Pair of Linear Equations in Two Variables

Which statement best summarizes the chapter "Pair of Linear Equations in Two Variables"?

1. Every pair has exactly one solution.
2. Pair of linear equations can be solved only graphically.
3. A pair of linear equations may have one solution, no solution, or infinitely many solutions depending on the relationship between the equations, and they can be solved using graphical and algebraic methods.
4. Linear equations always represent curves.

Which method uses determinants in Class 10?

1. Elimination method
2. Substitution method
3. Cross multiplication method
4. Factorization method

Which condition represents a consistent system?

1. At least one solution exists.
2. No solution exists.
3. Only parallel lines exist.
4. Coefficients are always equal.

Solve: 2x + y = 8 and x + y = 5

1. (2,3)
2. (3,2)
3. (4,1)
4. (5,0)

The graphical solution of two linear equations is:

1. Product of coefficients
2. Common point of intersection of the two lines
3. Midpoint of both lines
4. Origin always

Which pair has no solution?

1. Intersecting lines
2. Coincident lines
3. Parallel lines
4. Perpendicular lines

Which graph represents equations having infinitely many solutions?

1. Intersecting lines
2. Parallel lines
3. Coincident lines
4. Perpendicular lines

Which method eliminates one variable by adding or subtracting equations?

1. Graphical method
2. Elimination method
3. Factorization method
4. Division method

Solve: x + y = 7 and x - y = 3

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

Which method involves replacing one variable with its value from another equation?

1. Elimination method
2. Graphical method
3. Substitution method
4. Cross multiplication only

If a1/a2 = b1/b2 = c1/c2, then the equations are:

1. Parallel
2. Intersecting
3. Coincident
4. Perpendicular

If a1/a2 = b1/b2 ≠ c1/c2, the pair of equations has:

1. One solution
2. Infinitely many solutions
3. No solution
4. Two solutions

If a1/a2 ≠ b1/b2, then the pair of linear equations is:

1. Inconsistent
2. Consistent with a unique solution
3. Dependent
4. Has infinitely many solutions

A pair of linear equations has a unique solution when:

1. The lines are parallel.
2. The lines coincide.
3. The lines intersect at one point.
4. Both equations are identical.

Which of the following is a pair of linear equations in two variables?

1. x^2 + y = 5 and x - y = 2
2. 2x + 3y = 7 and 4x - y = 5
3. xy = 6 and x + y = 8
4. x^3 + y = 10 and x - y = 1

The graph of a linear equation in two variables is always a:

1. Curve
2. Circle
3. Straight line
4. Parabola

If y = mx + c, then m represents:

1. y-intercept
2. slope
3. origin
4. solution point

Which equation has infinitely many solutions?

1. x + y = 5
2. x − 2 = 0
3. x = 3 and y = 4
4. x² + y = 1

If x + y = 0, then the graph passes through:

1. (1,1)
2. (2,2)
3. (3,−3)
4. (4,4)

The equation x = −4 represents a line:

1. Parallel to x-axis
2. Parallel to y-axis
3. Passing through origin
4. Oblique line

If (2, k) is a solution of 3x + 4y = 18, then k is:

1. 2
2. 3
3. 4
4. 5

How many lines can pass through a single point?

1. One
2. Two
3. Finite
4. Infinite

The graph of x = 3 is a:

1. Straight line parallel to x-axis
2. Straight line parallel to y-axis
3. Parabola
4. Point

Which pair is a solution of x − y = 2?

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

If a line passes through (0,5), what is its y-intercept?

1. 0
2. 5
3. −5
4. 1

Which of the following is NOT a linear equation in two variables?

1. 3x + 2y = 7
2. x − y = 4
3. x² + y = 5
4. 2x + 5y = 0

If y = 2x + 3, what is y when x = −1?

1. 1
2. −1
3. −5
4. 5

How many solutions does x + y = 10 have?

1. 1
2. 2
3. Finite many
4. Infinite

Which point satisfies 2x + 3y = 12?

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

Which of the following is a linear equation in two variables?

1. x² + y = 5
2. x + y = 7
3. xy = 6
4. x/y = 3

Denominator is 7 more than numerator. Adding 5 to each gives 1/2. The original fraction is

1. 3/10
2. 5/12
3. 7/14
4. 4/11

The more convenient method for 4p+3q=23 and 4p+14q=10 is

1. elimination by addition
2. elimination by subtraction
3. substitution method
4. cross-multiplication method

The system x–y=2 and –2x+2y=–1 has

1. unique solution
2. infinitely many solutions
3. no solution
4. two solutions

Sheetal has Rs 2 and Rs 5 coins; 10 coins worth Rs 41. System representing this is

1. a+9b=9 and a+3b=5
2. a+b=10 and 2a+5b=41
3. a+b=10 and a+5b=41
4. a+b=10 and 2a+b=41

Dolly has 12 bags with either 2 or 3 balls; 30 balls total. System of equations representing this is

1. x+9y=20 and x+3y=25
2. x+y=12 and x+3y=20
3. x+y=12 and x+5y=25
4. x+y=12 and 3x+2y=30

If 5x+7y=3 and 15x+21y=k represent coincident lines, then k =

1. 10
2. 9
3. 8
4. 7

3x–4y=5 and 12x–16y=20 have

1. no common solution
2. exactly one common solution
3. exactly two common solutions
4. more than two common solutions

For 2x+ky=11 and 5x–7y=5 to have no solution, k =

1. 13/5
2. –13/5
3. –14/5
4. –16/5

If 4x+7y=10 and 10x+ky=25 represent coincident lines, then k =

1. 5
2. 17/2
3. 27/2
4. 35/2

Solution of (xy+x)/x = 2 and x+y=1 is

1. x=0,y=1
2. x=1,y=0
3. x=1/2,y=1/2
4. x=10,y=–9

If x+y=6 and 3x–y=4, then x–y =

1. –1
2. 0
3. 2
4. 4

Solution of 4x+3y=18xy and 4x–10y+8xy=0 is

1. x=1/2, y=1/3
2. x=1/4, y=1/3
3. x=–1/2, y=–1/3
4. x=–1, y=–3

The two lines y=5x+2 and y=6x–3 intersect at

1. (–1,7)
2. (5,27)
3. (8,14)
4. (11,–4)

For what value of p does 2x–py=0, 3x+4y=0 have nonzero solution?

1. p=–6
2. p=–8/3
3. p=8/3
4. p=6

Solution of (a+b)x+(2a–b)y=2a and (2a+b)x+2(a–b)y=2(a+b) is

1. (a,b)
2. (b,a)
3. (a/2,b/2)
4. x=0,y=0

For what value of p will 3x+y=1, (2p–1)x+(p–1)y=(2p+1) have no solution?

1. p=2
2. p=1/2
3. p=–2
4. p=2

Solution of (x+2)/3 + (y–1)/4 = 5/12 and (x+3)/4 + (y+2)/5 = 1 is

1. x=–2,y=8
2. x=4,y=–4
3. x=2,y=6
4. x=4,y=6

For coincident lines 2x+3y=7 and (a–b)x+(a+b)y=(3a+b–2), a and b are

1. a=5,b=–1
2. a=5,b=1
3. a=–5,b=–1
4. a=5,b=–1

Solution of (3x+y)/2 = 4 and x–(y–2)/2 = 3 is

1. x=1,y=2
2. x=0,y=0
3. x=–1,y=2
4. x=1,y=–2

Which system of equations is consistent?

1. 3x–y=1, 6x–2y=5
2. 4x+6y–7=0, 12x+18y–21=0
3. 4x+7y=3, 8x+14y=7
4. 6x+2y=3, 5x+6y=–2

A principal amounts to Rs 7400 in 4 years and Rs 9200 in 7 years at simple interest. The rate is

1. 4%
2. 8%
3. 12%
4. 14%

For equations 5x–6y=2 and 10x=12y+7

1. no solution
2. unique solution
3. two solutions
4. infinite solutions

If ax+by=a²–b² and bx+ay=a²+b², then (x,y) =

1. (a,b)
2. (–a,–b)
3. (b,–a)
4. (b,a)

A toy train crosses 210m and 122m tunnels in 25 and 17 seconds. Length of train is

1. 13 m
2. 65 m
3. 332 m
4. 88 m

Ratio of two-digit number to sum of its digits is 4:1, and units digit is 3 more than tens digit. The number is

1. 63
2. 36
3. 24
4. 53

If ratio of boys to girls is B and ratio of girls to boys is G, then B+G is

1. ≥1
2. greater than 1
3. less than 1
4. equal to 1

Incomes of P and Q are in ratio 3:2, expenses in ratio 5:3. Both save Rs 200. P's income is

1. Rs 700
2. Rs 1000
3. Rs 1200
4. Rs 900

Divide 62 into two parts such that 1/4 of first and 2/5 of second are in ratio 2:3

1. 24, 38
2. 32, 30
3. 16, 32
4. 40, 22

LCM of two numbers is 630, HCF is 9, sum is 153. Their difference is

1. 72
2. 27
3. 81
4. 18

Sum of digits of a two-digit number is 8. If digits are reversed, number decreases by 54. The number is

1. 35
2. 17
3. 71
4. 53

A man takes 3 hours upstream and half that time downstream. Speed of river is 5 km/h. Speed in still water is

1. 50 km/hr
2. 40 km/hr
3. 30 km/hr
4. 20 km/hr

A two-digit number is four times the sum and three times the product of its digits. The number is

1. 73
2. 71
3. 69
4. 67

Ten years ago a man was 12 times as old as his son. Ten years hence he will be twice as old. Their present ages are

1. 34 years, 12 years
2. 26 years, 24 years
3. 20 years, 18 years
4. 17 years, 15 years

The coordinates where 2(x–3)=y–8 meets the x-axis are

1. (3,0)
2. (2,0)
3. (–1,0)
4. (0,–1)

4kg sweets and 3kg apples cost Rs 233; 3kg sweets and 4kg apples cost Rs 194. Cost per kg of each is

1. Rs 50, Rs 11
2. Rs 40, Rs 12
3. Rs 30, Rs 18
4. Rs 28, Rs 16

The present age of a man equals the sum of ages of his five children. Twelve years hence the sum of children's ages will be twice the father's age. The father's present age is

1. 36 years
2. 37 years
3. 38 years
4. 39 years

If 5 is added to denominator and 5 subtracted from numerator the fraction becomes 1/3; if 3 is subtracted from numerator and 3 added to denominator it becomes 1/4. The fraction is

1. 7/9
2. 5/8
3. 11/16
4. 11/13

The coordinates where 3x–y=5 and 6x–y=10 meet the y-axis are

1. (0,–5),(0,–10)
2. (–5,0),(10,0)
3. (5,0),(0,–10)
4. (0,–5),(0,–10)

The solution of 2x+3y+5=0 and 3x–2y–12=0 is

1. x=–3,y=–2
2. x=2,y=–3
3. x=3,y=–2
4. x=12,y=13

If (x+y)/(xy) = 2 and (x–y)/(xy) = 6, then

1. x=a,y=b
2. x=b,y=a
3. x=a/2,y=b/2
4. x=b/2,y=a/2

If x/a + y/b = 2 and ax – by = a²–b², then

1. x=y=b
2. x=y=a
3. x=a,y=b
4. x=b,y=a

If 3ax + 2by = 5ab and 5ax – 3by = 2ab, then

1. x=b,y=a
2. x=a,y=b
3. x=y=a
4. x=y=b

If 2/x + 3/y = 9 and 4/x – 9/y = 3, then

1. x=3,y=1
2. x=1,y=3
3. x=1,y=1
4. x=2,y=2

If 4/x + 3y = 14 and 3/x – 4y = 23, then

1. x=1,y=2
2. x=1,y=1
3. x=1,y=1/2
4. x=1/2,y=1/2

If 1/(2x) + 1/(3y) = 2 and 1/(3x) + 1/(2y) = 13/6, then

1. x=1,y=2
2. x=3,y=4
3. x=2,y=2
4. x=3,y=3