A train travelling at \( 40 \mathrm{~km} / \mathrm{h} \) takes 3 hours for a journey. How long will it take the train omplete the same journey, travelling at \( 90 \mathrm{~km} / \mathrm{h} \) ?

B. Solve the following 1. If \( r \) varies directly as \( s \) and inversely as the square of \( u \), and \( r=2 \) when \( s=18 \) and \( u= \) find: a. \( r \) when \( u=3 \) and \( s=27 \). b. \( s \) when \( u=2 \) and \( r=4 \). c. \( u \) when \( r=1 \) and \( s=36 \). 2. \( p \) varies directly as \( q \) and the square of \( r \) and inversely as \( s \). a. Write the equation of the relation. b. Find \( k \) if \( p=40 \) when \( q=5, r=4 \) and \( s=6 \). c. Find \( p \) when \( q=8, r=6 \) and \( s=9 \). d. Find \( s \) when \( p=10, q=5 \) and \( r=2 \).

Si el ancho de cada rectángulo de cartón piedra mide \( x \) cm \( y v \) es el volumen de cada caja, medido en \( \mathrm{cm}^{3} \). ccuál de las siguientes expresiones modela \( v \) en función de \( x \) ? (A) \( V(x)=4 x(x+6) \) (B) \( V(x)=4(x+2)(x-4) \) (C) \( V(x)=4(x-8)(x-2) \) (D) \( V(x)=x(x-8)(x-2) \)

MTK มีขนาด TV̂K มีขนาด

Given \( V=x^{2}+y^{2}+z^{2} \), find in its simplest form \( x \frac{\partial V}{\partial x}+y \frac{\partial V}{\partial y}+z \frac{\partial V}{\partial z} \)

2. Use Theorem 4.2 .1 to determine which of the following are subspaces of \( M_{n n} \). (a) The set of all diagonal \( n \times n \) matrices. (b) The set of all \( n \times n \) matrices \( A \) such that \( \operatorname{det}(A)=0 \). (c) The set of all \( n \times n \) matrices \( A \) such that \( \operatorname{tr}(A)=0 \). (d) The set of all symmetric \( n \times n \) matrices. (e) The set of all \( n \times n \) matrices \( A \) such that \( A^{T}=-A \). (f) The set of all \( n \times n \) matrices \( A \) for which \( A \mathbf{x}=0 \) has only the trivial solution. (g) The set of all \( n \times n \) matrices \( A \) such that \( A B=B A \) for some fixed \( n \times n \) matrix \( B \).