Skip to main content
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.

Practice this chapter

Questions 151-200 of 281

  1. The set of all values of $\lambda$ for which the system of linear equations : $2x_1-2x_2+x_3=\lambda x_1$ $2x_1-3x_2+2x_3=\lambda x_2$ $-x_1+2x_2=\lambda x_3$ has a …
  2. If $\begin{vmatrix}a-b-c&2a&2a\\2b&b-c-a&2b\\2c&2c&c-a-b\end{vmatrix}=(a+b+c)(x+a+b+c)^{2}$, $x\neq0$ and $a+b+c\neq0$, then $x$ is equal to
  3. The number of real values of $\lambda$ for which the system of linear equations $2x+4y-\lambda z=0$, $4x+\lambda y+2z=0$ and $\lambda x+2y+2z=0$ has infinitely many …
  4. Let $\omega$ be a complex number such that $2\omega+1=z$ where $z=\sqrt{-3}$. If $\begin{vmatrix}1&1&1\\1&-\omega^2-1&\omega^2\\1&\omega^2&\omega^7\end{vmatrix}=3k$ …
  5. Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x+y+z=1$ $2x+Ny+2z=2$ $3x+3y+Nz=3$ has unique …
  6. Let the numbers $2,b,c$ be in an A.P. and $A=\begin{bmatrix}1&1&1\\2&b&c\\4&b^2&c^2\end{bmatrix}$. If $\det(A)\in[2,16]$, then $c$ lies in the interval :
  7. If the system of equations $2x+3y-z=0$, $x+ky-2z=0$ and $2x-y+z=0$ has a non-trivial solution $(x,y,z)$, then $\frac xy+\frac yz+\frac zx+k$ is equal to
  8. The set of all values of $t\in\mathbb R$, for which the matrix …
  9. The number of values of $\theta\in(0,\pi)$ for which the system of linear equations $x+3y+7z=0$ $-x+4y+7z=0$ $(\sin3\theta)x+(\cos2\theta)y+2z=0$ has a non-trivial …
  10. Let $a_1,a_2,a_3,\ldots,a_{10}$ be in G.P. with $a_i>0$ for $i=1,2,\ldots,10$ and $S$ be the set of pairs $(r,k)$, $r,k\in\mathbb N$ (the set of natural numbers) for …
  11. Let $\alpha,\beta\in\mathbb{R}$ be such that the system of linear equations $x+2y+z=5$, $2x+y+\alpha z=5$, $8x+4y+\beta z=18$ has no solution. Then …
  12. Let $A=[a_{ij}]$ and $B=[b_{ij}]$ be two $3\times3$ real matrices such that $b_{ij}=(3)^{(i+j-2)}a_{ij}$, where $i,j=1,2,3$. If the determinant of $B$ is $81$, then …
  13. Let $A$ be a $3\times3$ matrix such that $A^{T}\begin{bmatrix}1\\0\\1\end{bmatrix}=\begin{bmatrix}5\\2\\2\end{bmatrix}$ …
  14. Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $\lambda x+y+z=1$ $x+\lambda y+z=1$ $x+y+\lambda z=1$ is inconsistent, then …
  15. Let $A$ and $B$ be two invertible matrices of order $3\times3$. If $\det\left(ABA^{T}\right)=8$ and $\det\left(AB^{-1}\right)=8$, then $\det\left(BA^{-1}B^{T}\right)$ is …
  16. If the system of linear equations $2x+2ay+az=0$ $2x+3by+bz=0$ $2x+4cy+cz=0$, where $a,b,c\in R$ are non-zero and distinct; has a non-zero solution, then
  17. If $x=a$, $y=b$, $z=c$ is a solution of the system of linear equations $x+8y+7z=0$ $9x+2y+3z=0$ $x+y+z=0$ such that the point $(a,b,c)$ lies on the plane $x+2y+z=6$ …
  18. Let $S_1$ and $S_2$ be respectively the sets of all $a\in R-\{0\}$ for which the system of linear equations $ax+2ay-3az=1$ $(2a+1)x+(2a+3)y+(a+1)z=2$ …
  19. If $a,b,c$ are sides of a scalene triangle, then the value of $\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}$ is :
  20. The set of all values of $\lambda$ for which the system of linear equations $x-2y-2z=\lambda x$, $x+2y+z=\lambda y$, $-x-y=\lambda z$ has a non-trivial solution :
  21. Let $x,y,z>1$ and $A=\begin{bmatrix}1&\log_xy&\log_xz\\\log_yx&2&\log_yz\\\log_zx&\log_zy&3\end{bmatrix}$. Then …
  22. If $f(\theta)=\begin{vmatrix}1&\cos\theta&1\\-\sin\theta&1&-\cos\theta\\-1&\sin\theta&1\end{vmatrix}$ and $A$ and $B$ are respectively the maximum and the minimum values …
  23. The number of values of $k$ for which the system of linear equations $(k+2)x+10y=k$ and $kx+(k+3)y=k-1$ has no solution is
  24. An ordered pair $(\alpha,\beta)$ for which the system of linear equations $(1+\alpha)x+\beta y+z=2$ $\alpha x+(1+\beta)y+z=3$ $\alpha x+\beta y+2z=2$ has a unique …
  25. If …
  26. A value of $\theta\in\left(0,\frac\pi3\right)$, for which …
  27. If $S=\left\{x\in[0,2\pi]:\begin{vmatrix}0&\cos x&-\sin x\\\sin x&0&\cos x\\\cos x&\sin x&0\end{vmatrix}=0\right\}$, then $\sum_{x\in S}\tan\left(\frac\pi3+x\right)$ is …
  28. If the system of linear equations $x-2y+kz=1$ $2x+y+z=2$ $3x-y-kz=3$ has a solution $(x,y,z)$, $z\neq0$, then $(x,y)$ lies on the straight line whose equation is :
  29. The greatest value of $c\in R$ for which the system of linear equations $x-cy-cz=0$, $cx-y+cz=0$, $cx+cy-z=0$ has a non-trivial solution, is
  30. If $\alpha,\beta\neq0$, and $f(n)=\alpha^{n}+\beta^{n}$ and …
  31. If the system of linear equations $2x+2y+3z=a$, $3x-y+5z=b$, $x-3y+2z=c$, where $a,b,c$ are non-zero real numbers, has more than one solution, then
  32. If $\Delta_r=\begin{vmatrix}r&2r-1&3r-2\\\frac n2&n-1&a\\\frac12n(n-1)&(n-1)^{2}&\frac12(n-1)(3n+4)\end{vmatrix}$, then the value of $\sum_{r=1}^{n-1}\Delta_r$
  33. If the system of equations $x+y+z=5$, $x+2y+3z=9$, $x+3y+\alpha z=\beta$ has infinitely many solutions, then $\beta-\alpha$ equals
  34. If $A=\begin{bmatrix}1&\sin\theta&1\\-\sin\theta&1&\sin\theta\\-1&-\sin\theta&1\end{bmatrix}$, then for all $\theta\in\left(\frac{3\pi}{4},\frac{5\pi}{4}\right)$ …
  35. If $S$ is the set of distinct values of $b$ for which the following system of linear equations $x+y+z=1$ $x+ay+z=1$ $ax+by+z=0$ has no solution, then $S$ is :
  36. Let $M$ and $m$ respectively be the maximum and the minimum values of …
  37. Let $f(x)=\begin{vmatrix}1+\sin^2x&\cos^2x&\sin2x\\\sin^2x&1+\cos^2x&\sin2x\\\sin^2x&\cos^2x&1+\sin2x\end{vmatrix}$, $x\in\left[\frac\pi6,\frac\pi3\right]$. If $\alpha$ …
  38. For the system of linear equations $\alpha x+y+z=1$, $x+\alpha y+z=1$, $x+y+\alpha z=\beta$, which one of the following statements is NOT correct?
  39. Let $A=[a_{ij}]=\begin{bmatrix}\log_5128&\log_45\\\log_58&\log_425\end{bmatrix}$. If $A_{ij}$ is the cofactor of $a_{ij}$, $C_{ij}=\sum_{k=1}^{2}a_{ik}A_{jk}$ …
  40. If $A=\begin{bmatrix}e^{t}&e^{-t}\cos t&e^{-t}\sin t\\e^{t}&-e^{-t}\cos t-e^{-t}\sin t&-e^{-t}\sin t+e^{-t}\cos t\\e^{t}&2e^{-t}\sin t&-2e^{-t}\cos t\end{bmatrix}$, then …
  41. The system of linear equations $x+\lambda y-z=0$ $\lambda x-y-z=0$ $x+y-\lambda z=0$ has a non-trivial solution for
  42. If the system of equations $x+2y+3z=3$, $4x+3y-4z=4$ and $8x+4y-\lambda z=9+\mu$ has infinitely many solutions, then the ordered pair $(\lambda,\mu)$ is equal to
  43. Let $a,b,c,d$ be in arithmetic progression with common difference $\lambda$. If $\begin{vmatrix}x+a-c&x+b&x+a\\x-1&x+c&x+b\\x-b+d&x+d&x+c\end{vmatrix}=2$, then value of …
  44. For the system of linear equations $x+y+z=6$ $\alpha x+\beta y+7z=3$ $x+2y+3z=14$ which of the following is NOT true?
  45. Let $A$ be a $n\times n$ matrix such that $|A|=2$. If the determinant of the matrix $\operatorname{Adj}\left(2\cdot\operatorname{Adj}\left(2A^{-1}\right)\right)$ is …
  46. If $\Delta_1=\begin{vmatrix}x&\sin\theta&\cos\theta\\-\sin\theta&-x&1\\\cos\theta&1&x\end{vmatrix}$ and …
  47. Let $\alpha,\beta$ $(\alpha\neq\beta)$ be the values of $m$, for which the equations $x+y+z=1$; $x+2y+4z=m$ and $x+4y+10z=m^{2}$ have infinitely many solutions. Then the …
  48. If $A$ is a $3\times3$ matrix such that $|5\,\mathrm{adj}A|=5$, then $|A|$ is equal to
  49. Consider the system of equations : $x+ay=0$, $y+az=0$ and $z+ax=0$. Then the set of all real values of $a$ for which the system has a unique solution is :
  50. The values of $\alpha$, for which $\begin{vmatrix}1&\frac32&\alpha+\frac32\\1&\frac13&\alpha+\frac13\\2\alpha+3&3\alpha+1&0\end{vmatrix}=0$, lie in the interval :

Other chapters