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## STPM 2016 Mathematics (T) Term 3 Revision 1 Sample Solution

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June 2014/1

## STPM 2015 MT Chapter 8 Booklet A Page 20 Problem 11 Question 2

Given that $y=\sqrt{\cos{x}}$, show that $4y^3\frac{d^2y}{dx^2}+y^4+1=$0.

## STPM 2015 MT Chapter 8 Booklet A Page 20 Problem 11 Question 14

Given that $y=\sec^{-1}(e^x)$, prove that $\frac{dy}{dx}=\frac{1}{\sqrt{e^{2x}-1}}$.

## 2015 MT Chapter 8 Booklet A Page 20 Problem 11 Question 11

Given that $y=(\sin^{-1}x)^2$, show that $(1-x^2)\frac{d^2y}{dx^2}-x\frac{dy}{dx}-2=$0.

## 2015 MT Chapter 8 Booklet A Page 20 Question 6

Given that $y=\ln(1+x)-x+\frac{1}{2}x^2$ show that $\frac{dy}{dx}>=0$ for all values of $x>-1$.

## 2015 MT Chapter 8 Booklet A Page 20 Question 10

If $\sin y = 2 \sin x$, show that (a) $(\frac{dy}{dx})^2=1+3\sec^2 y$, (b) by differentiating (a) with respect to $x$, show that $\frac{d^2y}{dx^2}=3\sec^2 y \tan y$, and hence that $\cot y \frac{d^2y}{dx^2}-(\frac{dy}{dx})^2+1=$ $0$.

## 2015 MT Chapter 8 Booklet A Page 20 Question 8

Given $y=\ln(\sin px + \cos px)$. Prove that $\frac{d^2y}{dx^2}+(\frac{dy}{dx})^2+p^2=0$.

## MT Chapter 15 Booklet Page 23 Question 2

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