Mathematical Reasoning & Induction · JEE Main previous year questions

2026 3 questions

1. Q132026 · 8 Apr 2026 · Shift 2

   For the function $f(x)=e^{\sin |x|}-|x|$, $x\in \mathbf{R}$, consider the following statements : Statement I : $f$ is differentiable for all $x\in \mathbf{R}$. Statement II : $f$ is increasing in $\left(-\pi ,-\frac{\pi }{2}\right)$. In the light of the above statements, choose the correct answer from the options given below :

   1. A.Both Statement I and Statement II are true
   2. B.Both Statement I and Statement II are false
   3. C.Statement I is true but Statement II is false
   4. D.Statement I is false but Statement II is true
   Show answer

   Answer: A

2. Q42026 · 2 Apr 2026 · Shift 1

   Let $\mathrm{A}=\left[\begin{array}{cc}1& 2\\ 1& \alpha \end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}3& 3\\ \beta & 2\end{array}\right]$. If ${\mathrm{A}}^2-4\mathrm{A}+\mathrm{I}=\mathrm{O}$ and ${\mathrm{B}}^2-5\mathrm{B}-6\mathrm{I}=\mathrm{O}$, then among the two statements : (S1): $[(\mathrm{B}-\mathrm{A})(\mathrm{B}+\mathrm{A}){]}^{\mathrm{T}}=\left[\begin{array}{cc}13& 15\\ 7& 10\end{array}\right]$ and (S2): $\det (\mathrm{adj}(\mathrm{A}+\mathrm{B}))=-5$,

   1. A.only (S1) is correct
   2. B.only (S2) is correct
   3. C.both (S1) and (S2) are correct
   4. D.both (S1) and (S2) are wrong
   Show answer

   Answer: B

3. Q142026 · 23 Jan 2026 · Shift 1

   Among the statements : I: If $\left|\begin{array}{ccc}1& \cos \alpha & \cos \beta \\ \cos \alpha & 1& \cos \gamma \\ \cos \beta & \cos \gamma & 1\end{array}\right|=\left|\begin{array}{ccc}0& \cos \alpha & \cos \beta \\ \cos \alpha & 0& \cos \gamma \\ \cos \beta & \cos \gamma & 0\end{array}\right|$, then ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =\frac{3}{2}$, and II: If $\left|\begin{array}{ccc}{x}^2+x& x+1& x-2\\ 2x^2+3x-1& 3x& 3x-3\\ x^2+2x+3& 2x-1& 2x-1\end{array}\right|=\mathrm{p}x+\mathrm{q}$, then ${\mathrm{p}}^2=196{\mathrm{q}}^2$,

   1. A.both are true
   2. B.both are false
   3. C.only I is true
   4. D.only II is true
   Show answer

   Answer: B