Section II Part B — Full Practice Exam 1

Questions in this quiz (1)

1. (A) The functions $$g$$ and $$h$$ are given by$$g(x)=2\ln(x^7)-\left(3\ln x+\ln(8x^4)\right)$$$$h(x)=\frac{\sin 2x}{1-\cos^2 x}$$(i) Rewrite $$g(x)$$ as a single natural logarithm without negative exponents in any part of the expression. Your result should be of the form $$\ln(\text{expression})$$.(ii) Rewrite $$h(x)$$ as an expression in which $$\cot x$$ appears once and no other trigonometric functions are involved.(B) The functions $$j$$ and $$k$$ are given by$$j(x)=\tan^2 x-1$$ $$k(x)=5e^{x-2}$$(i) Solve $$j(x)=0$$ for all values of $$x$$ in the interval $$[0,\pi]$$.(ii) The solution to the equation $$k(x)=\frac{1}{e^{-3}}$$ for $$x$$ can be written in the form $$x=a-\ln a$$. State the value of $$a$$.(C) The function $$m$$ is given by$$m(x)=\cos 2x+3\sin x$$Find all input values in the domain of $$m$$ that yield an output value of $$-1$$.

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