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A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the
pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the
removed cards is k, then k ? 20 = ________
pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the
removed cards is k, then k ? 20 = ________
Correct answer is '5'. Can you explain this answer?
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A pack contains n cards numbered from 1 to n. Two consecutive numbered...
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A pack contains n cards numbered from 1 to n. Two consecutive numbered...
Problem: A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then k ? 20 = ________
Solution:
Let's assume that the two consecutive cards removed from the pack are x and x+1. Therefore, the sum of the remaining cards can be expressed as:
1 + 2 + 3 + ... + x-1 + x+2 + ... + n
= (1+2+3+...+n) - (x + x+1)
= (n(n+1)/2) - 2x-1
Given that the sum of the remaining cards is 1224, we have:
(n(n+1)/2) - 2x-1 = 1224
Simplifying this expression, we get:
n(n+1) - 4x - 3 = 2448
n^2 + n - 4x = 2451
Now, we can use the quadratic formula to solve for n:
n = (-1 ± sqrt(1 + 4(4x - 2451)))/2
n = (-1 ± sqrt(4x - 968))/2
Since n is a positive integer, we can ignore the negative solution. Therefore, we have:
n = (-1 + sqrt(4x - 968))/2
Next, we can use the fact that k is the smaller of the two removed numbers to write:
k + (k+1) = 2k+1
Since 2k+1 is odd, we know that the sum of the remaining cards must be even. Therefore, we can write:
(n(n+1)/2) - 2k-1 = 1224
n(n+1) - 4k - 3 = 2448
n^2 + n - 4k = 2451
Substituting the expression we derived for n, we get:
((-1 + sqrt(4x - 968))/2)^2 + (-1 + sqrt(4x - 968))/2 - 4k = 2451
Simplifying this expression, we get:
4x - 968 + 2sqrt(4x - 968) + 1 - 8k = 9804
2sqrt(4x - 968) - 8k = 8743 - 4x
Squaring both sides, we get:
16k^2 - 34kx + (4x^2 - 17486x + 761924) = 0
Now, we can use the quadratic formula to solve for k:
k = (34x ± sqrt(34^2x^2 - 4(16)(4x^2 - 17486x + 761924)))/(2(16))
k = (17x ± sqrt(577x - 38835))/16
Since k is a positive integer, we can ignore the negative solution. Therefore, we have:
k = (17x - sqrt(577x - 38835))/16
We know that k is between 1 and n-1, since the two removed cards are consecutive.
Solution:
Let's assume that the two consecutive cards removed from the pack are x and x+1. Therefore, the sum of the remaining cards can be expressed as:
1 + 2 + 3 + ... + x-1 + x+2 + ... + n
= (1+2+3+...+n) - (x + x+1)
= (n(n+1)/2) - 2x-1
Given that the sum of the remaining cards is 1224, we have:
(n(n+1)/2) - 2x-1 = 1224
Simplifying this expression, we get:
n(n+1) - 4x - 3 = 2448
n^2 + n - 4x = 2451
Now, we can use the quadratic formula to solve for n:
n = (-1 ± sqrt(1 + 4(4x - 2451)))/2
n = (-1 ± sqrt(4x - 968))/2
Since n is a positive integer, we can ignore the negative solution. Therefore, we have:
n = (-1 + sqrt(4x - 968))/2
Next, we can use the fact that k is the smaller of the two removed numbers to write:
k + (k+1) = 2k+1
Since 2k+1 is odd, we know that the sum of the remaining cards must be even. Therefore, we can write:
(n(n+1)/2) - 2k-1 = 1224
n(n+1) - 4k - 3 = 2448
n^2 + n - 4k = 2451
Substituting the expression we derived for n, we get:
((-1 + sqrt(4x - 968))/2)^2 + (-1 + sqrt(4x - 968))/2 - 4k = 2451
Simplifying this expression, we get:
4x - 968 + 2sqrt(4x - 968) + 1 - 8k = 9804
2sqrt(4x - 968) - 8k = 8743 - 4x
Squaring both sides, we get:
16k^2 - 34kx + (4x^2 - 17486x + 761924) = 0
Now, we can use the quadratic formula to solve for k:
k = (34x ± sqrt(34^2x^2 - 4(16)(4x^2 - 17486x + 761924)))/(2(16))
k = (17x ± sqrt(577x - 38835))/16
Since k is a positive integer, we can ignore the negative solution. Therefore, we have:
k = (17x - sqrt(577x - 38835))/16
We know that k is between 1 and n-1, since the two removed cards are consecutive.
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A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from thepack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on theremoved cards is k, then k ? 20 = ________Correct answer is '5'. Can you explain this answer?
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A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from thepack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on theremoved cards is k, then k ? 20 = ________Correct answer is '5'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from thepack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on theremoved cards is k, then k ? 20 = ________Correct answer is '5'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from thepack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on theremoved cards is k, then k ? 20 = ________Correct answer is '5'. Can you explain this answer?.
A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from thepack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on theremoved cards is k, then k ? 20 = ________Correct answer is '5'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from thepack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on theremoved cards is k, then k ? 20 = ________Correct answer is '5'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from thepack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on theremoved cards is k, then k ? 20 = ________Correct answer is '5'. Can you explain this answer?.
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A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from thepack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on theremoved cards is k, then k ? 20 = ________Correct answer is '5'. Can you explain this answer?, a detailed solution for A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from thepack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on theremoved cards is k, then k ? 20 = ________Correct answer is '5'. Can you explain this answer? has been provided alongside types of A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from thepack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on theremoved cards is k, then k ? 20 = ________Correct answer is '5'. Can you explain this answer? theory, EduRev gives you an
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