Simultaneous equations

In this question, \(x\), \(y\) and \(z\) are real numbers. Let \(\lfloor x \rfloor\) denote the largest integer that satisfies \(\lfloor x \rfloor \leqslant x\) and let \(\{x\}\) denote the fractional part of~\(x\), so that \(x = \lfloor x \rfloor + \{x\}\) and \(0 \leqslant \{x\} < 1\). For example, if \(x = 4.2\), then \(\lfloor x \rfloor = 4\) and \(\{x\} = 0.2\) and if \(x = -4.2\), then \(\lfloor x \rfloor = -5\) and \(\{x\} = 0.8\).

1. Solve the simultaneous equations \begin{align*} \lfloor x \rfloor + \{y\} &= 4.9, \\ \{x\} + \lfloor y \rfloor &= -1.4. \end{align*}
2. Given that \(x\), \(y\) and \(z\) satisfy the simultaneous equations \begin{align*} \lfloor x \rfloor + y + \{z\} &= 3.9, \\ \{x\} + \lfloor y \rfloor + z &= 5.3, \\ x + \{y\} + \lfloor z \rfloor &= 5, \end{align*} show that \(\{y\} + z = 3.2\) and solve the equations.
3. Solve the simultaneous equations \begin{align*} \lfloor x \rfloor + 2y + \{z\} &= 3.9, \\ \{x\} + 2\lfloor y \rfloor + z &= 5.3, \\ x + 2\{y\} + \lfloor z \rfloor &= 5. \end{align*}

1. The function f satisfies, for all \(x\), the equation \[ \f(x) + (1- x)\f(-x) = x^2\, . \] Show that \(\f(-x) + (1 + x)\f(x) = x^2\,\). Hence find \(\f(x)\) in terms of \(x\). You should verify that your function satisfies the original equation.
2. The function \({\rm K}\) is defined, for \(x\ne 1\), by \[{\rm K}(x) = \dfrac{x+1}{x-1}\,.\] Show that, for \(x\ne1\), \({\rm K(K(}x)) =x\,\). The function g satisfies the equation \[ \g(x)+ x\, \g\Big(\frac{ x+1 }{x-1}\Big) = x \ \ \ \ \ \ \ \ \ \ \ ( x\ne 1) \,. \] Show that, for \(x\ne1\), \(\g(x)= \dfrac{2x}{x^2+1}\,\).
3. Find \(\h(x)\), for \(x\ne0\), \(x\ne1\), given that \[ \h(x)+ \h\Big(\frac 1 {1-x}\Big)= 1-x -\frac1{1-x} \ \ \ \ \ \ ( x\ne0, \ \ x\ne1 ) \,. \]

It is given that the two curves \[ y=4-x^2 \text{ and } m x = k-y^2\,, \] where \(m > 0\), touch exactly once.

1. In each of the following four cases, sketch the two curves on a single diagram, noting the coordinates of any intersections with the axes:
   1. \(k < 0\, \);
   2. \(0 < k < 16\), \(k/m < 2\,\);
   3. \(k > 16\), \(k/m > 2\,\);
   4. \(k > 16\), \(k/m < 2\,\).
2. Now set \(m=12\). Show that the \(x\)-coordinate of any point at which the two curves meet satisfies \[ x^4-8x^2 +12x +16-k=0\,. \] Let \(a\) be the value of \(x\) at the point where the curves touch. Show that \(a\) satisfies \[ a^3 -4a +3 =0 \] and hence find the three possible values of \(a\). Derive also the equation \[ k= -4a^2 +9a +16\,. \] Which of the four sketches in part (i) arise?

Given that \[ 5x^{2}+2y^{2}-6xy+4x-4y\equiv a\left(x-y+2\right)^{2} +b\left(cx+y\right)^{2}+d\,, \] find the values of the constants \(a\), \(b\), \(c\) and \(d\). Solve the simultaneous equations \begin{align*} 5x^{2}+2y^{2}-6xy+4x-4y&=9\,, \\ 6x^{2}+3y^{2}-8xy+8x-8y&=14\,. \end{align*}

Two curves have equations \(\; x^4+y^4=u\;\) and \(\; xy = v\;\), where \(u\) and \(v\) are positive constants. State the equations of the lines of symmetry of each curve. The curves intersect at the distinct points \(A\), \(B\), \(C\) and \(D\) (taken anticlockwise from \(A\)). The coordinates of \(A\) are \((\alpha,\beta)\), where \(\alpha > \beta > 0\). Write down, in terms of \(\alpha\) and \(\beta\), the coordinates of \(B\), \(C\) and \(D\). Show that the quadrilateral \(ABCD\) is a rectangle and find its area in terms of \(u\) and \(v\) only. Verify that, for the case \(u=81\) and \(v=4\), the area is \(14\).

Show Solution

The curve \(x^4 + y^4 = u\) has lines of symmetry:

• \(y = 0\)
• \(x = 0\)
• \(y = x\)
• \(y = -x\)

The curve \(xy = v\) has lines of symmetry:

• \(y = x\)
• \(y = -x\)

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The points are \(A = (\alpha, \beta), B = (\beta, \alpha), C = (-\alpha, -\beta), D = (-\beta, -\alpha)\) \(AD\) has gradient \(\frac{\beta+\alpha}{\alpha+\beta} = 1\), \(BC\) has the same gradient. \(AB\) has gradient \(\frac{\alpha-\beta}{\beta-\alpha} = -1\), as does \(CD\). Therefore it has two sets of perpendicular and parallel sides, hence a rectangle. The area is \(|AD||AB| = \sqrt{2(\alpha+\beta)^2}\sqrt{2(\alpha-\beta)^2} = 2(\alpha^2-\beta^2)\) The squared area is \(4(\alpha^4+\beta^4 - 2 \alpha^2\beta^2) = 4(u - 2v^2)\) ie the area is \(2\sqrt{u-2v^2}\) When \(u = 81, v = 4\) we have the area is \(2 \sqrt{81 - 2 \cdot 16} = 14\) as required.

A transformation \(T\) of the real numbers is defined by \[ y=T(x)=\frac{ax-b}{cx-d}\,, \] where \(a,b,c\), \(d\) are real numbers such that \(ad\neq bc\). Find all numbers \(x\) such that \(T(x)=x.\) Show that the inverse operation, \(x=T^{-1}(y)\) expressing \(x\) in terms of \(y\) is of the same form as \(T\) and find corresponding numbers \(a',b',c'\),\(d'\). Let \(S_{r}\) denote the set of all real numbers excluding \(r\). Show that, if \(c\neq0,\) there is a value of \(r\) such that \(T\) is defined for all \(x\in S_{r}\) and find the image \(T(S_{r}).\) What is the corresponding result if \(c=0\)? If \(T_{1},\) given by numbers \(a_{1},b_{1},c_{1},d_{1},\) and \(T_{2},\) given by numbers \(a_{2},b_{2},c_{2},d_{2}\) are two such transformations, show that their composition \(T_{3},\) defined by \(T_{3}(x)=T_{2}(T_{1}(x)),\) is of the same form. Find necessary and sufficient conditions on the numbers \(a,b,c,d\) for \(T^{2}\), the composition of \(T\) with itself, to be the identity. Hence, or otherwise, find transformations \(T_{1},T_{2}\) and their composition \(T_{3}\) such that \(T_{1}^{2}\) and \(T_{2}^{2}\) are each the identity but \(T_{3}^{2}\) is not.

Let \(\mathrm{h}(x)=ax^{2}+bx+c,\) where \(a,b\) and \(c\) are constants, and \(a\neq0\). Give a condition which \(a,b\) and \(c\) must satisfy in order that \(\mathrm{h}(x)\) can be written in the form \[ a(x+k)^{2},\tag{*} \] where \(k\) is a constant. If \(\mathrm{f}(x)=3x^{2}+4x\) and \(\mathrm{g}(x)=x^{2}-2\), find the two constant values of \(\lambda\) such that \(\mathrm{f}(x)+\lambda\mathrm{g}(x)\) can be written in the form \((*)\). Hence, or otherwise, find constants \(A,B,C,D,m\) and \(n\) such that \begin{alignat*}{1} \mathrm{f}(x) & =A(x+m)^{2}+B(x+n)^{2}\\ \mathrm{g}(x) & =C(x+m)^{2}+D(x+n)^{2}. \end{alignat*} If \(\mathrm{f}(x)=3x^{2}+4x\) and \(\mathrm{g}(x)=x^{2}+\alpha\) and it is given by that there is only one value of \(\lambda\) for which \(\mathrm{f}(x)+\lambda\mathrm{g}(x)\) can be written in the form \((*)\), find \(\alpha\).

It is given that \(x,y\) and \(z\) are distinct and non-zero, and that they satisfy \[ x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}. \] Show that \(x^{2}y^{2}z^{2}=1\) and that the value of \(x+\dfrac{1}{y}\) is either \(+1\) or \(-1\).

The numbers \(x,y\) and \(z\) are non-zero, and satisfy \[ 2a-3y=\frac{\left(z-x\right)^{2}}{y}\quad\mbox{ and }\quad2a-3z=\frac{\left(x-y\right)^{2}}{z}, \] for some number \(a\). If \(y\neq z\), prove that \[ x+y+z=a, \] and that \[ 2a-3x=\frac{\left(y-z\right)^{2}}{x}. \] Determine whether this last equation holds only if \(y\neq z\).