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Yukti Vault

Algebra

Determinant: JEE Main & Advanced PYQs

281 previous year questions on Determinant from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 251-281 of 281

  1. Let $P$ be a matrix of order $3\times3$ such that all the entries in $P$ are from the set $\{-1,0,1\}$. Then, the maximum possible value of the determinant of $P$ is …
  2. Let $M$ be a $3\times3$ invertible matrix with real entries and let $I$ denote the $3\times3$ identity matrix. If $M^{-1}=\operatorname{adj}(\operatorname{adj}M)$, then …
  3. The system of equations $x+y+z=6$, $x+2y+5z=9$, $x+5y+\lambda z=\mu$, has no solution if
  4. The value of $|M|$ is ___ .
  5. If the point $P(a,b,c)$, with reference to (E), lies on the plane $2x+y+z=1$, then the value of $7a+b+c$ is
  6. Consider the following linear equations $$ax+by+cz=0$$ $$bx+cy+az=0$$ $$cx+ay+bz=0$$ Match the conditions/expressions in Column I with statements in Column II and …
  7. If $\alpha$ be a repeated root of a quadratic equation $f(x)=0$ and $A(x)$, $B(x)$ and $C(x)$ be polynomials of degree $3$, $4$ and $5$ respectively, then show that …
  8. Let $P$ and $Q$ be $3\times3$ matrices with $P\ne Q$. If $P^3=Q^3$ and $P^2Q=Q^2P$, then determinant of $\left(P^2+Q^2\right)$ is equal to :
  9. If $A$ is a $3\times3$ matrix and $|A|=\dfrac{1}{36}$, then the value of $\left|2\cdot\operatorname{adj}\left(3\cdot\operatorname{adj}(6A)\right)\right|$ is
  10. The system of equations $x+2y+pz=4$ $x+qy+z=5$ $2x+4y+z=8$ has
  11. The number of real solutions of the equation $\begin{vmatrix}x^{2}+\sin x\cos x&x(1+\sin x)\\x+\cos x&x+1\end{vmatrix}=0$ is $\_\_\_\_\_\_\_\_$ .
  12. If the system of equations $x+\left(\sqrt{2}\sin\alpha\right)y+\left(\sqrt{2}\cos\alpha\right)z=0$ $x+(\cos\alpha)y+(\sin\alpha)z=0$ $x+(\sin\alpha)y-(\cos\alpha)z=0$ …
  13. Let $\alpha\in(0,\infty)$ and $A=\begin{bmatrix}1&2&\alpha\\1&0&1\\0&1&2\end{bmatrix}$. If …
  14. If the system of equations $11x+y+\lambda z=-5$ $2x+3y+5z=3$ $8x-19y-39z=\mu$ has infinitely many solutions, then $\lambda^{4}-\mu$ is equal to :
  15. Let $A$ and $B$ be two square matrices of order $3$ such that $|A|=3$ and $|B|=2$. Then …
  16. Let $\alpha\beta\neq0$ and $A=\begin{bmatrix}\beta&\alpha&3\\\alpha&\alpha&\beta\\-\beta&\alpha&2\alpha\end{bmatrix}$. If …
  17. The values of $m,n$, for which the system of equations $x+y+z=4$, $2x+5y+5z=17$, $x+2y+mz=n$ has infinitely many solutions, satisfy the equation :
  18. If the system of linear equations $x+y+3z=0$ $x+3y+k^{2}z=0$ $3x+y+3z=0$ has a non-zero solution $(x,y,z)$ for some $k\in\mathbf{R}$, then $x+\left(\dfrac{y}{z}\right)$ …
  19. If $a+x=b+y=c+z+1$, where $a,b,c,x,y,z$ are non-zero distinct real numbers, then $\begin{vmatrix}x&a+y&x+a\\y&b+y&y+b\\z&c+y&z+c\end{vmatrix}$ is equal to :
  20. Let $m$ and $M$ be respectively the minimum and maximum values of …
  21. The values of $\lambda$ and $\mu$ for which the system of linear equations $x+y+z=2$ $x+2y+3z=5$ $x+3y+\lambda z=\mu$ has infinitely many solutions are, respectively :
  22. Let $A$ be a square matrix of order $3$ such that $\det(A)=-2$ and $\det\left(3\operatorname{adj}\left(-6\operatorname{adj}(3A)\right)\right)=2^{m+n}\cdot3^{mn}$, $m>n$. …
  23. For a $3\times3$ matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3\times3$ matrix such that $|A|=\dfrac{1}{2}$ and trace …
  24. If the system of linear equations : $x+y+2z=6$ $2x+3y+az=a+1$ $-x-3y+bz=2b$ where $a,b\in\mathbf{R}$, has infinitely many solutions, then $7a+3b$ is equal to :
  25. If the system of equations $(\lambda-1)x+(\lambda-4)y+\lambda z=5$ $\lambda x+(\lambda-1)y+(\lambda-4)z=7$ $(\lambda+1)x+(\lambda+2)y-(\lambda+2)z=9$ has infinitely many …
  26. For some $a,b$, let $f(x)=\begin{vmatrix}a+\dfrac{\sin x}{x}&1&b\\a&1+\dfrac{\sin x}{x}&b\\a&1&b+\dfrac{\sin x}{x}\end{vmatrix}$, $x\neq0$ …
  27. If the system of equations $x+2y-3z=2$ $2x+\lambda y+5z=5$ $14x+3y+\mu z=33$ has infinitely many solutions, then $\lambda+\mu$ is equal to :
  28. Let $A=\begin{bmatrix}5&5\alpha&\alpha\\0&\alpha&5\alpha\\0&0&5\end{bmatrix}$. If $\left|A^{2}\right|=25$, then $|\alpha|$ equals
  29. If $P=\begin{bmatrix}1&\alpha&3\\1&3&3\\2&4&4\end{bmatrix}$ is the adjoint of a $3\times3$ matrix $A$ and $|A|=4$, then $\alpha$ is equal to :
  30. Statement-1: The system of linear equations $x+(\sin\alpha)y+(\cos\alpha)z=0$, $x+(\cos\alpha)y+(\sin\alpha)z=0$, $x-(\sin\alpha)y-(\cos\alpha)z=0$ has a non-trivial …
  31. The value of $k\in\mathbf{R}$, for which the following system of linear equations $3x-y+4z=3$, $x+2y-3z=-2$, $6x+5y+kz=-3$, has infinitely many solutions, is :

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