Question
The diagram shows the curve\( y = 5sin^{2}xcox^{3}x \) for\( 0 ≤ x ≤\frac{1}{2}\pi \) , and its maximum point M. The shaded region R is bounded by the curve and the x-axis.
(i) Find the x-coordinate of M, giving your answer correct to 3 decimal places. [5]
(ii) Using the substitution u = sin x and showing all necessary working, find the exact area of R. [4]
Answer/Explanation
(i) Use product rule
Obtain correct derivative in any form Equate derivative to zero and obtain an equation in a single trig function
Obtain a correct equation, e.g.
Obtain answer x = 0.685
(ii) Use the given substitution and reach\( a\int (u^{2}-u^{4})du\) Obtain correct integral with a = 5 and limits 0 and 1
Use correct limits in an integral of the form \(a\left ( \frac{1}{3} u^{3}-\frac{1}{5}u^{5}\right )\)
Obtain answer \(\frac{2}{3}\)
Question
The parametric equations of a curve are
x = 2 sin 1 + sin 21, y = 2 cos 1 + cos 21, where 0 < 1 < 0.
(i) Obtain an expression for\(\frac{dy}{dx} \) in terms of \(\Theta \)
(ii) Hence find the exact coordinates of the point on the curve at which the tangent is parallel to the
y-axis.
Answer/Explanation
4(i) \(obtain \frac{dx}{d\Theta }=2cos\Theta +2cos2\Theta or \frac{dy}{d\Theta }=-2sin\Theta -2sin2\Theta\)
use\( dy/dx=dy/d\Theta \div dx/d\Theta\)
Obtain correct\( \frac{dy}{dx}e.g.-\frac{2sin\Theta +2sin2\Theta }{2cos\Theta +2cos2\Theta }\)
4(ii) Equate denominator to zero and use any correct double angle formula
Obtain correct 3-term quadratic in cosθ in any form Solve forθ
Obtain\( x = \sqrt[3]{3} and y=\frac{1}{2} y = 1\) 2 , or exact equivalents
Question
The parametric equations of a curve are
x = 2t + sin 2t, y = ln(1 − cos 2t). Show that\( \frac{dy}{dx}\)= cosec 2t
Answer/Explanation
.State \(\frac{dx}{dt}=2+2 cos2t\) Use the chain rule to find the derivative of y
Obtain \(\frac{dy}{dt}=\frac{2sin2t}{1-cos2t}\)
Use \(\frac{dy}{dx}=\frac{dy}{dt}\div \frac{dx}{dt}\)
Obtain \(\frac{dy}{dx}\) t x
= correctly
Question
Find the gradient of the curve \(x^{3}+3xy^{2}-y^{3}\)=1
at the point with coordinates(1, 3).
Answer/Explanation
State or imply \(3y^{2}+6xy\frac{\mathrm{d} y}{\mathrm{d} x}\) as derivative of \(3xy^{2}\)
State or imply \(3y^{2}\frac{\mathrm{d} y}{\mathrm{d} x}\) as derivative of \(y^{3}\)
Equate derivative of LHS to zero ,substitute(1,3) and find the gradient
Obtain final answer \(\frac{10}{3}\) or equivalent