 6 pens and 3 pencils is ₹ 84. One-third of the cost of one pen is equal to the cost of one pencil. What is the total cost of 4 pens and 5 pencils?

Accepted Answer

Concept & Strategy: This is a classic simultaneous linear equation problem. First, translate the word problem into two algebraic equations. Use substitution to find the individual cost of a pen and a pencil, then evaluate the final expression. Step-by-Step Solution: Let the cost of one pen be x. Let the cost of one pencil be y. Step 1: Form the equations From the first sentence: 6x + 3y = 84 --- (Equation 1) From the second sentence ("One-third of the cost of one pen is equal to the cost of one pencil"): (1/3)x = y Rearranging this to avoid fractions: x = 3y --- (Equation 2) Step 2: Solve for y (cost of a pencil) Substitute Equation 2 into Equation 1: 6(3y) + 3y = 84 18y + 3y = 84 21y = 84 y = (84/21) y = 4 So, the cost of one pencil is ₹ 4. Step 3: Solve for x (cost of a pen) Substitute the value of y back into Equation 2: x = 3(4) x = 12 So, the cost of one pen is ₹ 12. Step 4: Calculate the target cost The question asks for the total cost of 4 pens and 5 pencils. Total Cost = 4x + 5y Total Cost = 4(12) + 5(4) Total Cost = 48 + 20 Total Cost = 68 Exam Strategy & Shortcut: Direct Ratio Substitution: "One-third of a pen = one pencil" means 1 pen costs the same as 3 pencils. The first purchase is 6 pens and 3 pencils. Convert pens to pencils: 6 pens = 18 pencils. So, 18 pencils + 3 pencils = 21 pencils total. 21 pencils cost ₹ 84. Thus, 1 pencil = ₹ 4. The target purchase is 4 pens and 5 pencils. Convert pens to pencils: 4 pens = 12 pencils. Total target = 12 pencils + 5 pencils = 17 pencils. Target Cost = 17 × 4 = ₹ 68. This completely avoids setting up x and y algebra. Common Pitfall: A common translation error is writing the ratio equation backward, such as x = (1/3)y or 3x = y. Carefully re-read "One-third of... one pen... is equal to... one pencil" \implies (1/3) \cdot Pen = Pencil. Final Answer: Therefore, the correct answer is ₹ 68.

Question 14

What is the maximum number of half-pint bottles of cream that can be filled with a 4-gallon can of cream? (2pt. = 1qt. and 4qt. = 1gal.)

Accepted Answer

Concept & Strategy: To find the number of bottles that can be filled, you must establish unit consistency. Convert the total volume from gallons into pints before dividing by the volume of a single bottle. Total\ Bottles = (Total\ Volume\ (in\ pints)/Volume\ per\ Bottle\ (in\ pints)) Step-by-Step Solution: • Given: • Total volume = 4 gallons • Bottle capacity = 0.5 pints (half-pint) • 1 gallon = 4 quarts • 1 quart = 2 pints • Calculation: 1. First, find how many pints are in a single gallon: 1\ gallon = 4\ quarts × 2\ pints/quart = 8\ pints. 2. Next, calculate the total pints in the 4-gallon can: 4\ gallons × 8\ pints/gallon = 32\ pints. 3. Finally, divide the total pints by the capacity of one bottle: 32\ pints / 0.5\ pints = 64 bottles. Exam Strategy & Shortcut: You can calculate this purely using multiplier logic without writing equations. 1 gallon contains 8 pints, which equals 16 half-pints. If 1 gallon fills 16 bottles, then 4 gallons will fill 4 × 16 = 64 bottles. Common Pitfall: Students often divide by 2 instead of dividing by 0.5 when processing the phrase "half-pint". Dividing 32 by 2 gives 16, which is a distractor option. Remember that dividing by a half is the same as multiplying by 2. Final Answer: Therefore, the correct answer is 64.

Question 15

The sum of the weights of $A$ and $B$ is 80 kg. Half of the weight of $A$ is equal to $\frac{5}{6}$ times the weight of $B$. Find the weight of $B$.

Accepted Answer

Concept & Logic: Translate the textual relationships into a system of linear equations. Use substitution or the ratio method to isolate and solve for the target variable. Step-by-Step Solution: • Given: • A + B = 80 • (1/2)A = (5/6)B • Calculation: 1. Isolate A in the second equation by multiplying both sides by 2: A = 2 × ((5/6))B A = (10/6)B = (5/3)B 2. Substitute this expression for A into the first equation: (5/3)B + B = 80 3. Find a common denominator to add the terms: (5/3)B + (3/3)B = 80 (8/3)B = 80 4. Solve for B: B = 80 × (3/8) B = 10 × 3 = 30 kg Exam Strategy & Shortcut: The fastest way to solve this is using the Ratio Method. From (A/2) = (5B/6), we can deduce the ratio of their weights: (A/B) = (10/6) = (5/3). This means A is 5 parts and B is 3 parts. Total parts = 5 + 3 = 8 parts. Total weight is 80 kg, so 1 part = 80 / 8 = 10 kg. The weight of B is 3 parts, which equals 3 × 10 = 30 kg. Common Pitfall: A common trap is accidentally calculating the weight of A instead of B at the very end. 1 part is 10 kg, making A equal to 50 kg. If 50 was an option, many would rush to select it. Always double-check which variable the question is asking for. Final Answer: Therefore, the correct answer is 30 kg.

Question 16

How many pieces of 85 cm length can be cut from a rod 42.5 metres long?

Accepted Answer

Concept & Strategy: Before dividing a total length by a piece length, you must standardize the units. Both values must be in either meters or centimeters. Number\ of\ Pieces = (Total\ Length\ (in\ cm)/Length\ per\ Piece\ (in\ cm)) Step-by-Step Solution: • Given: • Rod length = 42.5 meters • Piece length = 85 cm • Calculation: 1. Convert the total rod length from meters to centimeters (since 1 meter = 100 cm): 42.5 × 100 = 4250 cm. 2. Divide the total length by the length of a single piece to find the count: 4250 / 85 = 50 pieces. 3. Look at the options provided: (a) 30, (b) 40, (c) 60. Since 50 is not listed, the correct choice is 'None of these'. Exam Strategy & Shortcut: Look at the numerical relationship between the significant digits. 85 is exactly double 42.5. So, 42.5 / 85 is 0.5. Because of the meter-to-centimeter conversion factor of 100, you just multiply 0.5 × 100 = 50. Since 50 isn't an option, select 'None of these' instantly. Common Pitfall: The primary mistake here is failing to convert units, leading students to divide 42.5 by 85 directly and getting 0.5, or multiplying incorrectly and landing on a distractor. Always harmonize units before operating on them. Final Answer: Therefore, the correct answer is None of these.

Question 17

The number of students in each section of a school is 24. After admitting new students, three new sections were started. Now, the total number of sections is 16 and there are 21 students in each section. The number of new students admitted is

Accepted Answer

Concept & Strategy: To find the number of newly admitted students, calculate the difference between the total number of students currently in the school and the original total number of students. New\ Students = Current\ Total\ Students - Original\ Total\ Students Step-by-Step Solution: • Given: • Original students per section = 24 • Current total sections = 16 • Current students per section = 21 • New sections added = 3 • Calculation: 1. Determine the original number of sections: Original\ Sections = 16 - 3 = 13 2. Calculate the original total number of students: 13\ sections × 24\ students/section = 312\ students 3. Calculate the current total number of students: 16\ sections × 21\ students/section = 336\ students 4. Find the number of new students admitted by taking the difference: 336 - 312 = 24\ students Exam Strategy & Shortcut: You can calculate this using a net change approach to minimize large multiplications. The current 16 sections have 21 students each. The original 13 sections lost 3 students each (dropping from 24 to 21). Total students lost from old sections = 13 × 3 = 39. The 3 new sections have 21 students each. Total students in new sections = 3 × 21 = 63. The net increase (new students) is the difference: 63 - 39 = 24. Common Pitfall: A common mistake is multiplying the 3 new sections by the old section size (24), yielding 72, or multiplying 3 by the new size (21) yielding 63, and assuming that is the answer. This ignores the fact that the original sections were rebalanced. Final Answer: Therefore, the correct answer is 24.

Question 18

A man earns ₹ 20 on the first day and spends ₹ 15 on the next day. He again earns ₹ 20 on the third day and spends ₹ 15 on the fourth day. If he continues to save like this, how soon will he have ₹ 60 in hand?

Accepted Answer

Concept & Logic: This is a variation of the classic "monkey climbing a pole" problem. Calculate the net savings over a complete cycle (2 days), but isolate the final step where the target is reached without a subsequent deduction. Net\ Cycle\ Savings = Earnings - Spendings Step-by-Step Solution: • Given: • Day 1 Earnings = ₹ 20 • Day 2 Spendings = ₹ 15 • Target Amount = ₹ 60 • Calculation: 1. Find the net savings for one complete 2-day cycle: Net\ Savings = 20 - 15 = 5 So, in 2 days, the man saves ₹ 5. 2. Identify the target amount needed *before* the final earning day. On the final day, he will earn ₹ 20 and not spend it yet. Target\ before\ final\ jump = 60 - 20 = 40 3. Calculate the time required to accumulate ₹ 40 using the 2-day cycles: Since ₹ 5 is saved every 2 days, the time to save ₹ 40 is (40 / 5) × 2 = 16\ days. 4. By the end of the 16th day, he has exactly ₹ 40 in hand. 5. On the 17th day, he earns ₹ 20. Total = 40 + 20 = 60. Exam Strategy & Shortcut: Always subtract the final positive jump from the total goal first. 60 - 20 = 40. Determine the rate of progress: ₹ 5 per 2 days. 40 / 5 = 8 cycles. 8 × 2 = 16 days. Add 1 day for the final jump. 16 + 1 = 17th day. Common Pitfall: The most common trap is simply dividing the total goal (60) by the net daily savings (5 per 2 days = 2.5/day). Doing 60 / 2.5 gives 24 days. This fails to account for the fact that on a specific earning day, he hits the goal before the next spending cycle begins. Final Answer: Therefore, the correct answer is On 17th day.

Question 19

It costs ₹ $x$ each to make the first thousand copies of a compact disc and ₹ $y$ to make each subsequent copy. If $z$ is greater than 1000, how much will it cost to make $z$ copies of the compact disc?

Accepted Answer

Concept & Logic: Split the total quantity into two distinct tiers and calculate the cost for each tier separately. Then, combine the costs and factor the algebraic expression to match the given options. Total\ Cost = (Cost\ of\ First\ Tier) + (Cost\ of\ Remaining\ Tier) Step-by-Step Solution: • Given: • Cost for first 1000 copies = ₹ x per copy • Cost for copies after 1000 = ₹ y per copy • Total copies = z (where z > 1000) • Calculation: 1. Calculate the total cost for the first 1000 copies: Cost_1 = 1000 × x = 1000x 2. Determine the number of subsequent copies made: Remaining\ copies = z - 1000 3. Calculate the cost for these remaining copies: Cost_2 = (z - 1000) × y = yz - 1000y 4. Add the two costs together for the total cost: Total\ Cost = 1000x + yz - 1000y 5. Rearrange and factor the expression to match the options: Factor out 1000 from the x and y terms: 1000(x - y) + yz Exam Strategy & Shortcut: When algebraic manipulation gets confusing, use the substitution method. Pick simple, concrete numbers. Let x = 10, y = 5, and z = 1001. Cost = (1000 × 10) + (1 × 5) = 10000 + 5 = 10005. Now plug these values into the options: (a) (1001×10) - (1001×5) = 10010 - 5005 = 5005 (Incorrect) (b) 1000(10) + (5×1001) = 10000 + 5005 = 15005 (Incorrect) (c) 1000(10 - 5) + (5×1001) = 5000 + 5005 = 10005 (Matches perfectly) Common Pitfall: Students often correctly formulate 1000x + y(z - 1000) but fail to recognize how it expands and regroups into option (c). Don't assume you made a mistake if your initial expression isn't listed; expand the options to check for equivalence. Final Answer: Therefore, the correct answer is 1000(x - y) + yz.

Question 20

Water boils at 212°F or 100°C and melts at 32°F or 0°C. If the temperature of a particular day is 35°C, it is equivalent to

Accepted Answer

Concept & Formula: The question provides the boiling and freezing points to help derive or recall the standard temperature conversion formula between Celsius and Fahrenheit. F = (9/5)C + 32 Step-by-Step Solution: • Given: • Celsius temperature (C) = 35°C • Calculation: 1. Substitute the given Celsius value into the conversion formula: F = (9/5)(35) + 32 2. Perform the multiplication/division first: F = 9 × 7 + 32 F = 63 + 32 3. Add to find the final Fahrenheit temperature: F = 95 Exam Strategy & Shortcut: A great mental math shortcut for Celsius to Fahrenheit is: "Double it and add 30." This gives a very close approximation. Double 35 is 70. Add 30 is 100°F. Since this is an approximation, the real answer is slightly lower. Looking at the options, 95°F is the closest logical answer. (The exact mental math correction is to double it, subtract 10% of that doubled value, then add 32. 70 - 7 + 32 = 95). Common Pitfall: Students often confuse the order of operations or the fraction, using 5/9 instead of 9/5, or adding 32 before multiplying. This will yield completely incorrect distractors. Always remember the fraction 9/5 is slightly less than 2. Final Answer: Therefore, the correct answer is 95°F.

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