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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 1-50 of 281

  1. If the system of equations $x+5y+6z=4$, $2x+3y+4z=7$, $x+6y+az=b$ has infinitely many solutions, then the point $(a,b)$ lies on the line
  2. The following system of linear equations $2x+3y+2z=9$ $3x+2y+2z=9$ $x-y+4z=8$
  3. The number of distinct real roots of $\begin{vmatrix}\sin x&\cos x&\cos x\\\cos x&\sin x&\cos x\\\cos x&\cos x&\sin x\end{vmatrix}=0$ in the interval …
  4. Consider the following system of equations : $x+2y-3z=a$ $2x+6y-11z=b$ $x-2y+7z=c$, where $a,b$ and $c$ are real constants. Then the system of equations :
  5. The system of equations $kx+y+z=1$, $x+ky+z=k$ and $x+y+zk=k^{2}$ has no solution if $k$ is equal to :
  6. If $f:\mathbf{N}\to\mathbf{Z}$ is defined by $f(n)=\begin{vmatrix}n&-1&-5\\-2n^{2}&3(2k+1)&2k+1\\-3n^{3}&3k(2k+1)&3k(k+2)+1\end{vmatrix}$, $k\in\mathbf{N}$, and …
  7. For the system of linear equations : $x-2y=1$, $x-y+kz=-2$, $ky+4z=6$, $k\in\mathbf{R}$, consider the following statements : (A) The system has unique solution if …
  8. The maximum value of $f(x)=\begin{vmatrix}\sin^{2}x&1+\cos^{2}x&\cos2x\\1+\sin^{2}x&\cos^{2}x&\cos2x\\\sin^{2}x&\cos^{2}x&\sin2x\end{vmatrix}$, $x\in\mathbf{R}$ is :
  9. Let $A$ be a $3\times3$ matrix with $\det(A)=4$. Let $R_i$ denote the $i^{\text{th}}$ row of $A$. If a matrix $B$ is obtained by performing the operation …
  10. The value of $\begin{vmatrix}(a+1)(a+2)&a+2&1\\(a+2)(a+3)&a+3&1\\(a+3)(a+4)&a+4&1\end{vmatrix}$ is :
  11. If the system of linear equations : $x+y+z=6$, $x+2y+5z=10$, $2x+3y+\lambda z=\mu$ has infinitely many solutions, then the value of $\lambda+\mu$ equals :
  12. Let $\alpha,\beta,\gamma$ be the real roots of the equation, $x^{3}+ax^{2}+bx+c=0$, $(a,b,c\in\mathbf{R}$ and $a,b\neq0)$. If the system of equations (in $u,v,w$) given …
  13. The values of $\lambda$ and $\mu$ such that the system of equations $x+y+z=6$, $3x+5y+5z=26$, $x+2y+\lambda z=\mu$ has no solution, are :
  14. Let the system of linear equations $4x+\lambda y+2z=0$ $2x-y+z=0$ $\mu x+2y+3z=0$, $\lambda,\mu\in\mathbf{R}$, has a non-trivial solution. Then which of the following is …
  15. For the system of linear equations $x+3y+2z=7$ $x+\lambda y+3z=8$ $x-2y+7z=\mu$, which of the following statements is NOT true?
  16. The system of linear equations $3x-2y-kz=10$ $2x-4y-2z=6$ $x+2y-z=5m$ is inconsistent if :
  17. Let $A=(a_{ij})$ be a $3\times3$ matrix, where $a_{ij}=7^{\max\{i,j\}}$. Then $\det A$ is equal to
  18. If $x,y,z$ are in arithmetic progression with common difference $d$, $x\neq3d$, and the determinant of the matrix …
  19. The sum of all possible values of $\theta\in[0,2\pi]$, for which the system of equations : $x\cos3\theta-8y-12z=0$ $x\cos2\theta+3y+3z=0$ $x+y+3z=0$ has a non-trivial …
  20. The values of $a$ and $b$, for which the system of equations $2x+3y+6z=8$, $x+2y+az=5$, $3x+5y+9z=b$ has no solution, are :
  21. Let $A=\begin{bmatrix}1&2&7\\4&-2&8\\3&8&-7\end{bmatrix}$ and $\det(A-\alpha I)=0$, where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$ …
  22. Let $A$ be a square matrix of order $3$ and $|A|=3$. If $|\operatorname{Adj}(3\operatorname{Adj}(4A))|=2^{m}\cdot3^{n}$, then the ordered pair $(m,n)$ is equal to
  23. If the system of equations: $x+y+z=5$ $x+2y+3z=9$ $x+3y+\lambda z=\mu$ has infinitely many solutions, then the value of $\lambda+\mu$ is:
  24. The solutions of the equation $\begin{vmatrix}1+\sin^{2}x&\sin^{2}x&\sin^{2}x\\\cos^{2}x&1+\cos^{2}x&\cos^{2}x\\4\sin2x&4\sin2x&1+4\sin2x\end{vmatrix}=0$, $(0<x<\pi)$ …
  25. Let $A=\begin{bmatrix}1&0&0\\0&\alpha&\beta\\0&\beta&\alpha\end{bmatrix}$ and $|2A|^{3}=2^{21}$ where $\alpha,\beta\in\mathbb{Z}$. Then a value of $\alpha$ is
  26. If $\alpha+\beta+\gamma=2\pi$, then the system of equations $x+(\cos\gamma)y+(\cos\beta)z=0$ $(\cos\gamma)x+y+(\cos\alpha)z=0$ $(\cos\beta)x+(\cos\alpha)y+z=0$ has :
  27. Let $[\lambda]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4$ …
  28. If $A$ is a $3\times3$ matrix and $|A|=2$, then $\left|3\operatorname{adj}\left(|3A|A^{2}\right)\right|$ is equal to
  29. Let the system of equations $x+5y-z=1$ $4x+3y-3z=7$ $24x+y+\lambda z=\mu$ $\lambda,\mu\in\mathbb{R}$, have infinitely many solutions. Then the number of the solutions of …
  30. Let $a\in\mathbf{R}$ and $A$ be a matrix of order $3\times3$ such that $\det(A)=-4$ and $A+I=\begin{bmatrix}1&a&1\\2&1&0\\a&1&2\end{bmatrix}$, where $I$ is the identity …
  31. The number of values of $\alpha$ for which the system of equations : $x+y+z=\alpha$ $\alpha x+2\alpha y+3z=-1$ $x+3\alpha y+5z=4$ is inconsistent, is
  32. Let the system of linear equations $-x+2y-9z=7$ $-x+3y+7z=9$ $-2x+y+5z=8$ $-3x+y+13z=\lambda$ has a unique solution $x=\alpha$, $y=\beta$, $z=\gamma$. Then the distance …
  33. Consider the system of linear equations $x+y+z=5$, $x+2y+\lambda^{2}z=9$, $x+3y+\lambda z=\mu$, where $\lambda,\mu\in\mathbb{R}$. Then, which of the following statement …
  34. Let $A=\begin{bmatrix}2&2+p&2+p+q\\4&6+2p&8+3p+2q\\6&12+3p&20+6p+3q\end{bmatrix}$. If $\det(\operatorname{adj}(\operatorname{adj}(3A)))=2^{m}\cdot3^{n}$ …
  35. If $f(x)=\begin{vmatrix}x^{3}&2x^{2}+1&1+3x\\3x^{2}+2&2x&x^{3}+6\\x^{3}-x&4&x^{2}-2\end{vmatrix}$ for all $x\in\mathbb{R}$, then $2f(0)+f'(0)$ is equal to
  36. For $\alpha,\beta\in\mathbb{R}$ and a natural number $n$, let $A_r=\begin{vmatrix}r&1&\frac{n^{2}}{2}+\alpha\\2r&2&n^{2}-\beta\\3r-2&3&\frac{n(3n-1)}{2}\end{vmatrix}$. …
  37. If $A$ is a square matrix of order $3$ such that $\det(A)=3$ and …
  38. The system of equations $-kx+3y-14z=25$ $-15x+4y-kz=3$ $-4x+y+3z=4$ is consistent for all $k$ in the set
  39. If $\begin{vmatrix}x+1&x&x\\x&x+\lambda&x\\x&x&x+\lambda^{2}\end{vmatrix}=\frac{9}{8}(103x+81)$, then $\lambda,\frac{\lambda}{3}$ are the roots of the equation
  40. Let $A$ be a $3\times3$ invertible matrix. If $|\operatorname{adj}(24A)|=|\operatorname{adj}(3\operatorname{adj}(2A))|$, then $|A|^{2}$ is equal to :
  41. If the system of linear equations $3x+y+\beta z=3$ $2x+\alpha y-z=-3$ $x+2y+z=4$ has infinitely many solutions, then the value of $22\beta-9\alpha$ is :
  42. If the system of equations $2x+\lambda y+3z=5$ $3x+2y-z=7$ $4x+5y+\mu z=9$ has infinitely many solutions, then $\left(\lambda^{2}+\mu^{2}\right)$ is equal to :
  43. Let $A=\begin{bmatrix}2&1&0\\1&2&-1\\0&-1&2\end{bmatrix}$. If $\left|\operatorname{adj}\left(\operatorname{adj}\left(\operatorname{adj}2A\right)\right)\right|=(16)^{n}$ …
  44. Consider the system of linear equations $x+y+z=4\mu$, $x+2y+2\lambda z=10\mu$, $x+3y+4\lambda^{2}z=\mu^{2}+15$, where $\lambda,\mu\in\mathbf{R}$. Which one of the …
  45. If the system of equations $2x+3y-z=5$ $x+\alpha y+3z=-4$ $3x-y+\beta z=7$ has infinitely many solutions, then $13\alpha\beta$ is equal to
  46. Let for $A=\begin{bmatrix}1&2&3\\\alpha&3&1\\1&1&2\end{bmatrix}$, $|A|=2$. If $|2\operatorname{adj}(2\operatorname{adj}(2A))|=32^{n}$, then $3n+\alpha$ is equal to
  47. Let $A$ be a matrix of order $3\times3$ and $|A|=5$. If $|2\operatorname{adj}(3A\operatorname{adj}(2A))|=2^{\alpha}\cdot3^{\beta}\cdot5^{\gamma}$ …
  48. For the system of linear equations $2x+4y+2az=b$ $x+2y+3z=4$ $2x-5y+2z=8$ which of the following is NOT correct?
  49. If the system of linear equations $7x+11y+\alpha z=13$ $5x+4y+7z=\beta$ $175x+194y+57z=361$ has infinitely many solutions, then $\alpha+\beta+2$ is equal to:
  50. If the system of linear equations $x+y+z=3$ $x+by+b^{2}z=7$ and $ax+a^{2}y+a^{3}z=1$ has infinitely many solutions, then the ordered pair $(a,b)$ is

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