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Home Question Bank Math Archive 2024 December 29

Math Questions from Dec 29,2024

Browse the Math Q&A Archive for Dec 29,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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What are dependent events in probability? What is a scatter plot? \( v=3-5 t \) Work out the value of \( v \) when \( t=4 \) Let \( f(x)=x^{2}+4 \) and \( h(x)=x+4 \), find the value of the following composite function. \( (f \circ h)(-4) \) \( (f \circ h)(-4)=\square \) (Simplify your answer.) 10 For positive real numbers \( a \), and \( b \), the lines \( l_{1} \) and \( l_{2} \) have equations given by \[ l_{1}: \mathbf{r}=\left(\begin{array}{l} 1 \\ 2 \\ 5 \end{array}\right)+\lambda\left(\begin{array}{c} -1 \\ 0 \\ 3 \end{array}\right) \text { for } \lambda \in \mathbb{R} \text { and } l_{2}: \mathbf{r}=\left(\begin{array}{l} a \\ 9 \\ 9 \end{array}\right)+\mu\left(\begin{array}{l} 1 \\ 7 \\ b \end{array}\right) \text { for } \mu \in \mathbb{R} \] It is given that the two lines intersect each other. (i) Show that \( 3 a=b+2 \). [2] (ii) Find the value of \( b \) if the angle between the two lines is \( \cos ^{-1}\left(\frac{11}{\sqrt{660}}\right) \) radians. The equation of the plane \( \Pi_{1} \) that contains both lines is now given by \( \mathbf{r} \cdot\left(\begin{array}{c}3 \\ -1 \\ 1\end{array}\right)=6 \). (iii) A second plane \( \Pi_{2} \) meets \( \Pi_{1} \) in the line \( l_{1} \) and contains the point \( (2,1,3) \). Find an equation for \( \Pi_{2} \) in scalar product form. [2] (iv) Hence find the angle between the planes \( \Pi_{1} \) and \( \Pi_{2} \). [2] (v) The plane \( \Pi_{3} \) is parallel to the plane \( \Pi_{1} \). Given that the distance between both planes is \( \frac{7}{\sqrt{11}} \) units and that \( \Pi_{3} \) is closer to the origin than \( \Pi_{1} \), find a cartesian equation for \( \Pi_{3} \). [3] \begin{tabular}{l|l} \hline 10 & (b) 4 (c) \( r \cdot\left(\begin{array}{l}3 \\ 1 \\ 1\end{array}\right)=10 \$ \). \end{tabular} (d) \( 0.613 \mathrm{rad}(3 \mathrm{sf}) \) or \( 35.1^{\circ}\left(\right. \) nearest \( \left.0.1^{\circ}\right) \) (e) \( 3 x-y+z=-1 \) For the polynomial function \( f(x) \), find a. \( f(-1), b . f(2) \), and c. \( f(0) \). a. \( f(-1)=7 x^{2}-11 x+2 \) (Simplify your answer. Type an integer or a fraction.) b. \( f(2)=\square \) (Simplify your answer. Type an integer or a fraction.) c. \( f(0)=\square \) (Simplify your answer. Type an integer or a fraction.) For the polynomial function \( f(x) \), find a. \( f(-1) \), b. \( f(2) \), and c. \( f(0) \). a. \( f(-1)=\square \) (Simplify your answer. Type an integer or a fraction. \( ) \) b. \( f(2)=\square \) (Simplify your answer. Type an integer or a fraction.) c. \( f(0)=\square \) (Simplify your answer. Type an integer or a fraction.) Forensic scientists use the lengths of certain bones to calculate the height of a person. Two such bones are the tibis (t), the bone from the ankle to the knee, and the femur ( r ), the bone from the knee to the hip socket. A person's height (h) in centimeters is determined from the lengths of these bones using the following functions. Answer parts (a) trough (d). For males For females \[ \begin{array}{ll} h(r)=69.09+2.24 r & h(t)=81.69+2.39 t \\ h(r)=61.41+2.32 r & h(t)=72.57+2.53 t \end{array} \] (a) Find the height of a male with a femur measuring 54 cm . \( \square \) cm (Simplify your answer. Type an integer or a decimal.) (b) Find the height of a male with a tibia measuring 38 cm . \( \square \) cm (Simplify your answer. Type an integer or a decimal.) (c) Find the height of a female with a femur measuring 50 cm . \( \square \) cm (Simplify your answer. Type an integer or a decimal.) (d) Find the height of a female with a tibia measuring 36 cm . إذا كان لديك صندوق يتمتع بحجم ثابت قدره 1000 سم³، احسب الأبعاد التي ستؤدي إلى الحد الأدنى من المساحة السطحية. questions. Statement Prove Archimedean Property: If \( x \in \mathbb{R} \), then there is \( n_{x} \in \mathbb{N} \) such that \( x \leq n_{x} \). If a sequence \( X=\left(x_{n}\right) \) of real numbers converges to \( L \), then prove that ar subsequence \( \left(x_{n_{k}}\right) \) of \( X \) also converges to \( L \). Use this result in part b) to prove that \( \left(x_{n}\right)=\left(c^{\frac{1}{n}}\right) \), where \( c>1 \) is fixed constan converges to 1. Prove that the sequence \( \left(\sin \frac{n \pi}{4}\right) \) is divergent. Prove that \( \left(x_{n}\right)=\left(\frac{1}{n}\right) \) is a Cauchy sequence. Discuss convergence/divergence of the series \( \sum_{n=1}^{\infty} \frac{1}{n(n+1)(n+2)} \) using its sequence partial sums. Prove that if \( \sum_{n=1}^{\infty} x_{n} \) is convergent, then necessarily \( \lim _{n \rightarrow \infty} x_{n}=0 \). Construct an alternating series which converges to 2 . Discuss convergence and absolute convergence of \( \sum_{n=1}^{\infty}(-1)^{n} \frac{(n+1)^{n}}{n^{n}} \). Consider the real function \( f: \mathbb{R} \rightarrow \mathbb{R} \) defined by \( f(x)=\sum_{n=1}^{\infty} n^{2}(x-2)^{n} \). Compute th domain of \( f \). Let \( A \subset \mathbb{R}, c \in \mathbb{R} \) is cluster point of \( A \) if for all \( \delta>0,(c-\delta, c+\delta) \cap(A \backslash\{c\}) \neq c \) Compute cluster points of the sets i) \( A=\left\{1-\frac{1}{n}, n \in \mathbb{N}\right\} \) ii) \( \mathbb{Q} \), set of rational number
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