System of Linear Equations

Q. Let $\theta \in (0,\frac{\pi }{2}).$ If the system of linear equations,
$\begin{array}{c}(1+{\cos }^2\theta )x+{\sin }^2\theta y+4\sin 3\theta z=0\\ {\cos }^2\theta x+(1+{\sin }^2\theta )y+4\sin 3\theta z=0\\ {\cos }^2\theta x+{\sin }^2\theta y+(1+4\sin 3\theta )z=0\end{array}$
has a non-trivial solution, then the value of $\theta$ is

1. $\frac{7\pi }{18}$
2. $\frac{\pi }{18}$
3. $\frac{4\pi }{9}$
4. $\frac{5\pi }{18}$

Q. The value of $\theta$ for which the system of equations $\left(\sin 3\theta \right)x-2y+3z=0,\left(\cos 2\theta \right)x+8y-7z=0$ and $2x+14y-11z=0$ has a non-trivial solution, is
($n\in Z)$

1. $n\pi$
2. $n\pi +(-1{)}^n\frac{\pi }{3}$
3. $n\pi +(-1{)}^n\frac{\pi }{8}$
4. $n\pi \pm \frac{\pi }{8}$

Q. Let $\alpha ,\beta$ and $\gamma$ be real numbers such that the system of linear equations
$x+2y+3z=\alpha$
$4x+5y+6z=\beta$
$7x+8y+9z=\gamma -1$
is consistent.
Let $|M|$ represent the determinant of the matrix $M=\left[\begin{array}{ccc}\alpha & 2& \gamma \\ \beta & 1& 0\\ -1& 0& 1\end{array}\right].$
Then the value of $|M|$ is

Q. Consider the following system of linear equations
$\left[\begin{array}{ccc}2& 1& -4\\ 4& 3& -12\\ 1& 2& -8\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}a\\ 5\\ 7\end{array}\right]$
Number of values of $a$ for which system has infinitely many solutions is

Q. The value of $\theta$ for which the system of equations $\left(\sin 3\theta \right)x-2y+3z=0,\left(\cos 2\theta \right)x+8y-7z=0$ and $2x+14y-11z=0$ has a non-trivial solution, is
($n\in Z)$

1. $n\pi$
2. $n\pi +(-1{)}^n\frac{\pi }{8}$
3. $n\pi +(-1{)}^n\frac{\pi }{3}$
4. $n\pi \pm \frac{\pi }{8}$

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

1. $\frac{3}{4}$
2. $-\frac{1}{4}$
3. $\frac{1}{2}$
4. $-4$

Q. If the system of equations ax + y + z = 0, x + by + z = 0 and x+y+cz = 0, where $a,b,c\ne 1$, has a nontrivial solution, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is

1. -1
2. 0
3. 1
4. None of these

Q. The system of linear equations
$x+y+z=2$
$2x+3y+2z=5$
$2x+3y+(a^2-1)z=a+1$

1. has infinitely many solutions for $a=4$
2. is incosistent when $a=4$
3. has a unique solution for $|a|=\sqrt{3}$
4. is inconsistent when $|a|=\sqrt{3}$

Q. If the following system of linear equations have a non-trivial solution
$x+4ay+az=0$
$x+3by+bz=0$ and
$x+2cy+cz=0$, then

1. $a,b,c$ are in $A.P.$
2. $a,b,c$ are in $G.P.$
3. $a,b,c$ are in $H.P.$
4. $a+b+c=0$

Q. If the system of equations
$2x+3y=7$
$2ax+(a+b)y=28$
has infinitely many solutions, then ______.

1. $a+2b=0$
2. $a=2b$
3. $b=2a$
4. $2a+b=0$

Q. Consider the following system of linear equations
$\left[\begin{array}{ccc}2& 1& -4\\ 4& 3& -12\\ 1& 2& -8\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}a\\ 5\\ 7\end{array}\right]$
Number of values of $a$ for which system has infinitely many solutions.

1. $1$
2. $2$
3. $3$
4. $0$

Q. Suppose $f\left(z\right)$ is a possibly complex valued function of a complex number z, which satisfies a function equation of the form $af\left(z\right)+bf\left(w^{2}z\right)=g\left(z\right)$ for all $zϵC,$ where a and b are some fixed complex numbers and $g\left(z\right)$ is some function of z and w is cube root of unity $(w\ne 1)$, then $f\left(z\right)$ can be determined uniquely if

1. a² + b² ≠ 0
2. a³ + b³ ≠ 0
3. a³ + b³ = 0
4. a + b = 0

Q. The system of linear equations
$x+y+z=154x+6y+7z=24k^{2}x+6y+7z=k+22$

Then the incorrect statement about the system of equations is

1. The system of equations has no solution for exactly one value of $k$.
2. For $k=0$ the system has unique solution.
3. For $k=\pm$ 2, the system has no solution
4. There exists $k\in R$ such that the system has no solution.

Q. Consider the following system of linear equations
$\left[\begin{array}{ccc}2& 1& -4\\ 4& 3& -12\\ 1& 2& -8\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}a\\ 5\\ 7\end{array}\right]$
Number of values of $a$ for which system has infinitely many solutions.

1. $1$
2. $2$
3. $3$
4. $0$

Q. If the quadratic equation $x^2+ax+b=0$ and $x^2+bx+a=0\left(a\ne b\right)$ have a common root, then find the numerical value of $a+b$

Q. Find the matrix $X$ so that $X\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]=\left[\begin{array}{ccc}-7& -8& -9\\ 2& 4& 6\end{array}\right]$

Q. Number of solutions for the system of equations $x+2y+z=0,3x+5y+2z=2$ and $2x+4y+2z=1$

1. $0$
2. $1$
3. $2$
4. infinity

Q. The system of equations
$x+3y+2z=6x+ay+2z=7x+3y+2z=b$
has

1. a unique solution, if $a=2$ and $b\ne 6$
2. infinitely many solutions, if $a=4$ and $b=6$
3. no solution, if $a=5$ and $b=7$
4. no solution, if $a=3$ and $b=5$

Q. $\left|\begin{array}{ccc}-a^2& ab& ac\\ ab& -b^2& bc\\ ac& bc& -c^2\end{array}\right|+\left|\begin{array}{ccc}{a}^2& -b^2& -c^2\\ -a^2& b^2& -c^2\\ -a^2& -b^2& c^2\end{array}\right|=$

1. $2$
2. $0$
3. $4$
4. $2a^2{b}^2{c}^2$

Q. Diffrentiate w.r.t. $t$: $t^{1/2}+sin5t\cos t$

Q. The system of equations
$x+2y+3z=4$
$2x+3y+4z=5$
$3x+4y+5z=6$ has

1. Unique solution
2. No solutions
3. None of these
4. Many solutions

Q. The system $\left[\begin{array}{ccc}1& -1& 2\\ 3& 5& -3\\ 2& 6& a\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}3\\ b\\ 2\end{array}\right]$ has no solution if

1. $a\ne -5,b\ne 5$
2. $a=-5,b\ne 5$
3. $a=-5,b=5$
4. $a\ne -5,b=5$

Q. ${\int }^{}\frac{\sin 2x}{\left(a^2{\sin }^{2}x+b^2{\cos }^{2}x\right)}dx$

Q. $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right],\text{then}{A}^3-4A^2-6A=$

1. $0$
2. $A$
3. $-A$
4. $I$

Q. Let $S$ be the set of all integer solutions $(x,y,z)$ of the system of equations
$x-2y+5z=0$
$-2x+4y+z=0$
$-7x+14y+9z=0,$
such that $15\le x^2+y^2+z^2\le 150$. Then, the number of elements in the set $S$ is equal to

Q. The system of linear equations
$3x-2y-kz=10$
$2x-4y-2z=6$
$x+2y-z=5m$
is inconsistent if:

1. $k=3,m=\frac{4}{5}$
2. $k\ne 3,m\in R$
3. $k\ne 3,m\ne \frac{4}{5}$
4. $k=3,m\ne \frac{4}{5}$

Q. The adjoint of the matric $A=\left[\begin{array}{ccc}1& 0& 2\\ 2& 1& 0\\ 0& 3& 1\end{array}\right]$ is

1. $\left[\begin{array}{ccc}-1& 6& 2\\ -2& 1& -4\\ 6& 3& 1\end{array}\right]$
2. $\left[\begin{array}{ccc}1& 6& -2\\ -2& 1& 4\\ 6& -3& 1\end{array}\right]$
3. $\left[\begin{array}{ccc}-6& 2& 1\\ 4& -2& 1\\ 3& 1& -6\end{array}\right]$
4. $\left[\begin{array}{ccc}6& 1& 2\\ 4& -1& 2\\ 6& 3& -1\end{array}\right]$

Q. If the system of equations $ax+y=3,x+2y=3$ and $3x+4y=7$ is consistent, then value of $a$ is

1. $2$
2. $1$
3. $-1$
4. $0$

Q. If a, b, c are all different and if $\left|\begin{array}{ccc}a& a^2& 1+a^3\\ b& b^2& 1+b^3\\ c& c^2& 1+c^3\end{array}\right|=0$ then $-abc=$

Q. Consider the following system of linear equations
$\left[\begin{array}{ccc}2& 1& -4\\ 4& 3& -12\\ 1& 2& -8\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}a\\ 5\\ 7\end{array}\right]$
Number of values of $a$ for which system has infinitely many solutions is