Start learning →

ISC Class 12 Probability & Statistics — practice questions — page 2

Explore 50 free Probability & Statistics questions in ISC Class 12 Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

Practise Probability & Statistics with Quill →

Probability & Statistics questions and exercises with answers

Showing questions 31–50 of 50.

$\int \dfrac{dx}{x^2-a^2}$ 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 →$\int \dfrac{x\,dx}{(x-1)(x-2)}$ equalsRead the answer and worked solution →$\int \dfrac{dx}{x(x^2+1)}$ equalsRead the answer and worked solution →The integration-by-parts formula $\int u\,\dfrac{dv}{dx}\,dx$ equalsRead the answer and worked solution →$\int x\cos x\,dx$ equalsRead the answer and worked solution →$\int \log x\,dx$ equalsRead the answer and worked solution →$\int x e^{x}\,dx$ equalsRead the answer and worked solution →$\int e^{x}\sin x\,dx$ equalsRead the answer and worked solution →$\int e^{x}\big[f(x)+f'(x)\big]\,dx$ equalsRead the answer and worked solution →$\int e^{x}\left(\tan^{-1}x+\dfrac{1}{1+x^2}\right)dx$ equalsRead the answer and worked solution →Higher or Lower: Probability (ISC 12)Read the answer and worked solution →Radians and degreesRead the answer and worked solution →True or false: reading the identitiesRead the answer and worked solution →

Worked practice questions

$\int \dfrac{dx}{x^2-a^2}$ equals

  1. $\dfrac{1}{a}\tan^{-1}\dfrac{x}{a}+C$
  2. $\dfrac{1}{2a}\log\left|\dfrac{a+x}{a-x}\right|+C$
  3. $\dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right|+C$
  4. $\log\left|x+\sqrt{x^2-a^2}\right|+C$
Show answer and solution

Correct answer: C. $\dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right|+C$

1. Factor: $\dfrac{1}{(x-a)(x+a)}=\dfrac{1}{2a}\left[\dfrac{1}{x-a}-\dfrac{1}{x+a}\right]$. 2. Integrate term by term: $\dfrac{1}{2a}[\log|x-a|-\log|x+a|]$. 3. Combine: $\dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right|+C$. 4. Option B is for $\dfrac{1}{a^2-x^2}$ (signs reversed). _Source: syllabus Mathematics Ch 7 "Integrals", p.306_

Open this question →

$\int \dfrac{dx}{\sqrt{a^2-x^2}}$ equals

  1. $\sin^{-1}\dfrac{x}{a}+C$
  2. $\dfrac{1}{a}\tan^{-1}\dfrac{x}{a}+C$
  3. $\log\left|x+\sqrt{a^2-x^2}\right|+C$
  4. $\cos^{-1}\dfrac{x}{a}+C$
Show answer and solution

Correct answer: A. $\sin^{-1}\dfrac{x}{a}+C$

1. Substitute $x=a\sin\theta$, so $dx=a\cos\theta\,d\theta$. 2. $\sqrt{a^2-x^2}=a\cos\theta$. 3. Integral $=\int d\theta=\theta=\sin^{-1}\dfrac{x}{a}+C$. 4. Check: $\dfrac{d}{dx}\sin^{-1}\dfrac{x}{a}=\dfrac{1}{\sqrt{a^2-x^2}}$. _Source: syllabus Mathematics Ch 7 "Integrals", p.307_

Open this question →

$\int \dfrac{dx}{x^2-16}$ equals

  1. $\dfrac{1}{4}\tan^{-1}\dfrac{x}{4}+C$
  2. $\dfrac{1}{8}\log\left|\dfrac{x-4}{x+4}\right|+C$
  3. $\dfrac{1}{8}\log\left|\dfrac{x+4}{x-4}\right|+C$
  4. $\sin^{-1}\dfrac{x}{4}+C$
Show answer and solution

Correct answer: B. $\dfrac{1}{8}\log\left|\dfrac{x-4}{x+4}\right|+C$

1. Here $x^2-16=x^2-4^2$, so $a=4$. 2. Use $\int \dfrac{dx}{x^2-a^2}=\dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right|+C$. 3. With $2a=8$: $\dfrac{1}{8}\log\left|\dfrac{x-4}{x+4}\right|+C$. 4. Option C inverts the fraction (wrong sign). _Source: syllabus Mathematics Ch 7 "Integrals", p.310_

Open this question →

How to practise

  1. Choose a Probability & Statistics question and try to work out your answer.
  2. Open the question to check the correct answer and read its worked solution.
  3. Continue in the app for bite-sized lessons, explanations and review with Quill. You can start without creating an account.

Put it into practice

Continue with Probability & Statistics practice for ISC Class 12. Read explanations as you go. No signup form needed to start.

Practise Probability & Statistics →

More Mathematics topics

Vector Algebra and 3D GeometryContinuity, Differentiability & DerivativesIntegrals and ApplicationsMatrices and DeterminantsRelations, Functions & Inverse TrigonometryDifferential EquationsLinear Programming