iran-konkur 2023 Q121

iran-konkur · Other · konkur-riazi_1402_general Not Maths
121-A Page 2
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  1. If the terms of a geometric sequence with common ratio $r$ are halved, you will obtain an arithmetic sequence with common difference $d$. What is the value of $r + d$?
    (1) zero (2) $1$ (3) $\sqrt{2}$ (4) $\dfrac{1}{2}$

  2. Points $A(3,y)$ and $B(-5,y)$ lie on a parabola whose vertex is at the origin and whose latus rectum equals 1. If this parabola cuts the $x$-axis with intercepts $\alpha$ and $\beta$, and $\alpha^2 + \beta^2 = 5$, at what $y$-intercept does this parabola cut the $y$-axis?
    (1) $-\dfrac{1}{3}$ (2) $-\dfrac{2}{3}$ (3) $\dfrac{1}{3}$ (4) $\dfrac{2}{3}$

  3. For the sets $A = \{a-2, 6, 2b+1, c\}$ and $B = \{\sqrt{d}, 5, -1\}$, suppose $A \times B = B \times A$. In how many cases is $a + b + c = 9$?
    (1) $1$ (2) $2$ (3) $3$ (4) zero

  4. According to the truth table below, which compound proposition can be a logical tautology for proposition $X$?
    $p$$q$$r$$X$
    دددن
    ددند
    دندد
    دننن
    نددن
    ندند
    نندد
    نننن

    (1) $(q \Rightarrow (p \vee r)) \Rightarrow ((p \vee \sim p) \wedge (\sim q \wedge r))$
    (2) $(r \Rightarrow (p \vee q)) \Rightarrow ((p \vee \sim p) \wedge (q \wedge \sim r))$
    (3) $[p \Rightarrow ((q \vee r) \Rightarrow (q \wedge r))] \Rightarrow (\sim (p \vee r) \wedge q)$
    (4) $(r \Rightarrow (p \vee q)) \Rightarrow [((p \Rightarrow r) \Rightarrow (\sim p \wedge r)) \wedge q]$

  5. If $\alpha$ and $\beta$ are the distinct roots of the equation $ax^2 - ax - b = 0$ and $40\beta^2 + 20\alpha^2 - 20\beta = 17$ and $40\beta^2 + 20\alpha^2 - 20\beta = 17$, what is the difference of the roots of this equation?
    (1) $\dfrac{1}{5}$ (2) $\dfrac{2}{5}$ (3) $\dfrac{1}{\sqrt{5}}$ (4) $\dfrac{2}{\sqrt{5}}$

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* Dear examinee, failure to fill in the information and sign the table below is considered as your absence from the exam session.

\vspace{0.2cm}
\noindent\fbox{\parbox{0.95\textwidth}{
\textbf{I, ............................, with candidate number ............................., confirm that my candidate number matches my seat number. The candidate number printed on my entry card to the session, on the answer sheet, and on the question booklet type and code have been verified and confirmed on the question booklet.}

\vspace{0.3cm}
\hfill \textbf{Signature:}
}}

\vspace{0.4cm}

\begin{enumerate}

\item If the terms of a geometric sequence with common ratio $r$ are halved, you will obtain an arithmetic sequence with common difference $d$. What is the value of $r + d$?

\vspace{0.2cm}
(1) zero \hfill (2) $1$ \hfill (3) $\sqrt{2}$ \hfill (4) $\dfrac{1}{2}$

\vspace{0.3cm}

\item Points $A(3,y)$ and $B(-5,y)$ lie on a parabola whose vertex is at the origin and whose latus rectum equals 1. If this parabola cuts the $x$-axis with intercepts $\alpha$ and $\beta$, and $\alpha^2 + \beta^2 = 5$, at what $y$-intercept does this parabola cut the $y$-axis?

\vspace{0.2cm}
(1) $-\dfrac{1}{3}$ \hfill (2) $-\dfrac{2}{3}$ \hfill (3) $\dfrac{1}{3}$ \hfill (4) $\dfrac{2}{3}$

\vspace{0.3cm}

\item For the sets $A = \{a-2, 6, 2b+1, c\}$ and $B = \{\sqrt{d}, 5, -1\}$, suppose $A \times B = B \times A$. In how many cases is $a + b + c = 9$?

\vspace{0.2cm}
(1) $1$ \hfill (2) $2$ \hfill (3) $3$ \hfill (4) zero

\vspace{0.3cm}

\item According to the truth table below, which compound proposition can be a logical tautology for proposition $X$?

\vspace{0.2cm}
\begin{tabular}{|c|c|c|c|}
\hline
$p$ & $q$ & $r$ & $X$ \\
\hline
د & د & د & ن \\
\hline
د & د & ن & د \\
\hline
د & ن & د & د \\
\hline
د & ن & ن & ن \\
\hline
ن & د & د & ن \\
\hline
ن & د & ن & د \\
\hline
ن & ن & د & د \\
\hline
ن & ن & ن & ن \\
\hline
\end{tabular}

\vspace{0.2cm}
(1) $(q \Rightarrow (p \vee r)) \Rightarrow ((p \vee \sim p) \wedge (\sim q \wedge r))$

(2) $(r \Rightarrow (p \vee q)) \Rightarrow ((p \vee \sim p) \wedge (q \wedge \sim r))$

(3) $[p \Rightarrow ((q \vee r) \Rightarrow (q \wedge r))] \Rightarrow (\sim (p \vee r) \wedge q)$

(4) $(r \Rightarrow (p \vee q)) \Rightarrow [((p \Rightarrow r) \Rightarrow (\sim p \wedge r)) \wedge q]$

\vspace{0.3cm}

\item If $\alpha$ and $\beta$ are the distinct roots of the equation $ax^2 - ax - b = 0$ and $40\beta^2 + 20\alpha^2 - 20\beta = 17$ and $40\beta^2 + 20\alpha^2 - 20\beta = 17$, what is the difference of the roots of this equation?

\vspace{0.2cm}
(1) $\dfrac{1}{5}$ \hfill (2) $\dfrac{2}{5}$ \hfill (3) $\dfrac{1}{\sqrt{5}}$ \hfill (4) $\dfrac{2}{\sqrt{5}}$

\end{enumerate}

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Paper Questions
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