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
2017 selectividad__madrid_matematicas-II_extraordinaria

4 maths questions

Q1 3 marks Differentiating Transcendental Functions Piecewise function analysis with transcendental components View
Given the function $f ( x ) = \left\{ \begin{array} { l l l } x e ^ { 2 x } & \text { if } & x < 0 \\ \frac { \ln ( x + 1 ) } { x + 1 } & \text { if } & x \geq 0 \end{array} \right.$, where $\ln$ means natural logarithm, it is requested:\ a) (1 point) Study the continuity and differentiability of $f ( x )$ at $x = 0$.\ b) (1 point) Calculate $\lim _ { x \rightarrow - \infty } f ( x )$ and $\lim _ { x \rightarrow + \infty } f ( x )$.\ c) (1 point) Calculate $\int _ { - 1 } ^ { 0 } f ( x ) d x$
Q2 3 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
Given the lines $r _ { 1 } \equiv \left\{ \begin{array} { l } 6 x - y - z = 1 , \\ 2 x - y + z = 1 \end{array} \quad \right.$ and $r _ { 2 } \equiv \left\{ \begin{array} { l } 3 x - 5 y - 2 z = 3 , \\ 3 x + y + 4 z = 3 \end{array} \right.$ it is requested:\ a) (1 point) Study the relative position of $r _ { 1 }$ and $r _ { 2 }$.\ b) (1 point) Calculate the distance between the two lines.\ c) (1 point) Find the equation of the plane that contains $r _ { 1 }$ and the point $\mathrm { P } ( 1,2,3 )$.
Q3 2 marks Simultaneous equations View
There are three alloys $\mathrm { A } , \mathrm { B }$ and C that contain, among other metals, gold and silver in the proportions indicated in the attached table.
Gold (\%)Silver (\%)
A1000
B7515
C6022

It is desired to obtain an ingot of 25 grams, with a proportion of $72 \%$ gold and a proportion of $16 \%$ silver, taking $x$ grams of $\mathrm { A } , y$ grams of B and $z$ grams of C . Determine the quantities $x$, $y , z$.
Q4 2 marks Independent Events View
Given two events, A and B, of a random experiment, with probabilities such that $p ( A ) = \frac { 4 } { 9 } , \quad p ( B ) = \frac { 1 } { 2 } y p ( A \cup B ) = \frac { 2 } { 3 }$, it is requested:\ a) (1 point) Check whether events A and B are independent or not.\ b) (1 point) Calculate $p ( \bar { A } / B )$, where $\bar { A }$ denotes the complementary event of A .