A Level H1 Mathematics Graphs Coordinate Geometry Quiz

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A-Level Maths H1 Quiz - Graphs Coordinate Geometry

Name: ____________________
Class: ____________________
Date: ____________________
Score: ________ / 55

Duration: 90 Minutes
Total Marks: 55
Instructions:

• Answer all questions.
• Use of a non-CAS Graphing Calculator (GC) is expected.
• Show all necessary working.
• Give your answers to 3 significant figures unless specified otherwise.

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Section A: Scatter Diagrams and Correlation (Questions 1–7)

1. A researcher collects data on the number of hours spent studying ($x$) and the test scores ($y$) of 10 students. (a) Describe the steps you would take on your GC to generate a scatter diagram for this data. [1] (b) If the scatter diagram shows points closely clustered around a straight line sloping upwards, what does this suggest about the correlation between $x$ and $y$? [1]

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2. Given a set of data where the product moment correlation coefficient $r = -0.87$. (a) Describe the strength and direction of the linear relationship. [2] (b) Does this value of $r$ imply that increasing $x$ causes $y$ to decrease? Explain your answer. [2]

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3. A scatter diagram is plotted for the relationship between the age of a car ($x$ in years) and its resale value ($y$ in dollars). (a) Sketch a typical scatter diagram for this relationship. [2] (b) State whether you expect the correlation coefficient $r$ to be positive or negative. [1]

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4. For a given dataset, the regression line of $y$ on $x$ is $y = 4.2x + 15.8$. (a) Interpret the meaning of the gradient $4.2$ in the context of the variables $x$ and $y$. [2] (b) Predict the value of $y$ when $x = 10$. [1]

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5. A student uses the regression line $y = 0.5x + 20$ to predict a value. The original data for $x$ ranged from 10 to 50. The student predicts $y$ for $x = 120$. (a) Is this an example of interpolation or extrapolation? [1] (b) Explain why the prediction for $x = 120$ might be unreliable. [2]

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6. Given the summary statistics $\sum x = 50, \sum y = 100, \sum x^2 = 300, \sum xy = 600$ for $n=10$ pairs of data. (a) Calculate the mean of $x$ and the mean of $y$. [2] (b) Calculate the gradient $m$ of the least squares regression line $y = mx + c$. [3]

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7. On a GC, a student plots a scatter diagram and finds that the points form a distinct curve rather than a straight line. (a) Is the product moment correlation coefficient $r$ a suitable measure of the relationship in this case? Justify your answer. [2]

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Section B: Regression and Linear Modelling (Questions 8–14)

8. The regression line of $y$ on $x$ is $y = -2.5x + 100$. (a) Find the value of $y$ when $x = 0$. [1] (b) Find the value of $x$ when $y = 50$. [2]

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9. A dataset has a regression line $y = 1.2x - 5$. (a) If every $y$-value in the original dataset is increased by 10, what is the new equation of the regression line? [2] (b) If every $x$-value is doubled, how does the gradient of the regression line change? [2]

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10. A researcher finds the regression line of $T$ (Temperature) on $H$ (Humidity) to be $T = -0.15H + 32$. (a) If the humidity increases by 10%, by how many units is the temperature expected to change? [2] (b) If the current humidity is 60%, predict the temperature. [2]

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11. Given the regression line $y = 3.1x + 12.4$, and the mean values $\bar{x} = 5$ and $\bar{y} = 27.9$. (a) Verify that the regression line passes through the point $(\bar{x}, \bar{y})$. [2] (b) If a new data point $(6, 30)$ is added, will the gradient $m$ necessarily increase? Explain. [2]

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12. A scatter diagram shows a very strong positive linear correlation ($r = 0.98$). (a) Describe the visual appearance of the points relative to the regression line. [2] (b) If $r$ were $0.10$, how would the scatter diagram look different? [2]

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13. For a set of data, the regression line of $x$ on $y$ is $x = 0.4y + 2$. (a) Is this the same line as the regression line of $y$ on $x$? Explain. [2] (b) Rearrange the equation to express $y$ in terms of $x$. [2]

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14. A company models the relationship between advertising spend ($x$) and sales ($y$). The regression line is $y = 5.2x + 100$. (a) What is the estimated sales if the advertising spend is zero? [1] (b) If the company wants to increase sales by 52 units, how much more should they spend on advertising? [2]

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Section C: Coordinate Geometry and Curve Analysis (Questions 15–20)

15. A curve $C$ has the equation $y = 2e^{0.5x}$. (a) Find the gradient of the curve at the point where $x = 0$. [2] (b) Find the equation of the tangent to the curve at $x = 0$ in the form $y = mx + c$. [3]

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16. A curve is defined by $y = \ln(x + 2)$. (a) Find the x-intercept of the curve. [2] (b) Determine the equation of the vertical asymptote of the curve. [2]

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17. Consider the curve $y = \frac{4}{x-1}$. (a) Find the coordinates of the point on the curve where the gradient is $-4$. [3] (b) Find the equation of the normal to the curve at this point. [3]

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18. A curve $C$ has the equation $y = x^2 - 4x + 7$. (a) Find the coordinates of the stationary point of the curve. [3] (b) Determine the nature of this stationary point using the second derivative test. [2]

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19. The region $R$ is bounded by the curve $y = e^x$, the x-axis, the line $x = 0$, and the line $x = 2$. (a) Set up the definite integral to find the area of $R$. [2] (b) Calculate the exact area of $R$. [2]

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20. A line $L$ passes through the points $(2, 5)$ and $(6, 13)$. (a) Find the equation of line $L$. [2] (b) Find the coordinates of the point where line $L$ intersects the curve $y = x^2 + 1$. [4]

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A-Level Maths H1 Quiz Answers - Graphs Coordinate Geometry

Section A: Scatter Diagrams and Correlation

1. (a) Enter data into lists (L1, L2) $\rightarrow$ Stat Plot $\rightarrow$ On $\rightarrow$ Select Scatter plot $\rightarrow$ Zoom Stat. [1] (b) Strong positive linear correlation. [1]
2. (a) Strong (since $|r|$ is close to 1) and negative (since $r < 0$). [2] (b) No. Correlation does not imply causation. [2]
3. (a) Points trending downwards from top-left to bottom-right. [2] (b) Negative. [1]
4. (a) For every 1 unit increase in $x$, $y$ is estimated to increase by 4.2 units on average. [2] (b) $y = 4.2(10) + 15.8 = 57.8$. [1]
5. (a) Extrapolation. [1] (b) The value $x=120$ is far outside the range of the original data (10-50); the linear trend may not continue. [2]
6. (a) $\bar{x} = 50/10 = 5$; $\bar{y} = 100/10 = 10$. [2] (b) $m = \frac{\sum xy - n\bar{x}\bar{y}}{\sum x^2 - n\bar{x}^2} = \frac{600 - 10(5)(10)}{300 - 10(5^2)} = \frac{600-500}{300-250} = \frac{100}{50} = 2$. [3]
7. (a) No. $r$ only measures the strength of a linear relationship. For a curved relationship, $r$ will underestimate the strength of the association. [2]

Section B: Regression and Linear Modelling

8. (a) $y = 100$. [1] (b) $50 = -2.5x + 100 \rightarrow -50 = -2.5x \rightarrow x = 20$. [2]
9. (a) $y = 1.2x + 5$ (The intercept $c$ increases by 10). [2] (b) The gradient $m$ is halved (since $y = m(2x) + c$ is not the case, but rather the spread of $x$ doubles, $m_{new} = m_{old}/2$). [2]
10. (a) $\Delta T = -0.15 \times 10 = -1.5$ units. [2] (b) $T = -0.15(60) + 32 = -9 + 32 = 23$. [2]
11. (a) $3.1(5) + 12.4 = 15.5 + 12.4 = 27.9$. Matches $\bar{y}$. [2] (b) Not necessarily. It depends on how far the point $(6, 30)$ is from the existing line and its influence on the sum of squares. [2]
12. (a) Points lie very close to the regression line. [2] (b) Points would be widely scattered with no clear linear trend. [2]
13. (a) No. The regression of $x$ on $y$ minimizes vertical distances to the $x$-axis, while $y$ on $x$ minimizes vertical distances to the $y$-axis. [2] (b) $0.4y = x - 2 \rightarrow y = 2.5x - 5$. [2]
14. (a) 100. [1] (b) $52 = 5.2 \times \Delta x \rightarrow \Delta x = 10$. [2]

Section C: Coordinate Geometry and Curve Analysis

15. (a) $y' = 2(0.5)e^{0.5x} = e^{0.5x}$. At $x=0, y' = e^0 = 1$. [2] (b) Point is $(0, 2)$. $y - 2 = 1(x - 0) \rightarrow y = x + 2$. [3]
16. (a) $0 = \ln(x+2) \rightarrow e^0 = x+2 \rightarrow 1 = x+2 \rightarrow x = -1$. [2] (b) $x+2 = 0 \rightarrow x = -2$. [2]
17. (a) $y' = -4(x-1)^{-2}$. Set $-4/(x-1)^2 = -4 \rightarrow (x-1)^2 = 1 \rightarrow x-1 = \pm 1$. $x=2$ or $x=0$. For $x=2, y=4$. For $x=0, y=-4$. (Either point acceptable). [3] (b) If $(2, 4)$, $m_{tangent} = -4 \rightarrow m_{normal} = 1/4$. $y - 4 = 0.25(x-2) \rightarrow y = 0.25x + 3.5$. [3]
18. (a) $y' = 2x - 4$. Set $2x-4=0 \rightarrow x=2$. $y = 2^2 - 4(2) + 7 = 3$. Point $(2, 3)$. [3] (b) $y'' = 2$. Since $y'' > 0$, it is a local minimum. [2]
19. (a) $\int_0^2 e^x dx$. [2] (b) $[e^x]_0^2 = e^2 - e^0 = e^2 - 1$. [2]
20. (a) $m = (13-5)/(6-2) = 8/4 = 2$. $y - 5 = 2(x-2) \rightarrow y = 2x + 1$. [2] (b) $x^2 + 1 = 2x + 1 \rightarrow x^2 - 2x = 0 \rightarrow x(x-2) = 0$. $x=0$ or $x=2$. Points $(0, 1)$ and $(2, 5)$. [4]