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 201-250 of 281

  1. If the system of linear equations $x+ay+z=3$ $x+2y+2z=6$ $x+5y+3z=b$ has no solution, then
  2. If the system of linear equations $x+y+z=5$, $x+2y+2z=6$, $x+3y+\lambda z=\mu$, ($\lambda,\mu\in\mathbb R$), has infinitely many solutions, then the value of …
  3. Let $S$ be the set of all real values of $k$ for which the system of linear equations $x+y+z=2$ $2x+y-z=3$ $3x+2y+kz=4$ has a unique solution. Then $S$ is
  4. The number of values of $k$ for which the linear equations $4x+ky+2z=0$, $kx+4y+z=0$ and $2x+2y+z=0$ possess a non-zero solution is :
  5. The number of distinct real roots of the equation $\begin{vmatrix}\cos x&\sin x&\sin x\\\sin x&\cos x&\sin x\\\sin x&\sin x&\cos x\end{vmatrix}=0$ in the interval …
  6. If $f(x)=\begin{vmatrix}\cos x&x&1\\2\sin x&x^{2}&2x\\\tan x&x&1\end{vmatrix}$, then $\lim_{x\to0}\frac{f'(x)}{x}$
  7. If the system of linear equations $x-4y+7z=g$; $3y-5z=h$; $-2x+5y-9z=k$ is consistent, then :
  8. Let for any three distinct consecutive terms $a,b,c$ of an A.P., the lines $ax+by+c=0$ be concurrent at the point $P$ and $Q(\alpha,\beta)$ be a point such that the …
  9. Let $A=\begin{bmatrix}2&1&2\\6&2&11\\3&3&2\end{bmatrix}$ and $P=\begin{bmatrix}1&2&0\\5&0&2\\7&1&5\end{bmatrix}$. The sum of the prime factors of …
  10. If $\begin{vmatrix}x^{2}+x&x+1&x-2\\2x^{2}+3x-1&3x&3x-3\\x^{2}+2x+3&2x-1&2x-1\end{vmatrix}=ax-12$, then $a$ is equal to :
  11. If $\begin{vmatrix}x-4&2x&2x\\2x&x-4&2x\\2x&2x&x-4\end{vmatrix}=(A+Bx)(x-A)^2$, then the ordered pair $(A,B)$ is equal to
  12. The system of linear equations $x+y+z=2$ $2x+3y+2z=5$ $2x+3y+\left(a^2-1\right)z=a+1$
  13. If the system of linear equations, $x+y+z=6$ $x+2y+3z=10$ $3x+2y+\lambda z=\mu$ has more than two solutions, then $\mu-\lambda^2$, is equal to
  14. For $\alpha,\beta\in\mathbb R$, suppose the system of linear equations $x-y+z=5$ $2x+2y+\alpha z=8$ $3x-y+4z=\beta$ has infinitely many solutions. Then $\alpha$ and …
  15. If the system of linear equations : $x_1+2x_2+3x_3=6$ $x_1+3x_2+5x_3=9$ $2x_1+5x_2+ax_3=b$ is consistent and has infinite number of solutions, then :
  16. If the system of linear equations $x+ky+3z=0$ $3x+ky-2z=0$ $2x+4y-3z=0$ has a non-zero solution $(x,y,z)$, then $\frac{xz}{y^2}$ is equal to:
  17. If the system of equations $2x-y+z=4$, $5x+\lambda y+3z=12$, $100x-47y+\mu z=212$ has infinitely many solutions, then $\mu-2\lambda$ is equal to
  18. Let $A$ be a $3\times3$ matrix such that $\left|\operatorname{adj}(\operatorname{adj}(\operatorname{adj}A))\right|=12^{4}$. Then $\left|A^{-1}\operatorname{adj}A\right|$ …
  19. Consider the following system of equations $\alpha x+2y+z=1$ $2\alpha x+3y+z=1$ $3x+\alpha y+2z=\beta$ For some $\alpha,\beta\in R$. Then which of the following is NOT …
  20. For what values of $m$ does the system of equations $$3x+my=m,\qquad 2x-5y=20$$ have a solution satisfying the conditions $x>0$, $y>0$?
  21. Find the solution set of the system $$x+2y+z=1;\quad 2x-3y-w=2;\quad x\ge0;\ y\ge0;\ z\ge0;\ w\ge0.$$
  22. The equations $x+2y+2z=1$ and $2x+4y+4z=9$ have
  23. Let $S=\left\{A=\begin{pmatrix}0&1&c\\1&a&d\\1&b&e\end{pmatrix}: a,b,c,d,e\in\{0,1\}\text{ and }|A|\in\{-1,1\}\right\}$, where $|A|$ denotes the determinant of $A$. Then …
  24. Let $\mathbb{R}^2$ denote $\mathbb{R}\times\mathbb{R}$. Let $S=\{(a,b,c): a,b,c\in\mathbb{R}$ and $ax^2+2bxy+cy^2>0$ for all $(x,y)\in\mathbb{R}^2-\{(0,0)\}\}$. Then …
  25. Let $R=\left\{\begin{pmatrix}a&3&b\\c&2&d\\0&5&0\end{pmatrix}: a,b,c,d\in\{0,3,5,7,11,13,17,19\}\right\}$. Then the number of invertible matrices in $R$ is
  26. The values of $\theta$ lying between $\theta=0$ and $\theta=\pi/2$ and satisfying the equation $$\begin{vmatrix}1+\sin^{2}\theta&\cos^{2}\theta&4\sin4\theta\\ …
  27. Let $f(x)=\begin{vmatrix}\sec x&\cos x&\sec^{2}x+\cot x\operatorname{cosec}x\\ \cos^{2}x&\cos^{2}x&\operatorname{cosec}^{2}x\\ 1&\cos^{2}x&\cos^{2}x\end{vmatrix}$. Then …
  28. If $f_r(x),g_r(x),h_r(x)$, $r=1,2,3$ are polynomials in $x$ such that $f_r(a)=g_r(a)=h_r(a)$, $r=1,2,3$ and …
  29. If $\begin{vmatrix}6i&-3i&1\\4&3i&-1\\20&3&i\end{vmatrix}=x+iy$, then
  30. Let $\omega=-\dfrac12+i\dfrac{\sqrt3}{2}$, then the value of the det. $\begin{vmatrix}1&1&1\\1&-1-\omega^{2}&\omega^{2}\\1&\omega^{2}&\omega^{4}\end{vmatrix}$ is
  31. The number of distinct real roots of $\begin{vmatrix}\sin x&\cos x&\cos x\\ \cos x&\sin x&\cos x\\ \cos x&\cos x&\sin x\end{vmatrix}=0$ in the interval …
  32. Let $|A|=6$, where $A$ is a $3\times3$ matrix. If …
  33. Let $P=[p_{ij}]$ and $Q=[q_{ij}]$ be two square matrices of order $3$ such that $q_{ij}=2^{(i+j-1)}p_{ij}$ and $\det(Q)=2^{10}$. Then the value of …
  34. Let $n$ be the number obtained on rolling a fair die. If the probability that the system $x-ny+z=6$ $x+(n-2)y+(n+1)z=8$ $(n-1)y+z=1$ has a unique solution is $\frac k6$ …
  35. If the system of equations $3x+y+4z=3$ $2x+\alpha y-z=-3$ $x+2y+z=4$ has no solution, then the value of $\alpha$ is equal to :
  36. The system of linear equations $x+y+z=6$ $2x+5y+az=36$ $x+2y+3z=b$ has :
  37. Among the statements : I : If …
  38. Let $f(x)=\int\frac{7x^{10}+9x^8}{\left(1+x^2+2x^9\right)^2}dx$, $x>0$, $\lim_{x\to0}f(x)=0$ and $f(1)=\frac14$. If …
  39. Let $\omega\neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $$\begin{bmatrix}1&a&b\\ \omega&1&c\\ …
  40. Let $p,q,r$ be nonzero real numbers that are, respectively, the $10^{th}$, $100^{th}$ and $1000^{th}$ terms of a harmonic progression. Consider the system of linear …
  41. Let $\alpha,\beta$ and $\gamma$ be real numbers. Consider the following system of linear equations $$x+2y+z=7$$ $$x+\alpha z=11$$ $$2x-3y+\beta z=\gamma$$ Match each …
  42. Let $(x,y,z)$ be points with integer coordinates satisfying the system of homogeneous equations: $$3x-y-z=0$$ $$-3x+z=0$$ $$-3x+2y+z=0.$$ Then the number of such points …
  43. The number of all possible values of $\theta$, where $0<\theta<\pi$, for which the system of equations $$(y+z)\cos3\theta=(xyz)\sin3\theta$$ …
  44. Let $\omega$ be the complex number $\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}$. Then the number of distinct complex numbers z satisfying …
  45. Let $k$ be a positive real number and let $$A=\begin{bmatrix}2k-1&2\sqrt k&2\sqrt k\\2\sqrt k&1&-2k\\-2\sqrt k&2k&-1\end{bmatrix}\text{ and …
  46. Let $P=[a_{ij}]$ be a $3\times3$ matrix and let $Q=[b_{ij}]$, where $b_{ij}=2^{i+j}a_{ij}$ for $1\le i,j\le 3$. If the determinant of $P$ is 2, then the determinant of …
  47. If the adjoint of a $3\times3$ matrix $P$ is $\begin{bmatrix} 1 & 4 & 4 \\ 2 & 1 & 7 \\ 1 & 1 & 3 \end{bmatrix}$, then the possible value(s) of the determinant of $P$ is …
  48. Which of the following values of $\alpha$ satisfy the equation …
  49. The total number of distinct $x\in\mathbb{R}$ for which $\begin{vmatrix}x&x^2&1+x^3\\2x&4x^2&1+8x^3\\3x&9x^2&1+27x^3\end{vmatrix}=10$ is
  50. For a real number $\alpha$, if the system …

Other chapters