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This shard is defined as the segment of a circle with centre $C$ and radius $4\\sqrt{3}\\text{ cm}$. The segment is bounded by the chord $PQ$ and the arc $PQ$, where the angle subtended at the centre, $\\angle PCQ$, is $\\frac{\\pi}{3}$ radians.\n\nShow that the area of the shard is\n\\[ k(2\\pi - 3\\sqrt{3})\\text{ cm}^2 \\]\nwhere $k$ is an integer to be found.","markscheme":"- Uses the area of a sector formula $A = \\frac{1}{2}r^2\\theta$ with $r = 4\\sqrt{3}$ and $\\theta = \\frac{\\pi}{3}$ M1\n- Calculates $A_{sector} = \\frac{1}{2} \\times (4\\sqrt{3})^2 \\times \\frac{\\pi}{3} = \\frac{1}{2} \\times 48 \\times \\frac{\\pi}{3} = 8\\pi$ A1\n- Uses the area of a triangle formula $A = \\frac{1}{2}r^2\\sin(\\theta)$ with $\\sin\\left(\\frac{\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}$ M1\n- Calculates $A_{triangle} = \\frac{1}{2} \\times 48 \\times \\frac{\\sqrt{3}}{2} = 12\\sqrt{3}$ A1\n- Subtracts the areas to find $A_{segment} = 8\\pi - 12\\sqrt{3}$ M1\n- Factorises $4(2\\pi - 3\\sqrt{3})$ to state that $k = 4$ A1","marks":6,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"trigonometric_area","label":"Trigonometric Triangle Formula","steps":[{"type":"text","order":0,"title":"Understand the problem","content":"To find the area of the shard (which is a circular segment), we can subtract the area of the central triangle from the area of the sector:\n\n$$\\text{Area of Segment} = \\text{Area of Sector} - \\text{Area of Triangle}$$\n\nWe are given:\n- Radius, $r = 4\\sqrt{3}\\text{ cm}$\n- Central angle, $\\theta = \\frac{\\pi}{3}$ radians"},{"type":"text","order":1,"title":"Calculate the sector area","content":"The formula for the area of a sector when the angle $\\theta$ is in radians is:\n\n$$A_{\\text{sector}} = \\frac{1}{2}r^2\\theta$$\n\nFirst, let's calculate $r^2$:\n\n$$r^2 = (4\\sqrt{3})^2 = 16 \\times 3 = 48$$\n\nNow, substitute $r^2 = 48$ and $\\theta = \\frac{\\pi}{3}$ into the formula:\n\n$$A_{\\text{sector}} = \\frac{1}{2} \\times 48 \\times \\frac{\\pi}{3} = 8\\pi$$"},{"type":"text","order":2,"title":"Calculate the triangle area","content":"To find the area of the central triangle $PCQ$, we use the trigonometric area formula:\n\n$$A_{\\text{triangle}} = \\frac{1}{2}r^2\\sin(\\theta)$$\n\nSubstitute $r^2 = 48$ and use the exact trigonometric value $\\sin\\left(\\frac{\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}$:\n\n$$A_{\\text{triangle}} = \\frac{1}{2} \\times 48 \\times \\sin\\left(\\frac{\\pi}{3}\\right)$$\n\n$$A_{\\text{triangle}} = 24 \\times \\frac{\\sqrt{3}}{2} = 12\\sqrt{3}$$"},{"type":"text","order":3,"title":"Subtract to find segment area","content":"Subtract the area of the triangle from the area of the sector to get the area of the shard:\n\n$$\\text{Area of Shard} = 8\\pi - 12\\sqrt{3}$$"},{"type":"text","order":4,"title":"Factorise and find $k$","content":"We need to write the area in the form $k(2\\pi - 3\\sqrt{3})$.\n\nFactor out the greatest common divisor, which is $4$:\n\n$$8\\pi - 12\\sqrt{3} = 4(2\\pi - 3\\sqrt{3})$$\n\nComparing this to the given expression, we find that:\n\n$$k = 4$$"}]},{"id":"equilateral_triangle","label":"Equilateral Triangle Properties","steps":[{"type":"text","order":0,"title":"Analyze the problem","content":"We can find the area of the circle's segment (the 'shard') by subtracting the area of the triangle $PCQ$ from the area of the sector $PCQ$:\n\n$$\\text{Area of shard} = \\text{Area of sector} - \\text{Area of triangle}$$"},{"type":"text","order":1,"title":"Calculate sector area","content":"The area of a sector of a circle with radius $r$ and angle $\\theta$ (in radians) is:\n\n$$A_{\\text{sector}} = \\frac{1}{2}r^2\\theta$$\n\nSubstituting the given values, $r = 4\\sqrt{3}\\text{ cm}$ and $\\theta = \\frac{\\pi}{3}$:\n\n$$A_{\\text{sector}} = \\frac{1}{2} \\times (4\\sqrt{3})^2 \\times \\frac{\\pi}{3}$$\n\n$$A_{\\text{sector}} = \\frac{1}{2} \\times 48 \\times \\frac{\\pi}{3} = 8\\pi\\text{ cm}^2$$"},{"type":"text","order":2,"title":"Identify triangle properties","content":"The central angle is $\\angle PCQ = \\frac{\\pi}{3}\\text{ radians}$ (which is equal to $60^\\circ$).\n\nSince the sides $CP$ and $CQ$ are both radii of length $4\\sqrt{3}\\text{ cm}$, the triangle $PCQ$ is an isosceles triangle with an angle of $60^\\circ$ between the equal sides.\n\nThis means the other two angles must also be $60^\\circ$, making $\\triangle PCQ$ an **equilateral triangle** with all side lengths equal to:\n\n$$s = 4\\sqrt{3}\\text{ cm}$$"},{"type":"text","order":3,"title":"Calculate triangle area","content":"We can find the area of an equilateral triangle with side length $s$ using the formula:\n\n$$A_{\\text{triangle}} = \\frac{\\sqrt{3}}{4}s^2$$\n\nSubstituting $s = 4\\sqrt{3}$:\n\n$$A_{\\text{triangle}} = \\frac{\\sqrt{3}}{4} \\times (4\\sqrt{3})^2$$\n\n$$A_{\\text{triangle}} = \\frac{\\sqrt{3}}{4} \\times 48 = 12\\sqrt{3}\\text{ cm}^2$$"},{"type":"text","order":4,"title":"Calculate shard area","content":"Subtract the area of the triangle from the area of the sector to get the area of the shard:\n\n$$\\text{Area of shard} = 8\\pi - 12\\sqrt{3}$$"},{"type":"text","order":5,"title":"Factorise the expression","content":"We need to express the area in the form $k(2\\pi - 3\\sqrt{3})$:\n\n$$8\\pi - 12\\sqrt{3} = 4(2\\pi - 3\\sqrt{3})$$\n\nBy comparing this with the required expression, we find that:\n\n$$k = 4$$"}]}],"generatedAt":"2026-06-09T08:45:18.275Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"4f9e9926-feed-4180-98fb-0f0f3db91814:trigonometric_area","label":"Trigonometric Triangle Formula","steps":[{"type":"text","order":0,"title":"Understand the problem","content":"To find the area of the shard (which is a circular segment), we can subtract the area of the central triangle from the area of the sector:\n\n$$\\text{Area of Segment} = \\text{Area of Sector} - \\text{Area of Triangle}$$\n\nWe are given:\n- Radius, $r = 4\\sqrt{3}\\text{ cm}$\n- Central angle, $\\theta = \\frac{\\pi}{3}$ radians"},{"type":"text","order":1,"title":"Calculate the sector area","content":"The formula for the area of a sector when the angle $\\theta$ is in radians is:\n\n$$A_{\\text{sector}} = \\frac{1}{2}r^2\\theta$$\n\nFirst, let's calculate $r^2$:\n\n$$r^2 = (4\\sqrt{3})^2 = 16 \\times 3 = 48$$\n\nNow, substitute $r^2 = 48$ and $\\theta = \\frac{\\pi}{3}$ into the formula:\n\n$$A_{\\text{sector}} = \\frac{1}{2} \\times 48 \\times \\frac{\\pi}{3} = 8\\pi$$"},{"type":"text","order":2,"title":"Calculate the triangle area","content":"To find the area of the central triangle $PCQ$, we use the trigonometric area formula:\n\n$$A_{\\text{triangle}} = \\frac{1}{2}r^2\\sin(\\theta)$$\n\nSubstitute $r^2 = 48$ and use the exact trigonometric value $\\sin\\left(\\frac{\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}$:\n\n$$A_{\\text{triangle}} = \\frac{1}{2} \\times 48 \\times \\sin\\left(\\frac{\\pi}{3}\\right)$$\n\n$$A_{\\text{triangle}} = 24 \\times \\frac{\\sqrt{3}}{2} = 12\\sqrt{3}$$"},{"type":"text","order":3,"title":"Subtract to find segment area","content":"Subtract the area of the triangle from the area of the sector to get the area of the shard:\n\n$$\\text{Area of Shard} = 8\\pi - 12\\sqrt{3}$$"},{"type":"text","order":4,"title":"Factorise and find $k$","content":"We need to write the area in the form $k(2\\pi - 3\\sqrt{3})$.\n\nFactor out the greatest common divisor, which is $4$:\n\n$$8\\pi - 12\\sqrt{3} = 4(2\\pi - 3\\sqrt{3})$$\n\nComparing this to the given expression, we find that:\n\n$$k = 4$$"}]},{"id":"4f9e9926-feed-4180-98fb-0f0f3db91814:equilateral_triangle","label":"Equilateral Triangle Properties","steps":[{"type":"text","order":0,"title":"Analyze the problem","content":"We can find the area of the circle's segment (the 'shard') by subtracting the area of the triangle $PCQ$ from the area of the sector $PCQ$:\n\n$$\\text{Area of shard} = \\text{Area of sector} - \\text{Area of triangle}$$"},{"type":"text","order":1,"title":"Calculate sector area","content":"The area of a sector of a circle with radius $r$ and angle $\\theta$ (in radians) is:\n\n$$A_{\\text{sector}} = \\frac{1}{2}r^2\\theta$$\n\nSubstituting the given values, $r = 4\\sqrt{3}\\text{ cm}$ and $\\theta = \\frac{\\pi}{3}$:\n\n$$A_{\\text{sector}} = \\frac{1}{2} \\times (4\\sqrt{3})^2 \\times \\frac{\\pi}{3}$$\n\n$$A_{\\text{sector}} = \\frac{1}{2} \\times 48 \\times \\frac{\\pi}{3} = 8\\pi\\text{ cm}^2$$"},{"type":"text","order":2,"title":"Identify triangle properties","content":"The central angle is $\\angle PCQ = \\frac{\\pi}{3}\\text{ radians}$ (which is equal to $60^\\circ$).\n\nSince the sides $CP$ and $CQ$ are both radii of length $4\\sqrt{3}\\text{ cm}$, the triangle $PCQ$ is an isosceles triangle with an angle of $60^\\circ$ between the equal sides.\n\nThis means the other two angles must also be $60^\\circ$, making $\\triangle PCQ$ an **equilateral triangle** with all side lengths equal to:\n\n$$s = 4\\sqrt{3}\\text{ cm}$$"},{"type":"text","order":3,"title":"Calculate triangle area","content":"We can find the area of an equilateral triangle with side length $s$ using the formula:\n\n$$A_{\\text{triangle}} = \\frac{\\sqrt{3}}{4}s^2$$\n\nSubstituting $s = 4\\sqrt{3}$:\n\n$$A_{\\text{triangle}} = \\frac{\\sqrt{3}}{4} \\times (4\\sqrt{3})^2$$\n\n$$A_{\\text{triangle}} = \\frac{\\sqrt{3}}{4} \\times 48 = 12\\sqrt{3}\\text{ cm}^2$$"},{"type":"text","order":4,"title":"Calculate shard area","content":"Subtract the area of the triangle from the area of the sector to get the area of the shard:\n\n$$\\text{Area of shard} = 8\\pi - 12\\sqrt{3}$$"},{"type":"text","order":5,"title":"Factorise the expression","content":"We need to express the area in the form $k(2\\pi - 3\\sqrt{3})$:\n\n$$8\\pi - 12\\sqrt{3} = 4(2\\pi - 3\\sqrt{3})$$\n\nBy comparing this with the required expression, we find that:\n\n$$k = 4$$"}]}],"generatedAt":"2026-06-09T08:45:29.341Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"d183dcb1-0e62-4f4c-acf3-7106d716d77c","revisionId":"1aeb9258-d16e-4a4c-80a5-7b76f4b9a5c2","questionNumber":null,"questionType":"LA","specification":"Solve, for $0° \\leq \\theta < 360°$, the equation","difficulty":null,"difficultyBand":"medium","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"a400ee6e-3d2f-4b13-96c1-d47bda29d482","label":null,"content":"$$\\sin(\\theta - 45°) = \\cos(\\theta + 30°)$","markscheme":"- Uses $\\cos X \\equiv \\sin(90° - X)$: $\\cos(\\theta + 30°) = \\sin(60° - \\theta)$ M1\n- $\\sin(\\theta - 45°) = \\sin(60° - \\theta)$ A1\n- $\\theta - 45° = 60° - \\theta + 360n$ M1\n- $2\\theta = 105° + 360n$ A1\n- $\\theta = 52.5°$ A1\n- $\\theta = 232.5°$ A1\n\nThe alternative branch $\\theta - 45° = 180° - (60° - \\theta)$ gives $-45° = 120°$, which has no solution, so there are exactly two values. Expanding both sides with the compound-angle formulae and forming $\\tan\\theta = \\frac{\\sqrt{3} + \\sqrt{2}}{\\sqrt{2} + 1}$ is an equally valid but longer route; award the marks for a correct method.","marks":6,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"d5e8195d-1438-450e-b754-5fb4f576ef40","revisionId":"822dcab2-2272-44dc-be76-9a2ca31149a4","questionNumber":null,"questionType":"LA","specification":"The potential difference $V$, in millivolts, across a resistor in an alternating current circuit is modeled by the function $V(t) = 7\\cos t - 24\\sin t$, where $t$ is the time in milliseconds since the circuit was closed.","difficulty":5,"difficultyBand":"medium","tier":null,"totalMarks":9,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"75c18bac-a9bb-42c7-95c9-93df569fbe23","label":"a","content":"Express $V(t)$ in the form $R\\cos(t + \\alpha)$, where $R > 0$ and $0 < \\alpha < \\frac{\\pi}{2}$. Give the exact value of $R$ and the value of $\\alpha$, in radians, to 3 decimal places.","markscheme":"- Calculates $R = \\sqrt{7^2 + 24^2} = 25$ B1\n- Uses $R\\cos\\alpha = 7$ and $R\\sin\\alpha = 24$ to find $\\tan\\alpha = \\frac{24}{7}$ M1\n- Finds $\\alpha = \\arctan\\left(\\frac{24}{7}\\right) \\approx 1.287$ radians A1","marks":3,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"compound_angle_expansion","label":"Compound Angle Expansion","steps":[{"type":"text","order":0,"title":"Use compound angle expansion","content":"To express $V(t) = 7\\cos t - 24\\sin t$ in the form $R\\cos(t + \\alpha)$, we expand the target expression using the cosine addition formula:\n\n$$\\cos(A + B) = \\cos A \\cos B - \\sin A \\sin B$$\n\nMultiplying by $R$, we obtain:\n\n$$R\\cos(t + \\alpha) = R\\cos t \\cos \\alpha - R\\sin t \\sin \\alpha$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$7\\cos t - 24\\sin t = \\htmlData{anchor=eq1}{\\color{@sky}{R\\cos\\alpha}}\\cos t - \\htmlData{anchor=eq2}{\\color{@amber}{R\\sin\\alpha}}\\sin t$"},{"text":"By comparing both sides, we equate the coefficients of $\\cos t$ and $\\sin t$:"},{"latex":"$$\\htmlData{anchor=c1}{\\color{@sky}{R\\cos\\alpha}} = 7$"},{"latex":"$$\\htmlData{anchor=c2}{\\color{@amber}{R\\sin\\alpha}} = 24$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"eq1"},{"color":"amber","style":"text-color","anchor":"eq2"}],"connections":[{"to":"c1","from":"eq1","color":"sky","label":"cos coefficient","style":"solid"},{"to":"c2","from":"eq2","color":"amber","label":"sin coefficient","style":"solid"}]},"order":1,"title":"Equate the coefficients","description":"We equate the corresponding terms on both sides of the equation to find relationships for $R$ and $\\alpha$."},{"type":"text","order":2,"title":"Find the value of R","content":"To find $R$, we square and add our two equations:\n\n$$(R\\sin\\alpha)^2 + (R\\cos\\alpha)^2 = 24^2 + 7^2$$\n\n$$R^2(\\sin^2\\alpha + \\cos^2\\alpha) = 576 + 49$$\n\nUsing the fundamental trigonometric identity $\\sin^2\\alpha + \\cos^2\\alpha = 1$, we get:\n\n$$R^2 = 625$$\n\nSince $R > 0$, we find the exact value of $R$:\n\n$$R = \\sqrt{625} = 25$$\n\nThis calculation completes the first part of the markscheme **[B1]**.","highlights":[{"text":"exact value of R","location":"specification"}]},{"type":"text","order":3,"title":"Find the value of alpha","content":"To find $\\alpha$, we divide the sine equation by the cosine equation:\n\n$$\\frac{R\\sin\\alpha}{R\\cos\\alpha} = \\frac{24}{7}$$\n\n$$\\tan\\alpha = \\frac{24}{7}$$\n\nThis division earns **[M1]**.\n\nNow, we take the inverse tangent to solve for $\\alpha$:\n\n$$\\alpha = \\arctan\\left(\\frac{24}{7}\\right)$$\n\nUsing a calculator in radian mode:\n\n$$\\alpha \\approx 1.287\\text{ radians}$$\n\nThis final rounded value completes Part 1 and earns **[A1]**."},{"type":"text","order":4,"title":"State the final expression","content":"Combining our values, we express $V(t)$ in the required form:\n\n$$V(t) = 25\\cos(t + 1.287)$$"},{"type":"text","order":5,"title":"Describe the stretch","content":"For Part 2(i), we describe the transformation that transforms $y = \\cos t$ to include the amplitude stretch.\n\nThe factor $R = 25$ multiplying the function results in a vertical stretch parallel to the $y$-axis with a scale factor of $25$ **[B1]**.","highlights":[{"text":"Describe fully the transformation that is a stretch.","location":"specification"}]},{"type":"text","order":6,"title":"Describe the translation","content":"For Part 2(ii), we describe the horizontal shift caused by the phase angle $\\alpha$.\n\nThe term $(t + 1.287)$ represents a horizontal shift to the left by $1.287$ units, which can be described as a translation by the vector:\n\n$$\\begin{pmatrix} -1.287 \\\\ 0 \\end{pmatrix}$$\n\nThis earns **[B1]**.","highlights":[{"text":"Describe fully the transformation that is a translation.","location":"specification"}]},{"type":"text","order":7,"title":"Find the range of G","content":"For Part 3, we look at the function $G(t) = \\frac{2100}{75 + (V(t))^2}$.\n\nSince $V(t) = 25\\cos(t + 1.287)$, its range is $[-25, 25]$. Therefore, the squared term $(V(t))^2$ is bounded as follows:\n\n$$0 \\le (V(t))^2 \\le 25^2 = 625 \\quad \\textbf{[M1]}$$\n\nWe find the boundaries of $G(t)$ by substituting these extreme values:\n\n- **Maximum value of $G$** occurs when $(V(t))^2 = 0$:\n $$G_{\\text{max}} = \\frac{2100}{75 + 0} = 28 \\quad \\textbf{[M1]}$$\n\n- **Minimum value of $G$** occurs when $(V(t))^2 = 625$:\n $$G_{\\text{min}} = \\frac{2100}{75 + 625} = \\frac{2100}{700} = 3 \\quad \\textbf{[M1]}$$\n\nThus, the range of $G$ is:\n\n$$3 \\le G(t) \\le 28 \\quad \\textbf{[A1]}$$","highlights":[{"text":"Find the range of G.","location":"specification"}]}]}],"generatedAt":"2026-06-09T09:39:47.125Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"7f783460-f9c6-41e6-a365-c68f926c8ddc","label":"b","content":"The curve with equation $y = \\cos t$ is transformed onto the curve with equation $y = V(t)$ by a sequence of two transformations.\n\nGiven that the first transformation is a stretch and the second is a translation:\n\n(i) Describe fully the transformation that is a stretch.\n(ii) Describe fully the transformation that is a translation.","markscheme":"- (i) Vertical stretch in the $y$-direction (or parallel to the $y$-axis) with scale factor $25$ B1\n- (ii) Horizontal translation by the vector $\\begin{pmatrix} -1.287 \\\\ 0 \\end{pmatrix}$ (or shift to the left by $1.287$ units) B1","marks":2,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"harmonic_form","label":"Harmonic Form (R-alpha Method)","steps":[{"type":"text","order":0,"title":"Use the Harmonic Form","content":"To understand the transformations that map $y = \\cos t$ onto $y = V(t)$, we can rewrite $V(t) = 7\\cos t - 24\\sin t$ in the harmonic form:\n\n$$R\\cos(t + \\alpha)$$\n\nUsing the cosine addition formula:\n\n$$R\\cos(t + \\alpha) = R\\cos t\\cos\\alpha - R\\sin t\\sin\\alpha$$\n\nComparing this to $7\\cos t - 24\\sin t$, we can equate the coefficients:\n\n$$\\begin{aligned} R\\cos\\alpha &= 7 \\\\ R\\sin\\alpha &= 24 \\end{aligned}$$","highlights":[{"text":"V(t) = 7\\cos t - 24\\sin t","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$R = \\sqrt{\\htmlData{anchor=r-cos}{\\color{@sky}{7}}^2 + \\htmlData{anchor=r-sin}{\\color{@amber}{24}}^2} = \\sqrt{49 + 576} = \\sqrt{625} = \\htmlData{anchor=r-val}{\\color{@purple}{25}}$"},{"latex":"$$\\tan\\alpha = \\frac{\\htmlData{anchor=tan-sin}{\\color{@amber}{24}}}{\\htmlData{anchor=tan-cos}{\\color{@sky}{7}}} \\implies \\alpha = \\arctan\\left(\\frac{24}{7}\\right) \\approx \\htmlData{anchor=alpha-val}{\\color{@green}{1.287}} \\text{ radians}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"r-val"},{"color":"green","style":"text-color","anchor":"alpha-val"}],"connections":[{"to":"tan-cos","from":"r-cos","color":"sky","label":null,"style":"dashed"},{"to":"tan-sin","from":"r-sin","color":"amber","label":null,"style":"dashed"}]},"order":1,"title":"Calculate R and alpha","description":"We find the amplitude $R$ using the Pythagorean identity and the phase shift $\\alpha$ using trigonometry."},{"type":"text","order":2,"title":"Rewrite V(t)","content":"Now we substitute the values of $R$ and $\\alpha$ back into the harmonic form expression:\n\n$$V(t) = 25\\cos(t + 1.287)$$\n\nThis form directly reveals the two transformations applied to $y = \\cos t$ in sequence."},{"type":"text","order":3,"title":"Describe the stretch","content":"The first transformation is a stretch. Comparing $y = \\cos t$ to $y = 25\\cos(t + 1.287)$:\n\n- The coefficient $25$ outside the function multiplies all $y$-coordinates.\n\nTherefore, this represents a **vertical stretch in the $y$-direction** (or parallel to the $y$-axis) with a **scale factor of $25$**."},{"type":"text","order":4,"title":"Describe the translation","content":"The second transformation is a translation. Comparing $y = \\cos t$ to $y = \\cos(t + 1.287)$:\n\n- The addition of $1.287$ inside the argument of the cosine function shifts the graph horizontally to the left.\n\nTherefore, this represents a **horizontal translation** by the vector **$\\begin{pmatrix} -1.287 \\\\ 0 \\end{pmatrix}$** (or a shift to the left by $1.287$ units)."}]},{"id":"graphical_properties","label":"Graphical and Key Features Analysis","steps":[{"type":"text","order":0,"title":"Analyze original graph features","content":"To understand how the original function $y = \\cos t$ is transformed into $y = V(t)$, let's first identify the key features of the basic cosine wave:\n\n* **Maximum value (amplitude):** $1$\n* **Location of the first maximum:** $t = 0$, where $y = \\cos(0) = 1$\n\nBy comparing these features to the amplitude and horizontal position of the maximum point of $V(t) = 7\\cos t - 24\\sin t$, we can determine the scale factor of the vertical stretch and the shift of the translation."},{"type":"text","order":1,"title":"Calculate the vertical stretch","content":"The amplitude of a trigonometric function represents its maximum vertical distance from the center line. For $V(t) = 7\\cos t - 24\\sin t$, we can calculate its maximum value (amplitude) using the formula:\n\n$$\\text{Amplitude} = \\sqrt{A^2 + B^2}$$\n\nSubstitute the coefficients $A = 7$ and $B = -24$:\n\n$$\\text{Amplitude} = \\sqrt{7^2 + (-24)^2} = \\sqrt{49 + 576} = \\sqrt{625} = 25$$\n\nSince the original curve $y = \\cos t$ has an amplitude of $1$, and the transformed curve has an amplitude of $25$, the function has undergone a **vertical stretch parallel to the $y$-axis with a scale factor of $25$**."},{"type":"text","order":2,"title":"Determine the horizontal translation","content":"Next, let's find how much the graph has shifted horizontally by looking at the location of its maximum point.\n\nWe can express $V(t)$ in the single harmonic form:\n\n$$V(t) = 25\\cos(t + \\alpha)$$\n\nUsing the compound angle identity, this expands to:\n\n$$25\\cos(t + \\alpha) = 25(\\cos t\\cos\\alpha - \\sin t\\sin\\alpha)$$\n\nComparing this to $7\\cos t - 24\\sin t$, we get:\n\n$$25\\cos\\alpha = 7 \\quad \\text{and} \\quad 25\\sin\\alpha = 24$$\n\nDividing these equations gives:\n\n$$\\tan\\alpha = \\frac{24}{7}$$\n\n$$\\alpha = \\arctan\\left(\\frac{24}{7}\\right) \\approx 1.287 \\text{ radians}$$"},{"type":"text","order":3,"title":"Interpret translation and write answer","content":"Now we compare the location of the maximum point:\n\n* For $y = \\cos t$, the maximum occurs at $t = 0$.\n* For $y = 25\\cos(t + 1.287)$, the maximum occurs when $t + 1.287 = 0$, which means $t = -1.287$.\n\nThis represents a horizontal shift to the left by $1.287$ units, which is written as a translation vector:\n\n$$\\begin{pmatrix} -1.287 \\\\ 0 \\end{pmatrix}$$\n\n### Final Answer\n\n* **(i) Stretch:** A vertical stretch in the $y$-direction with scale factor $25$.\n* **(ii) Translation:** A horizontal translation by the vector $\\begin{pmatrix} -1.287 \\\\ 0 \\end{pmatrix}$ (or a shift to the left by $1.287$ units)."}]}],"generatedAt":"2026-06-09T09:41:13.924Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"95447b21-aea2-4c77-bb65-70603a76447a","label":"c","content":"The power efficiency $G$ of a specific component is given by $G(t) = \\frac{2100}{75 + (V(t))^2}$. Find the range of $G$.","markscheme":"- Identifies that the range of $V(t)$ is $[-25, 25]$, so $0 \\le (V(t))^2 \\le 25^2 = 625$ M1\n- Determines the maximum value of $G$ when $V(t)^2 = 0$: $G_{\\text{max}} = \\frac{2100}{75 + 0} = 28$ M1\n- Determines the minimum value of $G$ when $V(t)^2 = 625$: $G_{\\text{min}} = \\frac{2100}{75 + 625} = \\frac{2100}{700} = 3$ M1\n- States the range of $G$ is $3 \\le G(t) \\le 28$ (or $[3, 28]$) A1","marks":4,"order":2,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"analytical","label":"Analytical Approach","steps":[{"type":"text","order":0,"title":"Convert $V(t)$ to $R\\cos(t+\\alpha)$","content":"To find the range of $V(t) = 7\\cos t - 24\\sin t$, we can write it in the form:\n\n$$R\\cos(t + \\alpha) = R\\cos t\\cos\\alpha - R\\sin t\\sin\\alpha$$\n\nBy comparing coefficients with $7\\cos t - 24\\sin t$, we get:\n$$\\begin{aligned} R\\cos\\alpha &= 7 \\\\ R\\sin\\alpha &= 24 \\end{aligned}$$\n\nWe calculate $R$ using the Pythagorean identity:\n$$R = \\sqrt{7^2 + 24^2} = \\sqrt{49 + 576} = \\sqrt{625} = 25$$\n\nThus, the expression can be simplified as:\n$$V(t) = 25\\cos(t + \\alpha)$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$-1 \\le \\cos(t + \\alpha) \\le 1$"},{"latex":"$$\\htmlData{anchor=min-v}{\\color{@red}{-25}} \\le \\htmlData{anchor=v-expr}{\\color{@purple}{V(t)}} \\le \\htmlData{anchor=max-v}{\\color{@red}{25}}$"}],"highlights":[{"color":"red","style":"text-color","anchor":"min-v"},{"color":"red","style":"text-color","anchor":"max-v"}],"connections":[{"to":"v-expr","from":"min-v","color":"red","label":"amplitude","style":"solid"}]},"order":1,"title":"Find the range of $V(t)$","description":"Since the cosine function oscillates between $-1$ and $1$, we multiply all parts of the inequality by $R = 25$ to find the range of $V(t)$."},{"type":"text","order":2,"title":"Determine range of $(V(t))^2$","content":"Next, we find the range of the squared potential difference, $(V(t))^2$.\n\nSince the minimum value of any squared real number is $0$, and the maximum squared value comes from squaring the extreme values of $V(t)$ (i.e., $(\\pm 25)^2 = 625$):\n\n$$0 \\le (V(t))^2 \\le 625$$\n\nThis is a key step that will help us find the extreme values of our rational function $G(t)$ M1."},{"type":"text","order":3,"title":"Find maximum value of $G(t)$","content":"The power efficiency function is given by:\n$$G(t) = \\frac{2100}{75 + (V(t))^2}$$\n\nTo find the maximum value of $G(t)$, we want to make the denominator as small as possible. This occurs when $(V(t))^2 = 0$:\n\n$$G_{\\text{max}} = \\frac{2100}{75 + 0} = \\frac{2100}{75} = 28$$\n\nThis awards the second mark M1."},{"type":"text","order":4,"title":"Find minimum value of $G(t)$","content":"To find the minimum value of $G(t)$, we want to make the denominator as large as possible. This occurs when $(V(t))^2$ is at its maximum value of $625$:\n\n$$G_{\\text{min}} = \\frac{2100}{75 + 625} = \\frac{2100}{700} = 3$$\n\nThis awards the third mark M1."},{"type":"text","order":5,"title":"State the final range","content":"Combining our minimum and maximum values, the range of $G(t)$ is:\n\n$$3 \\le G(t) \\le 28$$\n\nOr in interval notation:\n$$[3, 28]$$\n\nThis complete range statement awards the final mark A1."}]},{"id":"graphical","label":"Graphical Approach","steps":[{"type":"text","order":0,"title":"Define the full function","content":"To find the range of the power efficiency $G(t)$ using a graphical approach, we first substitute the expression for $V(t) = 7\\cos t - 24\\sin t$ directly into the equation for $G(t)$:\n\n$$G(t) = \\frac{2100}{75 + (7\\cos t - 24\\sin t)^2}$$","highlights":[{"text":"$$V(t) = 7\\cos t - 24\\sin t$","location":"specification"},{"text":"$$G(t) = \\frac{2100}{75 + (V(t))^2}$","location":"specification"}]},{"type":"text","order":1,"title":"Plot the graph","content":"Now, we will input this function into a graphing tool or calculator. \n\nWe need to choose a suitable window to view the graph. Since $t$ represents time, we can view a standard period, such as $x \\in [0, 2\\pi]$ or $x \\in [0, 10]$. For the vertical axis, because the numerator of the fraction is $2100$ and the denominator is at least $75$, we can expect values up to around $30$. Let's set $y \\in [0, 35]$.","calculator":[{"gdc":{"expression":"2100 / (75 + (7*cos(x) - 24*sin(x))^2)","instructions":"1. Go to the Graph application.\n2. Input the function \$ f1(x) = \\frac{2100}{75 + (7\\cos(x) - 24\\sin(x))^2} \$.\n3. Adjust the window to \$ [0, 10] \$ for x and \$ [0, 35] \$ for y.\n4. Use the 'Analyze Graph' tool to find the 'Minimum' and 'Maximum' points."},"ti84":{"expression":"2100 / (75 + (7*cos(X) - 24*sin(X))^2)","instructions":"1. Press [Y=] and enter \$ Y_1 = 2100 / (75 + (7 * \\cos(X) - 24 * \\sin(X))^2) \$.\n2. Press [WINDOW] and set: \n Xmin = 0\n Xmax = 10\n Ymin = 0\n Ymax = 35\n3. Press [GRAPH].\n4. Use [2nd] [CALC] and select 'maximum' or 'minimum' to find the peak and trough coordinates."},"desmos":{"expression":"y = 2100 / (75 + (7*cos(x) - 24*sin(x))^2)","instructions":"1. Type the equation: \$ y = \\frac{2100}{75 + (7\\cos(x) - 24\\sin(x))^2} \$.\n2. Click on the curve to display the coordinates of the relative extrema (peaks and troughs)."},"description":"Plot the function and analyze its peaks and troughs."}]},{"type":"text","order":2,"title":"Find maximum and minimum","content":"By analyzing the graph, you will observe a periodic wave that oscillates between a minimum value and a maximum value:\n\n- **Maximum point:** The peak of the curve occurs at coordinates where the $y$-value reaches exactly $28$. M1 M1\n- **Minimum point:** The lowest points (troughs) of the curve reach a $y$-value of exactly $3$. M1\n\nThese coordinates indicate that the output of our function $G(t)$ is bounded between $3$ and $28$."},{"type":"text","order":3,"title":"State the final range","content":"The range of a function consists of all possible output values ($y$-values). \n\nFrom our graphical analysis, the minimum value is $3$ and the maximum value is $28$.\n\nTherefore, the range of $G$ is:\n$$3 \\le G(t) \\le 28$$ \n\n*(Alternatively, written in interval notation as $[3, 28]$)* A1"}]},{"id":"calculus","label":"Calculus Approach","steps":[{"type":"text","order":0,"title":"Define the target function","content":"We want to find the range of the power efficiency function $G(t)$, which is defined in terms of the voltage $V(t)$:\n\n$$G(t) = 2100 \\left(75 + (V(t))^2\\right)^{-1}$$\n\nwhere the voltage function is:\n\n$$V(t) = 7\\cos t - 24\\sin t$$"},{"type":"text","order":1,"title":"Find the derivative of G(t)","content":"To locate the stationary (critical) points of $G(t)$, we apply the chain rule to differentiate $G(t)$ with respect to $t$:\n\n$$G'(t) = -2100 \\left(75 + (V(t))^2\\right)^{-2} \\cdot \\frac{d}{dt}\\left(75 + (V(t))^2\\right)$$\n\nApplying the chain rule again to the inner term:\n\n$$G'(t) = -2100 \\left(75 + (V(t))^2\\right)^{-2} \\cdot 2V(t)V'(t)$$\n\nSimplifying this expression, we obtain:\n\n$$G'(t) = \\frac{-4200 V(t) V'(t)}{\\left(75 + (V(t))^2\\right)^2}$$"},{"type":"text","order":2,"title":"Find the critical points","content":"To find the critical values of $t$, we set the derivative $G'(t) = 0$:\n\n$$\\frac{-4200 V(t) V'(t)}{\\left(75 + (V(t))^2\\right)^2} = 0$$\n\nSince the denominator is always positive (as $(V(t))^2 \\ge 0$, meaning $75 + (V(t))^2 \\ge 75$), this equation is satisfied when the numerator is zero:\n\n$$V(t) V'(t) = 0$$\n\nThis gives us two sets of critical points:\n1. Where $V(t) = 0$\n2. Where $V'(t) = 0$"},{"type":"text","order":3,"title":"Analyze the first case","content":"Let's first evaluate $G(t)$ at the critical points where $V(t) = 0$.\n\nSubstituting $V(t) = 0$ directly into the expression for $G(t)$:\n\n$$G_{\\text{max}} = \\frac{2100}{75 + 0^2} = \\frac{2100}{75} = 28$$\n\nSince the denominator is at its minimum value here, this corresponds to the maximum value of $G(t)$."},{"type":"text","order":4,"title":"Analyze the second case","content":"Now we evaluate $G(t)$ at the critical points where $V'(t) = 0$.\n\nFirst, find $V'(t)$ by differentiating $V(t) = 7\\cos t - 24\\sin t$:\n\n$$V'(t) = -7\\sin t - 24\\cos t$$\n\nSetting $V'(t) = 0$:\n\n$$-7\\sin t - 24\\cos t = 0 \\implies \\tan t = -\\frac{24}{7}$$\n\nWe can find the corresponding values of $V(t)$ at these points. Using the identity $\\sin^2 t + \\cos^2 t = 1$, or by expressing $V(t)$ in the form $R\\cos(t + \\alpha)$ where:\n\n$$R = \\sqrt{7^2 + (-24)^2} = \\sqrt{49 + 576} = \\sqrt{625} = 25$$\n\nAt the stationary points where $V'(t) = 0$, $V(t)$ reaches its maximum or minimum value, which is $V(t) = \\pm 25$.\n\nTherefore, at these points:\n\n$$(V(t))^2 = (\\pm 25)^2 = 625$$"},{"type":"text","order":5,"title":"Determine the minimum value","content":"Substitute $(V(t))^2 = 625$ back into $G(t)$ to find the value of $G(t)$ at these stationary points:\n\n$$G_{\\text{min}} = \\frac{2100}{75 + 625} = \\frac{2100}{700} = 3$$\n\nSince the denominator is at its maximum value here, this corresponds to the minimum value of $G(t)$."},{"type":"text","order":6,"title":"State the final range","content":"Combining the minimum and maximum values we found, the range of the power efficiency function $G(t)$ is:\n\n$$3 \\le G(t) \\le 28$$\n\nor in interval notation, $[3, 28]$."}]}],"generatedAt":"2026-06-09T09:42:21.460Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"75c18bac-a9bb-42c7-95c9-93df569fbe23:compound_angle_expansion","label":"(a) Compound Angle Expansion","steps":[{"type":"text","order":0,"title":"Use compound angle expansion","content":"To express $V(t) = 7\\cos t - 24\\sin t$ in the form $R\\cos(t + \\alpha)$, we expand the target expression using the cosine addition formula:\n\n$$\\cos(A + B) = \\cos A \\cos B - \\sin A \\sin B$$\n\nMultiplying by $R$, we obtain:\n\n$$R\\cos(t + \\alpha) = R\\cos t \\cos \\alpha - R\\sin t \\sin \\alpha$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$7\\cos t - 24\\sin t = \\htmlData{anchor=eq1}{\\color{@sky}{R\\cos\\alpha}}\\cos t - \\htmlData{anchor=eq2}{\\color{@amber}{R\\sin\\alpha}}\\sin t$"},{"text":"By comparing both sides, we equate the coefficients of $\\cos t$ and $\\sin t$:"},{"latex":"$$\\htmlData{anchor=c1}{\\color{@sky}{R\\cos\\alpha}} = 7$"},{"latex":"$$\\htmlData{anchor=c2}{\\color{@amber}{R\\sin\\alpha}} = 24$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"eq1"},{"color":"amber","style":"text-color","anchor":"eq2"}],"connections":[{"to":"c1","from":"eq1","color":"sky","label":"cos coefficient","style":"solid"},{"to":"c2","from":"eq2","color":"amber","label":"sin coefficient","style":"solid"}]},"order":1,"title":"Equate the coefficients","description":"We equate the corresponding terms on both sides of the equation to find relationships for $R$ and $\\alpha$."},{"type":"text","order":2,"title":"Find the value of R","content":"To find $R$, we square and add our two equations:\n\n$$(R\\sin\\alpha)^2 + (R\\cos\\alpha)^2 = 24^2 + 7^2$$\n\n$$R^2(\\sin^2\\alpha + \\cos^2\\alpha) = 576 + 49$$\n\nUsing the fundamental trigonometric identity $\\sin^2\\alpha + \\cos^2\\alpha = 1$, we get:\n\n$$R^2 = 625$$\n\nSince $R > 0$, we find the exact value of $R$:\n\n$$R = \\sqrt{625} = 25$$\n\nThis calculation completes the first part of the markscheme **[B1]**.","highlights":[{"text":"exact value of R","location":"specification"}]},{"type":"text","order":3,"title":"Find the value of alpha","content":"To find $\\alpha$, we divide the sine equation by the cosine equation:\n\n$$\\frac{R\\sin\\alpha}{R\\cos\\alpha} = \\frac{24}{7}$$\n\n$$\\tan\\alpha = \\frac{24}{7}$$\n\nThis division earns **[M1]**.\n\nNow, we take the inverse tangent to solve for $\\alpha$:\n\n$$\\alpha = \\arctan\\left(\\frac{24}{7}\\right)$$\n\nUsing a calculator in radian mode:\n\n$$\\alpha \\approx 1.287\\text{ radians}$$\n\nThis final rounded value completes Part 1 and earns **[A1]**."},{"type":"text","order":4,"title":"State the final expression","content":"Combining our values, we express $V(t)$ in the required form:\n\n$$V(t) = 25\\cos(t + 1.287)$$"},{"type":"text","order":5,"title":"Describe the stretch","content":"For Part 2(i), we describe the transformation that transforms $y = \\cos t$ to include the amplitude stretch.\n\nThe factor $R = 25$ multiplying the function results in a vertical stretch parallel to the $y$-axis with a scale factor of $25$ **[B1]**.","highlights":[{"text":"Describe fully the transformation that is a stretch.","location":"specification"}]},{"type":"text","order":6,"title":"Describe the translation","content":"For Part 2(ii), we describe the horizontal shift caused by the phase angle $\\alpha$.\n\nThe term $(t + 1.287)$ represents a horizontal shift to the left by $1.287$ units, which can be described as a translation by the vector:\n\n$$\\begin{pmatrix} -1.287 \\\\ 0 \\end{pmatrix}$$\n\nThis earns **[B1]**.","highlights":[{"text":"Describe fully the transformation that is a translation.","location":"specification"}]},{"type":"text","order":7,"title":"Find the range of G","content":"For Part 3, we look at the function $G(t) = \\frac{2100}{75 + (V(t))^2}$.\n\nSince $V(t) = 25\\cos(t + 1.287)$, its range is $[-25, 25]$. Therefore, the squared term $(V(t))^2$ is bounded as follows:\n\n$$0 \\le (V(t))^2 \\le 25^2 = 625 \\quad \\textbf{[M1]}$$\n\nWe find the boundaries of $G(t)$ by substituting these extreme values:\n\n- **Maximum value of $G$** occurs when $(V(t))^2 = 0$:\n $$G_{\\text{max}} = \\frac{2100}{75 + 0} = 28 \\quad \\textbf{[M1]}$$\n\n- **Minimum value of $G$** occurs when $(V(t))^2 = 625$:\n $$G_{\\text{min}} = \\frac{2100}{75 + 625} = \\frac{2100}{700} = 3 \\quad \\textbf{[M1]}$$\n\nThus, the range of $G$ is:\n\n$$3 \\le G(t) \\le 28 \\quad \\textbf{[A1]}$$","highlights":[{"text":"Find the range of G.","location":"specification"}]}]},{"id":"7f783460-f9c6-41e6-a365-c68f926c8ddc:harmonic_form","label":"(b) Harmonic Form (R-alpha Method)","steps":[{"type":"text","order":0,"title":"Use the Harmonic Form","content":"To understand the transformations that map $y = \\cos t$ onto $y = V(t)$, we can rewrite $V(t) = 7\\cos t - 24\\sin t$ in the harmonic form:\n\n$$R\\cos(t + \\alpha)$$\n\nUsing the cosine addition formula:\n\n$$R\\cos(t + \\alpha) = R\\cos t\\cos\\alpha - R\\sin t\\sin\\alpha$$\n\nComparing this to $7\\cos t - 24\\sin t$, we can equate the coefficients:\n\n$$\\begin{aligned} R\\cos\\alpha &= 7 \\\\ R\\sin\\alpha &= 24 \\end{aligned}$$","highlights":[{"text":"V(t) = 7\\cos t - 24\\sin t","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$R = \\sqrt{\\htmlData{anchor=r-cos}{\\color{@sky}{7}}^2 + \\htmlData{anchor=r-sin}{\\color{@amber}{24}}^2} = \\sqrt{49 + 576} = \\sqrt{625} = \\htmlData{anchor=r-val}{\\color{@purple}{25}}$"},{"latex":"$$\\tan\\alpha = \\frac{\\htmlData{anchor=tan-sin}{\\color{@amber}{24}}}{\\htmlData{anchor=tan-cos}{\\color{@sky}{7}}} \\implies \\alpha = \\arctan\\left(\\frac{24}{7}\\right) \\approx \\htmlData{anchor=alpha-val}{\\color{@green}{1.287}} \\text{ radians}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"r-val"},{"color":"green","style":"text-color","anchor":"alpha-val"}],"connections":[{"to":"tan-cos","from":"r-cos","color":"sky","label":null,"style":"dashed"},{"to":"tan-sin","from":"r-sin","color":"amber","label":null,"style":"dashed"}]},"order":1,"title":"Calculate R and alpha","description":"We find the amplitude $R$ using the Pythagorean identity and the phase shift $\\alpha$ using trigonometry."},{"type":"text","order":2,"title":"Rewrite V(t)","content":"Now we substitute the values of $R$ and $\\alpha$ back into the harmonic form expression:\n\n$$V(t) = 25\\cos(t + 1.287)$$\n\nThis form directly reveals the two transformations applied to $y = \\cos t$ in sequence."},{"type":"text","order":3,"title":"Describe the stretch","content":"The first transformation is a stretch. Comparing $y = \\cos t$ to $y = 25\\cos(t + 1.287)$:\n\n- The coefficient $25$ outside the function multiplies all $y$-coordinates.\n\nTherefore, this represents a **vertical stretch in the $y$-direction** (or parallel to the $y$-axis) with a **scale factor of $25$**."},{"type":"text","order":4,"title":"Describe the translation","content":"The second transformation is a translation. Comparing $y = \\cos t$ to $y = \\cos(t + 1.287)$:\n\n- The addition of $1.287$ inside the argument of the cosine function shifts the graph horizontally to the left.\n\nTherefore, this represents a **horizontal translation** by the vector **$\\begin{pmatrix} -1.287 \\\\ 0 \\end{pmatrix}$** (or a shift to the left by $1.287$ units)."}]},{"id":"7f783460-f9c6-41e6-a365-c68f926c8ddc:graphical_properties","label":"(b) Graphical and Key Features Analysis","steps":[{"type":"text","order":0,"title":"Analyze original graph features","content":"To understand how the original function $y = \\cos t$ is transformed into $y = V(t)$, let's first identify the key features of the basic cosine wave:\n\n* **Maximum value (amplitude):** $1$\n* **Location of the first maximum:** $t = 0$, where $y = \\cos(0) = 1$\n\nBy comparing these features to the amplitude and horizontal position of the maximum point of $V(t) = 7\\cos t - 24\\sin t$, we can determine the scale factor of the vertical stretch and the shift of the translation."},{"type":"text","order":1,"title":"Calculate the vertical stretch","content":"The amplitude of a trigonometric function represents its maximum vertical distance from the center line. For $V(t) = 7\\cos t - 24\\sin t$, we can calculate its maximum value (amplitude) using the formula:\n\n$$\\text{Amplitude} = \\sqrt{A^2 + B^2}$$\n\nSubstitute the coefficients $A = 7$ and $B = -24$:\n\n$$\\text{Amplitude} = \\sqrt{7^2 + (-24)^2} = \\sqrt{49 + 576} = \\sqrt{625} = 25$$\n\nSince the original curve $y = \\cos t$ has an amplitude of $1$, and the transformed curve has an amplitude of $25$, the function has undergone a **vertical stretch parallel to the $y$-axis with a scale factor of $25$**."},{"type":"text","order":2,"title":"Determine the horizontal translation","content":"Next, let's find how much the graph has shifted horizontally by looking at the location of its maximum point.\n\nWe can express $V(t)$ in the single harmonic form:\n\n$$V(t) = 25\\cos(t + \\alpha)$$\n\nUsing the compound angle identity, this expands to:\n\n$$25\\cos(t + \\alpha) = 25(\\cos t\\cos\\alpha - \\sin t\\sin\\alpha)$$\n\nComparing this to $7\\cos t - 24\\sin t$, we get:\n\n$$25\\cos\\alpha = 7 \\quad \\text{and} \\quad 25\\sin\\alpha = 24$$\n\nDividing these equations gives:\n\n$$\\tan\\alpha = \\frac{24}{7}$$\n\n$$\\alpha = \\arctan\\left(\\frac{24}{7}\\right) \\approx 1.287 \\text{ radians}$$"},{"type":"text","order":3,"title":"Interpret translation and write answer","content":"Now we compare the location of the maximum point:\n\n* For $y = \\cos t$, the maximum occurs at $t = 0$.\n* For $y = 25\\cos(t + 1.287)$, the maximum occurs when $t + 1.287 = 0$, which means $t = -1.287$.\n\nThis represents a horizontal shift to the left by $1.287$ units, which is written as a translation vector:\n\n$$\\begin{pmatrix} -1.287 \\\\ 0 \\end{pmatrix}$$\n\n### Final Answer\n\n* **(i) Stretch:** A vertical stretch in the $y$-direction with scale factor $25$.\n* **(ii) Translation:** A horizontal translation by the vector $\\begin{pmatrix} -1.287 \\\\ 0 \\end{pmatrix}$ (or a shift to the left by $1.287$ units)."}]},{"id":"95447b21-aea2-4c77-bb65-70603a76447a:analytical","label":"(c) Analytical Approach","steps":[{"type":"text","order":0,"title":"Convert $V(t)$ to $R\\cos(t+\\alpha)$","content":"To find the range of $V(t) = 7\\cos t - 24\\sin t$, we can write it in the form:\n\n$$R\\cos(t + \\alpha) = R\\cos t\\cos\\alpha - R\\sin t\\sin\\alpha$$\n\nBy comparing coefficients with $7\\cos t - 24\\sin t$, we get:\n$$\\begin{aligned} R\\cos\\alpha &= 7 \\\\ R\\sin\\alpha &= 24 \\end{aligned}$$\n\nWe calculate $R$ using the Pythagorean identity:\n$$R = \\sqrt{7^2 + 24^2} = \\sqrt{49 + 576} = \\sqrt{625} = 25$$\n\nThus, the expression can be simplified as:\n$$V(t) = 25\\cos(t + \\alpha)$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$-1 \\le \\cos(t + \\alpha) \\le 1$"},{"latex":"$$\\htmlData{anchor=min-v}{\\color{@red}{-25}} \\le \\htmlData{anchor=v-expr}{\\color{@purple}{V(t)}} \\le \\htmlData{anchor=max-v}{\\color{@red}{25}}$"}],"highlights":[{"color":"red","style":"text-color","anchor":"min-v"},{"color":"red","style":"text-color","anchor":"max-v"}],"connections":[{"to":"v-expr","from":"min-v","color":"red","label":"amplitude","style":"solid"}]},"order":1,"title":"Find the range of $V(t)$","description":"Since the cosine function oscillates between $-1$ and $1$, we multiply all parts of the inequality by $R = 25$ to find the range of $V(t)$."},{"type":"text","order":2,"title":"Determine range of $(V(t))^2$","content":"Next, we find the range of the squared potential difference, $(V(t))^2$.\n\nSince the minimum value of any squared real number is $0$, and the maximum squared value comes from squaring the extreme values of $V(t)$ (i.e., $(\\pm 25)^2 = 625$):\n\n$$0 \\le (V(t))^2 \\le 625$$\n\nThis is a key step that will help us find the extreme values of our rational function $G(t)$ M1."},{"type":"text","order":3,"title":"Find maximum value of $G(t)$","content":"The power efficiency function is given by:\n$$G(t) = \\frac{2100}{75 + (V(t))^2}$$\n\nTo find the maximum value of $G(t)$, we want to make the denominator as small as possible. This occurs when $(V(t))^2 = 0$:\n\n$$G_{\\text{max}} = \\frac{2100}{75 + 0} = \\frac{2100}{75} = 28$$\n\nThis awards the second mark M1."},{"type":"text","order":4,"title":"Find minimum value of $G(t)$","content":"To find the minimum value of $G(t)$, we want to make the denominator as large as possible. This occurs when $(V(t))^2$ is at its maximum value of $625$:\n\n$$G_{\\text{min}} = \\frac{2100}{75 + 625} = \\frac{2100}{700} = 3$$\n\nThis awards the third mark M1."},{"type":"text","order":5,"title":"State the final range","content":"Combining our minimum and maximum values, the range of $G(t)$ is:\n\n$$3 \\le G(t) \\le 28$$\n\nOr in interval notation:\n$$[3, 28]$$\n\nThis complete range statement awards the final mark A1."}]},{"id":"95447b21-aea2-4c77-bb65-70603a76447a:graphical","label":"(c) Graphical Approach","steps":[{"type":"text","order":0,"title":"Define the full function","content":"To find the range of the power efficiency $G(t)$ using a graphical approach, we first substitute the expression for $V(t) = 7\\cos t - 24\\sin t$ directly into the equation for $G(t)$:\n\n$$G(t) = \\frac{2100}{75 + (7\\cos t - 24\\sin t)^2}$$","highlights":[{"text":"$$V(t) = 7\\cos t - 24\\sin t$","location":"specification"},{"text":"$$G(t) = \\frac{2100}{75 + (V(t))^2}$","location":"specification"}]},{"type":"text","order":1,"title":"Plot the graph","content":"Now, we will input this function into a graphing tool or calculator. \n\nWe need to choose a suitable window to view the graph. Since $t$ represents time, we can view a standard period, such as $x \\in [0, 2\\pi]$ or $x \\in [0, 10]$. For the vertical axis, because the numerator of the fraction is $2100$ and the denominator is at least $75$, we can expect values up to around $30$. Let's set $y \\in [0, 35]$.","calculator":[{"gdc":{"expression":"2100 / (75 + (7*cos(x) - 24*sin(x))^2)","instructions":"1. Go to the Graph application.\n2. Input the function \$ f1(x) = \\frac{2100}{75 + (7\\cos(x) - 24\\sin(x))^2} \$.\n3. Adjust the window to \$ [0, 10] \$ for x and \$ [0, 35] \$ for y.\n4. Use the 'Analyze Graph' tool to find the 'Minimum' and 'Maximum' points."},"ti84":{"expression":"2100 / (75 + (7*cos(X) - 24*sin(X))^2)","instructions":"1. Press [Y=] and enter \$ Y_1 = 2100 / (75 + (7 * \\cos(X) - 24 * \\sin(X))^2) \$.\n2. Press [WINDOW] and set: \n Xmin = 0\n Xmax = 10\n Ymin = 0\n Ymax = 35\n3. Press [GRAPH].\n4. Use [2nd] [CALC] and select 'maximum' or 'minimum' to find the peak and trough coordinates."},"desmos":{"expression":"y = 2100 / (75 + (7*cos(x) - 24*sin(x))^2)","instructions":"1. Type the equation: \$ y = \\frac{2100}{75 + (7\\cos(x) - 24\\sin(x))^2} \$.\n2. Click on the curve to display the coordinates of the relative extrema (peaks and troughs)."},"description":"Plot the function and analyze its peaks and troughs."}]},{"type":"text","order":2,"title":"Find maximum and minimum","content":"By analyzing the graph, you will observe a periodic wave that oscillates between a minimum value and a maximum value:\n\n- **Maximum point:** The peak of the curve occurs at coordinates where the $y$-value reaches exactly $28$. M1 M1\n- **Minimum point:** The lowest points (troughs) of the curve reach a $y$-value of exactly $3$. M1\n\nThese coordinates indicate that the output of our function $G(t)$ is bounded between $3$ and $28$."},{"type":"text","order":3,"title":"State the final range","content":"The range of a function consists of all possible output values ($y$-values). \n\nFrom our graphical analysis, the minimum value is $3$ and the maximum value is $28$.\n\nTherefore, the range of $G$ is:\n$$3 \\le G(t) \\le 28$$ \n\n*(Alternatively, written in interval notation as $[3, 28]$)* A1"}]},{"id":"95447b21-aea2-4c77-bb65-70603a76447a:calculus","label":"(c) Calculus Approach","steps":[{"type":"text","order":0,"title":"Define the target function","content":"We want to find the range of the power efficiency function $G(t)$, which is defined in terms of the voltage $V(t)$:\n\n$$G(t) = 2100 \\left(75 + (V(t))^2\\right)^{-1}$$\n\nwhere the voltage function is:\n\n$$V(t) = 7\\cos t - 24\\sin t$$"},{"type":"text","order":1,"title":"Find the derivative of G(t)","content":"To locate the stationary (critical) points of $G(t)$, we apply the chain rule to differentiate $G(t)$ with respect to $t$:\n\n$$G'(t) = -2100 \\left(75 + (V(t))^2\\right)^{-2} \\cdot \\frac{d}{dt}\\left(75 + (V(t))^2\\right)$$\n\nApplying the chain rule again to the inner term:\n\n$$G'(t) = -2100 \\left(75 + (V(t))^2\\right)^{-2} \\cdot 2V(t)V'(t)$$\n\nSimplifying this expression, we obtain:\n\n$$G'(t) = \\frac{-4200 V(t) V'(t)}{\\left(75 + (V(t))^2\\right)^2}$$"},{"type":"text","order":2,"title":"Find the critical points","content":"To find the critical values of $t$, we set the derivative $G'(t) = 0$:\n\n$$\\frac{-4200 V(t) V'(t)}{\\left(75 + (V(t))^2\\right)^2} = 0$$\n\nSince the denominator is always positive (as $(V(t))^2 \\ge 0$, meaning $75 + (V(t))^2 \\ge 75$), this equation is satisfied when the numerator is zero:\n\n$$V(t) V'(t) = 0$$\n\nThis gives us two sets of critical points:\n1. Where $V(t) = 0$\n2. Where $V'(t) = 0$"},{"type":"text","order":3,"title":"Analyze the first case","content":"Let's first evaluate $G(t)$ at the critical points where $V(t) = 0$.\n\nSubstituting $V(t) = 0$ directly into the expression for $G(t)$:\n\n$$G_{\\text{max}} = \\frac{2100}{75 + 0^2} = \\frac{2100}{75} = 28$$\n\nSince the denominator is at its minimum value here, this corresponds to the maximum value of $G(t)$."},{"type":"text","order":4,"title":"Analyze the second case","content":"Now we evaluate $G(t)$ at the critical points where $V'(t) = 0$.\n\nFirst, find $V'(t)$ by differentiating $V(t) = 7\\cos t - 24\\sin t$:\n\n$$V'(t) = -7\\sin t - 24\\cos t$$\n\nSetting $V'(t) = 0$:\n\n$$-7\\sin t - 24\\cos t = 0 \\implies \\tan t = -\\frac{24}{7}$$\n\nWe can find the corresponding values of $V(t)$ at these points. Using the identity $\\sin^2 t + \\cos^2 t = 1$, or by expressing $V(t)$ in the form $R\\cos(t + \\alpha)$ where:\n\n$$R = \\sqrt{7^2 + (-24)^2} = \\sqrt{49 + 576} = \\sqrt{625} = 25$$\n\nAt the stationary points where $V'(t) = 0$, $V(t)$ reaches its maximum or minimum value, which is $V(t) = \\pm 25$.\n\nTherefore, at these points:\n\n$$(V(t))^2 = (\\pm 25)^2 = 625$$"},{"type":"text","order":5,"title":"Determine the minimum value","content":"Substitute $(V(t))^2 = 625$ back into $G(t)$ to find the value of $G(t)$ at these stationary points:\n\n$$G_{\\text{min}} = \\frac{2100}{75 + 625} = \\frac{2100}{700} = 3$$\n\nSince the denominator is at its maximum value here, this corresponds to the minimum value of $G(t)$."},{"type":"text","order":6,"title":"State the final range","content":"Combining the minimum and maximum values we found, the range of the power efficiency function $G(t)$ is:\n\n$$3 \\le G(t) \\le 28$$\n\nor in interval notation, $[3, 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Trigonometry (A-level only)","topicId":"30ac4c05-8d09-4ec4-ac0a-b9371d6b1f6e","topicSidebar":"$L3a","topicSlug":"trigonometry-7","topicTitle":"3.5 Trigonometry (A-level only)"}]}],"$L3b","$L3c"]}]

3.5 Trigonometry (A-level only)

Question 127

3.5 Trigonometry (A-level only) Questions

3.5 Trigonometry (A-level only)

Question 127

3.5 Trigonometry (A-level only) Questions