Equation of a Line Passing through 2 Points

Q. Let the system of linear equations
$4x+\lambda y+2z=0$
$2x-y+z=0$
$\mu +2y+3z=0,\lambda ,\mu \in R$
has a non-trivial solution. Then which of the following is true?

1. $\mu =6,\lambda \in R$
2. $\lambda =2,\mu \in R$
3. $\lambda =3,\mu \in R$
4. $\mu =-6,\lambda \in R$

Q.

Let $P$be a plane containing the line $\frac{[x-1]}{3}=\frac{[y+6]}{4}=\frac{[z+5]}{2}$ and parallel to the line $\frac{[x-3]}{4}=\frac{[y-2]}{-3}=\frac{[z+5]}{7}$ If the point $(1,-1,\alpha )$lies on the plane $P,$ then the value of $\left|5\alpha \right|$ is equal to

Q. For real numbers $\alpha$ and $\beta ,$ consider the following system of linear equations :
$x+y-z=2,x+2y+\alpha z=1,2x-y+z=\beta$. If the system has infinite solutions, then $\alpha +\beta$ is equal to

Q.

Find the equation of the line joining (3, 1) and (9, 3) using determinants.

Q.

Which of the following lines is concurrent with the lines $3\mathrm{x}+4\mathrm{y}+6=0$ and $6\mathrm{x}+5\mathrm{y}+9=0$?

1. $2\mathrm{x}+3\mathrm{y}+5=0$

2. $3\mathrm{x}+3\mathrm{y}+5=0$

3. $7\mathrm{x}+9\mathrm{y}+3=0$

4. None of these

Q.

If the coordinates of the vertices of an equilateral triangle with sides of length 'a' are $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$, then
${\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|}^2=\frac{3a^4}{4}$

Q. The points $(k-1,k+2),(k,k+1),(k+1,k)$ are collinear for

1. any value of $k$
2. $k=-\frac{1}{2}$ only
3. integral values of $k$ only
4. no value of $k$

Q.

If the area of the triangle on the complex plane formed by the points $z,iz,z+iz$ is $200$ sq. units then the value of $3\left|z\right|$ equals to:

1. $20$

2. $40$

3. $60$

4. $80$

Q.

Find the equation of the planes that passes through the sets of three points.
(1, 1, 0), (1, 2, 1), (-2, 2, -1)

Q. If $\frac{x}{c}+\frac{y}{d}=1$ be any line passing through point of intersection of lines $\frac{x}{a}+\frac{y}{b}=1\text{and}\frac{x}{b}+\frac{y}{a}=1$ then

1. $\frac{1}{c}-\frac{1}{d}=\frac{1}{a}-\frac{1}{b}$
2. $\frac{1}{c}+\frac{1}{d}=\frac{1}{a}+\frac{1}{b}$
3. $\frac{1}{c}-\frac{1}{b}=\frac{1}{a}-\frac{1}{d}$
4. $\frac{1}{d}-\frac{1}{c}=\frac{1}{a}-\frac{1}{b}$

Q. The sum of real value(s) of $\alpha$ for which the system of equations
$x+3y+5z=\alpha x$
$5x+y+3z=\alpha y$
$3x+5y+z=\alpha z$
has infinite number of solutions is

1. $1$
2. $3$
3. $9$
4. $15$

Q.

If $5A+B3=65$, then the value of $A$ and $B$ is

1. $A=2,B=3$

2. $A=3,B=2$

3. $A=2,B=1$

4. $A=1,B=2$

Q. If $a{x}_1^2+b{y}_1^2+c{z}_1^2=a{x}_2^2+b{y}_2^2+c{z}_2^2=a{x}_3^2+b{y}_3^2+c{z}_3^2=d$ and
$a{x}_2{x}_3+b{y}_2{y}_3+c{z}_2{z}_3=a{x}_3{x}_1+b{y}_3{y}_1+c{z}_3{z}_1=a{x}_1{x}_2+b{y}_1{y}_2+c{z}_1{z}_2=f,$ where $a,b,c,d,f>0$ and $d>2f,$ then the value of $\left|\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right|$ is:

1. $(d-f){\left\{\frac{(d+2f)}{abc}\right\}}^{\frac{1}{2}}$
2. $|f-d|{\left\{\frac{(d-2f)}{abc}\right\}}^{\frac{1}{2}}$
3. $(d-f){\left\{\frac{(d-2f)}{abc}\right\}}^{\frac{1}{2}}$
4. $(d-f){\left\{\frac{(d+f)}{abc}\right\}}^{\frac{1}{2}}$

Q.

Solve the equation. $\sqrt{\frac{1000}{10}}=n^2-54$ .

Q. The system of equations:
2x+y=3 and 4x+2y=5 is consistent.

1. True
2. False

Q. Let the system of linear equations $\left[\begin{array}{ccc}1& \lambda & {\lambda }^2+5\\ 1& \gamma & {\gamma }^2+5\\ 1& 5& 30\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}2\\ 4\\ -1\end{array}\right]$ has a unique solution. Then which of the following is TRUE?

1. $\lambda =5,\gamma =6$
2. $\lambda =7,\gamma =5$
3. $\lambda =\gamma$
4. $\lambda =7,\gamma =6$

Q. Find the equation of line joining (1, 2) and (3, 6) using determinants.

Q.

Let $X=1,2,3$ and $Y=4,5$. Find whether the following subsets of $X\times Y$ are functions from X to Y or not.

(i) $f=\left\{\right(1,4),(1,5),(2,4),(3,5\left)\right\}$

(ii)$g=\left\{\right(1,4),(2,4),(3,4\left)\right\}$

(iii) $h=\left\{\right(1,4),(2,5),(3,5\left)\right\}$

(iv) $k=\left\{\right(1,4),(2,5\left)\right\}$

Q. Let $\lambda \in R.$ The system of linear equations
$2x_1-4x_2+\lambda x_3=1$
$x_1-6x_2+x_3=2$
$\lambda x_1-10x_2+4x_3=3$
is inconsistent for:

1. exactly two values of $\lambda$.
2. exactly one negative value of $\lambda$.
3. every value of $\lambda$.
4. exactly one positive value of $\lambda$.

Q. If the system of equations
$x+y+z=22x+4y-z=63x+2y+\lambda z=\mu$
has infinitely many solutions, then

1. $\lambda -2\mu =-5$
2. $2\lambda +\mu =14$
3. $\lambda +2\mu =14$
4. 2$\lambda -\mu =5$

Q.

A straight line given by the equation  $\left|\begin{array}{ccc}x+3& y-1& 1\\ 7& -1& 1\\ -3& 9& 1\end{array}\right|=0$ passes through the point. 

1. (4, -1)

2. (4, 0)

3. (6, 10)

4. (-6, 0)

Q. Find the equation of the line joining $A(1,3)$ and $B(0,0)$ using determine and find $k$ of $D(k,0)$ is a point such that area of $△$ $ABD$ is $3$ sq. units.

Q. Find equation of line joining (1, 2) and (3, 6) using determinants.

Q. If $\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+y\end{array}\right|$ for $x\ne 0,y\ne 0$, then $\Delta$ is

1. divisible by both $x$ and $y$
2. divisible by $x$ but not $y$
3. divisible by $y$ but not $x$.
4. divisible by neither $x$ nor $y$.

Q. The points A (1, 2), B (3, 4), C (2, 5) and D (0, 3) in that order form a rectangle.

Q. If the lines $x+py+p=0,qx+y+q=0$ and $rx+ry+1=0(p,q,r$ being distinct and $\ne$ 1) are concurrent, then the value of
$\frac{p}{p-1}+\frac{q}{q-1}+\frac{r}{r-1}=$

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

Q. Numbers of ways in which 75600 can be resolved as product of two divisors which are relatively prime ?

1. $8$
2. $9$
3. $16$
4. 44

Q.

The value of $\left|\begin{array}{ccc}a-b& b+c& a\\ b-a& c+a& b\\ c-a& a+b& c\end{array}\right|$

(a) $a^3+b^3+c^3$
(b) $3bc$
(c) $a^3+b^3+c^3-3abc$
(d) None of these

Q. Using determinants show that points $A(a,b+c),B(b,c+a)$ and $C(c,a+b)$ are col-linear.

Q. The sum of real value(s) of $\alpha$ for which the system of equations
$x+3y+5z=\alpha x$
$5x+y+3z=\alpha y$
$3x+5y+z=\alpha z$
has infinite number of solutions is

1. $1$
2. $3$
3. $9$
4. $15$