Linear Equation in 2 Variables

Q.

The sum of distinct values of $\lambda$ for which the system of equations

$$\begin{gathered} \left(\lambda -1\right)x+\left(3\lambda +1\right)y+2\lambda z=0 \\ \left(\lambda -1\right)x+\left(4\lambda -2\right)y+\left(\lambda +3\right)z=0 \\ 2x+\left(3\lambda +1\right)y+3\left(\lambda -1\right)z=0 \end{gathered}$$

has no solutions is

Q.

How many linear equations can be satisfied by x = 2 and y = 3 ?

(a) only one

(b) only two

(c) only three

(d) infinitely many

Q.

If the system of equations

$$\begin{gathered} x+y+z=2 \\ 2x+4y–z=6 \\ 3x+2y+\lambda z=\mu \end{gathered}$$

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.

Who introduced the word polynomial?

Q.

Solve :

$2x-3=x+2$

Q.

Factorize the following expression: $x^4-{(y+z)}^4$.

Q.

If the system of equations

$$\begin{gathered} x-2y+3z=9 \\ 2x+y+z=b \\ x-7y+az=24, \end{gathered}$$

has infinitely many solutions, then $a-b$ is equal to:

Q.

The taxi fare in a city is as follows: For the first kilometre , the fare is RS. 20 and for the subsequent distance it is Rs.6 per km. Taking x km as the distance covered and RS. y as the total fare, write a linear equation for this information in the form of as+by+c=0 and indicate the values of a, b and c.

Q. Question 9 (v)
Solve the following pair of equations 43x + 67y = - 24 and 67x + 43y = 24.

Q.

A linear equation in two variables x and y is of the form ax + by + c = 0, where

(a) $a\ne 0,b\ne 0$.

(b) $a\ne 0,b=0$

(c) $a=0,b\ne 0$

(d) a = 0, c = 0

Q.

Find the equation of the line passing through the point $(2,3)$ and having slope $3$.

Q.

The cost of ball pen is Rs 5 less than half of the cost of fountain pen. Write this statement as a linear equation in two variables.

Q.

cost of 1 pen is Rs.x and that of 1 pencil is Rs.y.cost of 2 pens and 3 pencils together is Rs.18.write a linear equation to represent this information.

Q.

The linear equation 3x - 5y = 15 has

(a) a unique solution

(b) two solutions

(c) infinitely many solutions

(d) no solution

Q.

Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case.

(i) 3x + 5y = 7.5

(ii) $2x-\frac{y}{5}+6=0$

(iii) 3y - 2x = 6

(iv) 4x = 5y

(v)$\frac{x}{5}-\frac{y}{6}=1$

(vi) $\sqrt{2}x+\sqrt{3}y=5$

Q.

The number of girls x in a class is 5 less than twice the number of boys y of the same class. The linear equation in two variables for this information is

1. 2x - y = 5

2. x + 2y = -5

3. x + 5 = 2y

4. 2y - x = -5

Q.

The sum of the digits of a two-digit number is 7. If the number formed by interchanging the digits is less than the original number by 27, then find the original number.

1. 43

2. 61

3. 52

4. 25

Q.

The sum of three consecutive numbers is $66$ .Find the numbers .

Q. Question 14
The perpendicular bisector of a line segment joining points A(1, 5) and B(4, 6) cuts Y-axis at
(A) (0, 13)
(B) (0, –13)
(C) (0, 12)
(D) (13, 0)

Q.

The value of ${\left(1+i\right)}^8+{\left(1-i\right)}^8$ is

1. $16$

2. $-16$

3. $32$

4. $-32$

Q.

If $X$ and $Y$ are two non-empty sets, where $f:X\to Y$is function is defined such that $f\left(C\right)=\left\{f\left(x\right):x\in C\right\}$ for $C\subseteq X$ and $f^{-1}\left(D\right)=\left\{x:f\left(x\right)\in D\right\}$for $D\subseteq Y$, for any $A\subseteq Y$ and $B\subseteq Y$, then

1. $f^{-1}\left(f\left(A\right)\right)=A$

2. $f^{-1}\left(f\left(A\right)\right)=A$ only if $f\left(X\right)=Y$

3. $f\left(f^{-1}\left(B\right)\right)=B$, only if $B\subseteq X$

4. $f\left(f^{-1}\left(B\right)\right)=B$

Q. Determine the point on the graph of the equations $2x+5y=20$ whose x-coordinate is $\frac{5}{2}$ times its ordinate.

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

Q.

Point P(2, -3) lies on the line represented by the equation:

1. x + 2y = 0

2. 2x + 2y = 0

3. x + y = 1

4. 2x + y = 1

Q.

The linear equation y= x + 2 has-

1. Exactly one solution

2. Exactly two solutions

3. Infinitely many solutions

4. No solution

Q. The linear equation 3x − 5y = has
(a) a unique solution
(b) two solutions
(c) infinitely many solutions
(d) no solution

Q.

Places A and B are 200 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 4 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars? [4 MARKS]

Q.

The number of solutions to the equation $\sqrt{6-4x-x^2}=x+4$ is:

1. $0$

2. $1$

3. $2$

4. $4$

Q.

A three-wheeler scooter charges 15rupees for first km and 8rupees for every subsequent km. For a distance of x km, an amount of y rupees is paid. Write the linear equation representing the above information

Q.

When 5 times the larger of the two numbers is divided by the smaller , the quotient and the remainder is 2 and 9 respectively . Form a linear equation in two variables

Q. The equation of the line shown in the graph is.

1. $x+2y=0$
2. $2x+y=0$
3. $2x-y=0$