Condition for a Conic to Be a Pair of Straight Lines

Q. The value of k for which the system of equations:
2x + 3y - 2z = 0; 2x - y + 3z = 0 and 7x + ky - z = 0 has non-trivial solution, is .

1. $k=-\frac{57}{5}$
2. $k=\frac{57}{5}$
3. $k=-\frac{57}{10}$
4. $k=\frac{57}{10}$

Q. The shortest distance between the skew lines $\overrightarrow{r}=\overrightarrow{r_1}+t\overrightarrow{a_1}\text{and}\overrightarrow{r}=\overrightarrow{r_2}+n\overrightarrow{a_2,}\text{where}\overrightarrow{r_1}=9j+2k,\overrightarrow{a_1}=3i-j+k,\overrightarrow{r_2}=-6i-5j+10k,\overrightarrow{a_2}=-3i+2j+4k$ is $3\sqrt{a}$. Then the value of $a$ is:

Q.

If the equations $k{x}^2-2xy-y^2-2x+2y=0$represents a pair of lines, then $k$ is equal to

1. $2$

2. $-2$

3. $-5$

4. $3$

Q.

Examine the consistency of the system of equations.

3x − y − 2z = 2

2y − z = −1

3x − 5y = 3

Q. 16. The set of all values of λ for which the system of linear equations x-2y-2z=λ x, x+2y+z=λ y -x-y=λzhas a non-trivial solution. (A) contains more than two elements (B)is a singleton (C) is an empty set (D)contain exactly two elements.

Q.

Solve system of linear equations, using matrix method.

2x + 3y + 3z = 5

x − 2y + z = −4

3x − y − 2z = 3

Q. The value of $\lambda$ for which the equation$2x^2+7xy+3y^2+8x+14y+\lambda =0$ represents a pair of straight lines is:

1. $1$
2. $2$
3. $4$
4. $8$

Q. The second degree equation $x^2+3xy+2y^2-x-4y-6=0$ represent a pair of straight lines

1. True
2. False

Q. The value of k for which $2x^2+7\mathrm{xy}+3y^2+8x+14y+k=0$represents a pair of straight lines, is:

1. 2
2. 4
3. 8
4. 16

Q.

Solve system of linear equations, using matrix method.

x − y + z = 4

2x + y − 3z = 0

x + y + z = 2

Q.

Solve system of linear equations, using matrix method.

5x + 2y = 3

3x + 2y = 5

Q.

What is the condition for a conic $x^2+2xy+2y+kx+3y^2=0$ to represent a pair of straight lines.

1. $k=1$

2. $k=\frac{1}{2}$

3. $k=2$

4. $k=-\frac{1}{2}$

Q. The system of linear equations:
x + y + z = 2
2x + y − z = 3
3x + 2y + kz = 4 has a unique solution if
(a) k ≠ 0
(b) −1 < k < 1
(c) −2 < k < 2
(d) k = 0

Q. the condition for a conic $x^2+2xy+2y+kx+3y^2=0$ to represent a pair of straight lines

1. k=1
2. k=2
3. k=3
4. k=4

Q. Solve this:

Q. 13. Let S be the set of all real values of k for which the system of linear equations.
x + y +z =2
2x + y - z =3
3x +2y +kz =4
has a unique solution. Then S is :
Option
1. an empty set
2. equal to {0}
3. equal to R
4. equal to R - {0}

Q. The value of k for which $2x^2+7\mathrm{xy}+3y^2+8x+14y+k=0$represents a pair of straight lines, is:

1. 2
2. 4
3. 8
4. 16

Q. If $\frac{1}{x^4+x^2+1}=\frac{Ax+B}{x^2+x+1}+\frac{Cx+D}{x^2-x+1}$, then $C+D=$

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

Q. The system of equations:
x + y + z = 5
x + 2y + 3z = 9
x + 3y + λz = µ
has a unique solution, if
(a) λ = 5, µ = 13
(b) λ ≠ 5
(c) λ = 5, µ ≠ 13
(d) µ ≠ 13

Q.

Examine the consistency of the system of equations.

x + 3y = 5

2x + 6y = 8

Q. Find the real values of $\lambda$ for which the following system of linear equations has non-trivial solutions. Also, find the non-trivial solutions

$$\begin{gathered} 2\lambda x-2y+3z=0 \\ x+\lambda y+2z=0 \\ 2x+\lambda z=0 \end{gathered}$$

Q. Solve each of the following system of homogeneous linear equations.
3x + y + z = 0
x − 4y + 3z = 0
2x + 5y − 2z = 0

Q. If $8,2$ are the roots $x^2+ax+c=0$ and $3,3$ are the roots of $x^2+dx+b=0$ then the roots of the equation $x^2+ax+b=0$ are

1. $8,-1$
2. $-9,2$
3. $9,1$
4. $-8,-2$

Q. Investigate for what values of $\lambda ,\mu$ the simultaneous equation $x+y+z=6;x+2y+3z=10$ & $x+2y+\lambda z=\mu$ have a unique solution

1. $\lambda \ne 2$
2. $\lambda \ne 3$
3. $\lambda \ne 5$
4. $\lambda \ne 1$

Q. Solve each of the following system of homogeneous linear equations.
x + y − 2z = 0
2x + y − 3z = 0
5x + 4y − 9z = 0

Q. Show that each of the following systems of linear equations is consistent and also find their solutions:
(i) 6x + 4y = 2
9x + 6y = 3

(ii) 2x + 3y = 5
6x + 9y = 15

(iii) 5x + 3y + 7z = 4
3x + 26y + 2z = 9
7x + 2y + 10z = 5

(iv) x − y + z = 3
2x + y − z = 2
−x −2y + 2z = 1

(v) x + y + z = 6
x + 2y + 3z = 14
x + 4y + 7z = 30

(vi) 2x + 2y − 2z = 1
4x + 4y − z = 2
6x + 6y + 2z = 3

Q. If lines $px+qy+r=0,qx+ry+p=0$ and $rx+py+q=0$ are concurrent, then the value of $p+q+r$ is: (where $p,q,r$ are distinct).

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

Q. 3x + y − 2z = 0
x + y + z = 0
x − 2y + z = 0

Q. 20.The general solution of differential equation (cosx-y)dy=ysinx dx is

Q.

Equations x+y=2, 2x+2y=3 will have [UPSEAT 1999]​

1. Only one solution

2. No solution

3. None of these

4. Many finite solutions

Q. If $\left|\begin{array}{cc}x& 4\\ 3& 2\end{array}\right|=\left|\begin{array}{cc}-3& -5\\ 1& 3\end{array}\right|,$ then the value of $x$ is

$\left(a\right)-2\left(b\right)8\left(c\right)-4\left(d\right)4$