Let $(x,y,z)$ be points with integer coordinates satisfying the system of homogeneous equations: $$\begin{array}{r} 3x-y-z=0\\ -3x+z=0\\ -3x+2y+z=0 \end{array}$$ Then the number of such points for which $x^{2}+y^{2}+z^{2}\leq100$ is

The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system $A \left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$ has exactly two distinct solutions, is
A) 0
B) $2 ^ { 9 } - 1$
C) 168
D) 2

Let $P = \left[\begin{array}{ccc} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{array}\right]$, where $\alpha \in \mathbb{R}$. Suppose $Q = [q_{ij}]$ is a matrix such that $PQ = kI$, where $k \in \mathbb{R}, k \neq 0$ and $I$ is the identity matrix of order 3. If $q_{23} = -\frac{k}{8}$ and $\det(Q) = \frac{k^2}{2}$, then
(A) $\alpha = 0, k = 8$
(B) $4\alpha - k + 8 = 0$
(C) $\det(P\operatorname{adj}(Q)) = 2^9$
(D) $\det(Q\operatorname{adj}(P)) = 2^{13}$

Let $\alpha , \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations
$x + 2 y + z = 7$
$x + \alpha z = 11$
$2 x - 3 y + \beta z = \gamma$
Match each entry in List-I to the correct entries in List-II.
List-I
(P) If $\beta = \frac { 1 } { 2 } ( 7 \alpha - 3 )$ and $\gamma = 28$, then the system has
(Q) If $\beta = \frac { 1 } { 2 } ( 7 \alpha - 3 )$ and $\gamma \neq 28$, then the system has
(R) If $\beta \neq \frac { 1 } { 2 } ( 7 \alpha - 3 )$ where $\alpha = 1$ and $\gamma \neq 28$, then the system has
(S) If $\beta \neq \frac { 1 } { 2 } ( 7 \alpha - 3 )$ where $\alpha = 1$ and $\gamma = 28$, then the system has
List-II
(1) a unique solution
(2) no solution
(3) infinitely many solutions
(4) $x = 11 , y = - 2$ and $z = 0$ as a solution
(5) $x = - 15 , y = 4$ and $z = 0$ as a solution
The correct option is:
(A) $( P ) \rightarrow ( 3 ) \quad ( Q ) \rightarrow ( 2 ) \quad ( R ) \rightarrow ( 1 ) \quad ( S ) \rightarrow ( 4 )$
(B) $( P ) \rightarrow ( 3 ) \quad ( Q ) \rightarrow ( 2 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 4 )$
(C) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 1 ) \quad ( R ) \rightarrow ( 4 ) \quad ( S ) \rightarrow ( 5 )$
(D) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 1 ) \quad ( R ) \rightarrow ( 1 ) \quad ( S ) \rightarrow ( 3 )$

The number of values of $k$ for which the linear equations $4x+ky+2z=0$;\ $kx+4y+z=0$;\ $2x+2y+z=0$ possess a non-zero solution is
(1) 2
(2) 1
(3) 0
(4) 3

If the system of equations $$\begin{aligned} & x + y + z = 6 \\ & x + 2 y + 3 z = 10 \\ & x + 2 y + \lambda z = 0 \end{aligned}$$ has a unique solution, then $\lambda$ is not equal to
(1) 1
(2) 0
(3) 2
(4) 3

Statement 1: If the system of equations $x + k y + 3 z = 0, 3 x + k y - 2 z = 0, 2 x + 3 y - 4 z = 0$ has a nontrivial solution, then the value of $k$ is $\frac { 31 } { 2 }$. Statement 2: A system of three homogeneous equations in three variables has a non trivial solution if the determinant of the coefficient matrix is zero.
(1) Statement 1 is false, Statement 2 is true.
(2) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
(4) Statement 1 is true, Statement 2 is false.

Let $A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}$. If $u_{1}$ and $u_{2}$ are column matrices such that $Au_{1} = \begin{pmatrix}1\\0\\0\end{pmatrix}$ and $Au_{2} = \begin{pmatrix}0\\1\\0\end{pmatrix}$, then $u_{1}+u_{2}$ is equal to
(1) $\begin{pmatrix}-1\\1\\0\end{pmatrix}$
(2) $\begin{pmatrix}-1\\1\\-1\end{pmatrix}$
(3) $\begin{pmatrix}-1\\-1\\0\end{pmatrix}$
(4) $\begin{pmatrix}1\\-1\\-1\end{pmatrix}$

The number of values of $k$ for which the linear equations $4x+ky+2z=0$, $kx+4y+z=0$, $2x+2y+z=0$ possess a non-zero solution is
(1) 2
(2) 1
(3) zero
(4) 3

The system of linear equations \begin{align*} x + \lambda y - z &= 0 \lambda x - y - z &= 0 x + y - \lambda z &= 0 \end{align*} has a non-trivial solution for:
(1) infinitely many values of $\lambda$
(2) exactly one value of $\lambda$
(3) exactly two values of $\lambda$
(4) exactly three values of $\lambda$

The system of linear equations \begin{align*} x + \lambda y - z &= 0 \lambda x - y - z &= 0 x + y - \lambda z &= 0 \end{align*} has a non-trivial solution for: (1) infinitely many values of $\lambda$ (2) exactly one value of $\lambda$ (3) exactly two values of $\lambda$ (4) exactly three values of $\lambda$

The set of all values of $\lambda$ for which the system of linear equations $$x - 2 y - 2 z = \lambda x$$ $$x + 2 y + z = \lambda y$$ $$- x - y = \lambda z$$ has a non-trivial solution:
(1) is an empty set
(2) is a singleton
(3) contains two elements
(4) contains more than two elements

If the system of linear equations $x + k y + 3 z = 0$ $3 x + k y - 2 z = 0$ $2 x + 4 y - 3 z = 0$ has a non-zero solution $( x , y , z )$, then $\frac { x z } { y ^ { 2 } }$ is equal to:
(1) 30
(2) - 10
(3) 10
(4) - 30

If $\left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 1 & 3 \\ 0 & 1 \end{array} \right] \ldots \left[ \begin{array} { c c } 1 & n - 1 \\ 0 & 1 \end{array} \right] = \left[ \begin{array} { c c } 1 & 78 \\ 0 & 1 \end{array} \right]$, then the inverse of $\left[ \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right]$ is:
(1) $\left[ \begin{array} { c c } 1 & - 12 \\ 0 & 1 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 1 & 0 \\ 12 & 1 \end{array} \right]$
(3) $\left[ \begin{array} { c c } 1 & 0 \\ 13 & 1 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 1 & - 13 \\ 0 & 1 \end{array} \right]$

If the system of equations $2 x + 3 y - z = 0 , x + k y - 2 z = 0$ and $2 x - y + z = 0$ has a non-trivial solution $( x , y , z )$, then $\frac { x } { y } + \frac { y } { z } + \frac { z } { x } + k$ is equal to
(1) $- \frac { 1 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) $- 4$
(4) $\frac { 3 } { 4 }$

If $A = \begin{pmatrix} 2 & 2 \\ 9 & 4 \end{pmatrix}$ and $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$, then $10A^{-1}$ is equal to.
(1) $A - 4I$
(2) $6I - A$
(3) $A - 6I$
(4) $4I - A$

Let $S$ be the set of all $\lambda \in R$ 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 set $S$
(1) Contains more than two elements
(2) Is an empty set
(3) Is a singleton
(4) Contains exactly two elements

Let $A = \left\{ X = ( x , y , z ) ^ { T } : P X = 0 \text{ and } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1 \right\}$ where $P = \left[ \begin{array} { c c c } 1 & 2 & 1 \\ - 2 & 3 & - 4 \\ 1 & 9 & - 1 \end{array} \right]$ then the set $A$
(1) Is a singleton.
(2) Is an empty set.
(3) Contains more than two elements
(4) Contains exactly two elements

If the system of equations $x + y + z = 2$ $2 x + 4 y - z = 6$ $3 x + 2 y + \lambda z = \mu$ has infinitely many solutions, then:
(1) $\lambda + 2 \mu = 14$
(2) $2 \lambda - \mu = 5$
(3) $\lambda - 2 \mu = - 5$
(4) $2 \lambda + \mu = 14$

If the system of linear equations $$x + y + 3z = 0$$ $$x + 3y + k^2z = 0$$ $$3x + y + 3z = 0$$ has a non-zero solution $(x, y, z)$ for some $k \in \mathrm{R}$, then $x + \left(\frac{y}{z}\right)$ is equal to:
(1) $-3$
(2) $9$
(3) $3$
(4) $-9$

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
(1) 6 and 8
(2) 5 and 7
(3) 5 and 8
(4) 4 and 9

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 infinitely many solutions if $k = - 2$.

The system of linear equations $3 x - 2 y - k z = 10$ $2 x - 4 y - 2 z = 6$ $x + 2 y - z = 5 m$ is inconsistent if:
(1) $k = 3 , \quad m \neq \frac { 4 } { 5 }$
(2) $k = 3 , \quad m = \frac { 4 } { 5 }$
(3) $k \neq 3 , \quad m \in R$
(4) $k \neq 3 , \quad m \neq \frac { 4 } { 5 }$

Consider the following system of equations: $$\begin{aligned} & x + 2 y - 3 z = a \\ & 2 x + 6 y - 11 z = b \\ & x - 2 y + 7 z = c \end{aligned}$$ where $a , b$ and $c$ are real constants. Then the system of equations :
(1) has a unique solution when $5 a = 2 b + c$
(2) has no solution for all $a , b$ and $c$
(3) has infinite number of solutions when $5 a = 2 b + c$
(4) has a unique solution for all $a , b$ and $c$

The system of equations $kx + y + z = 1$, $x + ky + z = k$ and $x + y + zk = k^2$ has no solution if $k$ is equal to:
(1) 0
(2) 1
(3) $-1$
(4) $-2$