Question 1

The growth of a medicinal plant of height 1 metre, is 1 cm/day and it is growing straight vertically upward. A man cuts 6 cm from the top, at a regular interval of every 11 days. Its height will be 2 metres after

Accepted Answer

Concept & Strategy

This is a sequential growth and reduction problem. Care must be taken to distinguish between the continuous daily growth and the discrete periodic cuts. We must track the daily height to see exactly when it hits the threshold.

Step-by-Step Solution
• Given:
• Initial height = 1 m = 100 cm
• Target height = 2 m = 200 cm
• Growth rate = 1 cm/day
• Cut = 6 cm every 11 days.
• Deduction:
 In one 11-day period, the plant grows 11 cm and is then cut by 6 cm.
 	ext{Net growth per 11 days} = 11 - 6 = 5 	ext{ cm}
• Calculation:
 Find how many 11-day cycles it takes to get near 200 cm.
 After 19 cycles (19 	imes 11 = 209 days):
 	ext{Height} = 100 + 19(5) = 195 	ext{ cm}
 
 Now, track the daily growth from day 210 onwards (before the 20th cut happens on day 220):
• Day 210: 196 cm
• Day 211: 197 cm
• Day 212: 198 cm
• Day 213: 199 cm
• Day 214: 200 cm
 
 The plant reaches exactly 2 metres on the 214th day. Since 214 is not listed among the options, the correct choice is "None of these".

Exam Strategy & Shortcut

Evaluate the widely accepted "flawed" logic vs. the strict mathematical reality. 
Flawed logic: Target growth = 100 cm. Net rate = 5 cm / 11 days. Time = (100 \div 5) 	imes 11 = 220 days.
Strict math: At day 209, height is 195cm. It grows 1cm/day for 5 days to hit 200cm exactly at day 214. Since 214 is missing, confidently choose "None of these".

Common Pitfall

Blindly assuming the target height is only reached exactly at the end of a full cycle and calculating (100 \div 5) 	imes 11 = 220 days (Option a). This ignores the intra-cycle peak where the plant hits 200 cm organically on day 214 before the next cut occurs.

Final Answer

Therefore, the correct answer is None of these.

Question 2

If $ x = 5 $, $ y = 3 $, the value of $ \frac{x^3 - y^3}{x^2 - y^2} - \frac{3xy}{x + y} $ will be

Accepted Answer

Concept & Formula

This problem tests your ability to simplify rational algebraic expressions before inserting numerical values. The goal is to combine the two fractions into a much simpler form.

The necessary factorization formulas are:
 x^3 - y^3 = (x - y)(x^2 + xy + y^2) 
 x^2 - y^2 = (x - y)(x + y)

Step-by-Step Solution

Let's start with the first fraction in the expression:
 \frac{x^3 - y^3}{x^2 - y^2}

Factor both the numerator and the denominator using the standard formulas:
 = \frac{(x - y)(x^2 + xy + y^2)}{(x - y)(x + y)}

Cancel out the common (x - y) term from the top and bottom:
 = \frac{x^2 + xy + y^2}{x + y}

Now, bring back the second part of the original expression and subtract it:
 \frac{x^2 + xy + y^2}{x + y} - \frac{3xy}{x + y}

Because both fractions now share a common denominator ( x + y ), we can combine their numerators:
 = \frac{x^2 + xy + y^2 - 3xy}{x + y}

Combine the like terms ( +xy - 3xy = -2xy ):
 = \frac{x^2 - 2xy + y^2}{x + y}

Notice that the numerator is now a perfect square trinomial. Factor it:
 = \frac{(x - y)^2}{x + y}

The expression is fully simplified. Now substitute the given values: x = 5 and y = 3 .
 = \frac{(5 - 3)^2}{5 + 3} 
 = \frac{(2)^2}{8} 
 = \frac{4}{8} 
 = \frac{1}{2}

Exam Strategy & Shortcut

Instead of algebraic manipulation, you can use direct substitution ("Value Putting") immediately, as the numbers are very small.
Fraction 1: \frac{5^3 - 3^3}{5^2 - 3^2} = \frac{125 - 27}{25 - 9} = \frac{98}{16} = \frac{49}{8} 
Fraction 2: \frac{3(5)(3)}{5 + 3} = \frac{45}{8} 
Difference: \frac{49}{8} - \frac{45}{8} = \frac{4}{8} = \frac{1}{2} . 
For small integers, basic arithmetic is often faster than performing multi-step polynomial factorization.

Common Pitfall

When simplifying algebraically, students sometimes mistakenly cancel (x^2 - y^2) out of the denominator without properly factoring the numerator, or miscalculate the combination +xy - 3xy as +2xy , which leads to an entirely incorrect final value.

Final Answer

Therefore, the correct answer is \frac{1}{2} .

Question 3

The value of  $$\frac{(x - y)^3 + (y - z)^3 + (z - x)^3}{(x^2 - y^2)^3 + (y^2 - z^2)^3 + (z^2 - x^2)^3}$$ is

Accepted Answer

Concept & Formula

This fraction simplifies using the conditional identity for the sum of cubes: If A + B + C = 0, then A^3 + B^3 + C^3 = 3ABC.

Step-by-Step Solution
• Let the numerator components be A = x - y, B = y - z, and C = z - x.
• Notice that A + B + C = (x - y) + (y - z) + (z - x) = 0.
• Therefore, the numerator becomes 3(x - y)(y - z)(z - x).
• Now, let the denominator components be P = x^2 - y^2, Q = y^2 - z^2, and R = z^2 - x^2.
• Notice that P + Q + R = (x^2 - y^2) + (y^2 - z^2) + (z^2 - x^2) = 0.
• Therefore, the denominator becomes 3(x^2 - y^2)(y^2 - z^2)(z^2 - x^2).
• Expand the denominator using the difference of squares a^2 - b^2 = (a-b)(a+b):
3(x - y)(x + y)(y - z)(y + z)(z - x)(z + x)
• Divide the simplified numerator by the simplified denominator:
\frac{3(x - y)(y - z)(z - x)}{3(x - y)(x + y)(y - z)(y + z)(z - x)(z + x)}
• Cancel the common terms 3(x - y)(y - z)(z - x) from top and bottom:
= \frac{1}{(x + y)(y + z)(z + x)}
• Rewrite using a negative exponent:
= [(x + y)(y + z)(z + x)]^{-1}

Exam Strategy & Shortcut

Whenever you see a sum of three cubes in a pattern that looks cyclical (like x-y, y-z, z-x), immediately check if they sum to zero. If they do, instantly replace them with 3 times their product. This visual recognition avoids lengthy expansions.

Common Pitfall

A common error is to try expanding (x - y)^3 individually. This creates a massive, unmanageable polynomial that guarantees a loss of time and inevitable calculation errors.

Final Answer

Therefore, the correct answer is [(x + y) (y + z) (z + x)]^{-1}.

Question 4

If $a = 29$, $b = 24$, $c = 27$, the value of $a^3 + b^3 + c^3 - 3abc$ is

Accepted Answer

Concept & Formula

When the values of a, b, and c are large but relatively close to each other, computing their cubes directly is inefficient. Instead, use the specialized form of the cubic identity involving squares of differences:

a^3 + b^3 + c^3 - 3abc = \frac{1}{2}(a + b + c)[(a - b)^2 + (b - c)^2 + (c - a)^2]

Step-by-Step Solution
• Given: a = 29, b = 24, c = 27.
• First, calculate the sum of the variables:
a + b + c = 29 + 24 + 27 = 80
• Next, calculate the differences between each pair of variables:
a - b = 29 - 24 = 5
b - c = 24 - 27 = -3
c - a = 27 - 29 = -2
• Square these differences:
(a - b)^2 = 5^2 = 25
(b - c)^2 = (-3)^2 = 9
(c - a)^2 = (-2)^2 = 4
• Sum the squares of the differences:
25 + 9 + 4 = 38
• Plug everything into the specialized formula:
\frac{1}{2} 	imes (80) 	imes (38)
• Simplify the calculation:
= 40 	imes 38 = 1520

Exam Strategy & Shortcut

Using the difference-of-squares formula \frac{1}{2}(a+b+c)[(a-b)^2 + (b-c)^2 + (c-a)^2] is the ultimate shortcut here. It reduces large multi-digit cube calculations to basic addition and single-digit squaring.

Common Pitfall

The most common trap is attempting brute-force calculation (29^3 + 24^3 + 27^3 - \dots). This will consume several minutes and likely result in an arithmetic mistake.

Final Answer

Therefore, the correct answer is 1520.

Question 5

$(32)^3 + (79)^3 - (111)^3 + 3 	imes 32 	imes 79 	imes 111$ is equal to

Accepted Answer

Concept & Formula

This expression evaluates to zero based on a clever disguised application of the conditional identity: If a + b + c = 0, then a^3 + b^3 + c^3 - 3abc = 0.

Step-by-Step Solution
• Look closely at the structure of the expression: (32)^3 + (79)^3 + (-111)^3 - 3(32)(79)(-111).
• Notice that the negative signs have been distributed. The expression exactly matches the form a^3 + b^3 + c^3 - 3abc.
• Let a = 32, b = 79, and c = -111.
• Note that in the given expression, subtracting (111)^3 is mathematically identical to adding (-111)^3. 
• Similarly, adding + 3 	imes 32 	imes 79 	imes 111 is identical to subtracting -3 	imes (32) 	imes (79) 	imes (-111).
• Check the sum of the variables: 
a + b + c = 32 + 79 + (-111) = 111 - 111 = 0
• Because a + b + c = 0, the entire expression a^3 + b^3 + c^3 - 3abc must equal 0.

Exam Strategy & Shortcut

Any time an aptitude question asks you to add and subtract massive cubed numbers alongside a product of those same numbers, it is always testing the a+b+c=0 rule. Don't calculate anything; just verify that the base numbers sum to zero.

Common Pitfall

Failing to recognize that c is negative (-111) and incorrectly assuming the formula doesn't apply because the signs look slightly different than the standard textbook formula.

Final Answer

Therefore, the correct answer is 0.

Question 6

A man had two sons. To the elder, he left 5/11 of his property, to the younger 5/11 of the remainder, the rest to the widow. Find the share of the sons if the widow gets ₹ 3600.

Accepted Answer

Concept & Logic

This is a successive distribution problem. When a portion is given from the "remainder", the new fraction must be multiplied by the remaining fraction of the total, not simply subtracted from the total.

Remaining\ Fraction = 1 - Fraction\ Given

Step-by-Step Solution
• Given: Elder son gets \frac{5}{11} of the total property. Younger gets \frac{5}{11} of the remainder. Widow gets ₹ 3600. Let total property be P.
• Calculation: Elder son's share = \frac{5}{11}P.
• Deduction: The remainder of the property after the elder son's share is P - \frac{5}{11}P = \frac{6}{11}P.
• Calculation: Younger son's share is \frac{5}{11} of this remainder:
 Younger = \frac{5}{11} 	imes \frac{6}{11}P = \frac{30}{121}P.
• Deduction: The widow gets whatever is left from the remainder after the younger son is paid:
 Widow's fraction = \frac{6}{11}P - \frac{30}{121}P = \frac{66P - 30P}{121} = \frac{36}{121}P.
• Calculation: Equate the widow's fractional share to her actual amount:
 \frac{36}{121}P = 3600
 P = \frac{3600 	imes 121}{36} = 12100.
• Calculation: Find the exact share of both sons:
 Elder son = \frac{5}{11} 	imes 12100 = 5500.
 Younger son = \frac{30}{121} 	imes 12100 = 3000.
• Conclusion: The shares are ₹ 5500 and ₹ 3000, which do not match options (a), (b), or (c).

Exam Strategy & Shortcut

Work with ratios. The fractions involve 11, so let the total be a multiple of 11 	imes 11 = 121 units.
Total = 121 units.
Elder = \frac{5}{11} of 121 = 55 units. Remainder = 66 units.
Younger = \frac{5}{11} of 66 = 30 units. Remainder (Widow) = 36 units.
We know 36 units = ₹ 3600, so 1 unit = ₹ 100.
Thus, Elder = 5500, Younger = 3000. These are not in the options, confirming "None of these".

Common Pitfall

Treating "\frac{5}{11} of the remainder" as simply \frac{5}{11} of the total. Students who do this incorrectly calculate the widow's share as 1 - \frac{5}{11} - \frac{5}{11} = \frac{1}{11}, leading to completely wrong valuations for the property.

Final Answer

Therefore, the correct answer is None of these.

Question 7

Anand, Bijoy, Chetan and Dharma together have ₹ 47 with them. Anand and Bijoy together have ₹ 27; Chetan and Anand have ₹ 25 and Dharma and Anand have ₹ 23. How much money does Bijoy have?

Accepted Answer

Concept & Logic

This is a system of linear equations where one variable (Anand) acts as a common link across multiple subset sums. Summing the overlapping equations allows us to extract the value of the common variable efficiently.

Sum\ of\ Subsets = Multiplier 	imes Common\ Variable + Total\ Sum

Step-by-Step Solution
• Given: Let their money be A, B, C, D. 
 Total: A + B + C + D = 47
 Equation 1: A + B = 27
 Equation 2: C + A = 25
 Equation 3: D + A = 23
• Calculation: Add Equations 1, 2, and 3 together:
 (A + B) + (C + A) + (D + A) = 27 + 25 + 23
 3A + B + C + D = 75
• Substitution: We know that A + B + C + D = 47. We can rewrite our summed equation as:
 2A + (A + B + C + D) = 75
 2A + 47 = 75
• Calculation: Solve for A (Anand):
 2A = 75 - 47
 2A = 28
 A = 14
• Calculation: The question asks for Bijoy's money (B). Substitute A into Equation 1:
 14 + B = 27 \implies B = 13

Exam Strategy & Shortcut

Add the three partial sums: 27 + 25 + 23 = 75.
Notice that this sum counts Anand (A) three times, while counting everyone else (B, C, D) exactly once.
So, Sum = 2A + Total.
75 = 2A + 47 \implies 2A = 28 \implies A = 14.
Since A + B = 27, Bijoy (B) is simply 27 - 14 = 13. This can be done mentally in seconds.

Common Pitfall

Attempting to solve for each individual person (C and D) before answering the question. This wastes valuable exam time. Always focus strictly on finding the specific variable requested (in this case, B, which only requires finding A).

Final Answer

Therefore, the correct answer is ₹ 13.

Question 8

When a sum of money was distributed to 12 persons instead of 16 persons, each person got ₹ 400 more. What was the sum?

Accepted Answer

Concept & Logic

This problem involves inverse variation. The total sum is constant. The difference in the per-person share between the two scenarios forms a simple algebraic equation based on fractions of the total sum.

\frac{Total}{Smaller\ Group} - \frac{Total}{Larger\ Group} = Difference\ in\ Share

Step-by-Step Solution
• Given: Let the total sum of money be x.
 Original scenario: Divided among 16 persons. Share = \frac{x}{16}.
 New scenario: Divided among 12 persons. Share = \frac{x}{12}.
 Difference in share = ₹ 400.
• Calculation: Set up the equation representing the difference:
 \frac{x}{12} - \frac{x}{16} = 400
• Calculation: Find the Lowest Common Multiple (LCM) of 12 and 16, which is 48. Rewrite the fractions:
 \frac{4x}{48} - \frac{3x}{48} = 400
 \frac{x}{48} = 400
• Calculation: Solve for x:
 x = 400 	imes 48 = 19200.

Exam Strategy & Shortcut

Use ratio mapping to avoid algebra.
The ratio of people is 12 : 16, which simplifies to 3 : 4.
Because the sum is constant, the ratio of their individual shares is inversely proportional: 4 : 3.
The difference in shares is 1 unit (4 - 3 = 1 unit).
We are told this difference is ₹ 400. So, 1 unit = ₹ 400.
The share when divided among 12 people (the 4-unit share) is 4 	imes 400 = 1600.
Total sum = Share 	imes Number of people = 1600 	imes 12 = 19200.

Common Pitfall

Writing the equation backwards as \frac{x}{16} - \frac{x}{12} = 400, which results in a negative value for the total sum. Always remember that dividing by a smaller number yields a larger share, so \frac{x}{12} must come first.

Final Answer

Therefore, the correct answer is ₹ 19200.

Question 9

Vidushi and Sanya distributed ₹ 100 each in charity. Vidushi distributes money to 5 more people than Sanya and Sanya gives each ₹ 1 more than Vidushi. How many people are recipients of the charity?

Accepted Answer

Concept & Logic

This problem links the number of recipients to the amount given per recipient via an inverse relationship, forming a quadratic equation. The total amount given by each person is fixed at ₹ 100.

\frac{Total\ Amount}{Number\ of\ People} = Amount\ per\ Person

Step-by-Step Solution
• Given: Both distribute ₹ 100.
 Let the number of people Sanya distributes to be S.
 Then, the number of people Vidushi distributes to is S + 5.
• Deduction: Amount Sanya gives to each person = \frac{100}{S}.
 Amount Vidushi gives to each person = \frac{100}{S+5}.
• Calculation: The problem states Sanya gives ₹ 1 more per person than Vidushi.
 \frac{100}{S} - \frac{100}{S+5} = 1
• Calculation: Factor out 100 and find a common denominator:
 100 \left( \frac{(S+5) - S}{S(S+5)} \right) = 1
 100 \left( \frac{5}{S^2 + 5S} \right) = 1
 500 = S^2 + 5S
• Calculation: Solve the quadratic equation:
 S^2 + 5S - 500 = 0
 Find factors of -500 that add to +5. They are +25 and -20.
 (S + 25)(S - 20) = 0. Since S must be positive, S = 20.
• Conclusion: Sanya distributes to 20 people. 
 Vidushi distributes to 20 + 5 = 25 people.
 Total recipients = 20 + 25 = 45.

Exam Strategy & Shortcut

Bypass the quadratic equation by looking for two integer factors of 100 whose difference is related to the problem. 
We need two numbers of people that divide 100 perfectly (since usually amounts are integers in these contexts), and their difference must be 5.
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100.
Look at 20 and 25. Their difference is 5.
Check the amounts: \frac{100}{20} = 5 and \frac{100}{25} = 4.
The difference in amounts is 5 - 4 = 1. This perfectly matches all conditions in the question!
Total people = 20 + 25 = 45.

Common Pitfall

Solving for S (20) and selecting an option if it were available, or solving for Vidushi (25). Read the final question carefully: it asks "How many people are recipients of the charity?", meaning the *total combined* number of recipients (20 + 25 = 45).

Final Answer

Therefore, the correct answer is 45.

Question 10

In a classroom the number of boys is three times the number of girls. Which of the following numbers does not represent the total number of students in the classroom?

Accepted Answer

Concept & Logic

This problem relies on basic ratios and divisibility rules. Since the number of human beings (students) must be an integer, the total number of students must be a multiple of the sum of the ratio units.

Total\ Units = Ratio_1 + Ratio_2 
 Total\ Population = Integer 	imes Total\ Units

Step-by-Step Solution
• Given: Number of boys = 3 	imes Number of girls.
• Deduction: Let the number of girls be x.
 Then, the number of boys is 3x.
• Calculation: The total number of students in the classroom is the sum of boys and girls.
 Total = x + 3x = 4x.
• Conclusion: Since x (number of girls) must be a whole number, the total number of students (4x) must be a perfect multiple of 4.
• Evaluation: Check the given options against divisibility by 4:
 (a) 40 = 4 	imes 10 (Valid)
 (b) 42 = 4 	imes 10.5 (Invalid, results in a fractional person)
 (c) 44 = 4 	imes 11 (Valid)
 (d) 48 = 4 	imes 12 (Valid)
• Conclusion: 42 cannot represent the total number of students.

Exam Strategy & Shortcut

Simply identify the ratio: Boys : Girls = 3 : 1.
Sum of ratio parts = 3 + 1 = 4.
The correct answer must be the option that is not divisible by 4.
Looking at the options: 40, 44, and 48 are clearly multiples of 4. 42 is not. Done.

Common Pitfall

Overthinking the problem or trying to set up complex algebraic systems. Recognizing that populations must be discrete integers immediately unlocks the "divisibility by ratio sum" trick.

Final Answer

Therefore, the correct answer is 42.

Question 11

The expression $\frac{1}{x-1} - \frac{1}{x+1} - \frac{2}{x^2+1} - \frac{4}{x^4+1}$ is equal to

Accepted Answer

Concept & Formula

This involves stepwise algebraic simplification of rational expressions. Like the previous problem, it relies heavily on creating common denominators using the difference of squares.
Formula: (a - b)(a + b) = a^2 - b^2

Step-by-Step Solution
• Solve the expression in pairs from left to right. Take the first two terms:
\frac{1}{x-1} - \frac{1}{x+1}
• Find the common denominator (x-1)(x+1) = x^2 - 1:
\frac{(x+1) - (x-1)}{x^2 - 1} = \frac{2}{x^2 - 1}
• Substitute this result back into the expression, taking the next term:
\frac{2}{x^2 - 1} - \frac{2}{x^2 + 1}
• Find the common denominator (x^2 - 1)(x^2 + 1) = x^4 - 1:
\frac{2(x^2 + 1) - 2(x^2 - 1)}{x^4 - 1} = \frac{2x^2 + 2 - 2x^2 + 2}{x^4 - 1} = \frac{4}{x^4 - 1}
• Substitute this result back, taking the final term:
\frac{4}{x^4 - 1} - \frac{4}{x^4 + 1}
• Find the common denominator (x^4 - 1)(x^4 + 1) = x^8 - 1:
\frac{4(x^4 + 1) - 4(x^4 - 1)}{x^8 - 1} = \frac{4x^4 + 4 - 4x^4 + 4}{x^8 - 1} = \frac{8}{x^8 - 1}

Exam Strategy & Shortcut

Notice the predictable doubling pattern. The numerator doubles at every step (1 \rightarrow 2 \rightarrow 4 \rightarrow 8), and the denominator powers double with a minus sign (x^2-1 \rightarrow x^4-1 \rightarrow x^8-1). Knowing this structural behavior, you can directly leap to \frac{8}{x^8 - 1} without performing manual fraction subtractions.

Common Pitfall

A frequent mistake is mishandling the negative signs during numerator expansion (e.g., getting zero in the numerator). Always use parentheses when subtracting binomials in the numerator to ensure the negative distributes correctly across both terms.

Final Answer

Therefore, the correct answer is \frac{8}{x^8-1}.

Question 12

In a party 15 people shake their hands with each other. How many times did the hand-shakes take place?

Accepted Answer

Concept & Formula

When every person in a group shakes hands with every other person exactly once, the total number of handshakes is a basic combination problem. The number of unique pairs that can be formed from n items is given by:

ext{Total Handshakes} = \frac{n(n - 1)}{2}

Step-by-Step Solution
• Given: The total number of people, n = 15.
• Calculation: Substitute n = 15 into the combination formula.
• Total Handshakes = \frac{15 	imes (15 - 1)}{2}
• Total Handshakes = \frac{15 	imes 14}{2}
• Total Handshakes = 15 	imes 7 = 105

Exam Strategy & Shortcut

For handshake problems, do not waste time writing the formula. Immediately multiply the given number by the number just below it, and halve it. Mentally calculating 15 	imes 7 = 105 takes less than 3 seconds.

Common Pitfall

A very common mistake is using n(n - 1) which gives 210. This happens when students confuse handshakes (where A shaking B's hand is the same as B shaking A's hand) with gift exchanges (which are directional). Always divide by 2 for handshakes.

Final Answer

Therefore, the correct answer is 105.

Question 13

In a test, a candidate secured 336 marks out of maximum marks $x$. If the maximum marks $x$ had been converted into 400 marks, he would have secured 192 marks. What was the maximum marks of the test?

Accepted Answer

Concept & Formula

This problem requires setting up a direct proportion. The ratio of the marks obtained to the maximum marks remains constant, even if the maximum marks scale is converted.

\frac{	ext{Obtained Marks}_1}{	ext{Maximum Marks}_1} = \frac{	ext{Obtained Marks}_2}{	ext{Maximum Marks}_2}

Step-by-Step Solution
• Given:
• Initial state: Obtained = 336, Maximum = x.
• Converted state: Obtained = 192, Maximum = 400.
• Equation: Set up the proportion:
• \frac{336}{x} = \frac{192}{400}
• Calculation:
• Simplify the right side fraction first: \frac{192}{400} divides cleanly by 8 to give \frac{24}{50}, and dividing by 2 gives \frac{12}{25}.
• Now, \frac{336}{x} = \frac{12}{25}
• Cross-multiply to solve for x: 12x = 336 	imes 25
• x = \frac{336 	imes 25}{12}
• Divide 336 by 12: 336 / 12 = 28.
• x = 28 	imes 25
• x = 700

Exam Strategy & Shortcut

Recognize the multiplier scale. The new base is 400. The ratio is \frac{192}{400}. Observe that 192 is slightly less than half of 400. Thus, 336 must be slightly less than half of x. This means x is slightly more than double 336 (which is 672). The only logical option nearby is 700. Alternatively, multiplying 28 	imes 25 is quickly done as 28 	imes \frac{100}{4} = 7 	imes 100 = 700.

Common Pitfall

Getting bogged down in massive cross-multiplication without simplifying the fractions first. Doing 336 	imes 400 = 134400 and then trying to divide by 192 by hand wastes precious exam time and increases the likelihood of arithmetic errors. Always simplify fractions first.

Final Answer

Therefore, the correct answer is 700.

Question 14

The marks scored in an examination are converted from 50 to 10 for the purpose of internal assessment. The highest marks were 47 and the lowest were 14. The difference between the maximum and the minimum internal assessment scores is

Accepted Answer

Concept & Formula

When a set of values is scaled by a constant factor, the difference between any two values in the set is also scaled by that exact same factor.

ext{Scaled Difference} = 	ext{Original Difference} 	imes 	ext{Scale Factor}

Step-by-Step Solution
• Scale Factor: The base marks are converting from 50 to 10.
• Scale Factor = \frac{10}{50} = \frac{1}{5}. This means every score is divided by 5.
• Original Difference: Calculate the difference between the highest and lowest original marks.
• Original Maximum = 47, Original Minimum = 14.
• Original Difference = 47 - 14 = 33.
• Calculation: Apply the scale factor to the original difference.
• Scaled Difference = 33 	imes \frac{1}{5} = \frac{33}{5}
• Scaled Difference = 6.6

Exam Strategy & Shortcut

Instead of converting both the maximum and the minimum scores individually (\frac{47}{5} = 9.4 and \frac{14}{5} = 2.8) and then subtracting them (9.4 - 2.8 = 6.6), it is vastly faster to subtract the raw scores first (33) and divide by the scale factor once (\frac{33}{5} = 6.6).

Common Pitfall

Converting the individual numbers first. While mathematically correct, dealing with decimals like 9.4 and 2.8 under time pressure invites subtraction errors. Stick to integers as long as possible before dividing.

Final Answer

Therefore, the correct answer is 6.6.

Question 15

In an examination, a student scores 4 marks for every correct answer and loses 1 mark for every wrong answer. A student attempted all the 200 questions and scored in all 200 marks. The number of questions he answered correctly was

Accepted Answer

Concept & Strategy

This is a classic penalty scoring problem. It is best solved using the "Deviation Method" (or assumed total method), which calculates the total possible score if all questions were correct, and then divides the "loss" in marks by the penalty per wrong answer to find the number of incorrect answers.

ext{Wrong Answers} = \frac{	ext{Maximum Possible Score} - 	ext{Actual Score}}{	ext{Marks per Correct Answer} + 	ext{Penalty per Wrong Answer}}

Step-by-Step Solution
• Assume all are correct: If the student answered all 200 questions correctly, the maximum possible score would be 200 	imes 4 = 800 marks.
• Calculate the total loss: The student actually scored 200 marks. The marks lost due to wrong answers = 800 - 200 = 600 marks.
• Determine loss per wrong answer: Every wrong answer costs the student 4 marks (the marks they *would* have gotten) PLUS 1 mark (the negative penalty). Total loss per wrong answer = 4 + 1 = 5 marks.
• Find number of wrong answers: Total wrong answers = \frac{600}{5} = 120.
• Find number of correct answers: Total questions minus wrong answers = 200 - 120 = 80.

Exam Strategy & Shortcut

Using the algebraic method (4c - 1(200 - c) = 200) works, but the assumed total method is much faster mentally. You can simply think: "Max score is 800. He lost 600. Each mistake costs 5 points. 600 / 5 = 120 wrong. So 80 right." This eliminates algebra entirely.

Common Pitfall

The most common mistake is assuming the loss per wrong answer is just the penalty mark (1). This leads to calculating the wrong answers as 600 / 1 = 600, which makes no sense. You must remember that getting a question wrong also means forfeiting the positive marks it carried.

Final Answer

Therefore, the correct answer is 80.

Question 16

There are some cows, bulls and 45 hens in a group. One caretaker looks after 15 animals. The number of bulls is twice the number of cows. If the number of heads is less than the total number of feet by 186 (including the caretakers), how many careta…

Accepted Answer

Concept & Logic

This is an advanced algebraic puzzle requiring you to construct equations based on physical attributes (heads and feet) across multiple categories (animals and humans), while maintaining relational constraints (bulls to cows, caretakers to animals).

ext{Total Feet} - 	ext{Total Heads} = 186

Step-by-Step Solution
• Define Variables: Let the number of cows be c.
• Bulls (b) = 2c.
• Hens (h) = 45.
• Total Animals (A): A = c + 2c + 45 = 3c + 45.
• Caretakers (k): One caretaker for every 15 animals.
• k = \frac{3c + 45}{15} = \frac{c}{5} + 3.
• Total Heads (H): Every creature has 1 head.
• H = 	ext{Animals} + 	ext{Caretakers} = (3c + 45) + \left(\frac{c}{5} + 3ight) = 3.2c + 48.
• Total Feet (F): Cows and bulls have 4 feet, hens and caretakers have 2.
• F = 4(c) + 4(2c) + 2(45) + 2\left(\frac{c}{5} + 3ight)
• F = 4c + 8c + 90 + 0.4c + 6 = 12.4c + 96.
• Apply the Given Condition: Feet minus Heads = 186.
• (12.4c + 96) - (3.2c + 48) = 186
• 9.2c + 48 = 186
• 9.2c = 138
• c = \frac{138}{9.2} = 15.
• Final Calculation: Number of cows is 15. We need the number of caretakers (k).
• k = \frac{c}{5} + 3 = \frac{15}{5} + 3 = 3 + 3 = 6.

Exam Strategy & Shortcut

Notice the caretaker fraction: k = \frac{c}{5} + 3. For caretakers to be a whole number (you can't have a fraction of a person), the number of cows (c) *must* be a multiple of 5 (e.g., 5, 10, 15, 20). If you test c=15, then k = 6. Check the total feet and heads with these numbers to verify it equals 186. Guessing and checking logically is often faster than solving decimals algebraically.

Common Pitfall

Forgetting to include the caretakers in the head and feet count. The problem explicitly states "(including the caretakers)", but under pressure, students often calculate heads and feet only for the animals, leading to a completely unsolvable equation.

Final Answer

Therefore, the correct answer is 6.

Question 17

Mr. Shah decided to walk down the escalator of a tube station. He found that if he walks down 26 steps, he requires 30 seconds to reach the bottom. However, if he steps down 34 stairs he would only require 18 seconds to get to the bottom.   If the t…

Accepted Answer

Concept & Strategy

The total number of steps on an escalator is constant. The total steps is equal to the steps walked by the person plus the steps the escalator moves on its own during the time it took the person to reach the bottom.

ext{Total Steps} = 	ext{Steps Walked} + (	ext{Escalator Speed} 	imes 	ext{Time})

Step-by-Step Solution
• Set up the variables: Let N be the total number of steps, and e be the speed of the escalator in steps per second.
• Formulate Equation 1 (Case 1): He walks 26 steps in 30 seconds.
• N = 26 + 30e
• Formulate Equation 2 (Case 2): He walks 34 steps in 18 seconds.
• N = 34 + 18e
• Equate and Solve: Since N is the same in both cases:
• 26 + 30e = 34 + 18e
• 30e - 18e = 34 - 26
• 12e = 8
• e = \frac{8}{12} = \frac{2}{3} steps per second.
• Calculate Total Steps (N): Substitute e back into the first equation.
• N = 26 + 30 	imes \left(\frac{2}{3}ight)
• N = 26 + 20 = 46.

Exam Strategy & Shortcut

Observe the differences directly. 
Difference in steps walked = 34 - 26 = 8 steps.
Difference in time taken = 30 - 18 = 12 seconds.
This means the escalator covers those 8 steps in 12 seconds, giving an escalator speed of \frac{8}{12} or \frac{2}{3} steps per second. 
In the first scenario (30 seconds), the escalator contributes 30 	imes \left(\frac{2}{3}ight) = 20 steps. 
Total steps = 26 (	ext{walked}) + 20 (	ext{escalator}) = 46.

Common Pitfall

Assuming that speed equals \frac{	ext{Distance}}{	ext{Time}} for the person alone (e.g., trying to calculate 26/30). You must account for the fact that the escalator acts as a moving frame of reference, adding its own speed to the person's descent.

Final Answer

Therefore, the correct answer is 46.

Question 18

These questions are based on the following information:  85 children went to an amusement park where they could ride on merry-go-round, roller coaster and Ferris wheel. It was known that 20 of them took all the three rides and 55 of them have taken …

Accepted Answer

Concept & Logic

Following up on the previous calculation, the total number of children is the sum of children who took 0, 1, 2, and 3 rides. Once we have the subsets, finding the remainder is straightforward addition and subtraction.

Step-by-Step Solution
• Given: Total children = 85.
• From previous analysis, we know the breakdown of rides:
• Children taking exactly 3 rides: n(3) = 20
• Children taking exactly 2 rides: n(2) = 35
• Children taking exactly 1 ride: n(1) = 15
• Calculation: The sum of all categories must equal the total number of children.
n(0) + n(1) + n(2) + n(3) = 	ext{Total Children}
n(0) + 15 + 35 + 20 = 85
• Sum the known values:
n(0) + 70 = 85
• Isolate n(0):
n(0) = 85 - 70 = 15
• Thus, 15 children did not try any of the rides.

Exam Strategy & Shortcut

If you have already calculated that n(1) = 15 from the previous question, this step takes 5 seconds. 
Total children = 85. 
Active children = (Exactly 1) + (At least 2) = 15 + 55 = 70. 
Inactive children = 85 - 70 = 15.

Common Pitfall

Accidentally double counting the children who took 3 rides when trying to sum the groups. Remember that the "55" figure for "at least two rides" *already includes* the 20 who took three rides.

Final Answer

Therefore, the correct answer is 15.

Question 19

In a group of 52 persons, 16 drink tea but not coffee and 33 drink tea.  How many drink coffee but not tea?

Accepted Answer

Concept & Logic

This problem requires breaking down two sets into three distinct zones: Set A only, Set B only, and Both.

By definition, the total group (Union) is the sum of these three non-overlapping zones: (A 	ext{ only}) + (B 	ext{ only}) + (	ext{Both}).

Step-by-Step Solution
• Given: Total group size (assuming everyone drinks at least one), n(T \cup C) = 52
 Tea only, n(T - C) = 16
 Total Tea, n(T) = 33
• Calculation: * First, let's find how many people drink *both*. We know the total Tea drinkers consist of "Tea only" plus "Both".
n(T) = n(T - C) + n(T \cap C)
33 = 16 + n(T \cap C)
n(T \cap C) = 33 - 16 = 17
• Now, we want to find "Coffee only", which is n(C - T). We know the total group is 52.
	ext{Total} = (Tea 	ext{ only}) + (	ext{Both}) + (Coffee 	ext{ only})
52 = 16 + 17 + n(C - T)
52 = 33 + n(C - T)
• Solve for Coffee only:
n(C - T) = 52 - 33 = 19

Exam Strategy & Shortcut

Notice that the total Tea circle n(T) perfectly encompasses both "Tea only" and "Both". Therefore, if you subtract the entire Tea circle from the total population, you are left exclusively with "Coffee only".
	ext{Coffee Only} = 	ext{Total Population} - 	ext{Total Tea Drinkers}
	ext{Coffee Only} = 52 - 33 = 19
You do not even need the "16 drink tea but not coffee" data point—it is entirely redundant filler!

Common Pitfall

Getting distracted by the redundant "16 drink tea but not coffee" and spending time calculating the intersection (17), only to forget what the final goal is and mistakenly selecting 17 (Option c) as the final answer. Always keep your eye on the specific metric requested.

Final Answer

Therefore, the correct answer is 19.

Question 20

In a town with population of 4000, 3000 people are egg eaters, 2000 meat eaters and 1500 eat both eggs and meat.  How many are pure vegetarians?

Accepted Answer

Concept & Logic

This is a fundamental Venn diagram problem. "Pure vegetarians" implies people who eat *neither* eggs nor meat.

We must find the total number of non-vegetarians (the union of egg and meat eaters) and subtract it from the total population.

Step-by-Step Solution
• Given: Total Population = 4000
 Egg eaters, n(E) = 3000
 Meat eaters, n(M) = 2000
 Both, n(E \cap M) = 1500
• Calculation: * Find the total number of people who eat eggs, meat, or both using the inclusion-exclusion principle:
n(E \cup M) = n(E) + n(M) - n(E \cap M)
n(E \cup M) = 3000 + 2000 - 1500
n(E \cup M) = 5000 - 1500 = 3500
• This means 3500 people in the town are non-vegetarians (they consume animal products).
• To find the pure vegetarians, subtract this number from the total town population:
	ext{Vegetarians} = 	ext{Total Population} - n(E \cup M)
	ext{Vegetarians} = 4000 - 3500 = 500

Exam Strategy & Shortcut

Add the single categories together: 3000 + 2000 = 5000. 
Since the town only has 4000 people, the "excess" must represent the overlap (people counted twice). 
Wait, the overlap is given as 1500. 
So, distinct eaters = 5000 - 1500 = 3500.
Remaining population = 4000 - 3500 = 500.

Common Pitfall

The most frequent error is simply subtracting the two single categories from the total: 4000 - 3000 - 2000 = -1000. This fails because you are heavily double-subtracting the 1500 people who do both. Always establish the precise union before subtracting from the master total.

Final Answer

Therefore, the correct answer is 500.

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