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MP Board Class 12 (MPBSE) Calculus — practice questions

Explore 42 free Calculus questions in MP Board Class 12 (MPBSE) Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Calculus questions and exercises with answers

Showing questions 1–30 of 42.

The integrating factor of the linear differential equation $\dfrac{dy}{dx} + \dfrac{y}{x} = x^2$ is:Read the answer and worked solution →To solve the homogeneous differential equation $\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{xy}$, the appropriate substitution is:Read the answer and worked solution →The number of arbitrary constants in the general solution of a differential equation of order $n$ is:Read the answer and worked solution →$\displaystyle\int x^n\,dx$ (where $n \ne -1$) equals:Read the answer and worked solution →$\displaystyle\int \dfrac{1}{1+x^2}\,dx$ equals:Read the answer and worked solution →$\displaystyle\int \dfrac{1}{\sqrt{1 - x^2}}\,dx$ equals:Read the answer and worked solution →$\displaystyle\int \sec^2 x\,dx$ equals:Read the answer and worked solution →Using integration by parts, $\displaystyle\int x e^x\,dx$ equals:Read the answer and worked solution →$\displaystyle\int \log x\,dx$ equals:Read the answer and worked solution →$\displaystyle\int e^x\bigl(f(x) + f'(x)\bigr)\,dx$ equals:Read the answer and worked solution →Using the substitution $t = 1 + x^2$, $\displaystyle\int \dfrac{x}{1 + x^2}\,dx$ equals:Read the answer and worked solution →Using partial fractions, $\displaystyle\int \dfrac{1}{(x-1)(x-2)}\,dx$ equals:Read the answer and worked solution →$\displaystyle\int \dfrac{1}{x^2 - a^2}\,dx$ (with $|x| > a > 0$) equals:Read the answer and worked solution →Sort each inverse function into its domainRead the answer and worked solution →Tap every function whose domain is [−1, 1]Read the answer and worked solution →Match each inverse function to its domainRead the answer and worked solution →Match each inverse function to its principal value branchRead the answer and worked solution →Flip and pair: function with its principal rangeRead the answer and worked solution →Which principal ranges sit around zero, and which run 0 to π?Read the answer and worked solution →Tap every function whose principal range has a point removedRead the answer and worked solution →Name the principal value branchRead the answer and worked solution →Domain checkRead the answer and worked solution →Put the reasoning in order: why an inverse needs a restricted domainRead the answer and worked solution →True or false: existence of inversesRead the answer and worked solution →True or false: reading the tableRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Complete the summary tableRead the answer and worked solution →Complete the rule about restricting a domainRead the answer and worked solution →

Worked practice questions

The integrating factor of the linear differential equation $\dfrac{dy}{dx} + \dfrac{y}{x} = x^2$ is:

  1. $x$
  2. $\log x$
  3. $x^2$
  4. $e^x$
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Correct answer: A. $x$

Comparing with $\dfrac{dy}{dx} + P y = Q$ gives $P = 1/x$. **Hence $\text{IF} = e^{\int P\,dx} = e^{\int (1/x)\,dx} = e^{\log x} = x$.**

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To solve the homogeneous differential equation $\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{xy}$, the appropriate substitution is:

  1. $y - x = v$
  2. $x + y = v$
  3. $xy = v$
  4. $y = vx$
Show answer and solution

Correct answer: D. $y = vx$

The right side is a homogeneous function of degree zero in $x$ and $y$, so the equation has the form $dy/dx = F(y/x)$. Substituting $y = vx$ converts it into a variables-separable equation in $v$ and $x$ via $\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}$.

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The number of arbitrary constants in the general solution of a differential equation of order $n$ is:

  1. $n$
  2. $0$
  3. $1$
  4. $n - 1$
Show answer and solution

Correct answer: A. $n$

By definition, the general solution of an $n$-th order differential equation contains exactly $n$ independent arbitrary constants, matching the order of the equation. For example, $\dfrac{d^2y}{dx^2} + y = 0$ has the general solution $y = A\cos x + B\sin x$ with two constants $A$ and $B$, since the equation is of order 2.

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