Matrices
199 JEE Maths previous year questions on Matrices — options free on every question; 20 include the answer & explanation free, the rest unlock with PYQ Pass.
Let \(A=\left[\begin{matrix}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{matrix}\right]\). If \({A}^{3}=4{A}^{2}-A-21I\), where \(I\) is the identity matrix of order \(3\times 3\), then \(2a+3b\) is equal to
[JEE Main 2024, 8 Apr (Shift 1)]
-13
Given \(A^3-4A^2+A+21I=O\)
Therefore every eigenvalue of \(A\) satisfies
\(x^3-4x^2+x+21=0\)
\(=(x-3)(x^2-x-7)\)
Since the factors are distinct, the eigenvalues of \(A\) are
\(3,\ \dfrac{1+\sqrt{29}}{2},\ \dfrac{1-\sqrt{29}}{2}\)
The trace of \(A\) is
\(\operatorname{tr}(A)=2+3+b=5+b\)
Also, the sum of eigenvalues is
\(3+\dfrac{1+\sqrt{29}}{2}+\dfrac{1-\sqrt{29}}{2}=4\)
\(5+b=4\)
\(\Rightarrow b=-1\)
\(\det(A)=2(3b-5)-ab\)
Putting \(b=-1\),
\(\det(A)=-16+a\)
The product of eigenvalues is
\(3\cdot\dfrac{1+\sqrt{29}}{2}\cdot\dfrac{1-\sqrt{29}}{2}\)
\(=3\cdot\dfrac{1-29}{4}\)
\(=-21\)
\(-16+a=-21\)
\(\Rightarrow a=-5\)
\(2a+3b=2(-5)+3(-1)\)
\(=-10-3\)
\(=-13\)
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
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\(x-y=3\)
\( \Delta=\left|\begin{array}{lll}1 & 5 & 6 \\2 & 3 & 4 \\1 & 6 & a\end{array}\right|=0 \)
\( \Rightarrow(3 a-24)-5(2 a-4)+6(9)=0 \)
\( \Rightarrow-7 a+50=0 \)
\( a=\frac{50}{7}\)
Now \( \Delta_z=0 \Rightarrow\left|\begin{array}{lll}1 & 5 & 4 \\2 & 3 & 7 \\1 & 6 & b\end{array}\right|=0 \)
\(\Rightarrow(3 b-42)-5(2 b-7) +4(9)=0 \)
\( \Rightarrow-7 b+29=0\)
\(b=\frac{29}{7} \)
\( a-b=3 \)
\( \therefore x-y=3\)
If for the system of linear equations having infinite solutions
\(\begin{aligned}& (\lambda-4) x+(\lambda-2) y+\lambda z=0 \\& 2 x-3 y+5 z=0 \\& x+2 y+6 z=0\end{aligned}\)
then \(\lambda^2+\lambda\) is
\(\frac{9}{2}\)
\(\begin{aligned}&\text { Given system has infinite solution }\\&& \Rightarrow\left|\begin{array}{ccc}\lambda-4 & \lambda-2 & \lambda \\2 & -3 & 5 \\1 & 2 & 6\end{array}\right|=0 \\& \Rightarrow(\lambda-4)(-28)-(\lambda-2) 7+\lambda \times 7=0 \\& \Rightarrow-4(\lambda-4)-(\lambda-2)+d=0 \\& \Rightarrow-4 \lambda+16-\lambda+2+\lambda=0 \\& =-4 \lambda=-18 \quad\lambda=\frac{9}{2}\end{aligned}\)
The values of \(m, n\), for which the system of equations
\(x+y+z=4,\\ 2x+5y+5z=17,\\ x+2y+mz=n\)
has infinitely many solutions, satisfy the equation:
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\({m}^{2}+{n}^{2}-mn=39\)
For infinitely many solutions, Cramer's rule requires
\(D=0,\quad D_1=0,\quad D_2=0,\quad D_3=0\).
The coefficient determinant is
\(D=\begin{vmatrix}1\&1\&12\&5\&51\&2\&m\end{vmatrix}\).
Expanding along the first row,
\(D=\begin{vmatrix}5\&52\&m\end{vmatrix}-\begin{vmatrix}2\&51\&m\end{vmatrix}+\begin{vmatrix}2\&51\&2\end{vmatrix}\)
\(=(5m-10)-(2m-5)+(4-5)\)
\(=3m-6\).
Since \(D=0\),
\(3m-6=0\),
\(m=2\).
Now
\(D_1=\begin{vmatrix}4\&1\&117\&5\&5n\&2\&2\end{vmatrix}\).
Since the second and third columns are identical,
\(D_1=0\).
Next,
\(D_2=\begin{vmatrix}1\&4\&12\&17\&51\&n\&2\end{vmatrix}\).
Expanding along the first row,
\(D_2=\begin{vmatrix}17\&5n\&2\end{vmatrix}-4\begin{vmatrix}2\&51\&2\end{vmatrix}+\begin{vmatrix}2\&171\&n\end{vmatrix}\)
\(=(34-5n)-4(4-5)+(2n-17)\)
\(=21-3n\).
Since \(D_2=0\),
\(21-3n=0\),
\(n=7\).
Also,
\(D_3=\begin{vmatrix}1\&1\&42\&5\&171\&2\&7\end{vmatrix}\)
\(=\begin{vmatrix}1\&1\&42\&5\&171\&2\&7\end{vmatrix}\)
\(=0\)
after substituting \(n=7\), confirming consistency.
Thus
\(m=2,\quad n=7\).
Now,
\(m^2+n^2-mn=2^2+7^2-(2)(7)\)
\(=4+49-14\)
\(=39\).
Let \(A=\left[{a}_{ij}\right]\) be a matrix of order \(3 \times 3\), with \(a_{i j}=(\sqrt{2})^{i+j}\). If the sum of all the elements in the third row of \(A^2\) is \(\alpha+\beta \sqrt{2}, \alpha, \beta \in Z\), then \(\alpha+\beta\) is equal to :
[JEE Main 2025, 29 Jan (Shift 2)]
224
\(A=\left[\begin{matrix}{(\sqrt{2})}^{2} & {(\sqrt{2})}^{3} & {(\sqrt{2})}^{4} \\ {(\sqrt{2})}^{3} & {(\sqrt{2})}^{4} & {(\sqrt{2})}^{5} \\ {(\sqrt{2})}^{4} & {(\sqrt{2})}^{5} & {(\sqrt{2})}^{6}\end{matrix}\right]\)
\(A=\left[\begin{matrix}2 & 2\sqrt{2} & 4 \\ 2\sqrt{2} & 4 & 4\sqrt{2} \\ 4 & 4\sqrt{2} & 8\end{matrix}\right]\)
\({A}^{2}={2}^{2}\left[\begin{matrix}1 & \sqrt{2} & 2 \\ \sqrt{2} & 2 & 2\sqrt{2} \\ 2 & 2\sqrt{2} & 4\end{matrix}\right]\left[\begin{matrix}1 & \sqrt{2} & 2 \\ \sqrt{2} & 2 & 2\sqrt{2} \\ 2 & 2\sqrt{2} & 4\end{matrix}\right]\)
\(=4\left[\begin{matrix}− & − & − \\ − & − & − \\ (14) & (14\sqrt{2}) & (28)\end{matrix}\right]\)
Sum of the all elements of third row
\(=4(14+14\sqrt{2}+28)\\ =4(42+14\sqrt{2})\\ =168+56\sqrt{2}\)
\(=\alpha +\beta \sqrt{2}\\ ∴\alpha +\beta =168+56=224\)
Consider the system of linear equations \(x+y+z=5\), \(x+2 y+\lambda^2 z=9, x+3 y+\lambda z=\mu\), where \(\lambda, \mu \in R\). Then, which of the following statement is NOT correct?
System has unique solution if \(\lambda \neq 1\) and \(\mu \neq 13\)
\(∆=\left|\begin{matrix}1 & 1 & 1 \\ 1 & 2 & {\lambda }^{2} \\ 1 & 3 & \lambda \end{matrix}\right|=1\left(3-2\right)-{\lambda }^{2}\left(3-1\right)+\lambda \left(2-1\right)\\ =-\left(2{\lambda }^{2}-\lambda -1\right)\\ =-\left(\lambda -1\right)\left(2\lambda +1\right)\)
For unique solution \(\lambda \neq 1\) and \(\lambda \neq-\frac{1}{2}\)
If \(\lambda = 1\)
\({\Delta }_{x}=\left|\begin{matrix}5 & 1 & 1 \\ 9 & 2 & 1 \\ \mu & 3 & 1\end{matrix}\right|=-\mu +13\)
\({\Delta }_{y}=\left|\begin{matrix}1 & 5 & 1 \\ 1 & 9 & 1 \\ 1 & \mu & 1\end{matrix}\right|=0\)
\({\Delta }_{z}=\left|\begin{matrix}1 & 1 & 5 \\ 1 & 2 & 9 \\ 1 & 3 & \mu \end{matrix}\right|=\mu -13\)
Infinite solution \(\lambda =1\&\mu =13\)
For no solution\(\lambda =1\&\mu \neq 13\)
when \(\lambda =-\frac{1}{2}\) and \(\mu \neq 13\) gives no solution.
If the system of equations
\(\begin{aligned}& 2 x+3 y-z=5 \\& x+\alpha y+3 z=-4 \\& 3 x-y+\beta z=7\end{aligned}\)
has infinitely many solutions, then \(13 \alpha \beta\) is equal to
[JEE Main 2024, 1 Feb (Shift 1)]
1120
\(If the system of equations
2x+3y−z=5x+αy+3z=−43x−y+βz=7
has infinitely many solutions, then 13αβ is equal to
\begin{aligned}&2=\mathrm{k}_1+3 \mathrm{k}_2, 3=\mathrm{k}_1 \alpha-\mathrm{k}_2,-1=3 \mathrm{k}_1+\beta \mathrm{k}_2,-5=4 \mathrm{k}_1-7 \mathrm{k}_2\\&\text { On solving we get }\\&& k_2=\frac{13}{19}, k_1=\frac{-1}{19}, \alpha=-70, \beta=\frac{-16}{13} \\& 13 \alpha \beta=13(-70)\left(\frac{-16}{13}\right) \\& =1120\end{aligned}\)
If the system of linear equations :
\(\begin{aligned}& x+y+2 z=6 \\& 2 x+3 y+a z=a+1 \\& -x-3 y+b z=2 b\end{aligned}\)
where \(\mathrm{a}, \mathrm{b} \in \mathbf{R}\), has infinitely many solutions, then \(7 a+3 b\) is equal to :
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16
\(\text{using cramer's rule :}\\ \Delta =\left|\begin{matrix}1 & 1 & 2 \\ 2 & 3 & \mathrm{a} \\ -1 & -3 & \mathrm{b}\end{matrix}\right|=0\\ \Rightarrow 2\mathrm{a}+\mathrm{b}-6=0...\text{(1)}\\ {\Delta }_{3}=\left|\begin{matrix}1 & 1 & 6 \\ 2 & 3 & a+1 \\ -1 & -3 & 2b\end{matrix}\right|=0\\ \Rightarrow \mathrm{a}+\mathrm{b}-8=0...\text{(2)}\\ \text{from eqn}\left(1\right)\text{and}\left(2\right)\\ a=-2,b=10\\ \Rightarrow 7a+3b=16\)
Let the system of equations \(x+2 y+3 z=5,2 x+3 y+z=9,4 x+3 y+\lambda z=\mu\) have infinite number of solutions. Then \(\lambda+2 \mu\) is equal to :
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17
\(\begin{matrix} & x+2y+3z=5 \\ & 2x+3y+z=9 \\ & 4x+3y+\lambda z=\mu \end{matrix}\)
for infinite following \(\Delta ={\Delta }_{1}={\Delta }_{2}={\Delta }_{3}=0\)
\(\Delta =\left|\begin{matrix}1 & 2 & 3 \\ 2 & 3 & 1 \\ 4 & 3 & \lambda \end{matrix}\right|=0\Rightarrow \lambda =-13\)
\({\Delta }_{1}=\left|\begin{matrix}5 & 2 & 3 \\ 9 & 3 & 1 \\ \mu & 3 & -13\end{matrix}\right|=0\Rightarrow \mu =15\)
\({\Delta }_{2}=\left|\begin{matrix}1 & 5 & 3 \\ 2 & 9 & 1 \\ 4 & 15 & -13\end{matrix}\right|=0\)
\({\Delta }_{3}=\left|\begin{matrix}1 & 2 & 5 \\ 2 & 3 & 9 \\ 4 & 3 & 15\end{matrix}\right|=0\)
for \(\lambda =-13,\mu =15\) system of equation has infinite solution hence \(\lambda+2 \mu=17\).
Let \(\mathrm{A}=\left[\begin{matrix}\alpha & -1 \\ 6 & \beta \end{matrix}\right],\alpha >0\), such that \(\det (A)=0\) and \(\alpha +\beta =1\). If \(I\) denotes \(2\times 2\) identity matrix, then the matrix \((I+\mathrm{A}{)}^{8}\) is:
[JEE Main 2025, 2 Apr (Shift 1)]
\(\left[\begin{matrix}766 & -255 \\ 1530 & -509\end{matrix}\right]\)
We are given:
- \(A=[\begin{matrix}\alpha & −1 \\ 6 & \beta \end{matrix}]\) with \(\alpha >0\).
- \(\det (A)=0\text{ }⟹\text{ }\alpha \beta −(−1)(6)=0\text{ }\)\(⟹\text{ }\alpha \beta +6=0\text{ }⟹\text{ }\alpha \beta =−6\).
- \(\alpha +\beta =1\text{ }⟹\text{ }\beta =1−\alpha\).
Substitute \(\beta\) in the determinant equation:
\(\alpha (1−\alpha )=−6\) \(\alpha −{\alpha }^{2}=−6\) \({\alpha }^{2}−\alpha −6=0\)
Factoring the quadratic equation:
\((\alpha −3)(\alpha +2)=0\)
Since it's given that \(\alpha >0\), we must have \(\alpha =3\).
Then, \(\beta =1−3=−2\).
Thus, the matrix \(A\) is: \(A=[\begin{matrix}3 & −1 \\ 6 & −2\end{matrix}]\)
\({A}^{2}=[\begin{matrix}3 & −1 \\ 6 & −2\end{matrix}][\begin{matrix}3 & −1 \\ 6 & −2\end{matrix}]\)\(=[\begin{matrix}9−6 & −3+2 \\ 18−12 & −6+4\end{matrix}]=[\begin{matrix}3 & −1 \\ 6 & −2\end{matrix}]=A\)
Since \({A}^{2}=A\), matrix \(A\) is idempotent. .
Using the binomial expansion for matrices that commute (since \(I\) and \(A\) always commute):
\((I+A{)}^{8}={\sum }_{k=0}^{8}(\frac{8}{k}){I}^{8−k}{A}^{k}\)\(=I+{\sum }_{k=1}^{8}(\frac{8}{k}){A}^{k}\)
Because null for all \(k\geq 1\):
\((I+A{)}^{8}=I+({\sum }_{k=1}^{8}(\frac{8}{k}))A\)
We know that \({\sum }_{k=0}^{8}(\frac{8}{k})={2}^{8}=256\),
so \({\sum }_{k=1}^{8}(\frac{8}{k})={2}^{8}−(\frac{8}{0})=256−1=255\).
\((I+A{)}^{8}=I+255A\)
Substitute \(I\) and \(A\) back into the formula:
\((I+A{)}^{8}=[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}]+255[\begin{matrix}3 & −1 \\ 6 & −2\end{matrix}]\)
\((I+A{)}^{8}=[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}]+[\begin{matrix}765 & −255 \\ 1530 & −510\end{matrix}]\)
\((I+A{)}^{8}=[\begin{matrix}1+765 & −255 \\ 1530 & 1−510\end{matrix}]\)\(=[\begin{matrix}766 & −255 \\ 1530 & −509\end{matrix}]\)
Consider the system of linear equations in \(x, y, z:\)
\(x+2 y+t z=0\),
\(6 x+y+5 t z=0\),
\(3 x+t^2 y+f(t) z=0\),
where \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a differentiable function. If this system has infinitely many solutions for all \(t \in \mathbb{R}\), then \(f\):
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is strictly increasing on \(\mathbb{R}\)
\(D=\left|\begin{array}{ccc}1 & 2 & t \\ 6 & 1 & 5 t \\ 3 & t^2 & f(t)\end{array}\right|=0\)
\(\Rightarrow 1\left(\mathrm{f}(\mathrm{t})-5 \mathrm{t}^3\right)-2(6 \mathrm{f}(\mathrm{t})-15 \mathrm{t})+\mathrm{t}\left(6 \mathrm{t}^2-3\right)=0\)
\(f(t)=\frac{t^3+27 t}{11}\)
\(f^{\prime}(\mathrm{t})=\frac{1}{11}\left(3 \mathrm{t}^2+27\right)>0 \ \forall \mathrm{t} \in \mathrm{R}\)
\(\Rightarrow f(t)\) is strictly increasing on \(\mathbb{R}\)
Let \(\mathrm{A}=\left[\begin{matrix}\alpha & -1 \\ 6 & \beta \end{matrix}\right],\alpha >0\), such that \(\det (A)=0\) and \(\alpha +\beta =1\). If \(I\) denotes \(2\times 2\) identity matrix, then the matrix \((I+\mathrm{A}{)}^{8}\) is:
[JEE Main 2025, 2 Apr (Shift 1)]
\(\left[\begin{matrix}766 & -255 \\ 1530 & -509\end{matrix}\right]\)
1. Find \(\alpha\) and \(\beta\) :
Given \(\det (A)=0⟹\alpha \beta -(-1)(6)=0⟹\alpha \beta +6=0⟹\alpha \beta =-6\).
Given \(\alpha +\beta =1⟹\beta =1-\alpha\).
Substitute \(\beta\) into the first equation: \(\alpha (1-\alpha )=-6⟹\alpha -{\alpha }^{2}=-6⟹{\alpha }^{2}-\alpha -6=0\).
Factorizing the quadratic: \((\alpha -3)(\alpha +2)=0\).
Since \(\alpha >0\), we have \(\alpha =3\). Consequently, \(\beta =1-3=-2\).
Therefore, matrix \(A=\left[\begin{matrix}3 & -1 \\ 6 & -2\end{matrix}\right]\).
2. Check properties of A :
Let's compute \({A}^{2}\) :
\({A}^{2}=\left[\begin{matrix}3 & -1 \\ 6 & -2\end{matrix}\right]\left[\begin{matrix}3 & -1 \\ 6 & -2\end{matrix}\right]\\ =\left[\begin{matrix}9-6 & -3+2 \\ 18-12 & -6+4\end{matrix}\right]=\left[\begin{matrix}3 & -1 \\ 6 & -2\end{matrix}\right]=A.\)
Since \({A}^{2}=A\), matrix A is an idempotent matrix. This means \({A}^{n}=A\) for any positive integer n.
3. Compute \((I+A{)}^{8}\) :
Using binomial expansion: \((I+A{)}^{8}=I+\left(\frac{8}{1}\right)A+\left(\frac{8}{2}\right){A}^{2}+⋯+\left(\frac{8}{8}\right){A}^{8}\).
Since \({A}^{k}=A\) for all \(k\geq 1\) :
\((I+A{)}^{8}=I+A\left[\left(\frac{8}{1}\right)+\left(\frac{8}{2}\right)+⋯+\left(\frac{8}{8}\right)\right].\)
The sum of binomial coefficients \(\sum _{k=1}^{n}\left(\frac{n}{k}\right)={2}^{n}-1\).
So, \((I+A{)}^{8}=I+\left({2}^{8}-1\right)A=I+(256-1)A=I+255A\).
\((I+A{)}^{8}=\left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right]+255\left[\begin{matrix}3 & -1 \\ 6 & -2\end{matrix}\right]\\ =\left[\begin{matrix}1+765 & 0-255 \\ 0+1530 & 1-510\end{matrix}\right]=\left[\begin{matrix}766 & -255 \\ 1530 & -509\end{matrix}\right]\)
Consider the matrix \(P=\left(\begin{matrix}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{matrix}\right)\) Let the transpose of a matrix \(X\) be denoted by \(X^T\). Then the number of \(3 \times 3\) invertible matrices \(Q\) with integer entries, such that \({Q}^{-1}={Q}^{T}\text{ and }PQ=QP\) is
[JEE Advanced 2025]
16
Given
\(Q^{-1}=Q^T\)
So \(Q\) is an orthogonal matrix.
Since \(Q\) has integer entries, each row and each column must have exactly one non-zero entry equal to \(1\) or \(-1\).
Thus \(Q\) is a signed permutation matrix.
Now
\(PQ=QP\)
with
\(P=\operatorname{diag}(2,2,3)\).
Let
\(Q=(q_{ij})\).
Since \(PQ=QP\),
\((p_i-p_j)q_{ij}=0\)
for all \(i,j\),
where
\(p_1=p_2=2,\quad p_3=3\).
Hence
\(q_{13}=q_{31}=q_{23}=q_{32}=0\).
Therefore \(Q\) must have the form
\(\begin{pmatrix}*&*&0\\*&*&0\\0&0&*\end{pmatrix}\),
where \(Q\) is a signed permutation matrix.
The first two coordinates may be permuted in
\(2!\) ways.
Each non-zero entry can independently be assigned sign \(+1\) or \(-1\).
There are \(3\) non-zero entries, giving
\(2^3\) sign choices.
Therefore total number of such matrices
\(=2!\times2^3\)
\(=2\times8\)
\(=16\).
Let \(\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]\) and \(\mathrm{P}=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right], \theta>0\). If \(\mathrm{B}=\mathrm{PAP}^{\mathrm{T}}, \mathrm{C}=\mathrm{P}^{\mathrm{T}} \mathrm{B}^{10} \mathrm{P}\) and the sum of the diagonal elements of\(C\) is \(\frac{m}{n}\), where \(\operatorname{gcd}(m, n)=1\), then \(m+n\) is :
65
\(\mathrm{Given}:\mathrm{A}=\left[\begin{matrix}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{matrix}\right]\\ \text{and}P=\left[\begin{matrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{matrix}\right]\\ \text{Clearly}{\mathrm{P}}^{\mathrm{T}}\mathrm{P}=\text{P}{\mathrm{P}}^{\mathrm{T}}=\text{I}\\ \mathrm{B}={\mathrm{PAP}}^{T}\\ \mathrm{Pre}\mathrm{multiply}\mathrm{by}{\mathrm{P}}^{\mathrm{T}}\\ {\mathrm{P}}^{\mathrm{T}}\mathrm{B}={\mathrm{P}}^{\mathrm{T}}{\mathrm{PAP}}^{\mathrm{T}}={\mathrm{AP}}^{\mathrm{T}}\\ \mathrm{Now}\mathrm{post}\mathrm{multiply}\mathrm{by}\mathrm{P}\\ {\mathrm{P}}^{\mathrm{T}}\mathrm{BP}={\mathrm{AP}}^{\mathrm{T}}\mathrm{P}=\mathrm{A}\\ So{\mathrm{A}}^{2}=\left({\mathrm{P}}^{\mathrm{T}}\mathrm{BP}\right)\left({P}^{\mathrm{T}}\mathrm{BP}\right)\\ {\mathrm{A}}^{2}={\mathrm{P}}^{\mathrm{T}}{\mathrm{B}}^{2}\mathrm{P}\\ \mathrm{Similarly}{\mathrm{A}}^{10}={\mathrm{P}}^{\mathrm{T}}{\mathrm{B}}^{10}\mathrm{P}=\mathrm{C}\\ \mathrm{A}=\left[\begin{matrix}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{matrix}\right]\\ \Rightarrow {\mathrm{A}}^{2}=\left[\begin{matrix}\frac{1}{2} & -\sqrt{2}-2 \\ 0 & 1\end{matrix}\right]\\ \mathrm{Similarly}\mathrm{check}{\mathrm{A}}^{3}\mathrm{and}\mathrm{so}\mathrm{on}\\ \mathrm{since}\mathrm{C}={\mathrm{A}}^{10}\\ \mathrm{Sum}\mathrm{of}\mathrm{diagonal}\mathrm{elements}\mathrm{of}\mathrm{C}\mathrm{is}{\left(\frac{1}{\sqrt{2}}\right)}^{10}+1\\ =\frac{1}{32}+1=\frac{33}{32}=\frac{\mathrm{m}}{\mathrm{n}}\\ \mathrm{m}+\mathrm{n}=65\\\)
Let \(A, B\) and \(C\) be three \(2 \times 2\) matrices with real entries such that \(B={(I+A)}^{−1}\) and \(A+C=I\). If \(BC=\left[\begin{matrix}1 & −5 \\ −1 & 2\end{matrix}\right]\) and \(CB\left[\begin{matrix}{x}_{1} \\ {x}_{2}\end{matrix}\right]=\left[\begin{matrix}12 \\ −6\end{matrix}\right]\), then \({x}_{1}+{x}_{2}\) is
[JEE Main 2026, 28 Jan (Shift 1)]
0
\(B={(I+A)}^{−1},A+C=I\)
\(\Rightarrow B\left(I+A\right)=\left(I+A\right)B=I\)
\(\Rightarrow B+BA=B+AB\)
\(\Rightarrow B+B\left(I−C\right)=B+\left(I−C\right)B\)
\(\Rightarrow 2B−BC=2B−CB\)
\(\Rightarrow BC=CB\)
\(∴CB\left[\begin{matrix}{x}_{1} \\ {x}_{2}\end{matrix}\right]=\left[\begin{matrix}1 & −5 \\ −1 & 2\end{matrix}\right]\left[\begin{matrix}{x}_{1} \\ {x}_{2}\end{matrix}\right]=\left[\begin{matrix}12 \\ −6\end{matrix}\right]\)
\(\Rightarrow \left[\begin{matrix}{x}_{1} \\ {x}_{2}\end{matrix}\right]={\left[\begin{matrix}1 & −5 \\ −1 & 2\end{matrix}\right]}^{−1}\left[\begin{matrix}12 \\ −6\end{matrix}\right]=−\frac{1}{3}\left[\begin{matrix}2 & 5 \\ 1 & 1\end{matrix}\right]\left[\begin{matrix}12 \\ −6\end{matrix}\right]\)
\(\Rightarrow \left[\begin{matrix}{x}_{1} \\ {x}_{2}\end{matrix}\right]=\left[\begin{matrix}2 \\ −2\end{matrix}\right]\)
\(∴{x}_{1}+{x}_{2}=0\)
\(\begin{aligned}&\text { If system of equations }\\&& x+2 y-3 z=2 \\& 2 x+\lambda y+5 z=5 \\& 4 x+3 y+\mu z=33 \text { has infinite solutions, then } \lambda+\mu \text { is equal to }\end{aligned}\)
\( \frac{1334}{5}\)
Ans. (2)
Sol.
\(\begin{aligned}& \Delta=\left|\begin{array}{lll}1 & 2 & -3 \\2 & \lambda & 5 \\4 & 3 & \mu\end{array}\right|=0 \\& 12 \lambda+\lambda \mu-4 \mu+7=0\ldots \ldots (i) \\& \Delta z=\left|\begin{array}{lll}1 & 2 & 2 \\2 & \lambda & 5 \\4 & 3 & 33\end{array}\right|=0 \\& \lambda=\frac{19}{5} \\& \text { from (i) } \mu=263 \\& \lambda+\mu=\frac{19}{5}+263=\frac{1334}{5}\end{aligned}\)
If the system of equations
\(\begin{aligned}& x+2 y-3 z=2 \\& 2 x+\lambda y+5 z=5 \\& 14 x+3 y+\mu z=33\end{aligned}\)
has infinitely many solutions, then \(\lambda+\mu\) is equal to :
[JEE Main 2025, 24 Jan (Shift 2)]
12
Given
\(x+2y-3z=2\)
\(2x+\lambda y+5z=5\)
\(14x+3y+\mu z=33\)
For infinitely many solutions, the third equation must be a linear combination of the first two:
Checking the standard condition for infinitely many solutions using determinants,
\(\Delta=\begin{vmatrix}1\&2\&-32\&\lambda\&514\&3\&\mu\end{vmatrix}=0\)
\(\Delta_x=\Delta_y=\Delta_z=0\)
\(\lambda=4,\qquad \mu=8\)
\(\lambda+\mu=12\)
Consider the system of linear equations \(x+y+z=5\), \(x+2 y+\lambda^2 z=9, x+3 y+\lambda z=\mu\), where \(\lambda, \mu \in R\). Then, which of the following statement is NOT correct?
System has unique solution if \(\lambda \neq 1\) and \(\mu \neq 13\)
\(\left|\begin{matrix}1 & 1 & 1 \\ 1 & 2 & {\lambda }^{2} \\ 1 & 3 & \lambda \end{matrix}\right|=0\\ \begin{matrix} & \Rightarrow 2{\lambda }^{2}−\lambda −1=0 \\ & \lambda =1,−\frac{1}{2}\end{matrix}\\ \left|\begin{matrix}1 & 1 & 5 \\ 2 & {\lambda }^{2} & 9 \\ 3 & \lambda & \mu \end{matrix}\right|=0\Rightarrow \mu =13\)
Infinite solution \(\lambda =1\&\mu =13\)
For unique solution \(\lambda \neq 1\), \(\mu \neq 13\)
For no solution\(\lambda =1\&\mu \neq 13\)
Considering the case when\(\lambda =-\frac{1}{2}\) and \(\mu \neq 13\) this will generate no solution case.
Let \(A=\left[\begin{matrix}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{matrix}\right]\&P=\left[\begin{matrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{matrix}\right],\theta >0\). If \(B=PA{P}^{T},C={P}^{T}{B}^{10}P\&\) the sum of the diagonal element of ' C ' is \(\frac{m}{n}\) where \(gcd(m,n)=1\), then \((m+n)\) is
65
\(P=\left[\begin{matrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{matrix}\right]\\ P{P}^{T}={P}^{T}P=I\\ {B}^{2}=PA{P}^{T}PA{P}^{T}\\ {B}^{2}=P{A}^{2}{P}^{T}\\ \mathrm{Similarly},{B}^{10}=P{A}^{10}{P}^{T}\\ C={P}^{T}{B}^{10}P\\ ={P}^{T}P{A}^{10}{P}^{T}P={A}^{10}\\ {A}^{2}=\left[\begin{matrix}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{matrix}\right]\left[\begin{matrix}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{matrix}\right]=\left[\begin{matrix}{\left(\frac{1}{\sqrt{2}}\right)}^{2} & - \\ 0 & 1\end{matrix}\right]\\ {A}^{3}=\left[\begin{matrix}{\left(\frac{1}{\sqrt{2}}\right)}^{2} & - \\ 0 & 1\end{matrix}\right]\left[\begin{matrix}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{matrix}\right]=\left[\begin{matrix}{\left(\frac{1}{\sqrt{2}}\right)}^{3} & - \\ 0 & 1\end{matrix}\right]\\ .\\ .\\ .\\ .\\ \mathrm{Similarly}\mathrm{for}{A}^{10}=\left[\begin{matrix}{\left(\frac{1}{\sqrt{2}}\right)}^{10} & - \\ 0 & 1\end{matrix}\right]\\ \mathrm{Sum}\mathrm{of}\mathrm{diagonal}\mathrm{elements}\mathrm{of}C=\frac{1}{32}+1=\frac{33}{32}=\frac{m}{n}\\ \mathrm{and}m+n=33+32=65.\)
If the system of equations
\(11x+y+\lambda z=-5\\ 2x+3y+5z=3\\ 8x-19y-39z=\mu\)
has infinitely many solutions, then \({\lambda }^{4}-\mu\) is equal to :
[JEE Main 2024, 5 Apr (Shift 1)]
47
\(\begin{aligned}&11x+y+\lambda z=-5\\&2x+3y+5z=3\\&8x-19y-39z=\mu\end{aligned}\)
For infinite solutions.
\(\begin{aligned}&D=\left|\begin{array}{ccc}11 & 1 & \lambda\\2 & 3 & 5\\8 & -19 & -39\end{array}\right|=0\\&\Rightarrow 11(-117+95)-1(-78-40)+\lambda(-38-24)\\&\Rightarrow 11(-22)+118-\lambda(62)=0\\&\Rightarrow 62\lambda=118-242\\&\Rightarrow \lambda=\frac{-124}{62}=-2\\[4pt]&D_1=\left|\begin{array}{ccc}-5 & 1 & -2\\3 & 3 & 5\\\mu & -19 & -39\end{array}\right|=0\\&\Rightarrow -5(-117+95)-1(-117-5\mu)-2(-57-3\mu)=0\\&\Rightarrow -5(-22)+117+5\mu+114+6\mu=0\\&\Rightarrow 11\mu=-110-231=-341\\&\Rightarrow \mu=-31\\[4pt]&\lambda^4-\mu=(-2)^4-(-31)=16+31=47\end{aligned}\)
Let \(M\) be a \(3 \times 3\) matrix such that \(\mathrm{M}\left(\begin{matrix}1 \\ 0 \\ 0\end{matrix}\right)=\left(\begin{matrix}1 \\ 2 \\ 3\end{matrix}\right),\mathrm{M}\left(\begin{matrix}0 \\ 1 \\ 0\end{matrix}\right)=\left(\begin{matrix}0 \\ 1 \\ 2\end{matrix}\right)\) and \(\mathrm{M}\left(\begin{matrix}0 \\ 0 \\ 1\end{matrix}\right)=\left(\begin{matrix}-1 \\ 1 \\ 1\end{matrix}\right).\text{ If }\mathrm{M}\left(\begin{matrix}x \\ y \\ z\end{matrix}\right)=\left(\begin{matrix}1 \\ 7 \\ 11\end{matrix}\right)\), then \(x+y+z\) equals:
[JEE Main 2026, 5 Apr (Shift 2)]
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If \(\text{A}=\left[\begin{matrix}2 & 3 \\ 3 & 5\end{matrix}\right]\), then the determinant of the matrix \((A^{2025}-3A^{2024}+A^{2023})\) is
[JEE Main 2026, 22 Jan (Shift 1)]
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\(\text{ Let }{a}_{ij}=(\sqrt{2}{)}^{i+j},A={\left[{a}_{ij}\right]}_{3\times 3}\text{. If sum of third row}\\ \text{of }{A}^{2}\text{ is }\alpha +\beta \sqrt{2}\text{, then }\alpha +\beta \text{ is }\)
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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:
[JEE Main 2026, 8 Apr (Shift 2)]
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Let \(\alpha ,\beta \in R\) 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 \(\frac{\beta }{\alpha }\) is equal to:
[JEE Main 2026, 2 Apr (Shift 1)]
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Let \(\alpha\) be a solution of \({x}^{2}+x+1=0\), and for some a and b in \(\mathrm{ℝ},\left[\begin{matrix}4 & \mathrm{a} & \mathrm{b}\end{matrix}\right]\left[\begin{matrix}1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8\end{matrix}\right]=\left[\begin{matrix}0 & 0 & 0\end{matrix}\right].\) If \(\frac{4}{{\alpha }^{4}}+\frac{\mathrm{m}}{{\alpha }^{\mathrm{a}}}+\frac{\mathrm{n}}{{\alpha }^{\mathrm{b}}}=3,\)then m + n is equal to
[JEE Main 2025, 8 Apr (Shift 1)]
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Let A be a 3 \(\times\)3 real matrix such that
\(A\left(\begin{matrix}1 \\ 0 \\ 1\end{matrix}\right)=2\left(\begin{matrix}1 \\ 0 \\ 1\end{matrix}\right),A\left(\begin{matrix}-1 \\ 0 \\ 1\end{matrix}\right)=4\left(\begin{matrix}-1 \\ 0 \\ 1\end{matrix}\right),A\left(\begin{matrix}0 \\ 1 \\ 0\end{matrix}\right)=2\left(\begin{matrix}0 \\ 1 \\ 0\end{matrix}\right)\)
Then, the system \(\left(A-3I\right)\left(\begin{matrix}x \\ y \\ z\end{matrix}\right)=\left(\begin{matrix}1 \\ 2 \\ 3\end{matrix}\right)\text{ has }\)
[JEE Main 2024, 31 Jan (Shift 2)]
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Which one of the following matrices can be obtained by performing elementary row transformations on the \(3 \times 3\) identity matrix ?
[JEE Advanced 2026]
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If \(A=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right], P=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\) and \(X=A P A^T\), then \(A^T X^{50} A\) is:
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The system of equations
\(\begin{aligned}& x+y+z=6 \\& x+2 y+5 z=9 \\& x+5 y+\lambda z=\mu\end{aligned}\)
has no solution if
[JEE Main 2025, 23 Jan (Shift 2)]
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If \(\text{A}=\left[\begin{matrix}2 & 3 \\ 3 & 5\end{matrix}\right]\), then the determinant of the matrix \((A^{2025}-3A^{2024}+A^{2023})\) is
[JEE Main 2026, 22 Jan (Shift 1)]
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The system of equations
\(\begin{aligned}& x+y+z=6 \\& x+2 y+5 z=9 \\& x+5 y+\lambda z=\mu\end{aligned}\)
has no solution if
[JEE Main 2025, 23 Jan (Shift 2)]
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If the system of linear equations
\(3\mathrm{x}+\mathrm{y}+\mathrm{βz}=3\\ 2\mathrm{x}+\mathrm{αy}-\mathrm{z}=-3\\ \mathrm{x}+2\mathrm{y}+\mathrm{z}=4\)
has infinitely many solutions, then the value of \(22\beta -9\alpha\) is:
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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
[JEE Main 2025, 24 Jan (Shift 1)]
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If the system of equation
\(2\mathrm{x}+\mathrm{λy}+3\mathrm{z}=5\\ 3\mathrm{x}+2\mathrm{y}-\mathrm{z}=7\\ 4\mathrm{x}+5\mathrm{y}+\mathrm{μz}=9\)
has infinitely many solutions, then \(\left({\lambda }^{2}+{\mu }^{2}\right)\) is equal to :
[JEE Main 2025, 2 Apr (Shift 2)]
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Let \(A, B\) and \(C\) be three \(2 \times 2\) matrices with real entries such that \(B={(I+A)}^{−1}\) and \(A+C=I\). If \(BC=\left[\begin{matrix}1 & −5 \\ −1 & 2\end{matrix}\right]\) and \(CB\left[\begin{matrix}{x}_{1} \\ {x}_{2}\end{matrix}\right]=\left[\begin{matrix}12 \\ −6\end{matrix}\right]\), then \({x}_{1}+{x}_{2}\) is
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Let \(R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)\) be a non-zero \(3 \times 3\) matrix, where \(x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0, \theta \in(0,2 \pi)\). For a square matrix \(M\), let trace \((M)\) denote the sum of all the diagonal entries of \(M\). Then, among the statements:
(I) \(\operatorname{Trace}(R)=0\)
(II) If trace \((\operatorname{adj}(\operatorname{adj}(R))=0\), then \(R\) has exactly one non-zero entry.
[JEE Main 2024, 30 Jan (Shift 2)]
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If \(A=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right], P=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\) and \(X=A P A^T\), then \(A^T X^{50} A\) is:
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Let \(R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)\) be a non-zero \(3 \times 3\) matrix, where \(x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0, \theta \in(0,2 \pi)\). For a square matrix \(M\), let trace \((M)\) denote the sum of all the diagonal entries of \(M\). Then, among the statements:
(I) \(\operatorname{Trace}(R)=0\)
(II) If trace \((\operatorname{adj}(\operatorname{adj}(R))=0\), then \(R\) has exactly one non-zero entry.
[JEE Main 2024, 30 Jan (Shift 2)]
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Let \(\alpha, \beta(\alpha \neq \beta)\) be the values of m , for which the equations \((x+y+z=1;x+2y+4z=m)\) and \(x+4 y+10 z=m^2\) have infinitely many solutions. Then the value of \(\sum_{n=1}^{10}\left(n^\alpha+n^\beta\right)\) is equal to :
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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
[JEE Main 2025, 24 Jan (Shift 1)]
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If the system of equations
\(x+\left(\sqrt{2}\sin \alpha \right)y+\left(\sqrt{2}\cos \alpha \right)z=0\\ x+\left(\cos \alpha \right)y+\left(\sin \alpha \right)z=0\\ x+\left(\sin \alpha \right)y-\left(\cos \alpha \right)z=0\)
has a non-trivial solution, then \(\alpha \in \left(0,\frac{\pi }{2}\right)\) is equal to :
[JEE Main 2024, 4 Apr (Shift 1)]
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\(\begin{aligned}&\text { If system of equations }\\&& x+2 y-3 z=2 \\& 2 x+\lambda y+5 z=5 \\& 4 x+3 y+\mu z=33 \text { has infinite solutions, then } \lambda+\mu \text { is equal to }\end{aligned}\)
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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 \((\lambda ,\mu )\) and touching the line \(4x=3y\) is
[JEE Main 2025, 7 Apr (Shift 1)]
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\(\text{ Let }{a}_{ij}=(\sqrt{2}{)}^{i+j},A={\left[{a}_{ij}\right]}_{3\times 3}\text{. If sum of third row}\\ \text{of }{A}^{2}\text{ is }\alpha +\beta \sqrt{2}\text{, then }\alpha +\beta \text{ is }\)
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If the system of linear equations :
\(\begin{aligned}& x+y+2 z=6 \\& 2 x+3 y+a z=a+1 \\& -x-3 y+b z=2 b\end{aligned}\)
where \(\mathrm{a}, \mathrm{b} \in {R}\), has infinitely many solutions, then \(7 a+3 b\) is equal to :
[JEE Main 2025, 22 Jan (Shift 2)]
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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:
[JEE Main 2026, 4 Apr (Shift 2)]
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If the system of linear equations
\(3\mathrm{x}+\mathrm{y}+\mathrm{βz}=3\\ 2\mathrm{x}+\mathrm{αy}-\mathrm{z}=-3\\ \mathrm{x}+2\mathrm{y}+\mathrm{z}=4\)
has infinitely many solutions, then the value of \(22 \beta-9 \alpha\) is :
[JEE Main 2025, 2 Apr (Shift 1)]
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Let the system of equations
\(x+5y-z=1\\ 4x+3y-3z=7\\ 24x+y+\lambda z=\mu\)
\(\lambda ,\mu \in \mathrm{R}\), have infinitely many solutions.
Then the number of the solutions of this system,If \(x,y,z\) are integers and satisfy \(7\leq x+y+z\leq 77\), is
[JEE Main 2025, 7 Apr (Shift 2)]
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If the system of equations
\((\lambda -1)x+(\lambda -4)y+\lambda z=5\\ \lambda x+(\lambda -1)y+(\lambda -4)z=7\\ (\lambda +1)x+(\lambda +2)y-(\lambda +2)z=9\)
has infinitely many solutions, then \({\lambda }^{2}+\lambda\) is equal to
[JEE Main 2025, 23 Jan (Shift 1)]
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Let \(B=\left[\begin{matrix}1 & 3 \\ 1 & 5\end{matrix}\right]\) and A be a \(2\times 2\) matrix such that \(A{B}^{-1}={A}^{-1}\). If \(BC{B}^{-1}=A\) and \({C}^{4}+\alpha {C}^{2}+\beta I=O\), then \(2\beta -\alpha\) is equal to
[JEE Main 2024, 9 Apr (Shift 2)]
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Let \(\lambda, \mu \in R\). If the system of equations
\(\begin{aligned}& 3 x+5 y+\lambda z=3 \\& 7 x+11 y-9 z=2 \\& 97 x+155 y-189 z=\mu\end{aligned}\)
has infinitely many solutions, then \(\mu+2 \lambda\) is equal to:
[JEE Main 2024, 9 Apr (Shift 1)]
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Let \(\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]\) and \(\mathrm{P}=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right], \theta>0\). If \(\mathrm{B}=\mathrm{PAP}^{\mathrm{T}}, \mathrm{C}=\mathrm{P}^{\mathrm{T}} \mathrm{B}^{10} \mathrm{P}\) and the sum of the diagonal elements of \(C\) is \(\frac{m}{n}\), where \(\operatorname{gcd}(m, n)=1\), then \(m+n\) is :
[JEE Main 2025, 28 Jan (Shift 2)]
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If the system of equation
\(2\mathrm{x}+\mathrm{λy}+3\mathrm{z}=5\\ 3\mathrm{x}+2\mathrm{y}-\mathrm{z}=7\\ 4\mathrm{x}+5\mathrm{y}+\mathrm{μz}=9\)
has infinitely many solutions, then \(\left({\lambda }^{2}+{\mu }^{2}\right)\) is equal to :
[JEE Main 2025, 2 Apr (Shift 2)]
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Let \(\alpha\) be a solution of \({x}^{2}+x+1=0\), and for some \(a\) and \(b\) in \(\mathrm{ℝ},\) \(\left[\begin{matrix}4 & \mathrm{a} & \mathrm{b}\end{matrix}\right]\left[\begin{matrix}1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8\end{matrix}\right]=\left[\begin{matrix}0 & 0 & 0\end{matrix}\right].\) If \(\frac{4}{{\alpha }^{4}}+\frac{\mathrm{m}}{{\alpha }^{\mathrm{a}}}+\frac{\mathrm{n}}{{\alpha }^{\mathrm{b}}}=3,\) then \(m+n\) is equal to
[JEE Main 2025, 8 Apr (Shift 1)]
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If \(A =\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right], B =\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right], C = ABA ^{ T }\) and \(X = A ^{ T } C ^2 A\), then \(\operatorname{det} X\) is equal to :
[JEE Main 2024, 01 Feb (Shift 1)]
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If \(A =\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right], B =\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right], C = ABA ^{ T }\) and \(X = A ^{ T } C ^2 A\), then \(\operatorname{det} X\) is equal to :
[JEE Main 2024, 01 Feb (Shift 1)]
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Consider the system of linear equations \(x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15\), where \(\lambda, \mu \in R\). Which one of the following statements is NOT correct?
[JEE Main 2024, 30 Jan (Shift 1)]
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Consider the system of linear equations \(x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15\), where \(\lambda, \mu \in R\). Which one of the following statements is NOT correct?
[JEE Main 2024, 30 Jan (Shift 1)]
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Let \(\alpha, \beta(\alpha \neq \beta)\) be the values of m , for which the equations \(x+y+z=1,x+2y+4z=m\) and \(x+4 y+10 z=m^2\) have infinitely many solutions. Then the value of \(\sum_{n=1}^{10}\left(n^\alpha+n^\beta\right)\) is equal to :
[JEE Main 2025, 29 Jan (Shift 2)]
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Let \(A=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix such that \(\mathrm{A}\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], \mathrm{A}\left[\begin{array}{l}4 \\ 1 \\ 3\end{array}\right]=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]\) and \(\mathrm{A}\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\), then \(\mathrm{a}_{23}\) equals:
[JEE Main 2025, 23 Jan (Shift 2)]
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If the system of linear equations
\( x-2 y+z=-4 \)
\( 2 x+\alpha y+3 z=5 \)
\( 3 x-y+\beta z=3\)
has infinitely many solutions, then \(12 \alpha+13 \beta\) is equal to
[JEE Main 2024, 31 Jan (Shift 1)]
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If the system of equations
\(\begin{aligned}& x+2 y-3 z=2 \\& 2 x+\lambda y+5 z=5 \\& 14 x+3 y+\mu z=33\end{aligned}\)
has infinitely many solutions, then \(\lambda+\mu\) is equal to :
[JEE Main 2025, 24 Jan (Shift 2)]
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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 \(\frac{k}{6},\) then the sum of \(k\) and all possible values of \(n\) is:
[JEE Main 2026, 22 Jan (Shift 2)]
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The sum of all possible values of \(\theta \in[0,2 \pi]\), for which the system of equations:
\( x \cos 3 \theta-8 y-12 z=0 \)
\( x \cos 2 \theta+3 y+3 z=0 \)
\( x+y+3 z=0\)
has a non-trivial solution, is equal to:
[JEE Main 2026, 6 Apr (Shift 2)]
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Let \(A=\left[\begin{matrix}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{matrix}\right]\&P=\left[\begin{matrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{matrix}\right],\theta >0\). If \(B=PA{P}^{T},C={P}^{T}{B}^{10}P\&\) the sum of the diagonal element of ' \(c\) ' is \(\frac{m}{n}\) where \(gcd(m,n)=1\), then (m+n) is
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Let \(A=\left[{a}_{ij}\right]\) be a matrix of order \(3 \times 3\), with \(a_{i j}=(\sqrt{2})^{i+j}\). If the sum of all the elements in the third row of \(A^2\) is \(\alpha+\beta \sqrt{2}, \alpha, \beta \in Z\), then \(\alpha+\beta\) is equal to :
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Let \(\mathrm{A}=\left[\begin{matrix}\alpha & -1 \\ 6 & \beta \end{matrix}\right],\alpha >0\), such that \(\det (A)=0\) and \(\alpha +\beta =1\). If \(I\) denotes \(2\times 2\) identity matrix, then the matrix \((I+\mathrm{A}{)}^{8}\) is:
[JEE Main 2025, 2 Apr (Shift 1)]
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If the system of equations
\((\lambda -1)x+(\lambda -4)y+\lambda z=5\\ \lambda x+(\lambda -1)y+(\lambda -4)z=7\\ (\lambda +1)x+(\lambda +2)y-(\lambda +2)z=9\)
has infinitely many solutions, then \({\lambda }^{2}+\lambda\) is equal to
[JEE Main 2025, 23 Jan (Shift 1)]
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If the system of equations \(x+4 y-z=\lambda, 7 x+9 y+\mu z=-3,5 x+y+2 z=-1\) has infinitely many solutions, then \((2 \mu+3 \lambda)\) is equal to :
[JEE Main 2024, 08 Apr (Shift 2)]
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If the system of equations \(x+4 y-z=\lambda, 7 x+9 y+\mu z=-3,5 x+y+2 z=-1\) has infinitely many solutions, then \((2 \mu+3 \lambda)\) is equal to :
[JEE Main 2024, 08 Apr (Shift 2)]
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Let the matrix \(\mathrm{A}=\left[\begin{matrix}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{matrix}\right]\) satisfy \({\mathrm{A}}^{\mathrm{n}}={\mathrm{A}}^{\mathrm{n}-2}+{\mathrm{A}}^{2}-\mathrm{I}\) for \(\mathrm{n}\geq 3\). Then the sum of all the elements of \({\mathrm{A}}^{50}\) is :-
[JEE Main 2025, 4 Apr (Shift 2)]
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Let \(A=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix such that \(\mathrm{A}\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], \mathrm{A}\left[\begin{array}{l}4 \\ 1 \\ 3\end{array}\right]=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]\) and \(\mathrm{A}\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\), then \(\mathrm{a}_{23}\) equals:
[JEE Main 2025, 23 Jan (Shift 2)]
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If the system of linear equations
\( x-2 y+z=-4 \)
\( 2 x+\alpha y+3 z=5 \)
\( 3 x-y+\beta z=3\)
has infinitely many solutions, then \(12 \alpha+13 \beta\) is equal to
[JEE Main 2024, 31 Jan (Shift 1)]
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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 \(\frac{k}{6},\) then the sum of \(k\) and all possible values of \(n\) is:
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Let the system of equations
\(x+5y-z=1\\ 4x+3y-3z=7\\ 24x+y+\lambda z=\mu\)
\(\lambda ,\mu \in \mathrm{R}\), have infinitely many solutions. Then the number of the solutions of this system, If \(x,y,z\) are integers and satisfy \(7\leq x+y+z\leq 77\), is
[JEE Main 2025, 7 Apr (Shift 2)]
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If for the system of linear equations having infinite solutions
\(\begin{aligned}& (\lambda-4) x+(\lambda-2) y+\lambda z=0 \\& 2 x-3 y+5 z=0 \\& x+2 y+6 z=0\end{aligned}\)
then \(\lambda^2+\lambda\) is
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Let \(A\) be a \(3\times 3\) real matrix such that \({A}^{2}(A-2I)-4(A-I)=O\) , where \(I\) and \(O\) are the identity and null matrices, respectively. If \({A}^{5}=\alpha {A}^{2}+\beta A+\gamma I\), where \(\alpha ,\beta\) and \(\gamma\) are real constants, then \(\alpha +\beta +\gamma\) is equal to:
[JEE Main 2025, 2 Apr (Shift 2)]
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Let \(A=\left[\begin{matrix}1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1\end{matrix}\right]\) and \(B=\left[{b}_{ij}\right],1\leq i,j\leq 3\). If \(B={A}^{99}-I\), then the value of \(\frac{b_{31}-b_{21}}{b_{32}}\) is:
[JEE Main 2026, 6 Apr (Shift 2)]
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Let the system of equations \(x+2 y+3 z=5,2 x+3 y+z=9,4 x+3 y+\lambda z=\mu\) have infinite number of solutions. Then \(\lambda+2 \mu\) is equal to :
[JEE Main 2024, 1 Feb (Shift 2)]
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For \(\alpha, \beta \in R\), suppose the system of linear equations
\[\begin{aligned}& x-y+z=5 \\& 2 x+2 y+\alpha z=8 \\& 3 x-y+4 z=\beta\end{aligned}\]
has infinitely many solutions. Then \(\alpha\) and \(\beta\) are the roots of
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For the system of linear equations
\(\begin{aligned}& 2 x+4 y+2 a z=b \\& x+2 y+3 z=4 \\& 2 x-5 y+2 z=8\end{aligned}\)
which of the following is NOT correct?
[JEE Main 2023, 13 Apr (Shift 1)]
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\(\text{ If }P=\left[\begin{matrix}1 & 0 \\ \frac{1}{2} & 1\end{matrix}\right]\text{, then }{P}^{50}\text{ is : }\)
[JEE Main 2021, 25 Jul (Shift 2)]
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If the system of linear equations
\(\begin{aligned}& 2 x+2 a y+a z=0 \\& 2 x+3 b y+b z=0 \\& 2 x+4 c y+c z=0\end{aligned}\)
where \(a, b, c \in R\) are non-zero and distinct; has a non-zero solution, then
[JEE Main 2020, 7 Jan (Shift 1)]
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Let \(A=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0\end{array}\right)\). Then \(A^{2025}-A^{2020}\) is equal to:
[JEE Main 2021, 26 Aug (Shift 2)]
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Let \(\alpha\) and \(\beta\) be real numbers. Consider \(3 \times 3\) matrix \(A\) such that \(A^2=3 A+\alpha I\). If \(A^4=21 A+\beta I\), then
[JEE Main 2023, 29 Jan (Shift 1)]
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Let the system of linear equations
\[\begin{aligned}& -x+2 y-9 z=7 \\& -x+3 y+7 z=9 \\& -2 x+y+5 z=8 \\& -3 x+y+13 z=\lambda\end{aligned}\]
has a unique solution \(x=\alpha, y=\beta, z=\gamma\). Then the distance of the point \((\alpha, \beta, \gamma)\) from the plane \(2 x-2 y+z=\lambda\) is
[JEE Main 2023, 15 Apr (Shift 1)]
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If the system of linear equations
\[\begin{aligned}& 7 x+11 y+\alpha z=13 \\& 5 x+4 y+7 z=\beta \\& 175 x+194 y+57 z=361\end{aligned}\]
has infinitely many solutions, then \(\alpha+\beta+2\) is equal to
[JEE Main 2023, 11 Apr (Shift 2)]
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The number of symmetric matrices of order 3 , with all the entries from the set \(\{0,1,2,3,4,5,6,7,8,9\}\), is:
[JEE Main 2023, 13 Apr (Shift 1)]
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If the system of equations
\[\begin{aligned}& 2 x+y-z=5 \\& 2 x-5 y+\lambda z=\mu \\& x+2 y-5 z=7\end{aligned}\]
has infinitely many solutions, then \((\lambda+\mu)^2+(\lambda-\mu)^2\) is equal to
[JEE Main 2023, 13 Apr (Shift 2)]
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Let \(\lambda \in R\). The system of linear equations
\(\begin{aligned}& 2 x_1-4 x_2+\lambda x_3=1 \\& x_1-6 x_2+x_3=2 \\& \lambda x_1-10 x_2+4 x_3=3\end{aligned}\)
is inconsistent for
[JEE Main 2020, 5 Sep (Shift 1)]
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Let \( A \) be a \( 2 \times 2 \) real matrix with entries from \( \{0,1\} \) and \( |A| \neq 0 \). Consider the following two statements:
(P) If \( \mathrm{A} \neq \mathrm{I}_{2} \), then \( |\mathrm{A}|=-1 \)
(Q) If \( |A|=1 \), then \( \operatorname{tr}(\mathrm{A})=2 \),
where \( \mathrm{I}_{2} \) denotes \( 2 \times 2 \) identity matrix and \( \operatorname{tr}(\mathrm{A}) \) denotes the sum of the diagonal entries of \( A \).
Then:
[JEE Main 2020, 2 Sep (Shift 1)]
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The values of \(a\) and \(b\), for which the system of equations \(2 x+3 y+6 z=8, x+2 y+a z=5,3 x+5 y+9 z=b\) has no solution, are:
[JEE Main 2021, 25 Jul (Shift 1)]
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Let \(A=\left[\begin{matrix}2 & 3 \\ a & 0\end{matrix}\right],a\in R\) be written as P+Q where P is a symmetric matrix and Q is skew symmetric matrix. If \(\det (Q)=9\), then the modulus of the sum of all possible values of determinant of P is equal to :
[JEE Main 2021, 20 Jul (Shift 1)]
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The values of \(\lambda\) and \(\mu\) such that the system of equations \(x+y+z=6,3 x+5 y+5 z=26, x+2 y+\lambda z=\mu\) has no solution, are :
[JEE Main 2021, 22 Jul (Shift 2)]
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Let \(S _1\) and \(S _2\) be respectively the sets of all \(a \in R -\{0\}\) for which the system of linear equations \(\begin{aligned}&a x+2 a y-3 a z=1\\ &(2 a+1) x+(2 a+3) y+(a+1) z=2 \\& (3 a+5) x+(a+5) y+(a+2) z=3\end{aligned}\)
has unique solution and infinitely many solutions. Then
[JEE Main 2023, 25 Jan (Shift 1)]
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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?
[JEE Main 2023, 1 Feb (Shift 2)]
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Consider the following system of questions
\(\begin{aligned}& \alpha x+2 y+z=1 \\& 2 \alpha x+3 y+z=1 \\& 3 x+\alpha y+2 z=\beta\end{aligned}\)
For some \(\alpha, \beta \in R\). Then which of the following is NOT correct.
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If \(A\) and \(B\) are two non-zero \(n \times n\) matrics such that \(A^2+\) \(B=A^2 B\), then
[JEE Main 2023, 24 Jan (Shift 1)]
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The value of k\(\in\)R, for which the following system of linear equations 3x – y + 4z = 3,
x + 2y – 3z = –2, 6x + 5y + kz = –3 has infinitely many solutions, is
[JEE Main 2021, 20 Jul (Shift 2)]
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If a point \(P(\alpha, \beta, \gamma)\) satisfying \((\alpha \beta \gamma)\left(\begin{array}{ccc}2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8\end{array}\right)=\left(\begin{array}{lll}0 & 0 & 0\end{array}\right)\) lies on the plane \(2 x+\) \(4 y+3 z=5\), then \(6 \alpha+9 \beta+7 \gamma\) is equal to:
[JEE Main 2023, 31 Jan (Shift 2)]
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Let \(P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\) and \(Q=P A P^T\). If \(P^T Q^{2007} P=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\), then \(2 a+b-3 c-4 d\) equal to
[JEE Main 2023, 8 Apr (Shift 1)]
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Let \(S\) be the set of all values of \(\theta \in[-\pi, \pi]\) for which the system of linear equations
\(\begin{aligned}& x+y+\sqrt{3} z=0 \\& -x+(\tan \theta) y+\sqrt{7} z=0\\&x+y+(\tan \theta) z=0\end{aligned}\)
has non-trivial solution. Then \(\frac{120}{\pi} \sum_{\theta \in s} \theta\) is equal to
[JEE Main 2023, 8 Apr (Shift 2)]
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Let \(A\) and \(B\) are \(3 \times 3\) real matrices such that \(A\) is symmetric matrix and \(B\) is skew-symmetric matrix. Then the system of linear equations \(\left(A^2 B^2-B^2 A^2\right) X=O\), where \(X\) is a \(3 \times 1\) column matrix of unknown variables and \(O\) is a \(3 \times 1\) null matrix has:
[JEE Main 2021, 24 Feb (Shift 2)]
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Let \( A=\left[a_{i j}\right] \), be a real matrix of order \( 3 \times 3 \), such that \( a_{i 1}+a_{i 2}+a_{i 3}=1 \) for \( i=1,2,3 \). Then, the sum of all entries of the matrix \( A^{3} \) is equal to:
[JEE Main 2021, 22 Jul (Shift 2)]
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Let \(S\) be the set of all \(\lambda \in R\) for which the system of linear equations
\(\begin{aligned}& 2 x-y+2 z=2\\ & x-2 y+\lambda z=-4 \\& x+\lambda y+z=4\end{aligned}\)
has no solution. Then the set \(S \quad\)
[JEE Main 2020, 2 Sep (Shift 1)]
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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
\((k+1)x+(2k−1)y=7\)
\((2k+1)x+(k+5)y=10\text{ has :}\)
[JEE Main 2023, 30 Jan (Shift 1)]
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Let \(a, b, c \in R \) be all non- zero and satisfy \( a^{3}+b^{3}+c^{3}=2 \). If the matrix \(A=\left(\begin{array}{lll}a & b & c \\b & c & a \\c & a & b\end{array}\right)\)
satisfies \( \mathrm{A}^{\mathrm{T}} \mathrm{A}=\mathrm{I} \), then a value of \(abc\) can be :
[JEE Main 2020, 2 Sep (Shift 2)]
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If a point \(P(\alpha, \beta, \gamma)\) satisfying
\([\alpha \quad \beta \quad \gamma]\left[\begin{array}{ccc}2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0\end{array}\right]\) lies on the plane \(2 x+\) \(4 y+3 z=5\), then \(6 \alpha+9 \beta+7 \gamma\) is equal to:
[JEE Main 2023, 31 Jan (Shift 2)]
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Let \(S\) denote the set of all real values of \(\lambda\) such that the system of equations
\(\begin{aligned}& \lambda x+y+z=1 \\& x+\lambda y+z=1 \\& x+y+\lambda z=1\end{aligned} \)
is inconsistent, then\(\sum _{\lambda \in S}\left(|\lambda {|}^{2}+|\lambda |\right)\) is equal to
[JEE Main 2023, 1 Feb (Shift 1)]
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Suppose \(p,q,r\neq 0\) and system of equation
\((p+a)x+by+cz=0\),
\(ax+(q+b)y+cz=0\),
\(ax+by+(r+c)z=0\) has a non-trivial solution, then value of \(\frac{a}{p}+\frac{b}{q}+\frac{c}{r}=\)
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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 correct.
[JEE Main 2023, 29 Jan (Shift 1)]
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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 \(x+y+z=4,3 x+2 y+5 z=3,9 x+4 y+(28+[\lambda]) z=[\lambda]\) has a solution is:
[JEE Main 2021, 27 Aug (Shift 2)]
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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:
[JEE Main 2021, 31 Aug (Shift 2)]
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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?
[JEE Main 2023, 13 Apr (Shift 1)]
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Let \(A=\left[\begin{array}{ll}2 & 3 \\ a & 0\end{array}\right], a \in R\) be written as \(P+Q\) where \(P\) is a symmetric matrix and \(Q\) is skew symmetric matrix. If \(\operatorname{det}(Q)=9\), then the modulus of the sum of all possible values of determinant of \(P\) is equal to:
[JEE Main 2021, 20 Jul (Shift 1)]
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Consider the following system of equations
\(\begin{aligned}& \alpha x+2 y+z=1 \\& 2 \alpha x+3 y+z=1 \\& 3 x+\alpha y+2 z=\beta\end{aligned}\)
For some \(\alpha, \beta \in R\). Then which of the following is NOT correct.
[JEE Main 2023, 29 Jan (Shift 1)]
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Let \(A\) be a \( 3 \times 3 \) matrix such that \( \operatorname{adj} A=\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1\end{array}\right] \) and \( B=\operatorname{adj}(\operatorname{adj} A) \).If \( |\mathrm{A}|=\lambda \) and \( \left|\left(\mathrm{B}^{-1}\right)^{\mathrm{T}}\right|=\mu \), then the ordered pair, \( (|\lambda|, \mu) \) is equal to
[JEE Main 2020, 3 Sep (Shift 2)]
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Let \(\lambda \in R\). The system of linear equations
\[\begin{aligned}& 2 x_1-4 x_2+\lambda x_3=1 \\& x_1-6 x_2+x_3=2 \\& \lambda x_1-10 x_2+4 x_3=3\end{aligned}\]is inconsistent for
[JEE Main 2020, 5 Sep (Shift 1)]
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Let a and b be real numbers. Consider 3 × 3 matrix A such that A2 = 3A + aI. If A4 = 21A +βI, then
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Let \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{array}\right]\). Then the sum of the diagonal elements of the matrix \((A+I)^{11}\) is equal to:
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Let \(\theta \in\left(0, \frac{\pi}{2}\right)\). If the system of linear equations.
\(\begin{aligned}& \left(1+\cos ^2 \theta\right) x+\sin ^2 \theta y+4 \sin 3 \theta z=0 \\& \cos ^2 \theta x+\left(1+\sin ^2 \theta\right) y+4 \sin 3 \theta z=0 \\& \cos ^2 \theta x+\sin ^2 \theta y+(1+4 \sin 3 \theta) z=0\end{aligned}\)
has a non trivial solution, then the value of \(\theta\) is:
[JEE Main 2021, 26 Aug (Shift 1)]
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Let \(A, B, C\) be \(3 \times 3\) matrices such that \(A\) is symmetric and \(B\) and \(C\) are skew-symmetric.
Consider the statements
(S1) \(A^{13} B^{26}-B^{26} A^{13}\) is symmetric
(S2) \(A^{26} C^{13}-C^{13} A^{26}\) is symmetric
Then,
[JEE Main 2023, 25 Jan (Shift 2)]
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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:
[JEE Main 2021, 25 Jul (Shift 1)]
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For the system of linear equations
\(\begin{aligned}& 2 x-y+3 z=5 \\& 3 x+2 y-z=7 \\& 4 x+5 y+\alpha z=\beta\end{aligned}\)
Which of the following is NOT correct?
[JEE Main 2023, 10 Apr (Shift 1)]
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Let \(A=\left[\begin{array}{cc}1 & \frac{1}{51} \\ 0 & 1\end{array}\right]\). If \(B=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right] A\left[\begin{array}{cc}-1 & -2 \\ 1 & 1\end{array}\right]\), then the sum of all the elements of the matrix \(\sum_{n=1}^{50} B^n\) is equal to
[JEE Main 2023, 12 Apr (Shift 1)]
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For the system of equations
\(\begin{aligned}& x+y+z=6 \\& x+2 y+\alpha z=10\\&x+3 y+5 z=\beta\end{aligned}\)
, which one of the following is NOT true?
[JEE Main 2023, 6 Apr (Shift 2)]
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For the system of linear equations: \(x-2 y=1, x-y+k z=-2, k y+4 z=6, k \in R\). Consider the following statements:
(A) The system has unique solution if \(k \neq 2, k \neq-2\).
(B) The system has unique solution if \(k=-2\).
(C) The system has unique solution if \(k=2\).
(D) The system has no solution if \(k=2\).
(E) The system has infinite number of solutions if \(k \neq-2\). Which of the following statements are correct?
[JEE Main 2021, 24 Feb (Shift 2)]
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For the system of equations
\(\begin{aligned}& x+y+z=6 \\& x+2 y+\alpha z=10\\&x+3 y+5 z=\beta\end{aligned}\),
which one of the following is NOT true?
[JEE Main 2023, 6 Apr (Shift 2)]
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If for the matrix \(A=\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right], A A^{\top}=I_2\), then the value of \(\alpha^4+\beta^4\) is:
[JEE Main 2021, 25 Feb (Shift 2)]
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For the system of linear equations: \[x-2 y=1, x-y+k z=-2, k y+4 z=6, k \in R \text {, }\] Consider the following statements:
(A) The system has unique solution if \(k \neq 2, k \neq-2\).
(B) The system has unique solution if \(k=-2\).
(C) The system has unique solution if \(k=2\).
(D) The system has no solution if \(k=2\).
(E) The system has infinite number of solutions if \(k \neq-2\). Which of the following statements are correct?
[JEE Main 2021, 24 Feb (Shift 2)]
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The system of linear equations
\(\begin{aligned}& 3 x-2 y-k z=10 \\& 2 x-4 y-2 z=6 \\& x+2 y-z=5 m\end{aligned}\)
is inconsistent if
[JEE Main 2021, 24 Feb (Shift 1)]
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Let \( \alpha \) be a root of equation \( x^{2}+x+1=0 \) and the matrix \( A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & \alpha & \alpha^{2} \\ 1 & \alpha^{2} & \alpha^{4}\end{array}\right] \), then the matrix \( A^{31} \) is equal to :
[JEE Main 2020, 7 Jan (Shift 1)]
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Let \(A=\left[\begin{array}{cc}\frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}}\end{array}\right]\) and \(B=\left[\begin{array}{rr}1 & -i \\ 0 & 1\end{array}\right]\), where \(i=\sqrt{-1}\). If \(M=A^T B A\), then the inverse of the matrix \(A M^{2023} A^T\) is
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Let \( A \) be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of \( \mathrm{A}^{2} \) is 1 , then the possible number of such matrices is:
[JEE Main 2021, 26 Feb (Shift 1)]
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If the system of linear equations
\(x + y + 3z = 0\)
\(x + 3y + k^{2}z = 0\)
\(3x + y + 3z = 0\)
has a non-zero solution \((x,y,z)\) for some \(k \in R\) , then \(x + \left( \frac{y}{z} \right)\) is equal to : -
[JEE Main 2020, 5 Sep (Shift 2)]
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Let the system of linear equations
\(\begin{aligned}& x+y+k z=2 \\& 2 x+3 y-z=1 \\& 3 x+4 y+2 z=k\end{aligned}\)
have infinitely many solutions. Then the system
\(\begin{aligned}& (k+1) x+(2 k-1) y=7 \\& (2 k+1) x+(k+5) y=10 \text { has: }\end{aligned}\)
[JEE Main 2023, 30 Jan (Shift 1)]
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If the system of linear equations
\(\begin{aligned}& 7 x+11 y+\alpha z=13 \\& 5 x+4 y+7 z=\beta \\& 175 x+194 y+57 z=361\end{aligned}\)
has infinitely many solutions, then \(\alpha+\beta+2\) is equal to
[JEE Main 2023, 11 Apr (Shift 2)]
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Let the system of linear equations
\(\begin{aligned}& 4 x+\lambda y+2 z=0 \\& 2 x-y+z=0 \\& \mu x+2 y+3 z=0, \lambda, \mu \in R\end{aligned}\)
has a non-trivial solution. Then which of the following is true?
[JEE Main 2021, 18 Mar (Shift 2)]
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Let \(P\) be a square matrix such that \(P^2=I-P\). For \(\alpha, \beta, \gamma, \delta \in N\), if \(P^\alpha+P^\beta=\gamma I-29 P\) and \(P^\alpha-P^\beta=\) \(\delta I-13 P\), then \(\alpha+\beta+\gamma-\delta\) is equal to
[JEE Main 2023, 6 Apr (Shift 2)]
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If the following system of linear equations
\[ \begin{aligned} & 2 x+y+z=5 \\ & x-y+z=3 \\ & x+y+a z=b \end{aligned} \]
has no solution, then:
[JEE Main 2021, 31 Aug (Shift 1)]
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Let \(A=\left[a_{i j}\right]\) be a real matrix of order \(3 \times 3\), such that \(a_{i 1}+a_{i 2}+a_{i 3}=1\), for \(i=1,2,3\). Then, the sum of all the entries of the matrix \(A^3\) is equal to:
[JEE Main 2021, 22 Jul (Shift 2)]
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Let \(S\) be the set of all values of \(\theta \in[-\pi, \pi]\) for which the system of linear equations
\[\begin{aligned}& x+y+\sqrt{3} z=0 \\& -x+(\tan \theta) y+\sqrt{7} z=0 \\& x+y+(\tan \theta) z=0 \text { has non-trivial solution. Then } \\& \frac{120}{\pi} \sum_{\theta=s} \theta \text { is equal to }\end{aligned}\]
[JEE Main 2023, 08 Apr (Shift 2)]
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Let the matrix \(A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]\) and the matrix \(B_0=A^{49}+\) \(2 A^{98}\). If \(B_n=\operatorname{Adj}\left(B_{n-1}\right)\) for all \(n \geq 1\), then \(\operatorname{det}\left(B_4\right)\) is equal to
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If \(P=\left[\begin{array}{cc}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]\), then \(P^{50}\) is
[JEE Main 2021, 25 Jul (Shift 2)]
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If the system of equations
\(\begin{aligned}& x+2 y+3 z=3 \\& 4 x+3 y-4 z=4 \\& 8 x+4 y-\lambda z=9+\mu\end{aligned}\)
has infinitely many solutions, then the ordered pair \((\lambda, \mu)\) is equal to
[JEE Main 2023, 24 Jan (Shift 2)]
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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
\[x+y+z=4,3 x+2 y+5 z=3,9 x+4 y+(28+[\lambda]) z=[\lambda]\]
has a solution is:
[JEE Main 2021, 27 Aug (Shift 2)]
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If the matrix \(A=\left(\begin{array}{cc}0 & 2 \\ K & -1\end{array}\right)\), satisfied \(A\left(A^3+3 I\right)=2 I\), then the value of \(K\) is:
[JEE Main 2021, 27 Aug (Shift 1)]
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The number of square matrices of order 5 with entries from the set \(\{0,1\}\), such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1 , is
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For which of the following ordered pairs \((\mu, \delta)\), the system of linear equations
\(x+2 y+3 z=1\)
\(3 x+4 y+5 z=\mu\)
\(4 x+4 y+4 z=\delta\)
is inconsistent?
[JEE Main 2020, 8 Jan (Shift 1)]
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The system of equations \( kx+y+z=1, x+k y+z=k \) and \( x+y+z k=k^{2} \) has no solution if \( k \) is equal to :
[JEE Main 2021, 17 Mar (Shift 1)]
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Suppose \(\mathrm{p}, \mathrm{q}, \mathrm{r} \neq 0\) and system of equation
\( (p+a) x+b y+c z=0 \)
\(a x+(q+b) y+c z=0\)
\(a x+b y+(r+c) z=0\)
has a non-trivial solution, then value of \(\frac{\mathrm{a}}{\mathrm{p}}+\frac{\mathrm{b}}{\mathrm{q}}+\frac{\mathrm{c}}{\mathrm{r}}\) is
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If the system of equations
\(\begin{aligned}& x+y+a z=b \\& 2 x+5 y+2 z=6 \\& x+2 y+3 z=3\end{aligned}\)
has infinitely many solutions, then \(2 a+3 b\) is equal to
[JEE Main 2023, 6 Apr (Shift 1)]
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If the system of equations
\(\begin{aligned}& 2 x+y-z=5 \\& 2 x-5 y+\lambda z=\mu \\& x+2 y-5 z=7\end{aligned}\)
has infinitely many solutions, then \((\lambda+\mu)^2+(\lambda-\mu)^2\) is equal to
[JEE Main 2023, 13 Apr (Shift 2)]
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Two fair dice are thrown. The numbers on them are taken as \(\lambda\) and \(\mu\), and a system of linear equations \(x+y+z=5\), \(x+2 y+3 z=\mu\) and \(x+3 y+\lambda z=1\) is constructed. If \(p\) is the probability that the system has a unique solution \(q\) is the probability that the system has no solution, then
[JEE Main 2021, 26 Aug (Shift 2)]
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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 :
[JEE Main 2023, 6 Apr (Shift 1)]
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Let \(A, B, C\) be \(3 \times 3\) matrices such that \(A\) is symmetric and \(B\) and \(C\) are skew-symmetric.
Consider the statements
(S1) \(A^{13} B^{26}-B^{26} A^{13}\) is symmetric
(S2) \(A^{26} C^{13}-C^{13} A^{26}\) is symmetric
Then,
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For the system of linear equations
\(\begin{aligned}& x+y+z=6 \\& \alpha x+\beta y+7 z=3 \\& x+2 y+3 z=14\end{aligned}\)
which of the following is NOT true?
[JEE Main 2023, 31 Jan (Shift 1)]
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If the system of equations
\(\begin{aligned}& x+y+z=2 \\& 2 x+4 y-z=6 \\& 3 x+2 y+\lambda z=\mu\end{aligned}\)
has infinitely many solutions, then:
[JEE Main 2020, 4 Sep (Shift 2)]
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If \(\mathbf{A}=\left[\begin{array}{cc}0 & 2 \\ 3 & -4\end{array}\right]\) and \(\mathbf{k} \mathbf{A}=\left[\begin{array}{cc}0 & 3 \mathrm{a} \\ 2 \mathrm{~b} & 24\end{array}\right]\), then the values of \(\mathbf{k}\), \(\mathbf{a}\) and \(\mathbf{b}\) respectively are:
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The values of \( \lambda \) and \( \mu \) for which the system of linear equations
\( x+y+z=2 \)
\( x+2 y+3 z=5 \)
\( x+3 y+\lambda z=\mu \)
has infinitely many solutions are, respectively :
[JEE Main 2020, 6 Sep (Shift 1)]
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The system of linear equations \(\lambda x+2 y+2 z=5\)
\(\begin{aligned} & 2 \lambda x+3 y+5 z=8 \\ & 4 x+\lambda y+6 z=10 \text { has: }\end{aligned}\)
[JEE Main 2020, 8 Jan (Shift 2)]
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The system of linear equations
\(\lambda x+2 y+2 z=5\)
\( 2 \lambda x+3 y+5 z=8 \)
\( 4 x+\lambda y+6 z=10 \)
[JEE Main 2020, 8 Jan (Shift 2)]
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The following system of linear equations.
\(\begin{aligned}& 2 x+3 y+2 z=9 \\& 3 x+2 y+2 z=9 \\& x-y+4 z=8\end{aligned}\)
[JEE Main 2021, 25 Feb (Shift 2)]
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Let \( \theta=\frac{\pi}{5} \) and \( A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right] \). If \( \mathrm{B}=\mathrm{A}+\mathrm{A}^{4} \), then \( \operatorname{det}(B) \) :
[JEE Main 2020, 6 Sep (Shift 2)]
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Let \( A+2 B=\left[\begin{array}{ccc}1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1\end{array}\right] \) and \( 2 A-B=\left[\begin{array}{ccc}2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2\end{array}\right] \). If \( \operatorname{tr}(\mathrm{A}) \) denotes the sum of all diagonal elements of the matrix \( A \), then \( \operatorname{tr}(\mathrm{A})-\operatorname{tr}(\mathrm{B}) \) has value equal to:
[JEE Main 2021, 18 Mar (Shift 1)]
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If \(A=\frac{1}{2}\left[\begin{matrix}1 & \sqrt{3} \\ −\sqrt{3} & 1\end{matrix}\right],\) then
[JEE Main 2023, 1 Feb (Shift 2)]
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If \( \alpha+\beta+\gamma=2 \pi \), then the system of equations
\(\begin{array}{l}x+(\cos \gamma) y+(\cos \beta) z=0 \\(\cos \gamma) x+y+(\cos \alpha) z=0 \\(\cos \beta) x+(\cos \alpha) y+z=0\end{array}\)
has:
[JEE Main 2021, 31 Aug (Shift 2)]
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If \(A=\frac{1}{2}\left[\begin{array}{cc}1 & \sqrt{3} \\ -\sqrt{3} & 1\end{array}\right]\), then:
[JEE Main 2023, 1 Feb (Shift 2)]
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Let \(S\) denote the set of all real values of \(\lambda\) such that the system of equations
\(\begin{aligned}& \lambda x+y+z=1 \\& x+\lambda y+z=1 \\& x+y+\lambda z=1\end{aligned} \)
is inconsistent, then \(\sum_{\lambda \in S}\left(|\lambda|^2+|\lambda|\right)\) is equal to
[JEE Main 2023, 1 Feb (Shift 1)]
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Let \(S\) denote the set of all real values of \(\lambda\) such that the system of equations
\[\begin{aligned}& \lambda x+y+z=1 \\& x+\lambda y+z=1 \\& x+y+\lambda z=1\end{aligned}\]
is inconsistent, then \(\sum_{\lambda \in S}\left(|\lambda|^2+|\lambda|\right)\) is equal to
[JEE Main 2023, 1 Feb (Shift 1)]
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If \(A=\left(\begin{array}{cc}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}}\end{array}\right), B=\left(\begin{array}{ll}1 & 0 \\ i & 1\end{array}\right), i =\sqrt{-1}\), and \(Q=A^T B A\), then the inverse of the matrix \(A Q^{2021} A^T\) is equal to
[JEE Main 2021, 26 Aug (Shift 1)]
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Let \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{array}\right]\). Then the sum of the diagonal elements of the matrix \((A+I)^{11}\) is equal to:
[JEE Main 2023, 31 Jan (Shift 1)]
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For the system of linear equations
\(\begin{aligned}& x+y+z=6 \\& \alpha x+\beta y+7 z=3 \\& x+2 y+3 z=14\end{aligned}\)
which of the following is NOT true?
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If the following system of linear equations \(2 x+y+z=5\), \(x-y+z=3\) and \(x+y+a z=b\) has no solution, then:
[JEE Main 2021, 31 Aug (Shift 1)]
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For the system of linear equations
\[\begin{aligned}& x+y+z=6 \\& \alpha x+\beta y+7 z=3 \\& x+2 y+3 z=14\end{aligned}\]
which of the following is NOT true?
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Let \(A=\left[a_{i j}\right]_{2 \times 2}\) where \(a_{i j} \neq 0\) for all \(i, j\) and \(A^2=I\). Let \(a\) be the sum of all diagonal elements of \(A\) and \(b=|A|\), then \(3 a^2+4 b^2\) is equal to
[JEE Main 2023, 6 Apr (Shift 1)]
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The following system of linear equations
\(\begin{aligned} & 7 x+6 y-2 z=0 \\ & 3 x+4 y+2 z=0 \\ & x-2 y-6 z=0, \text { has }\end{aligned}\)
[JEE Main 2020, 9 Jan (Shift 2)]
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If \( A=\left[\begin{array}{cc}\cos \theta & i \sin \theta \\ i \sin \theta & \cos \theta\end{array}\right],\left(\theta=\frac{\pi}{24}\right) \) and \( A^{5}=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \) where \( \mathrm{i}=\sqrt{-1} \), then which one of the following is not true?
[JEE Main 2020, 4 Sep (Shift 1)]
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The sum of distinct values of \( \lambda \) for which the system of equations
\( (\lambda-1) x+(3 \lambda+1) y+2 \lambda z=0 \)
\( (\lambda-1) x+(4 \lambda-2) y+(\lambda+3) z=0 \)
\( 2 \mathrm{x}+(3 \lambda+1) \mathrm{y}+3(\lambda-1) \mathrm{z}=0 \),
has non-zero solutions, is.............
[JEE Main 2020, 6 Sep (Shift 2)]
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If \(A\) and \(B\) are two non-zero \(n \times n\) matrices such that \(A^2+\) \(B=A^2 B\), then
[JEE Main 2023, 24 Jan (Shift 1)]
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Let \(A=\left[\begin{array}{cc}i & -i \\ -i & i\end{array}\right], i=\sqrt{-1}\). Then, the system of linear equation \(A^8\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{c}8 \\ 64\end{array}\right]\) has:
[JEE Main 2021, 16 Mar (Shift 1)]
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Let the system of linear equations
\(4x+\lambda y+2z=0\)
\(2x−y+z=0\)
\(\mu x+2y+3z=0,\lambda ,\mu \in R\)
has a non-trivial solution. Then which of the following is true?
[JEE Main 2021, 18 Mar (Shift 2)]
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Consider the following system of equations:
\(\begin{array}{l}x+2 y-3 z=a \\2 x+6 y-11 z=b \\x-2 y+7 z=c\end{array}\)
where \( \mathrm{a}, \mathrm{b} \) and \( c \) are real constants. Then the system of equations:
[JEE Main 2021, 26 Feb (Shift 2)]
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For \(\alpha, \beta \in R\), suppose the system of linear equations
\(\begin{aligned}& x-y+z=5 \\& 2 x+2 y+\alpha z=8 \\& 3 x-y+4 z=\beta\end{aligned}\)
has infinitely many solutions. Then \(\alpha\) and \(\beta\) are the roots of
[JEE Main 2023, 30 Jan (Shift 2)]
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If \(A=\left[\begin{matrix}1 & −\sin \alpha \\ \sin \alpha & 1\end{matrix}\right]\) and \(B=\left[\begin{matrix}1 & −\sin \beta \\ \sin \beta & 1\end{matrix}\right]\), then the correct relation is
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Let \(A=\left[\begin{array}{cc}1 & \frac{1}{51} \\ 0 & 1\end{array}\right]\). If \(B=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right] A\left[\begin{array}{cc}-1 & -2 \\ 1 & 1\end{array}\right]\), then the sum of all the elements of the matrix \(\sum_{n=1}^{50} B^n\) is equal to
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Let \(A+2 B=\left[\begin{array}{ccc}1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1\end{array}\right]\) and \(2 A-B=\left[\begin{array}{ccc}2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2\end{array}\right]\)
If \(\operatorname{Tr}(A)\) denotes the sum of all diagonal elements of the matrix \(A\), then \(\operatorname{Tr}(A)-\operatorname{Tr}(B)\) has value equal to:
[JEE Main 2021, 18 Mar (Shift 1)]
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If the system of equations
\[\begin{aligned}& x+y+z=2 \\& 2 x+4 y-z=6 \\& 3 x+2 y+\lambda z=\mu\end{aligned}\]has infinitely many solutions, then:
[JEE Main 2020, 4 Sep (Shift 2)]
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\(\text { The set of all values of } t \in R \text {, for which the matrix }\)\(\left[\begin{array}{ccc}e^t & e^{-t}(\sin t-2 \cos t) & e^{-t}(-2 \sin t-\cos t) \\e^t & e^{-t}(2 \sin t+\cos t) & e^{-t}(\sin t-2 \cos t) \\e^t & e^{-t} \cos t & e^{-t} \sin t\end{array}\right]\)\(\text { Invertible, is }\)
[JEE Main 2023, 29 Jan (Shift 2)]
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Let \( \mathrm{A}=\left\{\mathrm{X}=[\mathrm{x}, \mathrm{y}, \mathrm{z}]^{\mathrm{T}}: \mathrm{PX}=0\right. \) and \( \left.\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=1\right\} \), where \( P=\left[\begin{array}{ccc}1 & 2 & 1 \\ -2 & 3 & -4 \\ 1 & 9 & -1\end{array}\right] \), then the set A:
[JEE Main 2020, 2 Sep (Shift 2)]
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If \( \mathrm{A}=\left(\begin{array}{ll}2 & 2 \\ 9 & 4\end{array}\right) \) and \( \mathrm{I}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right) \), then \( 10 \mathrm{~A}^{-1} \) is equal to:
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The value of \(k \in R\), for which the following system of linear equations
\(\begin{aligned}& 3 x-y+4 z=3 \\& x+2 y-3 z=-2 \\& 6 x+5 y+k z=-3\end{aligned}\)
has infinitely many solutions, is :
[JEE Main 2021, 20 Jul (Shift 2)]
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\(\text { If P= }\left[\begin{array}{cc}1 & 0 \\\frac{1}{2} & 1\end{array}\right] \text { then } P ^{50} \text { is }\)
[JEE Main 2021, 25 Jul (Shift 2)]
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If a point \(P(\alpha, \beta, \gamma)\) satisfying
\((\alpha \quad \beta \quad \gamma)\left(\begin{array}{ccc}2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8\end{array}\right)=\left(\begin{array}{lll}0 & 0 & 0\end{array}\right)\) lies on the plane \(2 x+\) \(4 y+3 z=5\), then \(6 \alpha+9 \beta+7 \gamma\) is equal to:
[JEE Main 2023, 31 Jan (Shift 2)]
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Suppose the vectors \( x_{1}, x_{2} \) and \( x_{3} \) are the solutions of the system of linear equations, \( \mathrm{Ax}=\mathrm{b} \) when the vector \( \mathrm{b} \) on the right side is equal to \( \mathrm{b}_{1}, \mathrm{~b}_{2} \) and \( \mathrm{b}_{3} \) respectively. If
\( \mathrm{x}_{1}=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], \mathrm{x}_{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], \mathrm{x}_{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], \mathrm{b}_{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right], \mathrm{b}_{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{array}\right] \) and \( \mathrm{b}_{3} \) \( =\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right] \), then the determinant of \( \mathrm{A} \) is equal to :
[JEE Main 2020, 4 Sep (Shift 2)]
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For the system of equations
\[\begin{aligned}& x+y+z=6 \\& x+2 y+\alpha z=10\end{aligned}\]
\(x+3 y+5 z=\beta\), which one of the following is NOT true?
[JEE Main 2023, 6 Apr (Shift 2)]
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The system of equation \(\mathrm{kx}+\mathrm{y}+\mathrm{z}=1,\mathrm{x}+\mathrm{ky}+\mathrm{z}=\mathrm{k}\) and \(\mathrm{x}+\mathrm{y}+\mathrm{zk}={\mathrm{k}}^{2}\) has no solution if k is equal to:
[JEE Main 2021, 17 Mar (Shift 1)]
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The system of linear equations
\[\begin{aligned}& 3 x-2 y-k z=10 \\& 2 x-4 y-2 z=6 \\& x+2 y-z=5 m\end{aligned}\] is inconsistent if
[JEE Main 2021, 24 Feb (Shift 1)]
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