Continuity & Differentiability: JEE Main Mathematics Previous Year Questions

371 previous year questions from Continuity & Differentiability. Here are the 25 most important ones; open any question for the correct answer and a full step-by-step solution.

1. 1.If $f\text{(}x\text{)}=\frac{e^{x^2}-\text{cos}\text{⁡}x}{x^2}$, for $x\ne 0$ is continuous at $x=0$, then value of $f\text{(}0\text{)}$ is
2. 2.$f\text{(}x\text{)}=x^x$ has stationary point at
3. 3.If ${\text{sin}}^{-1}\text{⁡}a$ is the acute angle between the curves $x^2+y^2=4x$ and $x^2+y^2=8$ at $\text{(}2\text{,}2\text{)}$, then $a=$
4. 4.The derivative of ${\text{tan}}^{-1}\left(\frac{\text{sin}x-\text{cos}x}{\text{sin}x+\text{cos}x}\right)$, with respect to $\frac{x}{2}$, where…
5. 5.If $f\text{(}x\text{)}=x^2-2x+4$, then the set of values of $x$ satisfying $f\text{(}x-1\text{)}=f\text{(}x+1\text{)}$ is
6. 6.The function $f\text{(}x\text{)}=x^2+\frac{54}{x}$
7. 7.Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is :
8. 8.If the function $f\text{(}x\text{)}=x\text{(}x+3\text{)}{e}^{\frac{-x}{2}}$ satisfies all the conditions of Rolle's theorem in $\text{[}-3\text{,}0\text{]}$, then a roo…
9. 9.If $y={\left({\text{sin}}^{-1}\text{⁡}x\right)}^2+{\left({\text{cos}}^{-1}\text{⁡}x\right)}^2$, then $(1-x^2)y_2-x{y}_1=$
10. 10.Let $a$ and $b$ be real constants such that the function $f$ defined by $f\text{(}x\text{)}=(x^2+3x+a\text{,}x\le 1\text{bx}+2\text{,}x>1)$ be differentiable on…
11. 11.If…
12. 12.If $f\text{(}x\text{)}=(\frac{\text{log}\text{⁡}x}{x-1}\text{,}\text{if}x\ne 1k\text{,}\text{if}x=1)$ is continuous at $x=1$, then the value of $k$ is
13. 13.The value of $c$ of Lagrange's mean value theorem for $f\text{(}x\text{)}=\sqrt{25-x^2}$ on $\text{[}1\text{,}5\text{]}$ is
14. 14.If ${\text{log}}_{10}\text{⁡}\left(\frac{x^2-y^2}{x^2+y^2}\right)=2$, then $\frac{\text{dy}}{\text{dx}}=$
15. 15.If…
16. 16.If $f\text{(}a\text{)}=\text{log}\text{⁡}\left(\frac{1-a}{1+a}\right)$ for $a\ne \text{-1,1}$, then the set of values of all ' $a$ ', for which…
17. 17.If $f\left(x\right)=\text{sin}\text{(}{\text{cos}}^{-1}\left(\frac{1-2^{2x}}{1+2^{2x}}\right)$and its first derivative with respect to x is…
18. 18.Let $f\text{(}x\text{)}$ be a polynomial of degree $6$ in $x$, in which the coefficient of $x^6$is unity and it has extrema at $x=-1$ and $x=1$ . If…
19. 19.If $x^y=e^{x-y}$, then $\text{dy}/\text{dx}$ at $x=1$ is
20. 20.If…
21. 21.If $x+y=k\text{,}x>0\text{,}y>0$ then $x^2+y^2$ is minimum, if
22. 22.Let f be any continuous function on [0,2] and twice differentiable on (0,2). If f(0)=0,f(a)=1 and f(b)=2, then
23. 23.If $f\left(x\right)=(\frac{x^2-(a+2)x+a}{x-2}\text{,}x\ne 22\text{,}x=2)$is continuous at $x=2$, then the value of $a$ is
24. 24.If $f\text{:}N\to R$ is defined by $f\text{(}1\text{)}=-1$ and $f\text{(}n+1\text{)}=3f$ $\text{(}n\text{)}+2$ for $\ge 1$, then $f$ is
25. 25.If $x=e^{\left(\frac{x}{y}\right)}$, then $\frac{\text{dy}}{\text{dx}}=$