Cramer's Rule

Q. There are $200$ individuals with a skin disorder, $120$ had been exposed to the chemical $C_1,50$ to chemical $C_2$, and $30$ to both the chemicals $C_1$ and $C_2$. Find the number of individuals exposed to:
$\left(i\right)$ Chemical $C_1$ but not Chemical $C_2$
$\left(ii\right)$ Chemical $C_2$ but not Chemical $C_1$
$\left(iii\right)$ Chemical $C_1$ or Chemical $C_2$

Q. find the equation of line perpendicular to x-7y+5=0 and having x intercept=3

Q.

Let a, b, c, be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x= cy +bz, y=az + cx and z = bx +ay. Then , a² + b²+ c²+ 2abc is equal to

1. -1

2. 0

3. 1

4. 2

Q. If a, b, c are non – zero real numbers and if the equations (a-1) x = y + z, (b -1)y = z + x, (c - 1)z = x + y has a non trivial solution, then ab + bc + ca is equal to

1. a +b + c
2. abc
3. 1
4. None of these

Q. If $A=\left[\begin{array}{cc}3& x-1\\ 2x+3& x+2\end{array}\right]$ is a symmetric matrix, then the value of $x$ is

[1 mark]

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

Q. Some properties of idempotent matrix and involutory matrix?

Q.

If A is a square matrix of order n such that |adj (adj A)|$=|A{|}^9$, then the value of n can be

1. 4

2. 2

3. either 4 or 2

4. None of these

Q.

Examine the consistency of the system of equations.

x + y + z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4

Q. What is "arbitrary constant" ? What is the meaning of arbitrary?

Q.

Find the value of p so that the three lines 3x + y – 2 = 0, px + 2y – 3 = 0 and 2x – y – 3 = 0 may intersect at one point.

Q. The system of equations
$px+y+z=px+py+z=px+y+pz=p$
has no solution for

1. three distinct values of $p$
2. two distinct values of $p$
3. only one value of $p$
4. no such $p$ exists

Q. If the system of linear equations
$\begin{array}{c}x-4y+7z=7g\\ 3y-5z=h\\ -2x+5y-9z=k\end{array}$

is consistent, then :

1. $2g+h+k=0$
2. $g+2h+k=0$
3. $g+h+2k=0$
4. $g+h+k=0$

Q. For which of the following ordered pairs $(\mu ,\delta )$, the system of linear equations
$x+2y+3z=1$
$3x+4y+5z=\mu$
$4x+4y+4z=\delta$
is inconsistent?

1. $(3,4)$
2. $(1,0)$
3. $(4,3)$
4. $(4,6)$

Q. Solve the following system of equations by matrix method:
(i) x + y − z = 3
2x + 3y + z = 10
3x − y − 7z = 1

(ii) x + y + z = 3
2x − y + z = − 1
2x + y − 3z = − 9

(iii) 6x − 12y + 25z = 4
4x + 15y − 20z = 3
2x + 18y + 15z = 10

(iv) 3x + 4y + 7z = 14
2x − y + 3z = 4
x + 2y − 3z = 0

(v)
$$\begin{gathered} \frac{2}{x}-\frac{3}{y}+\frac{3}{z}=10 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=10 \\ \frac{3}{x}-\frac{1}{y}+\frac{2}{z}=13 \end{gathered}$$

(vi) 5x + 3y + z = 16
2x + y + 3z = 19
x + 2y + 4z = 25

(vii) 3x + 4y + 2z = 8
2y − 3z = 3
x − 2y + 6z = −2

(viii) 2x + y + z = 2
x + 3y − z = 5
3x + y − 2z = 6

(ix) 2x + 6y = 2
3x − z = −8
2x − y + z = −3

(x) x − y + z = 2
2x − y = 0
2y − z = 1

(xi) 8x + 4y + 3z = 18
2x + y +z = 5
x + 2y + z = 5

(xii) x + y + z = 6
x + 2z = 7
3x + y + z = 12

(xiii) $\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4,\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1,\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2;x,y,z\ne 0$

(xiv) x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12

Q.

Find: $4.8÷10$

Q.

The differential equation for the line y=mx+c is (where c is arbitrary constant)

1. 
   

2. None of these

3. 
   

4. 
   

Q. Why every function is relation but every relation not have function

Q.

1. 1

2. 0

3. -1

4. 2

Q.

Examine the consistency of the system of equations.

x + 2y = 2

2x + 3y = 3

Q. $x+ky-z=0,3x-ky-z=0$and$x-3y+z=0$ has non-zero solution for k =

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

Q.

The order and degree of the differential equation: (y")² + (y")³ + (y')⁴ + y⁵ = 0 is:

1. 3, 5

2. 2, 5

3. 2, 4

4. 2, 3

Q. The ordered triad $(x,y,z)$ which satisfies the system of linear equations is:
$x+y+z=6$
$x-y+z=2$
$3x+2y-4z=-5$

1. $(1,2,3)$
2. $(1,2,5)$
3. $(2,3,4)$
4. $(1,3,4)$

Q. Solve the equation ${\cos }^{-1}\frac{a}{x}-{\cos }^{-1}\frac{b}{x}={\cos }^{-1}\frac{1}{b}-{\cos }^{-1}\frac{1}{a}$

Q.

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

x = a cos θ, y = b cos θ

Q. Find the general solution of $(x+2y^3)\frac{dy}{dx}=y$

Q. If the system of linear equation $x+2ay+az=0,x+3by+bz=0,x+4cy+cz=0$ has a non zero solution, then a, b, c

1. Are in A.P.
2. Are in H. P.
3. Satisfy a +2b + 3c = 0
4. Are in G. P.

Q.

Which of the following cases may lead to non trivial solutions in case of system of linear equations according to Cramer's rule Convention?

1. D1 = 0, D2 = 0, D3 = 0

2. D1 = 0, D2 = 0, D3 != 0

3. D1 != 0, D2 != 0, D3 = 0

4. D1 != 0, D2 != 0, D3 != 0

Q. The system of equations
$x+y-2z=0$
$2x-3y+z=0$
$x-5y+4z=k$

is consistent for

1. all non-zero real values of $k$
2. all real values of $k$
3. two real values of $k$
4. $k=0$ only

Q.

Find the general solution of the equation

Q. Is a relation from a empty set to another empty or non-empty set defined?