Home / CIE A level -Pure Mathematics 3 : Topic : 3.4 Differentiation – derivatives of ex, ln x, sin x, cos x, tan x, tan–1 xph : Exam Style Questions Paper 3

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

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