JEE Main · 2022 · 25 Jul 2022 · Shift 1 · Determinants

More JEE Main questions on Determinants

1. Let n be the number obtained on rolling a fair die. If the probability that the system
   $x-ny+z=6$
   $x+(n-2)y+(n+1)z=8$
   $(n-1)y+z=1$
   has a unique solution is $\frac{k}{6}$, then the sum of k and all possible values of n is :
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2. Among the statements :
   I : If …
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3. If $\mathrm{f}:\mathrm{N}\to \mathrm{Z}$ is defined by
   $f(n)=\left|\begin{array}{ccc}n& -1& -5\\ -2n^2& 3(2k+1)& 2k+1\\ -3n^3& 3k(2k+1)& 3k(k+2)+1\end{array}\right|,k\in N$, and ${\sum }_{n=1}^{k}f(n)=98$, then $k$ is equal to:
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4. The sum of all possible values of $\theta \in [0,2\pi ]$, for which the system of equations :
   

   $$\begin{array}{rl}& x\cos 3\theta -8y-12z=0\\ & x\cos 2\theta +3y+3z=0\\ & x+y+3z=0\end{array}$$

   
   has a non-trivial solution, is equal to:
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5. Let $\mathrm{A}=\left|\begin{array}{ccc}1& 3& -1\\ 2& 1& \alpha \\ 0& 1& -1\end{array}\right|$ be a singular matrix. Let $f(x)={\int }_0^x\left(t^2+2t+3\right)dt,x\in [1,\alpha ]$. If $M$ and $m$ are respectively the maximum and the minimum values of $f$ in …
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6. Let $\alpha ,\beta (\alpha \ne \beta )$ be the values of $m$, for which the equations $\mathrm{x}+\mathrm{y}+\mathrm{z}=1;\mathrm{x}+2\mathrm{y}+4\mathrm{z}=\mathrm{m}$ and $x+4y+10z=m^2$ have infinitely many solutions. Then the value of …
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