Question 1

How many terms are there in the G.P. 3, 6, 12, 24, …, 384?

Accepted Answer

Given data Geometric progression (G.P.): 3, 6, 12, 24, …, 384 First term a = 3, common ratio r = 2, last term = 384 Concept / Approach n-th term of a G.P.: Tn = a · r^{n−1} Set Tn = 384 and solve for n. Step-by-step calculation 384 = 3 · 2^{n−1}⇒ 2^{n−1} = 384 ÷ 3 = 128 = 2^7⇒ n − 1 = 7 ⇒ n = 8 Verification Terms: 3,6,12,24,48,96,192,384 — there are 8 terms. Common pitfalls Mistaking r as 3 (it is 2). Using arithmetic progression formula (not applicable). Final Answer Number of terms = 8.

Question 2

Evaluate (963 + 476)^2 + (963 − 476)^2.

Accepted Answer

Given data Compute S = (a + b)^2 + (a − b)^2 for a = 963, b = 476. Concept / Approach Identity: (a + b)^2 + (a − b)^2 = 2(a^2 + b^2) Step-by-step calculation a^2 = 963^2 = (1000 − 37)^2 = 1000000 − 74000 + 1369 = 927369b^2 = 476^2 = (500 − 24)^2 = 250000 − 24000 + 576 = 226576a^2 + b^2 = 927369 + 226576 = 1153945S = 2(a^2 + b^2) = 2 × 1153945 = 2307890 Alternative check (963 + 476) = 1439, (963 − 476) = 487 ⇒ 1439^2 + 487^2 = 2(963^2 + 476^2) (same identity). Common pitfalls Expanding both squares manually (time-consuming and error-prone). Forgetting the factor 2 in the identity result. Final Answer Value = 2307890.

Question 3

Which of the following numbers is divisible by 24?

Accepted Answer

Given data Divisibility by 24 requires divisibility by 3 and by 8 (since 24 = 3 × 8 and gcd(3,8)=1). Checks 312: last three digits 312 ⇒ 312 ÷ 8 = 39 (integer), digit sum 3+1+2=6 ⇒ /3 yes ⇒ /24 yes.444: 444 ÷ 8 = 55 remainder 4 ⇒ not /8 ⇒ not /24.520: 520 ÷ 8 = 65 (ok) but digit sum 5+2+0=7 ⇒ not /3 ⇒ not /24.458: 458 ÷ 8 not integer; digit sum 17 ⇒ not /3 ⇒ not /24. Final Answer Number divisible by 24 is 312.

Question 4

If the three-digit number 52 is divisible by 6, which of the following values of * works?

Accepted Answer

Given data Number: 52 must be divisible by 6. Concept / Approach Divisible by 6 ⇔ divisible by 2 and by 3. Ends with 2 ⇒ already divisible by 2. Divisible by 3 ⇒ digit sum multiple of 3. Step-by-step condition Digit sum = 5 + * + 2 = 7 + *Require 7 + * ≡ 0 (mod 3) ⇒ * ≡ 2 (mod 3)Digits 0–9 satisfying * ≡ 2 (mod 3): {2, 5, 8}Therefore the acceptable values are 2, 5 and 8. Common pitfalls Forgetting the divisibility-by-3 condition (checking only evenness). Final Answer Valid digits: 2, 5 and 8.

Question 5

A number leaves remainder 3 when divided by 5. What is the remainder when its square is divided by 5?

Accepted Answer

Given data n ≡ 3 (mod 5) Concept / Approach Use modular arithmetic: square both sides of a congruence. Step-by-step calculation n^2 ≡ 3^2 = 9 ≡ 4 (mod 5) Check via examples n = 3, 8, 13, … ⇒ n^2 = 9, 64, 169, … ⇒ remainders mod 5 are 4. Final Answer Remainder = 4.

Question 6

Compute: 35 + 15 × 1.5

Accepted Answer

Given data & approach Apply order of operations (BODMAS/PEMDAS): multiplication before addition. Step-by-step arithmetic 15 × 1.5 = 22.535 + 22.5 = 57.5 Common pitfalls Adding 35 + 15 first, which is incorrect. Final Answer Value = 57.5.

Question 7

Evaluate 1^2 + 2^2 + 3^2 + … + 10^2.

Accepted Answer

Given data Sum S = 1^2 + 2^2 + … + 10^2 Concept / Approach Use the formula: 1^2 + 2^2 + … + n^2 = n(n + 1)(2n + 1)/6 Step-by-step calculation S = 10 × 11 × 21 / 6= (10 × 11 = 110); 110 × 21 = 23102310 ÷ 6 = 385 Verification Manual partial sums (1^2+…+5^2 = 55; 6^2+…+10^2 = 330) ⇒ total 385. Final Answer Sum = 385.

Question 8

Find ?: 8597 − ? = 7429 − 4358

Accepted Answer

Given data 8597 − ? = 7429 − 4358 Concept / Approach Evaluate the right-hand side (RHS) first, then solve for the unknown. Step-by-step calculation RHS = 7429 − 4358 = 3071So 8597 − ? = 3071 ⇒ ? = 8597 − 3071? = 5526 Quick check 8597 − 5526 = 3071 (matches RHS). Final Answer 5526

Question 9

A number leaves remainder 3 when divided by 6. What is the remainder when its square is divided by 6?

Accepted Answer

Given data n ≡ 3 (mod 6) Concept / Approach Square the congruence to obtain n^2 modulo 6. Step-by-step calculation n^2 ≡ 3^2 = 9 ≡ 3 (mod 6) Insight If n is of the form 6k + 3 (i.e., odd multiple of 3), then n^2 ≡ 3 (mod 6) always. Final Answer Remainder = 3.

Question 10

Evaluate: 287^2 + 269^2 − 2 × 287 × 269

Accepted Answer

Given data Expression: a^2 + b^2 − 2ab with a = 287, b = 269. Concept / Approach Identity: a^2 + b^2 − 2ab = (a − b)^2 Step-by-step calculation a − b = 287 − 269 = 18(a − b)^2 = 18^2 = 324 Verification Expanding (a − b)^2 gives a^2 − 2ab + b^2, matching the given expression. Final Answer Value = 324.

Question 11

A 3-digit number N leaves the same remainder when dividing 2272 and 875. Find the sum of the digits of N.

Accepted Answer

Given data Remainders on dividing 2272 and 875 by the same 3-digit N are equal. Concept / Approach If two numbers leave the same remainder upon division by N, then N divides their difference. Step-by-step Difference = 2272 − 875 = 1397So N must be a 3-digit divisor of 1397.Factor 1397: 1397 = 11 × 127The only 3-digit divisor is N = 127Sum of digits of N = 1 + 2 + 7 = 10 Verification 2272 mod 127 = 112; 875 mod 127 = 112. Remainders match. Common pitfalls Choosing 11 which is not 3-digit. Not using the equal-remainder property to pass to a difference. Final Answer Sum of digits = 10.

Question 12

If x and y are positive integers and 3x + 7y is a multiple of 11, then which expression is divisible by 11?

Accepted Answer

Given data 3x + 7y ≡ 0 (mod 11) Concept / Approach Work modulo 11. Replace 7 with −4 and solve for x in terms of y. Step-by-step 3x + 7y ≡ 3x − 4y ≡ 0 (mod 11)3x ≡ 4y (mod 11)Multiply both sides by the inverse of 3 modulo 11. Since 3 × 4 ≡ 12 ≡ 1 (mod 11), inv(3) = 4.x ≡ 4 × 4y ≡ 16y ≡ 5y (mod 11)Thus x − 5y ≡ 0 (mod 11)Hence x − 5y is divisible by 11. Common pitfalls Assuming only even or odd values work. The relation is linear in modulo arithmetic. Final Answer x − 5y is divisible by 11.

Question 13

Identify the smallest prime number.

Accepted Answer

Given data Definition: a prime has exactly two distinct positive divisors, 1 and itself. Reasoning 0 is not prime. 1 has only one positive divisor, so not prime.2 has divisors {1, 2} only and is the least such integer.Therefore the smallest prime is 2. Common pitfalls Thinking 1 is prime. It is not by modern definition. Final Answer 2

Question 14

Find the missing number: 9548 + 7314 = 8362 + ?

Accepted Answer

Given data 9548 + 7314 = 8362 + ? Concept / Approach Transpose to isolate the unknown. Step-by-step ? = 9548 + 7314 − 8362= 16862 − 8362= 8500 Verification 8362 + 8500 = 16862 which equals 9548 + 7314. Final Answer 8500

Question 15

Which number will completely divide 4^61 + 4^62 + 4^63 + 4^64?

Accepted Answer

Given data Sum: 4^61 + 4^62 + 4^63 + 4^64 Concept / Approach Factor out the lowest power. Recognize a geometric sum. Step-by-step 4^61 + 4^62 + 4^63 + 4^64 = 4^61(1 + 4 + 16 + 64)Inside sum = 85 = 5 × 17Thus the whole expression = 4^61 × 85It is divisible by 85. The other options 51 = 3 × 17 and 57 = 3 × 19 and 95 = 5 × 19 require a factor 3 or 19 which is not present. Common pitfalls Missing the factorization and attempting long division. Final Answer Divisible by 85.

Question 16

How many terms of the series 6 + 12 + 18 + 24 + … sum to 1800?

Accepted Answer

Given data Arithmetic progression with a = 6 and d = 6; sum S = 1800. Concept / Approach Use AP sum: Sn = n/2 [2a + (n − 1)d]. Step-by-step Sn = n/2 [12 + 6(n − 1)] = n/2 [6n + 6] = 3n(n + 1)Set 3n(n + 1) = 1800 ⇒ n(n + 1) = 600n^2 + n − 600 = 0 ⇒ Discriminant = 1 + 2400 = 2401 = 49^2n = (−1 + 49)/2 = 24 Verification S = 3 × 24 × 25 = 1800 ✔ Common pitfalls Forgetting to divide by 2 in the sum formula. Final Answer Number of terms = 24.

Question 17

Evaluate: 666 ÷ 6 ÷ 3 (left to right).

Accepted Answer

Given data Successive division is evaluated from left to right. Step-by-step First: 666 ÷ 6 = 111Then: 111 ÷ 3 = 37 Common pitfalls Interpreting as 666 ÷ (6 ÷ 3) which would change the value. Final Answer Result = 37.

Question 18

Which number is completely divisible by 99?

Accepted Answer

Given data Divisible by 99 means divisible by 9 and by 11. Checks 2178: digit sum = 2+1+7+8 = 18 ⇒ divisible by 9; 11 test: (2 − 1 + 7 − 8) = 0 ⇒ divisible by 11 ⇒ divisible by 99.1452: digit sum = 12 (not a multiple of 9) ⇒ not by 9 ⇒ not by 99.1364: digit sum = 14 ⇒ not by 9.1639: digit sum = 19 ⇒ not by 9. Final Answer Number divisible by 99: 2178.

Question 19

Evaluate the sum of the consecutive integers: Compute the value of the series 51 + 52 + 53 + ... + 100.

Accepted Answer

Introduction: This question involves finding the sum of a sequence of consecutive integers. Recognizing this as an arithmetic series allows us to use the standard formula for the sum of the first n terms of an arithmetic progression (A.P.), making the calculation quick and reliable. Given Data / Assumptions: We must find S = 51 + 52 + 53 + ... + 100. The terms increase by 1 each time, so this is an arithmetic progression. We assume all operations are performed on exact integers. Concept / Approach: For an arithmetic progression with first term a, last term l, and number of terms n, the sum is given by: S = n * (a + l) / 2. Our task is to determine a, l, and n for this series. The numbers from 51 to 100 are consecutive, so finding n is only a matter of counting how many integers lie in that closed interval. Step-by-Step Solution: First term a = 51. Last term l = 100. Number of terms n = (last − first) + 1 = (100 − 51) + 1. Compute n: 100 − 51 = 49, and 49 + 1 = 50, so n = 50. Use the arithmetic series sum formula: S = n * (a + l) / 2 = 50 * (51 + 100) / 2. Compute a + l = 51 + 100 = 151. So S = 50 * 151 / 2. Simplify: 50 / 2 = 25, so S = 25 * 151. Now compute 25 * 151: 25 * 150 = 3750 and 25 * 1 = 25, total 3750 + 25 = 3775. Therefore, the sum S is 3775. Verification / Alternative check: Another way is pairing terms: (51 + 100), (52 + 99), (53 + 98), and so on. Each pair sums to 151. Since there are 50 terms total, there are 25 such pairs. Sum = 25 * 151 = 3775, confirming our previous result. Why Other Options Are Wrong: The options 2525, 2975, and 3225 are typical incorrect results that might come from miscounting the number of terms or miscomputing the average. None of these match the exact calculation using the correct arithmetic series formula. Common Pitfalls: Students often forget to add 1 when computing the number of terms using (last − first) + 1. Another frequent error is to mistakenly use the average of the first few terms instead of the first and last term to form the mean. Carefully applying the standard formula avoids these mistakes. Final Answer: The value of the sum 51 + 52 + 53 + ... + 100 is 3775.

Question 20

In the six digit numeral 967452, the digit 7 is in the thousands place. What is the difference between the place value of this digit and its face value?

Accepted Answer

Introduction: This question checks your understanding of the concepts of face value and place value of a digit in a numeral. These are basic ideas in number systems and positional notation, which are often tested in aptitude exams. Given Data / Assumptions: The numeral is 967452. We focus on the digit 7 in this numeral. We must find the difference: (place value of 7) − (face value of 7). Concept / Approach: The face value of a digit is simply the digit itself, regardless of where it appears in the number. The place value, however, depends on the position of the digit. In a six digit number, the positions from right to left are units, tens, hundreds, thousands, ten thousands, and lakhs. The value of a digit is the digit multiplied by the positional multiplier corresponding to its place. Step-by-Step Solution: Write the number with place labels: 967452. From right to left: 2 is units, 5 is tens, 4 is hundreds, 7 is thousands, 6 is ten thousands, 9 is lakhs. So the digit 7 is in the thousands place. Face value of 7 = 7. Place value of 7 = 7 × 1000 = 7000. We need the difference between place value and face value: Difference = 7000 − 7 = 6993. Verification / Alternative check: You can also expand the full number: 967452 = 9 × 100000 + 6 × 10000 + 7 × 1000 + 4 × 100 + 5 × 10 + 2. Here the contribution of the digit 7 is clearly 7000. Subtracting its face value 7 again gives 6993, confirming our earlier computation. Why Other Options Are Wrong: The values 6393, 5831, and 6339 would correspond to incorrect place values such as 7000 being misread, wrong subtraction, or confusion about the position of the digit in the number. None of these arise from correctly identifying 7 as the thousands digit and performing the subtraction 7000 − 7. Common Pitfalls: Learners sometimes mistake face value for place value and answer 7000 or 7 directly, ignoring the requested difference. Others misidentify the place, thinking 7 is in the ten thousands or hundreds place. Carefully counting positions from the right helps avoid such errors. Final Answer: The difference between the place value and face value of the digit 7 in 967452 is 6993.

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