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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 51-100 of 281

  1. If the system of equations $x+y+az=b$ $2x+5y+2z=6$ $x+2y+3z=3$ has infinitely many solutions, then $2a+3b$ is equal to
  2. For the system of linear equations $(k-3)x+y+z=0$ $x+(k-3)y+z=0$ $x+y+(k-3)z=0$ the number of distinct values of $k$, for which it has a non-trivial solution, is
  3. Let the system of linear equations $x+y+\alpha z=2$ $3x+y+z=4$ $x+2z=1$ have a unique solution $(x^{*},y^{*},z^{*})$. If $(\alpha,x^{*})$, $(y^{*},\alpha)$ and …
  4. If $\alpha\neq a$, $\beta\neq b$, $\gamma\neq c$ and $\begin{vmatrix}\alpha&b&c\\a&\beta&c\\a&b&\gamma\end{vmatrix}=0$, then …
  5. Let $\theta\in\left(0,\frac{\pi}{2}\right)$. If the system of linear equations. $(1+\cos^{2}\theta)x+\sin^{2}\theta\,y+4\sin3\theta\,z=0$ …
  6. Let $A=\begin{pmatrix}[x+1]&[x+2]&[x+3]\\ [x]&[x+3]&[x+3]\\ [x]&[x+2]&[x+4]\end{pmatrix}$, where $[t]$ denotes the greatest integer less than or equal to $t$. If …
  7. Let the system of equations $x+2y+3z=5$, $2x+3y+z=9$, $4x+3y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda+2\mu$ is equal to :
  8. Let $S=\{\sqrt{n}:1\le n\le50$ and $n$ is odd$\}$. Let $a\in S$ and $A=\begin{bmatrix}1&0&a\\-1&1&0\\-a&0&1\end{bmatrix}$. If …
  9. If $A=\frac{1}{5!\,6!\,7!}\begin{bmatrix}5!&6!&7!\\6!&7!&8!\\7!&8!&9!\end{bmatrix}$, then $\left|\operatorname{adj}\left(\operatorname{adj}(2A)\right)\right|$ is equal to
  10. If the system of equations $2x+y-z=5$ $2x-5y+\lambda z=\mu$ $x+2y-5z=7$ has infinitely many solutions, then $(\lambda+\mu)^{2}+(\lambda-\mu)^{2}$ is equal to
  11. Let the system of equations $2x+3y+5z=9$, $7x+3y-2z=8$, $12x+3y-(4+\lambda)z=16-\mu$, have infinitely many solutions. Then the radius of the circle centred at …
  12. For the system of equations $x+y+z=6$ $x+2y+\alpha z=10$ $x+3y+5z=\beta$, which one of the following is NOT true?
  13. If the system of equations $x+4y-z=\lambda$, $7x+9y+\mu z=-3$, $5x+y+2z=-1$ has infinitely many solutions, then $(2\mu+3\lambda)$ is equal to :
  14. If $A=\begin{bmatrix}\sqrt{2}&1\\-1&\sqrt{2}\end{bmatrix}$, $B=\begin{bmatrix}1&0\\1&1\end{bmatrix}$, $C=ABA^{T}$ and $X=A^{T}C^{2}A$, then $\det X$ is equal to :
  15. For the system of linear equations $2x-y+3z=5$ $3x+2y-z=7$ $4x+5y+\alpha z=\beta$, which of the following is NOT correct?
  16. Let $A$ be a $3\times3$ matrix and $|A|=-1$. A matrix $B$ is obtained from the matrix $A$ by applying the following elementary row operations : $R_2\to R_2+3R_1$ …
  17. Let $\lambda,\mu\in\mathbf{R}$. If the system of equations $3x+5y+\lambda z=3$ $7x+11y-9z=2$ $97x+155y-189z=\mu$ has infinitely many solutions, then $\mu+2\lambda$ is …
  18. Let $A$ be a $3\times3$ matrix such that $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj}A))|=81$. If …
  19. Let $S$ be the set of all values of $\theta\in[-\pi,\pi]$ for which the system of linear equations $x+y+\sqrt{3}z=0$ $-x+(\tan\theta)y+\sqrt{7}z=0$ $x+y+(\tan\theta)z=0$ …
  20. The ordered pair $(a,b)$, for which the system of linear equations $3x-2y+z=b$ $5x-8y+9z=3$ $2x+y+az=-1$ has no solution, is :
  21. If the system of linear equations $x-2y+z=-4$ $2x+\alpha y+3z=5$ $3x-y+\beta z=3$ has infinitely many solutions, then $12\alpha+13\beta$ is equal to
  22. Let the determinant of a square matrix $A$ of order $m$ be $m-n$, where $m$ and $n$ satisfy $4m+n=22$ and $17m+4n=93$. If …
  23. If $f(x)=\begin{vmatrix}2\cos^{4}x&2\sin^{4}x&3+\sin^{2}2x\\3+2\cos^{4}x&2\sin^{4}x&\sin^{2}2x\\2\cos^{4}x&3+2\sin^{4}x&\sin^{2}2x\end{vmatrix}$, then $\frac{1}{5}f'(0)$ …
  24. If the following system of linear equations $2x+y+z=5$ $x-y+z=3$ $x+y+az=b$ has no solution, then :
  25. The positive value of the determinant of the matrix $A$, whose $\operatorname{Adj}(\operatorname{Adj}(A))=\begin{pmatrix}14&28&-14\\-14&14&28\\28&-14&14\end{pmatrix}$ …
  26. The system of linear equations $\lambda x+2y+2z=5$, $2\lambda x+3y+5z=8$, $4x+\lambda y+6z=10$ has :
  27. Let $A$ be a matrix of order $3\times3$ and $\det(A)=2$. Then $\det\left(\det(A)\operatorname{adj}\left(5\operatorname{adj}\left(A^{3}\right)\right)\right)$ is equal to
  28. Let $A$ be a $3\times3$ real matrix. If $\det\left(2\operatorname{Adj}\left(2\operatorname{Adj}\left(\operatorname{Adj}(2A)\right)\right)\right)=2^{41}$, then the value …
  29. If the system of linear equations $2x+y-z=7$ $x-3y+2z=1$ $x+4y+\delta z=k$, where $\delta,k\in\mathbf{R}$ has infinitely many solutions, then $\delta+k$ is equal to :
  30. Let $A=\begin{bmatrix}2&0&1\\1&1&0\\1&0&1\end{bmatrix}$, $B=[B_1,B_2,B_3]$, where $B_1,B_2,B_3$ are column matrices, and $AB_1=\begin{bmatrix}1\\0\\0\end{bmatrix}$ …
  31. Let $A$ and $B$ be two $3\times3$ real matrices such that $(A^2-B^2)$ is invertible matrix. If $A^5=B^5$ and $A^3B^2=A^2B^3$, then the value of the determinant of the …
  32. Let $a-2b+c=1$. If $f(x)=\begin{vmatrix}x+a&x+2&x+1\\x+b&x+3&x+2\\x+c&x+4&x+3\end{vmatrix}$, then :
  33. The number of $\theta\in(0,4\pi)$ for which the system of linear equations $3(\sin3\theta)x-y+z=2$, $3(\cos2\theta)x+4y+3z=3$, $6x+7y+7z=9$ has no solution, is :
  34. Let $A$ and $B$ be two $3\times3$ matrices such that $AB=I$ and $|A|=\frac{1}{8}$. Then $|\operatorname{adj}(B\operatorname{adj}(2A))|$ is equal to
  35. If the system of linear equations $2x+3y-z=-2$ $x+y+z=4$ $x-y+|\lambda|z=4\lambda-4$ where $\lambda\in\mathbf{R}$, has no solution, then
  36. Let for some real numbers $\alpha$ and $\beta$, $a=\alpha-i\beta$. If the system of equations $4ix+(1+i)y=0$ and …
  37. Let $A$ and $B$ be two square matrices of order $2$. If $\det(A)=2$, $\det(B)=3$ and $\det\left(\left(\det(5(\det A)B)\right)A^{2}\right)=2^{a}3^{b}5^{c}$ for some …
  38. Let $S$ be the set of all $\lambda\in\mathbf{R}$ for which the system of linear equations $2x-y+2z=2$, $x-2y+\lambda z=-4$, $x+\lambda y+z=4$ has no solution. Then the …
  39. Let $A=\{X=(x,y,z)^T:PX=0$ and $x^2+y^2+z^2=1\}$, where $P=\begin{bmatrix}1&2&1\\-2&3&-4\\1&9&-1\end{bmatrix}$, then the set $A$ :
  40. Let $S$ be the set of all integer solutions, $(x,y,z)$, of the system of equations $x-2y+5z=0$, $-2x+4y+z=0$, $-7x+14y+9z=0$ such that $15\le x^2+y^2+z^2\le150$. Then …
  41. Let $\alpha\beta\gamma=45$; $\alpha,\beta,\gamma\in\mathbb{R}$. If $x(\alpha,1,2)+y(1,\beta,2)+z(2,3,\gamma)=(0,0,0)$ for some $x,y,z\in\mathbb{R}$, $xyz\neq0$, then …
  42. If for some $\alpha$ and $\beta$ in $\mathbf{R}$, the intersection of the following three planes $x+4y-2z=1$, $x+7y-5z=\beta$, $x+5y+\alpha z=5$ is a line in …
  43. If $1$, $\log_{10}\left(4^{x}-2\right)$ and $\log_{10}\left(4^{x}+\frac{18}{5}\right)$ are in arithmetic progression for a real number $x$, then the value of the …
  44. If the minimum and the maximum values of the function $f:\left[\frac\pi4,\frac\pi2\right]\to\mathbf{R}$, defined by …
  45. The number of real values of $\lambda$, such that the system of linear equations $2x-3y+5z=9$, $x+3y-z=-18$, $3x-y+(\lambda^2-|\lambda|)z=16$ has no solutions, is
  46. Let $A$ be a $3\times3$ matrix and $\det(A)=2$. If …
  47. Let $I$ be the identity matrix of order $3\times3$ and for the matrix $A=\begin{bmatrix}\lambda&2&3\\4&5&6\\7&-1&2\end{bmatrix}$, $|A|=-1$. Let $B$ be the inverse of the …
  48. Let $f(x)=\begin{vmatrix}a&-1&0\\ax&a&-1\\ax^{2}&ax&a\end{vmatrix}$, $a\in\mathbf{R}$. Then the sum of the squares of all the values of $a$, for which …
  49. Let $f(x)=\begin{vmatrix}\sin^2x&-2+\cos^2x&\cos2x\\2+\sin^2x&\cos^2x&\cos2x\\\sin^2x&\cos^2x&1+\cos2x\end{vmatrix}$, $x\in[0,\pi]$. Then the maximum value of $f(x)$ is …
  50. Let $S$ be the set of values of $\lambda$, for which the system of equations $6\lambda x-3y+3z=4\lambda^{2}$, $2x+6\lambda y+4z=1$, $3x+2y+3\lambda z=\lambda$ has no …

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