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AP Intermediate 1st Year Probability & Statistics — practice questions

Explore 34 free Probability & Statistics questions in AP Intermediate 1st Year Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Probability & Statistics questions and exercises with answers

Showing questions 1–30 of 34.

$\int \dfrac{1}{x}\,dx$ equalsRead the answer and worked solution →$\int a^{x}\,dx$ (with $a>0,\ a\neq 1$) equalsRead the answer and worked solution →$\int \sec^2 x\,dx$ equalsRead the answer and worked solution →$\int \dfrac{x^3-1}{x^2}\,dx$ equalsRead the answer and worked solution →$\int \operatorname{cosec} x\,(\operatorname{cosec} x+\cot x)\,dx$ equalsRead the answer and worked solution →$\int \dfrac{1-\sin x}{\cos^2 x}\,dx$ equalsRead the answer and worked solution →If $F$ is the antiderivative of $f(x)=4x^3-6$ with $F(0)=3$, then $F(x)$ isRead the answer and worked solution →The antiderivative of $\left(\sqrt{x}+\dfrac{1}{\sqrt{x}}\right)$ equalsRead the answer and worked solution →Using inspection, an antiderivative of $\cos 2x$ isRead the answer and worked solution →$\int 2x\,\sin(x^2+1)\,dx$ equalsRead the answer and worked solution →$\int \sin mx\,dx$ equalsRead the answer and worked solution →$\int \tan x\,dx$ equalsRead the answer and worked solution →$\int \cot x\,dx$ equalsRead the answer and worked solution →$\int \sec x\,dx$ equalsRead the answer and worked solution →$\int \dfrac{2x}{1+x^2}\,dx$ equalsRead the answer and worked solution →$\int \dfrac{(\log x)^2}{x}\,dx$ equalsRead the answer and worked solution →$\int \dfrac{e^{\tan^{-1}x}}{1+x^2}\,dx$ equalsRead the answer and worked solution →$\int \dfrac{10x^9+10^{x}\log_e 10}{x^{10}+10^{x}}\,dx$ equalsRead the answer and worked solution →$\int \dfrac{dx}{\sin^2 x\cos^2 x}$ equalsRead the answer and worked solution →$\int \cos^2 x\,dx$ equalsRead the answer and worked solution →$\int \sin^3 x\,dx$ equalsRead the answer and worked solution →$\int \dfrac{dx}{x^2+a^2}$ equalsRead the answer and worked solution →$\int \dfrac{dx}{x^2-a^2}$ equalsRead the answer and worked solution →$\int \dfrac{dx}{\sqrt{a^2-x^2}}$ equalsRead the answer and worked solution →$\int \dfrac{dx}{x^2-16}$ equalsRead the answer and worked solution →$\int \dfrac{dx}{\sqrt{2x-x^2}}$ equalsRead the answer and worked solution →$\int \dfrac{dx}{x^2-6x+13}$ equalsRead the answer and worked solution →$\int \dfrac{dx}{x^2+2x+2}$ equalsRead the answer and worked solution →$\int \dfrac{dx}{(x+1)(x+2)}$ equalsRead the answer and worked solution →In decomposing $\dfrac{1}{(x+1)(x+2)}=\dfrac{A}{x+1}+\dfrac{B}{x+2}$, the constants areRead the answer and worked solution →

Worked practice questions

$\int \dfrac{1}{x}\,dx$ equals

  1. $-\dfrac{1}{x^2}+C$
  2. $x\log|x|+C$
  3. $\dfrac{x^2}{2}+C$
  4. $\log|x|+C$
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Correct answer: D. $\log|x|+C$

1. The power rule fails for $n=-1$. 2. Recall $\dfrac{d}{dx}(\log|x|)=\dfrac{1}{x}$ for $x\neq 0$. 3. Hence $\int \dfrac{1}{x}\,dx=\log|x|+C$. _Source: syllabus Mathematics Ch 7 "Integrals", p.290_

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$\int a^{x}\,dx$ (with $a>0,\ a\neq 1$) equals

  1. $a^{x}\log a+C$
  2. $x\,a^{x-1}+C$
  3. $\dfrac{a^{x}}{\log a}+C$
  4. $\dfrac{a^{x+1}}{x+1}+C$
Show answer and solution

Correct answer: C. $\dfrac{a^{x}}{\log a}+C$

1. From the standard list, $\dfrac{d}{dx}\left(\dfrac{a^{x}}{\log a}\right)=a^{x}$. 2. Therefore $\int a^{x}\,dx=\dfrac{a^{x}}{\log a}+C$. 3. Option A is the derivative, not the integral. _Source: syllabus Mathematics Ch 7 "Integrals", p.290_

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$\int \sec^2 x\,dx$ equals

  1. $\cot x+C$
  2. $-\cot x+C$
  3. $\sec x+C$
  4. $\tan x+C$
Show answer and solution

Correct answer: D. $\tan x+C$

1. We need a function whose derivative is $\sec^2 x$. 2. Since $\dfrac{d}{dx}(\tan x)=\sec^2 x$, we get $\int \sec^2 x\,dx=\tan x+C$. 3. Note $\int \operatorname{cosec}^2 x\,dx=-\cot x+C$, which is option B, a distractor. _Source: syllabus Mathematics Ch 7 "Integrals", p.289_

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