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Mathematics · 2026 · 4 marks
CBSE 2026 · Region 2 · Set 2 · Q37
Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes. The…
Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes.
The equation of one such track is given as follows : \[\mathrm{f}(x)=\left\{\begin{array}{cc} x^{4}-4 x^{2}+4, & 0 \leq x<3 \\ x^{2}+40, & x \geq 3 \end{array}\right. \] Based on given information, answer the following questions :
(i)
Find $\displaystyle \mathrm{f}^{\prime}(x)$ for $\displaystyle 0<x<3$. $\displaystyle 1$
(ii)
Find $\displaystyle \mathrm{f}^{\prime}(4)$ $\displaystyle 1$
(iii)
(a)
Test for continuity of $\displaystyle \mathrm{f}(x)$ at $\displaystyle x=3$.
OR
(b)
Test for differentiability of $\displaystyle \mathrm{f}(x)$ at $\displaystyle x=3$.
Marking-scheme solution
(i)
For $\displaystyle 0<x<3, f(x)=x^{4}-4 x^{2}+4$
$\displaystyle \therefore f^{\prime}(x)=4 x^{3}-8 x$
(ii)
For $\displaystyle x \geq 3, f(x)=x^{2}+40$
$\displaystyle \therefore f^{\prime}(x)=2 x$
Hence, $\displaystyle f^{\prime}(4)=2 \times 4=8$
(iii)
(a)
Here $\displaystyle f(3)=49$
LHL : $\displaystyle \lim\limits_{x \rightarrow 3^{-}}\left(x^{4}-4 x^{2}+4\right)=49$
RHL : $\displaystyle \lim\limits_{x \rightarrow 3^{+}}\left(x^{2}+40\right)=49$
Since $\displaystyle \lim\limits_{x \rightarrow 3^{-}} f(x)=\lim\limits_{x \rightarrow 3^{+}} f(x)=f(3)$, so $\displaystyle f(x)$ is continuous at $\displaystyle x=3$.
OR
(b)
Here $\displaystyle f(3)=49$
getting LHD $\displaystyle =84$
getting RHD $\displaystyle =6$
Since $\displaystyle \mathrm{LHD} \neq \mathrm{RHD}$ at $\displaystyle x=3$, so $\displaystyle f(x)$ is not differentiable at $\displaystyle x=3$.
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CBSE Class 12 Mathematics past-paper question from the 2026 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.