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Difference between revisions of "User:Geometry285"

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Posting here until I find a place for an upcoming mock I’m creating
 
Posting here until I find a place for an upcoming mock I’m creating
  
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{{G285 Mock Problems|year=2021|type=A}}
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{{G285 MC10A}}
  
 
==Problem 1==
 
==Problem 1==
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What value of <math>x</math> minimizes <math>|2^x - 1|</math>?
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What is the smallest value of <math>x</math> that minimizes <math>|||2^{|x^2|} - 4|-4|-8|</math>?
  
−
<math>\textbf{(A)}\ -1\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \infty</math>
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<math>\textbf{(A)}\ -2\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2</math>
  
 
[[G285 MC10A Problems/Problem 1|Solution]]
 
[[G285 MC10A Problems/Problem 1|Solution]]
  
 
==Problem 2==
 
==Problem 2==
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Suppose Mark wanted to arrange <math>5</math> books onto a bookshelf, <math>3</math> of which are math books and <math>2</math> of which are science. If both science and math books are indistinguishable, in how many ways can Mark arrange the books on the shelf?
+
Suppose the set <math>S</math> denotes <math>S = \{1,2,3 \cdots n\}</math>. Then, a subset of length <math>1<k<n</math> is chosen. All even digits in the subset <math>k</math> are then are put into group <math>k_1</math>, and the odd digits are put in <math>k_2</math>. Then, one number is selected at random from either <math>k_1</math> or <math>k_2</math> with equal chances. What is the probability that the number selected is a perfect square, given <math>n=4</math>?
  
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<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15</math>
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<math>\textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{3}{11}\qquad\textbf{(C)}\ \frac{6}{11}\qquad\textbf{(D)}\ \frac{7}{13}\qquad\textbf{(E)}\ \frac{3}{5}</math>
  
 
[[G285 MC10A Problems/Problem 2|Solution]]
 
[[G285 MC10A Problems/Problem 2|Solution]]
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[[G285 MC10A Problems/Problem 4|Solution]]
 
[[G285 MC10A Problems/Problem 4|Solution]]
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 +
==Problem 5==
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Let a recursive sequence be denoted by <math>a_n</math> such that <math>a_0 = 1</math> and <math>a_1 = k</math>. Suppose <math>a_{n-1} = n+a_n</math> for <math>n>1</math>. Let an infinite arithmetic sequence <math>P</math> be such that <math>P=\{k+1, k-p+1, k-2p+1 \cdots\}</math>. If <math>k</math> is prime, for what value of <math>p</math> will <math>k_{2021} = k-2022p+1</math>?
 +
 +
<math>\textbf{(A)}\ 1011\qquad\textbf{(B)}\ \frac{1011}{2}\qquad\textbf{(C)}\ 2021\qquad\textbf{(D)}\ \frac{2021}{2}\qquad\textbf{(E)}\ 4042</math>
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 +
[[G285 MC10A Problems/Problem 5|Solution]]
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 +
==Problem 6==
 +
Find <cmath>\sum_{j=1}^{50} s^3 \sum_{h=3}^{10} {4h+5}</cmath>
 +
 +
<math>\textbf{(A)}\ 323400\qquad\textbf{(B)}\ 336600\qquad\textbf{(C)}\ 673200\qquad\textbf{(D)}\ 646800\qquad\textbf{(E)}\ 2124150</math>
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[[G285 MC10A Problems/Problem 6|Solution]]
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==Problem 7==

Revision as of 23:09, 11 May 2021

Posting here until I find a place for an upcoming mock I’m creating

Template:G285 MC10A

Problem 1

What is the smallest value of $x$ that minimizes $|||2^{|x^2|} - 4|-4|-8|$?

$\textbf{(A)}\ -2\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2$

Solution

Problem 2

Suppose the set $S$ denotes $S = \{1,2,3 \cdots n\}$. Then, a subset of length $1<k<n$ is chosen. All even digits in the subset $k$ are then are put into group $k_1$, and the odd digits are put in $k_2$. Then, one number is selected at random from either $k_1$ or $k_2$ with equal chances. What is the probability that the number selected is a perfect square, given $n=4$?

$\textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{3}{11}\qquad\textbf{(C)}\ \frac{6}{11}\qquad\textbf{(D)}\ \frac{7}{13}\qquad\textbf{(E)}\ \frac{3}{5}$

Solution

Problem 3

Let $ABCD$ be a unit square. If points $E$ and $F$ are chosen on $AB$ and $CD$ respectively such that the area of $\triangle AEF = \frac{3}{2} \triangle CFE$. What is $EF^2$?

$\textbf{(A)}\ \frac{13}{9}\qquad\textbf{(B)}\ \frac{8}{9}\qquad\textbf{(C)}\ \frac{37}{36}\qquad\textbf{(D)}\ \frac{5}{4}\qquad\textbf{(E)}\ \frac{13}{36}$

Solution

Problem 4

What is the smallest value of $k$ for which \[2^{18k} \equiv 76 \mod 100\]

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 20$

Solution

Problem 5

Let a recursive sequence be denoted by $a_n$ such that $a_0 = 1$ and $a_1 = k$. Suppose $a_{n-1} = n+a_n$ for $n>1$. Let an infinite arithmetic sequence $P$ be such that $P=\{k+1, k-p+1, k-2p+1 \cdots\}$. If $k$ is prime, for what value of $p$ will $k_{2021} = k-2022p+1$?

$\textbf{(A)}\ 1011\qquad\textbf{(B)}\ \frac{1011}{2}\qquad\textbf{(C)}\ 2021\qquad\textbf{(D)}\ \frac{2021}{2}\qquad\textbf{(E)}\ 4042$

Solution

Problem 6

Find \[\sum_{j=1}^{50} s^3 \sum_{h=3}^{10} {4h+5}\]

$\textbf{(A)}\ 323400\qquad\textbf{(B)}\ 336600\qquad\textbf{(C)}\ 673200\qquad\textbf{(D)}\ 646800\qquad\textbf{(E)}\ 2124150$

Solution

Problem 7

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