Mathematical Reasoning & Induction · JEE Main previous year questions

Let A and B denote the statements A: $\cos \alpha +\cos \beta +\cos \gamma =0$ B: $\sin \alpha +\sin \beta +\sin \gamma =0$ If $\cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta )=-\frac{3}{2}$, then:

1. A.A is false and B is true
2. B.both A and B are true
3. C.both A and B are false
4. D.A is true and B is false

Let $f(x)=(x+1{)}^2-1,x\ge -1$ Statement - 1 : The set $\{x:f(x)=f^{-1}(x)\}=\{0,-1\}$. Statement - 2 : $f$ is a bijection.

1. A.Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for Statement - 1
2. B.Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1
3. C.Statement - 1 is true, Statement - 2 is false
4. D.Statement - 1 is false, Statement - 2 is true

Statement-1 : $\sim (p\leftrightarrow \sim q)$ is equivalent to $p\leftrightarrow q$. Statement-2 : $\sim (p\leftrightarrow \sim q)$ is a tautology.

1. A.Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
2. B.Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1
3. C.Statement-1 is true, Statement-2 is false
4. D.Statement-1 is false, Statement-2 is true

If $A=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then which one of the following holds for all $n\ge 1$, by the principle of mathematical induction?

1. A.$A^n=nA-(n-1)\mathrm{I}$
2. B.$A^n=2^{n-1}A-(n-1)\mathrm{I}$
3. C.$A^n=nA+(n-1)\mathrm{I}$
4. D.$A^n=2^{n-1}A+(n-1)\mathrm{I}$

If $a_n=\sqrt{7+\sqrt{7+\sqrt{7+\dots \dots }}}$ having $n$ radical signs then by methods of mathematical induction which is true

1. A.$a_n>7\forall n\ge 1$
2. B.$a_n<7\forall n\ge 1$
3. C.$a_n<4\forall n\ge 1$
4. D.$a_n>3\forall n\ge 1$