UPSC CSE Prelims
Number System Previous Year Questions (PYQs)
Showing solved Previous Year Questions for Chapter: Number System
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$X$, $Y$ and $Z$ jump forward $4'$, $6'$ and $5'$, respectively. At 8 AM, they all land on mark $199'$. How many times will they all land on the same mark (need not be at the same moment) between mark $195'$ and $1000'$, if all of them cross mark $1000'$ by 9 AM?
Detailed Explanation:
At 8 AM, all three are at mark 199'. Since X jumps 4', Y jumps 6', and Z jumps 5', they will all land on the same mark again when the distance from 199' is a multiple of LCM(4, 6, 5) = 60'.
Common landing marks are 199 + 60k where k = 0, 1, 2, ... Within the range 195' < m < 1000', we get: 199, 259, 319, 379, 439, 499, 559, 619, 679, 739, 799, 859, 919, 979. The next mark (1039') exceeds 1000'.
Therefore, counting from k=0 to k=13 gives 14 common marks, making Option 4 the correct answer.
If $x$ and $y$ are integers, then is $x$ even? Statement I : $x^2 y^2$ is even. Statement II : $1 + x^2 + y^2$ is odd.
Detailed Explanation:
We need to determine if x is even.
- Statement I (x²y² is even): At least one of x or y is even, but x could be odd (y even) or even (y odd/even) — insufficient alone.
- Statement II (1 + x² + y² is odd): x² + y² must be even, so x and y have the same parity (both even or both odd) — insufficient alone.
- Combining both: At least one is even (Statement I) and both have same parity (Statement II) → both x and y must be even.
- Therefore, both statements together uniquely determine that x is even, making Option 3 the correct answer.
If $x$, $y$ and $z$ are integers, each greater than $1$, then is $x$ a prime number? Statement I : $xy^2 = 116$ Statement II : $xz = 261$
Detailed Explanation:
Statement I: $xy^2 = 116 = 2^2 \times 29$. Since $y > 1$ is an integer, $y^2$ must be a perfect square divisor of $116$. The only such divisor is $4$, so $y^2 = 4$ and $y = 2$. Thus $x = 116/4 = 29$, which is prime. Statement I alone is sufficient.
Statement II: $xz = 261 = 3^2 \times 29$. Possible values for $x$ are $3, 9, 29, 87$. Since $x = 3$ or $29$ gives prime, while $x = 9$ or $87$ gives composite, we cannot determine definitively. Statement II alone is insufficient.
Therefore, the question can be answered using Statement I alone but not Statement II alone, making Option 1 the correct answer.
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How many words can one form by shuffling the letters of the word QUEUE, if Q is always followed by U? The words thus formed need not necessarily have any meaning.
Detailed Explanation:
The word QUEUE has letters: Q, U, E, U, E. Since Q must always be followed by U, treat QU as a single block. The remaining letters are U, E, E.
We now arrange 4 objects: {QU, U, E, E}. Since the two E's are identical, the number of arrangements = 4!/2! = 24/2 = 12.
Therefore, 12 words can be formed, making Option 4 the correct answer.
If $10^m \times 1000 \times n = 75^{25} \times 25^{32} \times 32^{75}$, where $n$ is not divisible by $10$, then the value of $m$ is
Detailed Explanation:
Prime factorize the RHS: $75^{25} = (3 \times 5^2)^{25} = 3^{25} \times 5^{50}$, $25^{32} = 5^{64}$, and $32^{75} = 2^{375}$. Combined: $2^{375} \times 3^{25} \times 5^{114}$.
Maximum power of 10: Since $10 = 2 \times 5$, the limiting factor is $\min(375, 114) = 114$. Thus, RHS = $10^{114} \times 2^{261} \times 3^{25}$, where $(2^{261} \times 3^{25})$ is not divisible by $10$ (no factor of $5$).
LHS comparison: $10^m \times 1000 \times n = 10^{m+3} \times n$. Since $n$ is not divisible by $10$, we equate: $m + 3 = 114 \Rightarrow m = 111$.
Therefore, m = 111, making Option 2 the correct answer.
Three variables $x$, $y$ and $z$ take values $2, 3, 4$ or $5$ such that their values are always distinct. If $M$ and $N$ denote the largest possible value and the smallest possible value, respectively, for the expression ${(x \times y) + z}$; then $M - N$ is
Detailed Explanation:
To maximize $(x \times y) + z$, choose the two largest values for the product: $x = 5, y = 4$ gives $5 \times 4 = 20$, and the largest remaining value $z = 3$ gives $M = 20 + 3 = 23$.
To minimize $(x \times y) + z$, choose the two smallest values for the product: $x = 2, y = 3$ gives $2 \times 3 = 6$, and the smallest remaining value $z = 4$ gives $N = 6 + 4 = 10$.
Therefore, $M - N = 23 - 10 = 13$, making Option 3 the correct answer.
If $x$ and $y$ are two digits and the number $4x5y790$ is divisible by $11$, then what is the remainder, if $x+y$ is divided by $11$?
Detailed Explanation:
For the number 4x5y790 to be divisible by 11, apply the alternating digit sum rule: (sum of odd-position digits from right) − (sum of even-position digits from right) must be divisible by 11.
Odd positions (from right): 0, 7, 5, 4 → sum = 16. Even positions: 9, y, x → sum = x + y + 9. Thus, 16 − (x + y + 9) = 7 − (x + y) must be ≡ 0 (mod 11).
Since x, y are digits (0–9), x + y ∈ [0, 18], so 7 − (x + y) ∈ [−11, 7]. The only multiples of 11 in this range are 0 and −11. If 7 − (x + y) = 0 → x + y = 7; if 7 − (x + y) = −11 → x + y = 18. In both cases, (x + y) mod 11 = 7.
Therefore, the remainder when x + y is divided by 11 is 7, making Option 4 the correct answer.
In an objective type question paper, $5$ marks are awarded for a correct answer and $2$ marks are deducted for a wrong answer. A student attempted all the questions and got a score of $69$. Had he been awarded $4$ marks for a correct answer and $1$ mark deducted for a wrong answer, he would have scored $84$. How many questions were there in the question paper?
Detailed Explanation:
Let correct answers = x, wrong answers = y. First condition: 5x - 2y = 69. Second condition: 4x - y = 84.
From second equation, y = 4x - 84. Substituting into first: 5x - 2(4x - 84) = 69 → 5x - 8x + 168 = 69 → -3x = -99 → x = 33. Thus y = 132 - 84 = 48.
Total questions = 33 + 48 = 81, making Option 2 the correct answer.
If the product of the HCF and LCM of two distinct numbers is the cube of one of the numbers, then which of the following statements is/are correct?
I. The difference of the numbers is an even number. II. One of the numbers is a perfect square.
Select the answer using the code given below.
Detailed Explanation:
Let two distinct numbers be a and b. Using HCF × LCM = a × b, and given this product equals the cube of one number (say a³): a × b = a³, so b = a².
Statement I: Difference = a² − a = a(a − 1), the product of two consecutive integers, which is always even. ✓
Statement II: One number is a², which is a perfect square. ✓
Therefore, both statements I and II are correct, making Option 3 the correct answer.
How many three-digit numbers can be expressed as an integral power of $2$?
Detailed Explanation:
Three-digit numbers range from 100 to 999. We need to find integral powers of 2 in this range.
Calculate: 2⁷ = 128, 2⁸ = 256, 2⁹ = 512 (all three-digit). 2⁶ = 64 (two-digit) and 2¹⁰ = 1024 (four-digit) fall outside the range.
Therefore, exactly 3 three-digit numbers (128, 256, 512) are integral powers of 2, making Option 3 the correct answer.
The digit in the unit place of the number $6^{129} \times 7^{307}$ is
Detailed Explanation:
Unit digit of 6^129: Any power of 6 always ends in 6 (since 6×6=36, 36×6=216, etc.). Thus, unit digit = 6.
Unit digit of 7^307: Powers of 7 cycle every 4: 7¹→7, 7²→9, 7³→3, 7⁴→1. Divide 307 by 4: remainder = 3. So unit digit matches 7³ = 3.
Final unit digit: Multiply 6 × 3 = 18, giving unit digit 8.
Therefore, the unit digit is 8, making Option 3 the correct answer.
How many times does $5$ appear in all two-digit positive integers?
Detailed Explanation:
Count the digit 5 in tens place: appears in 50, 51, 52, …, 59 — that's 10 occurrences.
Count the digit 5 in units place: appears in 15, 25, 35, 45, 55, 65, 75, 85, 95 — that's 9 occurrences.
Therefore, total = 10 + 9 = 19, making Option 2 the correct answer.
The ratio of male to female workers in two companies $A$ and $B$ is $13 : 10$ and $7 : 5$, respectively. If both the companies have the same number of female workers, then what is the ratio of the total number of workers in $A$ to those in $B$?
Detailed Explanation:
Company A has male : female = 13 : 10, and company B has male : female = 7 : 5. Since both companies have the same number of female workers, equalize the female ratio parts by multiplying B's ratio by 2 → B becomes 14 : 10.
Now both companies have 10 units of female workers. Total workers in A = 13 + 10 = 23 units; total workers in B = 14 + 10 = 24 units.
Therefore, the ratio of total workers in A to B is 23 : 24, making Option 2 the correct answer.
In a sequence of numbers, each number other than the first two is the sum of the two immediately preceding numbers from it. If the first two numbers in the sequence are 4 and 7, then the sixth number is
Detailed Explanation:
This is a Fibonacci-like sequence where each term equals the sum of the two preceding terms.
Starting with a₁ = 4 and a₂ = 7: a₃ = 4 + 7 = 11; a₄ = 7 + 11 = 18; a₅ = 11 + 18 = 29; a₆ = 18 + 29 = 47.
Therefore, the sixth number is 47, making Option 4 the correct answer.
There are three types of rectangular tiles : 3' * 3', 3' * 7' and 3' * 11'. An area of rectangular shape of dimensions 3' * 100' is to be covered using these tiles without breaking them. If x and y are the maximum and minimum numbers of tiles of various sizes, respectively, that can be used to cover the area exactly, then x - y is
Detailed Explanation:
The area 3' × 100' must be covered by tiles 3' × 3', 3' × 7', and 3' × 11'. Since all tiles have width 3', they fit along the length, giving the equation 3a + 7b + 11c = 100.
Maximum tiles (x): Use smallest tiles. 31 tiles of 3' give 93', leaving 7' for 1 tile of 7'. Total = 32 tiles. Minimum tiles (y): Use largest tiles. 8 tiles of 11' give 88', leaving 12' for 4 tiles of 3'. Total = 12 tiles.
Therefore, x − y = 32 − 12 = 20, making Option 1 the correct answer.
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