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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 101-150 of 281

  1. The following system of linear equations $7x+6y-2z=0$, $3x+4y+2z=0$, $x-2y-6z=0$, has :
  2. If $A=\begin{bmatrix}2&3\\0&-1\end{bmatrix}$, then the value of $\det\left(A^{4}\right)+\det\left(A^{10}-\left(\operatorname{Adj}(2A)\right)^{10}\right)$ is equal to …
  3. Let $A=\begin{bmatrix}1&-2&\alpha\\\alpha&2&-1\end{bmatrix}$ and $B=\begin{bmatrix}2&\alpha\\-1&2\\4&-5\end{bmatrix}$, $\alpha\in\mathbf{C}$. Then the absolute value of …
  4. If the system of linear equations $2x-3y=\gamma+5$, $\alpha x+5y=\beta+1$, where $\alpha,\beta,\gamma\in\mathbf{R}$ has infinitely many solutions, then the value of …
  5. If the system of equations $2x+7y+\lambda z=3$ $3x+2y+5z=4$ $x+\mu y+32z=-1$ has infinitely many solutions, then $(\lambda-\mu)$ is equal to $\_\_\_\_\_\_$ .
  6. Let $D_k=\begin{vmatrix}1&2k&2k-1\\n&n^{2}+n+2&n^{2}\\n&n^{2}+n&n^{2}+n+2\end{vmatrix}$. If $\sum_{k=1}^{n}D_k=96$, then $n$ is equal to $\_\_\_\_\_\_$ .
  7. Let $p$ and $p+2$ be prime numbers and let $\Delta=\begin{vmatrix}p!&(p+1)!&(p+2)!\\(p+1)!&(p+2)!&(p+3)!\\(p+2)!&(p+3)!&(p+4)!\end{vmatrix}$. Then the sum of the maximum …
  8. If the system of linear equations $8x+y+4z=-2$, $x+y+z=0$, $\lambda x-3y=\mu$ has infinitely many solutions, then the distance of the point …
  9. If $\Delta=\begin{vmatrix}x-2&2x-3&3x-4\\2x-3&3x-4&4x-5\\3x-5&5x-8&10x-17\end{vmatrix}=Ax^3+Bx^2+Cx+D$, then $B+C$ is equal to :
  10. If the system of equations $x+y+z=6$, $2x+5y+\alpha z=\beta$, $x+2y+3z=14$ has infinitely many solutions, then $\alpha+\beta$ is equal to
  11. Suppose the vectors $x_1,x_2$ and $x_3$ are the solutions of the system of linear equations, $Ax=b$ when the vector $b$ on the right side is equal to $b_1,b_2$ and $b_3$ …
  12. For real numbers $\alpha$ and $\beta$, consider the following system of linear equations : $x+y-z=2$, $x+2y+\alpha z=1$, $2x-y+z=\beta$. If the system has infinite …
  13. If the system of equations $x+y+z=2$, $2x+4y-z=6$, $3x+2y+\lambda z=\mu$ has infinitely many solutions, then :
  14. For which of the following ordered pairs $(\mu,\delta)$, the system of linear equations $x+2y+3z=1$, $3x+4y+5z=\mu$, $4x+4y+4z=\delta$ is inconsistent ?
  15. If the system of linear equations $2x+y-z=3$ $x-y-z=\alpha$ $3x+3y+\beta z=3$ has infinitely many solution, then $\alpha+\beta-\alpha\beta$ is equal to $\_\_\_\_\_\_$ .
  16. Let $A$ be a non-singular matrix of order $3$. If $\det(3\operatorname{adj}(2\operatorname{adj}((\det A)A)))=3^{-13}\cdot2^{-10}$ and …
  17. Consider the matrices: $A=\begin{bmatrix}2&-5\\3&m\end{bmatrix}$, $B=\begin{bmatrix}20\\m\end{bmatrix}$ and $X=\begin{bmatrix}x\\y\end{bmatrix}$. Let the set of all $m$ …
  18. If the system of equations $\alpha x+y+z=5$, $x+2y+3z=4$, $x+3y+5z=\beta$ has infinitely many solutions, then the ordered pair $(\alpha,\beta)$ is equal to :
  19. Let the system of linear equations $x+2y+z=2$, $\alpha x+3y-z=\alpha$, $-\alpha x+y+2z=-\alpha$ be inconsistent. Then $\alpha$ is equal to :
  20. If the matrices $A=\begin{bmatrix}1&1&2\\1&3&4\\1&-1&3\end{bmatrix}$, $B=\mathrm{adj}\,A$ and $C=3A$, then $\frac{|\mathrm{adj}\,B|}{|C|}$ is equal to :
  21. If the system of linear equations $\lambda x+y-z=-1$, $x-y-3z=2$, $-x+y+z=\mu$ has infinitely many solutions, then the equation of the line passing through the points …
  22. If the system of equations $kx+y+2z=1$ $3x-y-2z=2$ $-2x-2y-4z=3$ has infinitely many solutions, then $k$ is equal to $\_\_\_\_\_\_$ .
  23. Let $\lambda\in\mathbf{R}$. The system of linear equations $2x_1-4x_2+\lambda x_3=1$, $x_1-6x_2+x_3=2$, $\lambda x_1-10x_2+4x_3=3$ is inconsistent for :
  24. The sum of distinct values of $\lambda$ for which the system of equations $(\lambda-1)x+(3\lambda+1)y+2\lambda z=0$, $(\lambda-1)x+(4\lambda-2)y+(\lambda+3)z=0$ …
  25. If the system of equations $x-2y+3z=9$, $2x+y+z=b$, $x-7y+az=24$, has infinitely many solutions, then $a-b$ is equal to ________.
  26. Let $a,b,c$ be any real numbers. Suppose that there are real numbers $x,y,z$ not all zero such that $x=cy+bz$, $y=az+cx$, and $z=bx+ay$. Then $a^{2}+b^{2}+c^{2}+2abc$ is …
  27. Consider the system of linear equations: $x_1+2x_2+x_3=3$ $2x_1+3x_2+x_3=3$ $3x_1+5x_2+2x_3=1$ The system has :
  28. Let $a,b,c$ be such that $b(a+c)\neq0$. If …
  29. If $a_1,a_2,a_3,\ldots,a_n,\ldots$ are in G.P., then the value of the determinant …
  30. If $\begin{vmatrix}a&a^{2}&1+a^{3}\\b&b^{2}&1+b^{3}\\c&c^{2}&1+c^{3}\end{vmatrix}=0$ and vectors $(1,a,a^{2})$, $(1,b,b^{2})$ and $(1,c,c^{2})$ are non-coplanar, then …
  31. The system of equations $\alpha x+y+z=\alpha-1$ $x+\alpha y+z=\alpha-1$ $x+y+\alpha z=\alpha-1$ has no solutions, if $\alpha$ is :
  32. If $a^{2}+b^{2}+c^{2}=-2$ and $f(x)=\begin{vmatrix}1+a^{2}x&(1+b^{2})x&(1+c^{2})x\\(1+a^{2})x&1+b^{2}x&(1+c^{2})x\\(1+a^{2})x&(1+b^{2})x&1+c^{2}x\end{vmatrix}$, then …
  33. If the system of linear equations $x+2ay+az=0$; $x+3by+bz=0$; $x+4cy+cz=0$; has a non-zero solution, then $a,b,c$.
  34. If $a,b,c$ are non-zero real numbers and if the system of equations $(a-1)x=y+z$, $(b-1)y=x+z$, $(c-1)z=x+y$ has a non-trivial solution, then $ab+bc+ca$ equals :
  35. If $1,\omega,\omega^{2}$ are the cube roots of unity, then …
  36. If $D=\begin{vmatrix}1&1&1\\1&1+x&1\\1&1&1+y\end{vmatrix}$ for $x\neq0$, $y\neq0$, then $D$ is :
  37. If $a>0$ and discriminant of $ax^{2}+2bx+c$ is $-ve$, then $\begin{vmatrix}a&b&ax+b\\b&c&bx+c\\ax+b&bx+c&0\end{vmatrix}$ is equal to
  38. $l,m,n$ are the $p^{th}$, $q^{th}$ and $r^{th}$ term of a G.P. all positive, then $\begin{vmatrix}\log l&p&1\\\log m&q&1\\\log n&r&1\end{vmatrix}$ equals
  39. If $a_1,a_2,a_3,\ldots,a_n,\ldots$ are in G.P., then the determinant …
  40. Let $d\in R$, and $A=\begin{bmatrix}-2&4+d&(\sin\theta)-2\\1&(\sin\theta)+2&d\\5&(2\sin\theta)-d&(-\sin\theta)+2+2d\end{bmatrix}$, $\theta\in[0,2\pi]$. If the minimum …
  41. Let $A=\begin{bmatrix}2&b&1\\b&b^2+1&b\\1&b&2\end{bmatrix}$, where $b>0$. Then the minimum value of $\frac{\det(A)}{b}$ is :
  42. The sum of the real roots of the equation $\begin{vmatrix}x&-6&-1\\2&-3x&x-3\\-3&2x&x+2\end{vmatrix}=0$ is equal to :
  43. Consider the system of linear equations $-x+y+2z=0$ $3x-ay+5z=1$ $2x-2y-az=7$ Let $S_1$ be the set of all $a\in R$ for which the system is inconsistent and $S_2$ be the …
  44. Let $\lambda$ be a real number for which the system of linear equations $x+y+z=6$, $4x+\lambda y-\lambda z=\lambda-2$ and $3x+2y-4z=-5$ has infinitely many solutions. …
  45. Let $A=[a_{ij}]$ be a $3\times3$ matrix, where $a_{ij}=\begin{cases}1, & \text{if } i=j\\ -x, & \text{if } |i-j|=1\\ 2x+1, & \text{otherwise}\end{cases}$. Let a function …
  46. Consider the system of linear equations in $x,y,z$: $x+2y+tz=0$, $6x+y+5tz=0$, $3x+t^{2}y+f(t)z=0$, where $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function. If …
  47. If $[x]$ denotes the greatest integer $\le x$, then the system of linear equations $[\sin\theta]x+[-\cos\theta]y=0$, $[\cot\theta]x+y=0$
  48. Let the system of linear equations $x+y+kz=2$ $2x+3y-z=1$ $3x+4y+2z=k$ have infinitely many solutions. Then the system $(k+1)x+(2k-1)y=7$ $(2k+1)x+(k+5)y=10$ has:
  49. Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}+x+1=0$. Then for $y\neq0$ in $R$ …
  50. The number of values of $k$, for which the system of equations : $(k+1)x+8y=4k$ $kx+(k+3)y=3k-1$ has no solution, is

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