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Calculus

Each subtopic includes About section, revision page link, 10 preview questions, and practice CTAs.

Continuity and Differentiability

Subtopic

Continuity and Differentiability under Calculus for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    What is the value of lim⁡x→0ex−1x\lim_{x \to 0} \frac{e^x - 1}{x}?

    A.

    0

    B.

    ∞\infty

    C.

    1

    D.

    ee

  2. 2.

    Find ddx(x2)\frac{d}{dx}(\frac{x}{2}).

    A.

    12\frac{1}{2}

    B.

    1

    C.

    2

    D.

    xx

  3. 3.

    The derivative of ln⁡(x3)\ln(x^3) is:

    A.

    3x\frac{3}{x}

    B.

    1x3\frac{1}{x^3}

    C.

    3x23x^2

    D.

    3x2\frac{3}{x^2}

Download the worksheet for Calculus - Continuity and Differentiability to practice offline. It includes additional chapter-level practice questions.

Derivatives of Composite, Implicit, Inverse Trigonometric, Exponential and Logarithmic Functions

Subtopic

Derivatives of Composite, Implicit, Inverse Trigonometric, Exponential and Logarithmic Functions under Calculus for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Find dydx\frac{dy}{dx} for y=tan⁡−1(x2)y = \tan^{-1}(x^2).

    A.

    2x1+x4\frac{2x}{1 + x^4}

    B.

    11+x4\frac{1}{1 + x^4}

    C.

    2x1+x2\frac{2x}{1 + x^2}

    D.

    x21+x4\frac{x^2}{1 + x^4}

  2. 2.

    If y=sec⁡(x2)y = \sec(x^2), then dydx\frac{dy}{dx} is:

    A.

    sec⁡(x2)tan⁡(x2)\sec(x^2) \tan(x^2)

    B.

    2xsec⁡(x2)tan⁡(x2)2x \sec(x^2) \tan(x^2)

    C.

    2xsec⁡(x2)2x \sec(x^2)

    D.

    x2sec⁡(x2)tan⁡(x2)x^2 \sec(x^2) \tan(x^2)

  3. 3.

    Find the derivative of y=(1−x2)4y = (1 - x^2)^4.

    A.

    4(1−x2)34(1 - x^2)^3

    B.

    −8x(1−x2)3-8x(1 - x^2)^3

    C.

    8x(1−x2)38x(1 - x^2)^3

    D.

    −2x(1−x2)3-2x(1 - x^2)^3

Download the worksheet for Calculus - Derivatives of Composite, Implicit, Inverse Trigonometric, Exponential and Logarithmic Functions to practice offline. It includes additional chapter-level practice questions.

Logarithmic and Parametric Differentiation

Subtopic

Logarithmic and Parametric Differentiation under Calculus for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    If x=2tx = 2t and y=2/ty = 2/t, then the value of dydx\frac{dy}{dx} is:

    A.

    1/t21/t^2

    B.

    −1/t2-1/t^2

    C.

    −4/t2-4/t^2

    D.

    −2/t2-2/t^2

  2. 2.

    If y=log⁡(ex⋅x2)y = \log(e^x \cdot x^2), find dydx\frac{dy}{dx}.

    A.

    ex+2xe^x + 2x

    B.

    x+2log⁡xx + 2\log x

    C.

    1exx2\frac{1}{e^x x^2}

    D.

    1+2x1 + \frac{2}{x}

  3. 3.

    If x=acos⁡tx = a \cos t and y=bcos⁡ty = b \cos t, then find dydx\frac{dy}{dx}.

    A.

    −b/a-b/a

    B.

    a/ba/b

    C.

    b/ab/a

    D.

    −a/b-a/b

Download the worksheet for Calculus - Logarithmic and Parametric Differentiation to practice offline. It includes additional chapter-level practice questions.

Second Order Derivatives

Subtopic

Second Order Derivatives under Calculus for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Find the second order derivative of y=log⁡(ax+b)y = \log(ax+b).

    A.

    aax+b\frac{a}{ax+b}

    B.

    −a2(ax+b)2-\frac{a^2}{(ax+b)^2}

    C.

    a2(ax+b)2\frac{a^2}{(ax+b)^2}

    D.

    −a(ax+b)2-\frac{a}{(ax+b)^2}

  2. 2.

    If y=excos⁡xy = e^x \cos x, then d2ydx2\frac{d^2y}{dx^2} is:

    A.

    −2exsin⁡x-2e^x \sin x

    B.

    2excos⁡x2e^x \cos x

    C.

    −2excos⁡x-2e^x \cos x

    D.

    ex(cos⁡x−sin⁡x)e^x (\cos x - \sin x)

  3. 3.

    If y=exsin⁡xy = e^x \sin x, then y′′y'' is:

    A.

    ex(sin⁡x+cos⁡x)e^x (\sin x + \cos x)

    B.

    2exsin⁡x2e^x \sin x

    C.

    2excos⁡x−exsin⁡x2e^x \cos x - e^x \sin x

    D.

    2excos⁡x2e^x \cos x

Download the worksheet for Calculus - Second Order Derivatives to practice offline. It includes additional chapter-level practice questions.

Applications of Derivatives: Rate of Change, Increasing/Decreasing Functions, Tangents and Normals, Maxima and Minima

Subtopic

Applications of Derivatives: Rate of Change, Increasing/Decreasing Functions, Tangents and Normals, Maxima and Minima under Calculus for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    If y=2x3y = 2x^3, what is the rate of change of yy with respect to xx when x=1x = 1?

    A.

    22

    B.

    33

    C.

    66

    D.

    11

  2. 2.

    Find the maximum value of f(x)=−(x−1)2+3f(x) = -(x-1)^2 + 3.

    A.

    11

    B.

    33

    C.

    −1-1

    D.

    00

  3. 3.

    What is the slope of the normal to the curve y=x+1y = \sqrt{x+1} at x=3x = 3?

    A.

    1/41/4

    B.

    −1/4-1/4

    C.

    44

    D.

    −4-4

Download the worksheet for Calculus - Applications of Derivatives: Rate of Change, Increasing/Decreasing Functions, Tangents and Normals, Maxima and Minima to practice offline. It includes additional chapter-level practice questions.

Indefinite Integrals: Integration by Substitution, by Parts, and by Partial Fractions

Subtopic

Indefinite Integrals: Integration by Substitution, by Parts, and by Partial Fractions under Calculus for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Find ∫ex/3dx\int e^{x/3} dx.

    A.

    13ex/3+C\frac{1}{3}e^{x/3} + C

    B.

    3ex/3+C3e^{x/3} + C

    C.

    ex/3+Ce^{x/3} + C

    D.

    e3x+Ce^{3x} + C

  2. 2.

    Evaluate ∫cos⁡(x/2)dx\int \cos(x/2) dx.

    A.

    12sin⁡(x/2)+C\frac{1}{2}\sin(x/2) + C

    B.

    2sin⁡(x/2)+C2\sin(x/2) + C

    C.

    −2sin⁡(x/2)+C-2\sin(x/2) + C

    D.

    sin⁡(x/2)+C\sin(x/2) + C

  3. 3.

    Using partial fractions, evaluate ∫dx(x−3)(x−4)\int \frac{dx}{(x-3)(x-4)}.

    A.

    ln⁡∣x−3x−4∣+C\ln|\frac{x-3}{x-4}| + C

    B.

    ln⁡∣x−4x−3∣+C\ln|\frac{x-4}{x-3}| + C

    C.

    ln⁡∣(x−3)(x−4)∣+C\ln|(x-3)(x-4)| + C

    D.

    1x−4−1x−3+C\frac{1}{x-4} - \frac{1}{x-3} + C

Download the worksheet for Calculus - Indefinite Integrals: Integration by Substitution, by Parts, and by Partial Fractions to practice offline. It includes additional chapter-level practice questions.

Definite Integrals and their Properties

Subtopic

Definite Integrals and their Properties under Calculus for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Evaluate the definite integral ∫0πcos⁡(x2)dx\int_{0}^{\pi} \cos(\frac{x}{2}) dx.

    A.

    11

    B.

    00

    C.

    22

    D.

    −2-2

  2. 2.

    Find the value of ∫12exdx\int_{1}^{2} e^{x} dx.

    A.

    e2e^{2}

    B.

    e2−ee^{2} - e

    C.

    e−e2e - e^{2}

    D.

    ee

  3. 3.

    Evaluate ∫01x1/3dx\int_{0}^{1} x^{1/3} dx.

    A.

    34\frac{3}{4}

    B.

    14\frac{1}{4}

    C.

    23\frac{2}{3}

    D.

    11

Download the worksheet for Calculus - Definite Integrals and their Properties to practice offline. It includes additional chapter-level practice questions.

Fundamental Theorem of Calculus

Subtopic

Fundamental Theorem of Calculus under Calculus for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Find the value of ∫01(1−x2)dx\int_0^1 (1 - x^2) dx.

    A.

    13\frac{1}{3}

    B.

    23\frac{2}{3}

    C.

    1

    D.

    0

  2. 2.

    Evaluate the integral ∫142xdx\int_1^4 \frac{2}{\sqrt{x}} dx.

    A.

    2

    B.

    8

    C.

    4

    D.

    6

  3. 3.

    Calculate ddx∫x5cos⁡tdt\frac{d}{dx} \int_x^5 \cos t dt.

    A.

    sin⁡x\sin x

    B.

    cos⁡x\cos x

    C.

    −sin⁡x-\sin x

    D.

    −cos⁡x-\cos x

Download the worksheet for Calculus - Fundamental Theorem of Calculus to practice offline. It includes additional chapter-level practice questions.

Differential Equations: Order, Degree, General and Particular Solutions

Subtopic

Differential Equations: Order, Degree, General and Particular Solutions under Calculus for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    The number of arbitrary constants in a particular solution of a second-order differential equation is:

    A.

    2

    B.

    1

    C.

    0

    D.

    None of these

  2. 2.

    What is the degree of d2ydx2=1+dydx3\frac{d^2y}{dx^2} = \sqrt[3]{1 + \frac{dy}{dx}}?

    A.

    1

    B.

    2

    C.

    3

    D.

    1/3

  3. 3.

    The order of the differential equation y=c1+c2cos⁡x+c3sin⁡xy = c_1 + c_2 \cos x + c_3 \sin x is:

    A.

    1

    B.

    2

    C.

    3

    D.

    4

Download the worksheet for Calculus - Differential Equations: Order, Degree, General and Particular Solutions to practice offline. It includes additional chapter-level practice questions.

Solving Differential Equations: Separation of Variables, Homogeneous, Linear form

Subtopic

Solving Differential Equations: Separation of Variables, Homogeneous, Linear form under Calculus for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Which of the following is a homogeneous differential equation?

    A.

    dydx=x+yx2\frac{dy}{dx} = \frac{x+y}{x^2}

    B.

    dydx=x2+y2xy\frac{dy}{dx} = \frac{x^2+y^2}{xy}

    C.

    dydx=x2+y\frac{dy}{dx} = x^2 + y

    D.

    dydx=sin⁡x+y\frac{dy}{dx} = \sin x + y

  2. 2.

    Find the Integrating Factor for dydx−2yx=x\frac{dy}{dx} - \frac{2y}{x} = x.

    A.

    1x2\frac{1}{x^2}

    B.

    x2x^2

    C.

    e−2xe^{-2x}

    D.

    ln⁡(x−2)\ln(x^{-2})

  3. 3.

    What is the general solution of dydx=2x+1\frac{dy}{dx} = 2x + 1?

    A.

    y=x2+Cy = x^2 + C

    B.

    y=2x2+x+Cy = 2x^2 + x + C

    C.

    y=x2+2x+Cy = x^2 + 2x + C

    D.

    y=x2+x+Cy = x^2 + x + C

Download the worksheet for Calculus - Solving Differential Equations: Separation of Variables, Homogeneous, Linear form to practice offline. It includes additional chapter-level practice questions.

Calculus - ICSE Class 12 Maths Notes & Revision | Krit.club