Q.If $y = ae^{nx} + be^{-nx}$, then prove that $y'' = n^2 y$.

Showing the 12 most recent of 62 on this concept.

1. CBSE 2026Set ANNUAL1 markMCQ
   Q.Derivative of $\cot x$ is

   (a) $\sec^2 x$

   (b) $-\csc^2 x$

   (c) $\csc^2 x$

   (d) $\csc x \cot x$

   ›Reveal solutionSolution

   This is a standard trigonometric derivative, derivable from $\cot x = \cos x/\sin x$ via the quotient rule.

   Write $\cot x = \dfrac{\cos x}{\sin x}$. By the quotient rule: …

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2. CBSE 2026Set ANNUAL1 mark
   Q.$\frac{d}{dx}(x+a) =$ ______.

   ›Reveal solutionSolution

   $\frac{d}{dx}(x+a) = 1$.

   Using the sum rule of differentiation: $\frac{d}{dx}(x+a) = \frac{d}{dx}(x) + \frac{d}{dx}(a)$.

   …

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3. CBSE 2026Set ANNUAL1 mark
   Q.Find: d/dx (sin x cos x + 5x + 3)

   ›Reveal solutionSolution

   Rewrite sin x cos x as (1/2)sin 2x before differentiating, then differentiate term by term.

   Use the identity $\sin x\cos x = \dfrac12\sin 2x$:

   $f(x) = \frac12\sin2x + 5x + 3$

   Differentiate term by term:

   $f'(x) = \frac12\cdot2\cos2x + 5 + 0 = \cos2x + 5$

   …

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4. CBSE 2026Set ANNUAL1 markMCQ
   Q.The derivative of $\tan x$ w.r.t. 'x' is:

   (a) $\sec x$

   (b) $\sec^3 x$

   (c) $\sec^2 x$

   (d) $-\sec x$

   ›Reveal solutionSolution

   Differentiating $\tan x = \dfrac{\sin x}{\cos x}$ using the quotient rule gives $\sec^2 x$.

   Write $\tan x = \dfrac{\sin x}{\cos x}$. By the quotient rule, …

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5. CBSE 2026Set ANNUAL1 markMCQ
   Q.Assertion (A): The derivative of a constant function is zero. Reason (R): A constant function does not change, hence its rate of change is zero.

   (a) Both Assertion (A) and Reason (R) are correct and Reason (R) is the correct explanation of Assertion (A).

   (b) Both Assertion (A) and Reason (R) are correct, but Reason (R) is not the correct explanation of Assertion (A).

   (c) Assertion (A) is correct, but Reason (R) is incorrect.

   (d) Assertion (A) is incorrect, but Reason (R) is correct.

   ›Reveal solutionSolution

   Both statements are true, and (R) correctly explains why (A) holds: a constant function's value never changes, so its derivative (rate of change) is zero.

   Assertion (A): For $f(x)=c$ (a constant), $f'(x) = \lim_{h\to0}\dfrac{f(x+h)-f(x)}{h} = \lim_{h\to0}\dfrac{c-c}{h} = \lim_{h\to0} 0 = 0$. So (A) is TRUE.

   …

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6. CBSE 2026Set ANNUAL1 markMCQ
   Q.Match the columns — Column A: $\dfrac{d}{dx}\sin x$. Choose its correct equivalent from Column B.

   (a) $\dfrac{1-\tan^2 x}{1+\tan^2 x}$

   (b) $\dfrac{2\tan x}{1-\tan^2 x}$

   (c) $\dfrac{2\tan x}{1+\tan^2 x}$

   (d) $-\sin x$

   (e) $\sin x$

   (f) $-\cos x$

   (g) $\cos x$

   ›Reveal solutionSolution

   $\dfrac{d}{dx}\sin x = \cos x$, a standard derivative result.

   …

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7. CBSE 2026Set ANNUAL1 mark
   Q.Write derivative of $\cos x$.

   ›Reveal solutionSolution

   $\dfrac{d}{dx}\cos x = -\sin x$, a standard derivative result.

   By first principles, $\dfrac{d}{dx}\cos x = \lim_{h\to0}\dfrac{\cos(x+h)-\cos x}{h}$. Using $\cos(x+h)-\cos x = -2\sin\left(x+\tfrac{h}{2}\right)\sin\tfrac{h}{2}$, …

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8. CBSE 2026Set ANNUAL1 markMCQ
   Q.if $y = \log\left(\frac{e^x}{x^2}\right)$ then $\frac{dy}{dx}$ = ______.

   (a) $\frac{2 - x}{x}$

   (b) $\frac{x - 2}{x}$

   (c) $\frac{e - x}{e^x}$

   (d) $\frac{x - e}{ex}$

   ›Reveal solutionSolution

   Expand $\log\!\left(\dfrac{e^x}{x^2}\right)$ with log rules to get $x - 2\log x$, then differentiate; the answer is $\dfrac{x-2}{x}$ — option (ii).

   Simplify using $\log\dfrac{A}{B} = \log A - \log B$ and $\log e^x = x$:

   $y = \log\!\left(\frac{e^x}{x^2}\right) = \log e^x - \log x^2 = x - 2\log x.$

   …

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9. CBSE 2026Set ANNUAL1 mark
   Q.If $y = e^{ax}$, then $x \cdot \frac{dy}{dx}$ = ______.

   ›Reveal solutionSolution

   $\dfrac{dy}{dx}=a e^{ax}$, so $x\cdot\dfrac{dy}{dx}=ax\,e^{ax}=axy$.

   Differentiate $y = e^{ax}$ using $\dfrac{d}{dx}(e^{ax}) = a e^{ax}$:

   $\frac{dy}{dx} = a e^{ax}.$

   Multiply by $x$:

   $x\cdot\frac{dy}{dx} = x\cdot a e^{ax} = ax\,e^{ax}.$

   …

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10. CBSE 2026Set MARCH1 markMCQ
    Q.If $y = ax + b$, $a$ and $b$ are constant then what will be $\dfrac{dy}{dx}$?

    (a) $a$

    (b) $b$

    (c) $a + b$

    (d) $0$

    ›Reveal solutionSolution

    $\dfrac{d}{dx}(ax+b)=a$.

    Using the sum and constant-multiple rules, $\dfrac{d}{dx}(ax)=a$ and $\dfrac{d}{dx}(b)=0$ (derivative of a cons …

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11. CBSE 2025Set ANNUAL1 markMCQ
    Q.$\dfrac{d}{dx}(x^2 + 25x) =$

    (a) $2x$

    (b) 25

    (c) $2x + 25$

    (d) 0

    ›Reveal solutionSolution

    $\dfrac{d}{dx}(x^2+25x)=2x+25$.

    …

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12. CBSE 2025Set ANNUAL1 markMCQ
    Q.$\dfrac{d}{dx}(\tan x) =$

    (a) $\sec x$

    (b) $\cot x$

    (c) $\cot^2 x$

    (d) $\sec^2 x$

    ›Reveal solutionSolution

    $\dfrac{d}{dx}(\tan x)=\sec^2x$.

    Write $\tan x=\dfrac{\sin x}{\cos x}$ and apply the quotient rule: …

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