Question 1

Find the value of   $$ \frac{1}{1 + \frac{1}{3 - \frac{4}{2 + \frac{1}{3 - \frac{1}{2}}}}} + \frac{3}{3 - \frac{4}{3 + \frac{1}{2 - \frac{1}{2}}}} $$

Accepted Answer

Concept & Strategy: Treat this massive expression as two independent continued fraction problems. Simplify each major fraction independently from the bottom layer up, taking extreme care with subtraction signs, before adding the final two rational numbers together. Step-by-Step Solution: • Calculation for the First Term: 1. Deepest denominator: 3 - (1/2) = (5/2) 2. Next layer: 2 + (1/\frac{5){2}} = 2 + (2/5) = (12/5) 3. Next layer: 3 - (4/\frac{12){5}} = 3 - (4 × (5/12)) = 3 - (5/3) = (9/3) - (5/3) = (4/3) 4. Next layer: 1 + (1/\frac{4){3}} = 1 + (3/4) = (7/4) 5. Final First Term value: (1/\frac{7){4}} = (4/7) • Calculation for the Second Term: 1. Deepest denominator: 2 - (1/2) = (3/2) 2. Next layer: 3 + (1/\frac{3){2}} = 3 + (2/3) = (11/3) 3. Next layer: 3 - (4/\frac{11){3}} = 3 - (4 × (3/11)) = 3 - (12/11) = (33/11) - (12/11) = (21/11) 4. Final Second Term value: (3/\frac{21){11}} = 3 × (11/21) = (11/7) • Final Addition: 1. Sum = First Term + Second Term 2. Sum = (4/7) + (11/7) = (15/7) Exam Strategy & Shortcut: Break complex problems into distinct modular zones. Write "Left" and "Right" on your scratchpad. Execute the reciprocal flips mentally and only write down the resulting simplified fraction for each vertical step. For example, for the Right side, write: 3/2 \rightarrow flip add 3 \rightarrow 11/3 \rightarrow multiply 4, flip, subtract from 3 \rightarrow 3 - 12/11 = 21/11 \rightarrow multiply 3, flip \rightarrow 33/21 = 11/7. This keeps your workspace clean and prevents transposition errors. Common Pitfall: The most dangerous step is 3 - (4/\frac{12){5}}. Students often invert the fraction incorrectly, or multiply 4 into both the numerator and denominator, resulting in 20/48 instead of 20/12. Remember that a whole number multiplied by a fraction only scales the numerator. Final Answer: Therefore, the correct answer is (15/7).

Question 2

$\frac{2}{2 + \frac{2}{3 + \frac{2}{3 + \frac{2}{3}}}} 	imes 0.39$ is simplified to

Accepted Answer

Concept & Strategy: This problem features a deeply nested continued fraction alongside decimal multiplication. Important Note on Typography: In typical aptitude tests, a multiplier placed alongside the main denominator bar (like the × 0.39 here) implicitly applies to that entire evaluated denominator grouping. Thus, the expression we are evaluating is mathematically represented as (2/(2 + \dots) × 0.39). The strategy is to simplify the continued fraction first, multiply the denominator by 0.39, and finally divide the numerator. Step-by-Step Solution: Let's simplify the continued fraction in the denominator from the bottom up. • Step 1: Lowest level 3 + (2/3) = (9 + 2/3) = (11/3) • Step 2: Second level Substitute and invert: (2/\frac{11){3}} = (6/11) Add to 3: 3 + (6/11) = (33 + 6/11) = (39/11) • Step 3: Third level Substitute and invert: (2/\frac{39){11}} = (22/39) Add to 2 (completing the bracketed denominator): 2 + (22/39) = (78 + 22/39) = (100/39) • Step 4: Apply the Decimal Multiplication Now, multiply this evaluated denominator by 0.39. It is easier to convert 0.39 to a fraction: 0.39 = (39/100) Multiplying them together: (100/39) × (39/100) = 1 • Step 5: Final Evaluation Substitute the final denominator value back under the main numerator: (2/1) = 2 Exam Strategy & Shortcut: Look for cancellation opportunities! Once you see the denominator simplifying to (100/39) and notice the multiplier is 0.39, you should immediately recognize that 0.39 = 39/100. These numbers are perfectly designed to cross-cancel to 1. Spotting these relationships prevents unnecessary and error-prone decimal division. Common Pitfall: Misinterpreting the position of × 0.39. If you multiply the *entire evaluated fraction* by 0.39 (i.e., treating it as (2/100/39) × 0.39), you would get 0.78 × 0.39 = 0.3042, which isn't an option. Always look closely at the horizontal alignment of the arithmetic operators in poorly typeset exams. Final Answer: Therefore, the correct answer is 2.

Question 3

Simplify: $\frac{1}{1 + \frac{\frac{2}{3}}{1 + \frac{2}{3} + \frac{\frac{8}{9}}{1 - \frac{2}{3}}}}$

Accepted Answer

Concept & Strategy: This is a multi-level complex fraction. While it looks intimidating, it is solved using the standard bottom-up simplification strategy. The key is to carefully isolate and evaluate the deepest nested terms first before combining them with adjacent terms. Step-by-Step Solution: Let's break the problem down into manageable chunks. • Step 1: Simplify the deepest term on the right Evaluate the fraction (\frac{8/9)}{1 - (2/3)}: Denominator: 1 - (2/3) = (1/3) Now divide: (\frac{8/9)}{(1/3)} = (8/9) × (3/1) = (24/9) = (8/3) • Step 2: Simplify the denominator of the middle layer The denominator is 1 + (2/3) + (Result from Step 1): 1 + (2/3) + (8/3) Since the fractions share a common denominator, combine them: 1 + (10/3) = (3 + 10/3) = (13/3) • Step 3: Evaluate the middle fraction Divide the numerator (2/3) by the result from Step 2: (\frac{2/3)}{(13/3)} = (2/3) × (3/13) = (2/13) • Step 4: Final calculation Substitute the result from Step 3 back into the main expression: (1/1 + \frac{2){13}} Simplify the denominator: 1 + (2/13) = (13 + 2/13) = (15/13) Invert for the final answer: (1/\frac{15){13}} = (13/15) Exam Strategy & Shortcut: When dividing fractions with the same denominators (like in Step 3 where both have a denominator of 3), you can ignore the denominators entirely and just divide the numerators. So, (2/3/13/3) immediately becomes (2/13). This mental shortcut saves valuable seconds during an exam. Common Pitfall: A frequent mistake is getting lost in the arithmetic when copying terms from one step to the next. Students often miscalculate 1 - (2/3) or forget to invert fractions when dividing. Keep your scratchpad organized by clearly drawing boxes around completed sub-components. Final Answer: Therefore, the correct answer is (13/15).

Question 4

If $\frac{2}{2 + \frac{2}{2 + \frac{2}{x}}} = 3$, then $x$ is equal to

Accepted Answer

Concept & Strategy: Unlike standard continued fraction problems where you simplify from the bottom up, this is an Inverse Continued Fraction Equation. Because the unknown x is buried at the very bottom, but the total value of the expression is known (it equals 3), the fastest strategy is to solve from the top down. Step-by-Step Solution: We will isolate the terms layer by layer, starting from the outermost shell. • Step 1: Isolate the main denominator The equation is (2/Denominator) = 3. Rearranging this gives: Denominator = (2/3) So, 2 + (2/2 + \frac{2){x}} = (2/3) • Step 2: Isolate the next nested fraction Subtract 2 from both sides: (2/2 + \frac{2){x}} = (2/3) - 2 (2/2 + \frac{2){x}} = (2 - 6/3) = -(4/3) • Step 3: Isolate the next denominator We have (2/Denominator) = -(4/3). Rearranging gives Denominator = 2 × (-(3/4)) = -(6/4) = -(3/2). So, 2 + (2/x) = -(3/2) • Step 4: Solve for x Subtract 2 from both sides: (2/x) = -(3/2) - 2 (2/x) = (-3 - 4/2) = -(7/2) Invert both sides and multiply to find x: (x/2) = -(2/7) x = 2 × (-(2/7)) = -(4/7) Exam Strategy & Shortcut: The top-down approach *is* the shortcut. If you try to simplify this from the bottom up (by finding a common denominator for 2 + 2/x, then inverting, then adding 2, etc.), you will create massive, algebraically heavy expressions that are incredibly prone to silly mistakes. Working backwards with simple integers is always faster. Common Pitfall: The biggest danger here is losing track of negative signs. In Step 2, subtracting 2 from 2/3 yields a negative number (-4/3). If you accidentally write positive 4/3, you will end up with an incorrect positive fraction for x. Always double-check your sign arithmetic when moving terms across the equals sign. Final Answer: Therefore, the correct answer is -(4/7).

Question 5

$\frac{\left[\left(1 + \frac{1}{10 + \frac{1}{10}}\right)\left(1 + \frac{1}{10 + \frac{1}{10}}\right) - \left(1 - \frac{1}{10 + \frac{1}{10}}\right)\left(1 - \frac{1}{10 + \frac{1}{10}}\right)\right]}{\left[\left(1 + \frac{1}{10 + \frac{1}{10}}\righ…

Accepted Answer

Concept & Strategy: This is a classic Algebraic Identity simplification problem masked by bulky fractional terms. The core strategy is to replace the repeating complex expressions with simple variables to reveal a fundamental algebraic structure, specifically the difference of squares: a^2 - b^2. Step-by-Step Solution: • Step 1: Abstract the complex terms Let a = 1 + (1/10 + \frac{1){10}} Let b = 1 - (1/10 + \frac{1){10}} • Step 2: Rewrite the expression Substituting a and b into the original expression gives: ([a \cdot a - b \cdot b]/[a + b]) (a^2 - b^2/a + b) • Step 3: Apply the algebraic identity We know that a^2 - b^2 = (a - b)(a + b). Substitute this into the numerator: ((a - b)(a + b)/a + b) The (a + b) terms cancel out, leaving simply: a - b • Step 4: Substitute the original values back Now we evaluate a - b: a - b = (1 + (1/10 + \frac{1){10}}) - (1 - (1/10 + \frac{1){10}}) The 1 and -1 cancel out. The fractional parts add together: a - b = (2/10 + \frac{1){10}} • Step 5: Final Evaluation Simplify the denominator: 10 + (1/10) = (100 + 1/10) = (101/10) Now divide 2 by this result: (2/\frac{101){10}} = 2 × (10/101) = (20/101) Exam Strategy & Shortcut: Train your eyes to see the pattern ((x^2 - y^2)/(x + y)). The moment you spot this, you know the entire massive expression immediately reduces to just (x - y). You can skip all the intermediate writing and directly calculate the difference of the two bracketed terms in your head. Common Pitfall: A major trap is attempting to simplify the inner fraction (10 + 1/10) right at the beginning and substituting 101/10 into every single instance across the huge numerator. This leads to heavy, error-prone arithmetic (1 + 10/101, squaring it, etc.). Always simplify the algebraic structure *before* evaluating the numbers. Final Answer: Therefore, the correct answer is (20/101).

Question 6

Which of the following values of $x$ and $y$ satisfy the following equations I and II?  I. $3x + y = 19$  II. $x - y = 9$

Accepted Answer

Concept & Strategy: This is a fundamental Simultaneous Linear Equations problem. The most efficient strategy here is the Elimination Method. Because the coefficients of y in the two equations are exact opposites (+1 and -1), adding the two equations together will instantly eliminate y, allowing us to solve for x immediately. Step-by-Step Solution: • Step 1: Align the equations Eq I: 3x + y = 19 Eq II: x - y = 9 • Step 2: Add the equations vertically Add the left sides together and the right sides together: (3x + x) + (y - y) = 19 + 9 4x + 0 = 28 4x = 28 • Step 3: Solve for x Divide both sides by 4: x = (28/4) = 7 • Step 4: Solve for y Substitute the value of x back into the simpler equation (Eq II): 7 - y = 9 -y = 9 - 7 -y = 2 y = -2 Exam Strategy & Shortcut: For multiple-choice questions with simple linear equations, Option Verification (Back-solving) is often the fastest method. Just plug the options into Equation II (x - y = 9): (a) -7 - (-2) = -5 (Incorrect) (b) -7 - 2 = -9 (Incorrect) (c) 7 - (-2) = 9 (Correct!) You found the answer in seconds without doing any formal algebra. Just quickly verify it with Eq I: 3(7) + (-2) = 21 - 2 = 19. Done. Common Pitfall: A common mistake is arithmetic errors with negative signs when substituting x back in to find y. Many students reach 7 - y = 9 and incorrectly calculate y = 2 instead of y = -2, which leads them straight to the trap Option (d). Always verify your final x and y in *both* original equations. Final Answer: Therefore, the correct answer is 7, -2.

Question 7

If $4x = p(x + 3) + q(x - 1)$ is an identity, then the values of $p$ and $q$ are

Accepted Answer

Concept & Strategy: This question involves a Polynomial Identity. An identity means the equation holds true for *any* possible value of x. We can solve this using two main strategies: 1. Coefficient Equating: Expand the right side and equate the coefficients of x and the constant terms to the left side. 2. Strategic Substitution: Choose clever values for x that make terms cancel out (become zero), allowing instant evaluation of p and q. (This is the faster method). Step-by-Step Solution (Substitution Method): Since the equation is an identity, it works for all values of x. We pick values that make the brackets zero. • Step 1: Solve for p by eliminating q To make the q term disappear, we set x - 1 = 0, which means x = 1. Substitute x = 1 into the entire equation: 4(1) = p(1 + 3) + q(1 - 1) 4 = p(4) + 0 4 = 4p p = 1 • Step 2: Solve for q by eliminating p To make the p term disappear, we set x + 3 = 0, which means x = -3. Substitute x = -3 into the entire equation: 4(-3) = p(-3 + 3) + q(-3 - 1) -12 = p(0) + q(-4) -12 = -4q q = (-12/-4) = 3 Exam Strategy & Shortcut: The Strategic Substitution method shown above *is* the ultimate exam shortcut for identity problems. It bypasses the need to create and solve simultaneous equations entirely. The moment you see (x-1) and (x+3), you should mentally plug in x=1 (yielding 4=4p \rightarrow p=1) and look at the options. Options (a), (b), and (c) all have p=1. Then quickly plug in x=-3 (yielding -12=-4q \rightarrow q=3). The answer is (b). Common Pitfall: Students often default to expanding the expression: 4x = px + 3p + qx - q, then grouping: 4x = (p+q)x + (3p-q). This creates a system of equations: p+q=4 and 3p-q=0. While this method is mathematically sound, it is time-consuming and introduces more opportunities for basic addition/subtraction errors compared to strategic substitution. Final Answer: Therefore, the correct answer is 1, 3.

Question 8

If $x = y = 2z$ and $xyz = 256$, then $x$ is equal to

Accepted Answer

Concept & Strategy: This is an Algebraic Substitution problem. The strategy is to express all variables (y and z) in terms of the target variable (x) using the given linear equality. This converts the multi-variable product equation into a single-variable cubic equation. Step-by-Step Solution: • Step 1: Express variables in terms of x We are given x = y = 2z. From this, we can isolate y and z in terms of x: y = x 2z = x \Rightarrow z = (x/2) • Step 2: Substitute into the product equation We are given xyz = 256. Substitute y and z with their x-equivalents: x \cdot (x) \cdot ((x/2)) = 256 • Step 3: Solve for x Multiply the terms: (x^3/2) = 256 Multiply both sides by 2: x^3 = 512 Take the cube root of both sides. Since 8 × 8 × 8 = 512: x = 8 Exam Strategy & Shortcut: For equations with neat integer options, Option Verification is incredibly fast. Test the options directly: If x = 8 (Option c), then y = 8. Since 2z = 8, z = 4. Check the product: 8 × 8 × 4 = 64 × 4 = 256. It matches perfectly! You can bypass the algebraic solving entirely. Common Pitfall: A very common mistake is solving for the wrong variable. Students might manipulate the equations to solve for z, find z = 4, see '4' in the options (Option b), and instantly select it without realizing the question explicitly asked for x. Always double-check what the question is requesting. Final Answer: Therefore, the correct answer is 8.

Question 9

If $3x + 7y = 75$ and $5x - 5y = 25$, then what is the value of $x + y$?

Accepted Answer

Concept & Strategy: This is a standard System of Linear Equations problem. The most effective strategy is to look for common factors in the equations to simplify them before attempting elimination or substitution. Step-by-Step Solution: • Step 1: Simplify the second equation Look closely at 5x - 5y = 25. All coefficients are multiples of 5. Divide the entire equation by 5: x - y = 5 From this, we can easily isolate x: x = y + 5 • Step 2: Substitute into the first equation The first equation is 3x + 7y = 75. Substitute x = (y + 5) into this equation: 3(y + 5) + 7y = 75 3y + 15 + 7y = 75 Combine the y terms: 10y + 15 = 75 10y = 60 y = 6 • Step 3: Solve for x Substitute y = 6 back into our simplified equation x = y + 5: x = 6 + 5 x = 11 • Step 4: Calculate the final required value The question asks for x + y: x + y = 11 + 6 = 17 Exam Strategy & Shortcut: Always look for an equation that can be reduced! If you immediately jump to cross-multiplying (e.g., multiplying the first equation by 5 and the second by 3), you will end up dealing with large, error-prone numbers (15x + 35y = 375). Spotting that 5x - 5y = 25 reduces to x - y = 5 cuts the calculation time in half. Common Pitfall: A common error is stopping the calculation prematurely. After working through the algebra to find x = 11 or y = 6, students sometimes forget that the question specifically asks for their sum (x + y). Always re-read the final line of the question before marking your choice. Final Answer: Therefore, the correct answer is 17.

Question 10

If $a + b = 5$ and $3a + 2b = 20$, then $(3a + b)$ will be:

Accepted Answer

Concept & Strategy: This problem requires solving Simultaneous Linear Equations. We can use the Elimination Method to quickly find the values of a and b, and then substitute them into the target expression (3a + b). Step-by-Step Solution: • Step 1: Set up the equations Equation 1: a + b = 5 Equation 2: 3a + 2b = 20 • Step 2: Equate coefficients for elimination Let's eliminate b. Multiply Equation 1 by 2 so the coefficients of b match: 2 × (a + b) = 2 × 5 2a + 2b = 10 (This is our new Equation 3) • Step 3: Subtract to find a Subtract Equation 3 from Equation 2: (3a + 2b) - (2a + 2b) = 20 - 10 3a - 2a + 2b - 2b = 10 a = 10 • Step 4: Find b Substitute a = 10 back into Equation 1: 10 + b = 5 b = 5 - 10 b = -5 • Step 5: Evaluate the target expression The question asks for the value of 3a + b: 3(10) + (-5) 30 - 5 = 25 Exam Strategy & Shortcut: Look for algebraic manipulations that directly yield the target expression without finding individual variables. Notice that we want (3a + b). We are given (3a + 2b) = 20. If we subtract b from 20, we get our answer. How do we get b? We know (a + b) = 5, which implies 2a + 2b = 10. Subtracting this from 3a + 2b = 20 instantly gives a = 10. Then b must be -5. The target is 3(10) - 5 = 25. While this still requires solving for the variables, actively looking for these structural relationships builds strong algebraic intuition. Common Pitfall: The main pitfall is dealing with negative numbers. When substituting a = 10 back into a + b = 5, students frequently make a sign error, calculating b = 5 instead of b = -5. This would lead to evaluating 3(10) + 5 = 35, which is incorrect. Always track your signs carefully when moving terms across the equals sign. Final Answer: Therefore, the correct answer is 25.

Question 11

If $2p + 3q = 18$ and $2p - q = 2$, then $2p + q$ is

Accepted Answer

Concept & Strategy: This is an Algebraic Pattern Recognition problem masked as a standard simultaneous equations question. While you can solve for p and q individually using standard elimination or substitution, a highly optimized strategy involves manipulating the entire equations to instantly produce the target expression (2p + q). Step-by-Step Solution (Standard Method): • Step 1: Set up the equations Eq 1: 2p + 3q = 18 Eq 2: 2p - q = 2 • Step 2: Eliminate p to find q Since both equations have a 2p term, subtract Eq 2 from Eq 1: (2p + 3q) - (2p - q) = 18 - 2 2p + 3q - 2p + q = 16 4q = 16 q = 4 • Step 3: Substitute to find 2p Substitute q = 4 into Eq 2: 2p - 4 = 2 2p = 6 *(Note: No need to find p=3, as our target expression needs 2p)* • Step 4: Calculate the target expression The question asks for 2p + q: 6 + 4 = 10 Exam Strategy & Shortcut: The Equation Addition Trick: Always check if adding or subtracting the given equations directly creates the target expression. Let's add the two equations together: Eq 1 + Eq 2: (2p + 3q) + (2p - q) = 18 + 2 This simplifies to: 4p + 2q = 20 Now, look at what we need: 2p + q. Simply divide the combined equation by 2: (4p + 2q/2) = (20/2) 2p + q = 10 You arrived at the answer in one step, completely bypassing the need to solve for individual variables! Common Pitfall: The biggest mistake is wasting time. Many students will systematically solve for q=4, then p=3, and then substitute both back into 2p + q = 2(3) + 4 = 10. While mathematically correct, this brute-force approach eats up precious seconds in a competitive exam where shortcuts like equation addition are designed to be used. Final Answer: Therefore, the correct answer is 10.

Question 12

If $2x + y = 5$ and $3x - 4y = 2$, then the value of $2xy$ is

Accepted Answer

Concept & Formula: To solve a system of two linear equations, we use substitution or elimination to find the individual values of x and y. Once identified, we simply substitute them into the target expression, 2xy. ax + by = c Step-by-Step Solution: Given: 1. 2x + y = 5 2. 3x - 4y = 2 Calculation: • From equation (1), isolate y: y = 5 - 2x • Substitute this expression for y into equation (2): 3x - 4(5 - 2x) = 2 • Distribute the -4: 3x - 20 + 8x = 2 • Combine like terms and solve for x: 11x = 22 x = 2 • Substitute x = 2 back into our isolated y equation: y = 5 - 2(2) y = 5 - 4 = 1 • Finally, compute the requested value of 2xy: 2xy = 2(2)(1) = 4 Exam Strategy & Shortcut: Instead of substitution, the Elimination Method is often faster for avoiding sign errors. Multiply the first equation by 4 to quickly eliminate y: 8x + 4y = 20 3x - 4y = 2 Add them together vertically: 11x = 22 \Rightarrow x = 2. This requires fewer writing steps on rough paper during high-pressure exams. Common Pitfall: A major trap is solving for x = 2 or y = 1 and frantically looking for those numbers in the options, or forgetting to multiply them by the coefficient 2 in the final step (2xy). Always re-read the exact variable expression the question is asking for. Final Answer: Therefore, the correct answer is 4.

Question 13

If $3x - 4y + z = 7$; $2x - z + 3y = 19$; $x + 2y + 2z = 24$, then what is the value of $z$?

Accepted Answer

Concept & Formula: Solving a 3-variable system of linear equations requires systematic elimination. The strategy is to eliminate one variable completely using two pairs of equations, reducing the problem to a standard 2-variable system. Step-by-Step Solution: Given: 1. 3x - 4y + z = 7 2. 2x + 3y - z = 19 (Note: terms rearranged for alignment) 3. x + 2y + 2z = 24 Calculation: • Step 1: Eliminate z from Eqs (1) and (2) Simply add Equation 1 and Equation 2 because they have +z and -z: (3x + 2x) + (-4y + 3y) + (z - z) = 7 + 19 5x - y = 26 \implies y = 5x - 26 --- (Eq A) • Step 2: Eliminate z from Eqs (2) and (3) Multiply Equation 2 by 2 so its z term matches Equation 3: 2(2x + 3y - z = 19) \implies 4x + 6y - 2z = 38 Add this to Equation 3 (x + 2y + 2z = 24): (4x + x) + (6y + 2y) + (-2z + 2z) = 38 + 24 5x + 8y = 62 --- (Eq B) • Step 3: Solve for x and y Substitute Eq A (y = 5x - 26) into Eq B: 5x + 8(5x - 26) = 62 5x + 40x - 208 = 62 45x = 270 \implies x = 6 Find y: y = 5(6) - 26 = 30 - 26 = 4 • Step 4: Solve for z Substitute x=6, y=4 into Equation 1: 3(6) - 4(4) + z = 7 18 - 16 + z = 7 \implies 2 + z = 7 \implies z = 5 Exam Strategy & Shortcut: If you are only asked for z, and the algebra looks dense, check if you can cleverly multiply the equations to cancel x and y simultaneously. If no obvious multiplier exists, execute systematic elimination rapidly. Never mix up the pairs of equations you are eliminating from. Common Pitfall: Sign errors are the #1 reason for missed marks here. When multiplying an entire equation (like Eq 2 by 2), students often forget to multiply the constant on the right side of the equals sign (e.g., writing 19 instead of 38). Final Answer: Therefore, the correct answer is 5.

Question 14

At a certain fast food restaurant, Brian can buy 3 burgers, 7 shakes and one order of fries for ₹ 120 exactly. At the same place, it would cost ₹ 164.50 for 4 burgers, 10 shakes and one order of fries.   How much would it cost for an ordinary meal o…

Accepted Answer

Concept & Strategy: When asked for a specific combination of items (1 burger, 1 shake, 1 fries) from a system that has more variables (3) than equations (2), you cannot find the individual price of each item. Instead, you must find a multiplier for each equation so that when added together, they perfectly produce the target expression. Step-by-Step Solution: Given: Let Burger = b, Shake = s, Fries = f. 1. 3b + 7s + 1f = 120 2. 4b + 10s + 1f = 164.5 Target: Find 1b + 1s + 1f. Calculation: • We need to find two multipliers, let's call them x and y, such that: x(Equation 1) + y(Equation 2) = Target x(3b + 7s + 1f) + y(4b + 10s + 1f) = 1b + 1s + 1f • This gives us a mini-system of coefficients to solve: For f: 1x + 1y = 1 \implies x = 1 - y For b: 3x + 4y = 1 • Substitute x: 3(1 - y) + 4y = 1 3 - 3y + 4y = 1 \implies y = -2 • If y = -2, then x = 1 - (-2) = 3. • Let's verify these multipliers work for s (the shakes): 7x + 10y = 7(3) + 10(-2) = 21 - 20 = 1. It matches perfectly! • Apply these multipliers to the actual costs: Target Cost = 3(120) - 2(164.5) Target Cost = 360 - 329 = 31 Exam Strategy & Shortcut: An intuitive shortcut for finding combinations is to eliminate the most "awkward" variable first. Subtracting Eq 1 from Eq 2 gives: 1b + 3s = 44.5 Now, how do we get 1b + 1s + 1f? Notice that if we subtract our new equation (1b + 3s) from a multiple of the original equations, we can shape it. Actually, a faster visual trick: Eq 1 has 1f. We need 1f. If we take 3 times Eq 1, we get 3f. If we subtract 2 times Eq 2, we get -2f, leaving 1f. Let's try it: 3(3b, 7s, 1f) - 2(4b, 10s, 1f) = (9-8)b + (21-20)s + (3-2)f = 1b + 1s + 1f. This visual matching is much faster than setting up the x,y multiplier system. Common Pitfall: The most common mistake is choosing "Cannot be determined". Students learn in basic algebra that you need 3 equations to solve for 3 variables. While true for individual variables, it is a dangerous trap here because the question only asks for a *linear combination* of those variables, which is perfectly solvable. Final Answer: Therefore, the correct answer is ₹ 31.

Question 15

If $2x^2 + 12x + 18 = 0$, what is the value of $x$?

Accepted Answer

Concept & Formula: To solve a quadratic equation of the form ax^2 + bx + c = 0, we can simplify the equation by dividing by the greatest common divisor of the coefficients, and then factorize it or use the perfect square formula. Step-by-Step Solution: Given equation: 2x^2 + 12x + 18 = 0 Divide the entire equation by 2: x^2 + 6x + 9 = 0 Recognize the expression as a perfect square (a + b)^2 = a^2 + 2ab + b^2: (x + 3)^2 = 0 Taking the square root on both sides: x + 3 = 0 \implies x = -3 Exam Strategy & Shortcut: Instead of solving completely, look at the signs. Since all signs in the equation are positive (+12x, +18), the value of x must be negative to cancel them out. This immediately eliminates choices (c) and (d). Test x = -3: 2(-3)^2 + 12(-3) + 18 = 18 - 36 + 18 = 0 This confirms the answer efficiently. Common Pitfall: Students often confuse the sign during factorization, writing (x - 3)^2 = 0 instead of (x + 3)^2 = 0, leading to an incorrect answer of 3. Always double-check by expanding your factored expression. Final Answer: Therefore, the correct answer is - 3.

Question 16

The value of $\left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right)\cdots\left(1 - \frac{1}{m}\right)$ is

Accepted Answer

Concept & Formula: This problem involves a telescoping product where successive terms simplify and lead to widespread cancellation between the numerators and denominators. Step-by-Step Solution: Simplify each individual factor in the product: 1 - (1/2) = (1/2) 1 - (1/3) = (2/3) 1 - (1/4) = (3/4) \vdots 1 - (1/m) = (m - 1/m) Substitute these simplified fractions back into the product: ((1/2)) × ((2/3)) × ((3/4)) × \cdots × ((m-2/m-1)) × ((m-1/m)) Notice the cancellation pattern: the denominator of each fraction cancels out with the numerator of the next fraction. The only terms remaining after this cascading cancellation are the first numerator and the last denominator: (1/m) Exam Strategy & Shortcut: For telescoping products of the type (1 - (1/k)), substitute a small value for m to find the pattern. Let m = 3: (1 - (1/2))(1 - (1/3)) = (1/2) × (2/3) = (1/3) Since substituting m = 3 yields (1/3), the general formula must be (1/m). Common Pitfall: Students sometimes write the remaining terms in reverse as (m/1), forgetting that the cancellation pattern eliminates the final numerator, leaving the denominator intact. Final Answer: Therefore, the correct answer is 1/m.

Question 17

The expression $\left(1 + \frac{1}{3}\right)\left(1 + \frac{1}{4}\right)\left(1 + \frac{1}{5}\right)\cdots\left(1 + \frac{1}{n}\right)$ simplifies to

Accepted Answer

Concept & Formula: This represents an additive telescoping product. When terms are simplified, the numerator of each fraction cancels out with the denominator of the subsequent fraction. Step-by-Step Solution: Simplify each term in the product: 1 + (1/3) = (4/3) 1 + (1/4) = (5/4) 1 + (1/5) = (6/5) \vdots 1 + (1/n) = (n + 1/n) Write the complete product using these simplified forms: ((4/3)) × ((5/4)) × ((6/5)) × \cdots × ((n/n-1)) × ((n+1/n)) Observe the cancellation pattern: the numerator of the first term (4) cancels with the denominator of the second term (4), and so on. The only terms that survive the cancellation are the first denominator and the last numerator: (n + 1/3) Exam Strategy & Shortcut: Test the series with a small value, say n = 4: (1 + (1/3))(1 + (1/4)) = (4/3) × (5/4) = (5/3) Substitute n = 4 into the options: Option (a): (4 + 1/3) = (5/3) This perfectly matches our tested result. Common Pitfall: Ensure you track whether it is the numerator or the denominator that survives at each end. For positive terms (1 + (1/k)), the larger value survives on top, and the smaller value survives on the bottom. Final Answer: Therefore, the correct answer is (n + 1)/3.

Question 18

$\frac{3}{4}\left(1 + \frac{1}{3}\right)\left(1 + \frac{2}{3}\right)\left(1 - \frac{2}{5}\right)\left(1 + \frac{6}{7}\right)\left(1 - \frac{12}{13}\right) = x$

Accepted Answer

Concept & Formula: To solve an expression involving a chain product of fractions, first evaluate the expression inside each set of parentheses, then write out the continuous product and cancel common factors across numerators and denominators. Step-by-Step Solution: Simplify every individual bracketed term first: 1 + (1/3) = (4/3) 1 + (2/3) = (5/3) 1 - (2/5) = (3/5) 1 + (6/7) = (13/7) 1 - (12/13) = (1/13) Now, substitute these values back into the primary expression: (3/4) × (4/3) × (5/3) × (3/5) × (13/7) × (1/13) Group and cancel terms sequentially: • (3/4) × (4/3) = 1 • (5/3) × (3/5) = 1 • (13/7) × (1/13) = (1/7) Combining all of these results gives: 1 × 1 × (1/7) = (1/7) Exam Strategy & Shortcut: Scan the entire expression for reciprocal pairs. We can clearly see that (3/4) and (1+(1/3)) = (4/3) are reciprocals, so they cancel out completely. Similarly, (1+(2/3)) = (5/3) and (1-(2/5)) = (3/5) are reciprocals and cancel out. We are immediately left with just the final two components: (13/7) × (1/13) = (1/7). Common Pitfall: Avoid trying to find a common denominator for the whole expression or performing cross-multiplication too early. Simplify each component cleanly first to let the cancellations happen naturally. Final Answer: Therefore, the correct answer is 1/7.

Question 19

What is the value of the following expression?  $\frac{1}{(2^2 - 1)} + \frac{1}{(4^2 - 1)} + \frac{1}{(6^2 - 1)} + \cdots + \frac{1}{(20^2 - 1)}$

Accepted Answer

Concept & Formula: This series can be solved by recognizing it as a telescoping sum. We first expand the denominator using the difference of squares a^2 - b^2 = (a - b)(a + b), and then use partial fractions to split the terms. T_k = (1/(2k)^2 - 1) = (1/(2k - 1)(2k + 1)) Using partial fractions, this splits into: (1/(2k - 1)(2k + 1)) = (1/2) [ (1/2k - 1) - (1/2k + 1) ] Step-by-Step Solution: Apply the partial fraction decomposition to each term in the series: First term (k=1): (1/2^2 - 1) = (1/1 × 3) = (1/2) ( (1/1) - (1/3) ) Second term (k=2): (1/4^2 - 1) = (1/3 × 5) = (1/2) ( (1/3) - (1/5) ) Third term (k=3): (1/6^2 - 1) = (1/5 × 7) = (1/2) ( (1/5) - (1/7) ) Last term (k=10): (1/20^2 - 1) = (1/19 × 21) = (1/2) ( (1/19) - (1/21) ) Now, add all the terms together. We can factor out the (1/2) from the entire sum: Sum = (1/2) [ ((1/1) - (1/3)) + ((1/3) - (1/5)) + \dots + ((1/19) - (1/21)) ] Observe the interior terms cancelling out: -(1/3) cancels +(1/3), and so on. The only terms remaining are the first and the last: Sum = (1/2) [ 1 - (1/21) ] Sum = (1/2) [ (20/21) ] = (10/21) Exam Strategy & Shortcut: When faced with a series of fractions (1/a × b) + (1/b × c) + \dots + (1/y × z) where the difference in the denominator factors (b - a) is constant (let us call this difference D), the sum is simply: Sum = (1/D) × ( (1/First Factor) - (1/Last Factor) ) Here, the factors are 1 × 3, 3 × 5, \dots, 19 × 21. The difference D = 2. First factor is 1, last factor is 21. Sum = (1/2) ( (1/1) - (1/21) ) = (10/21) This takes less than 15 seconds to execute. Common Pitfall: Forgetting to multiply by the factor (1/2) (or (1/D) in general) is the most frequent mistake. Students correctly identify the first and last terms but compute 1 - (1/21) = (20/21), completely missing the (1/2) scaling factor. Final Answer: Therefore, the correct answer is 10/21.

Question 20

The value of $999\frac{995}{999} 	imes 999$ is

Accepted Answer

Concept & Strategy: When a mixed fraction is multiplied by its exact denominator, separating the integer part from the fractional part makes the calculation straightforward. You can apply the distributive property (a + b)c = ac + bc. A(B/C) × C = (A + (B/C)) × C = (A × C) + B Step-by-Step Solution: Write the mixed fraction as an addition: (999 + (995/999)) × 999 Distribute the multiplication by 999: = (999 × 999) + ((995/999) × 999) = 999^2 + 995 To evaluate 999^2 easily, rewrite 999 as (1000 - 1): 999^2 = (1000 - 1)^2 = 1000^2 - 2(1000)(1) + 1^2 = 1000000 - 2000 + 1 = 998001 Add the remaining 995 to the squared value: 998001 + 995 = 998996 Exam Strategy & Shortcut: An alternate trick for calculating 999^2 is using the base 1000. The number is 1 less than 1000. Subtract 1 from the number itself: 999 - 1 = 998. Write the square of the difference (1^2 = 1) in a 3-digit format: 001. Concatenate: 998001. Then simply add 995 to get 998996. Common Pitfall: A very common mistake is assuming the 999 in the integer part and the 999 multiplier cancel out entirely, wrongly leading to simple addition. The multiplier must be distributed to BOTH the integer and the fractional component. Final Answer: Therefore, the correct answer is 998996.

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