JEE Exam > JEE Questions > A pack contains n cards numbered from 1 to n....
Start Learning for Free
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?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
A pack contains n cards numbered from 1 to n. Two consecutive numbered...
Most Upvoted Answer
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.
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
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?
Question Description
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?.
Solutions 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? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning 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? defined & explained in the simplest way possible. Besides giving the explanation 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?, 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
ample number of questions to practice 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? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup