spain-selectividad

Papers (16)
2025
selectividad__matematicas-II_modelo 7
2024
selectividad__matematicas-II_modelo 7
2023
selectividad__madrid_matematicas-II 8 selectividad__madrid_matematicas-II_extraordinaria 8
2022
selectividad__madrid_matematicas-II 8 selectividad__madrid_matematicas-II_extraordinaria 8
2021
selectividad__madrid_matematicas-II_extraordinaria 8 selectividad__madrid_matematicas-II_ordinaria 8
2020
selectividad__madrid_matematicas-II_extraordinaria 8 selectividad__madrid_matematicas-II_ordinaria 7
2019
selectividad__madrid_matematicas-II_extraordinaria 4 selectividad__madrid_matematicas-II_ordinaria 4
2018
selectividad__madrid_matematicas-II_extraordinaria 4 selectividad__madrid_matematicas-II_ordinaria 4
2017
selectividad__madrid_matematicas-II_extraordinaria 4 selectividad__madrid_matematicas-II_ordinaria 4
2018 selectividad__madrid_matematicas-II_extraordinaria

4 maths questions

Q1 2.5 marks 3x3 Matrices Linear System Existence and Uniqueness via Determinant View
Given the matrices $A = \left( \begin{array} { c c c } 14 & 0 & 10 \\ 0 & 7 & 5 \\ 3 & 4 & 5 \alpha \end{array} \right) , \quad X = \left( \begin{array} { l } x \\ y \\ z \end{array} \right) \text{ and } \quad B = \left( \begin{array} { c } 2 \\ 37 / 2 \\ 11 \end{array} \right)$, it is requested:
a) (1.25 points) Discuss the rank of matrix A, as a function of the values of the parameter $\alpha$.
b) (0.75 points) For $\alpha = 0$, calculate, if possible, $A ^ { - 1 }$.
c) (0.5 points) Solve, if possible, the system $A X = B$, in the case $\alpha = 0$.
Q2 2.5 marks Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Consider the function $f ( x ) = \left\{ \begin{array} { l l l } 8 e ^ { 2 x - 4 } & \text { if } & x \leq 2 \\ \frac { x ^ { 3 } - 4 x } { x - 2 } & \text { if } & x > 2 \end{array} \right.$ and it is requested:
a) (0.75 points) Study the continuity of $f$ at $x = 2$.
b) (1 point) Calculate the asymptotes of $f ( x )$. Is there any vertical asymptote?
c) (0.75 points) Calculate $\int _ { 0 } ^ { 2 } f ( x ) d x$
Q3 2.5 marks Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Consider the vectors $\vec { u } = ( - 1,2,3 ) , \quad \vec { v } = ( 2,0 , - 1 )$ and the point $\mathrm { A } ( - 4,4,7 )$. It is requested:
a) (1 point) Determine the vector $\vec { w } _ { 1 }$ that is orthogonal to $\vec { u }$ and $\vec { v }$, unitary, and with negative third coordinate.
b) (0.75 points) Find the non-zero vector $\overrightarrow { w _ { 2 } }$ that is a linear combination of $\vec { u }$ and $\vec { v }$ and orthogonal to $\vec { v }$.
c) (0.75 points) Determine the vertices of the parallelogram whose sides have the directions of vectors $\vec { u }$ and $\vec { v }$ and one of its diagonals is the segment $\overrightarrow { O A }$.
Q4 2.5 marks Conditional Probability Bayes' Theorem with Diagnostic/Screening Test View
According to data from the Diabetes Foundation, 13.8\% of Spanish people over 18 years old have diabetes, although 43\% of them do not know they have it. A Spanish person over 18 years old is chosen at random.
a) (1 point) What is the probability that they are diabetic and know it? What is the probability that they are not diabetic or do not know they are?
b) (1.5 points) A certain test correctly diagnoses 96\% of positive diabetes cases, but gives 2\% false positives. If a Spanish person over 18 years old tests positive, what is the probability that they are actually diabetic?