Moderate Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) What is the graphical representation of a pair of inconsistent equations? Two lines that never intersectintersect once

Perfect! Inconsistent equations have no solution, so their graphs are parallel lines.

(2) State the condition for a pair of linear equations to have a unique solution.

a1a2 ≠= b1b2

Correct! This means the lines intersect at exactly one point.

(3) If the pair of equations is dependent and consistent, how will their graphs look?

The graphs will be coincidentintersecting lines

Excellent! Dependent equations represent the same line (infinite solutions).

(4) Find the solution of the equation x + 2y = 8 if y = 2. x =

Perfect! Substitution is the simplest method when one variable is known.

(5) Find the value of y if the point (3, y) lies on the line 2x - y = 4. y =

Excellent! Points on a line satisfy the line's equation.

Short Answer Questions (2 Marks Each)

(1) Solve the pair: 2x + 3y = 12, 3x - y = 5 using the substitution method.

x = and y =

Excellent substitution method! Solution: (2711, 2611).

(2) Solve by elimination: 4x + 5y = 19, 3x - 2y = 4. x = and y =

Good elimination method! Always verify by substitution.

(3) Draw the graphs of x + y = 7 and 2x - y = 4, and find the point of intersection.

Point of intersection: (x,y) = ( , )

Perfect! Graphical and algebraic methods give the same answer.

(4) A number consists of two digits. The digit at the ten's place is 1 more than the digit at the one's place. If the digits are reversed, the new number is 9 less than the original number. Find the number.

The number is

With tens digit 1 more than units digit, the number is 43.

(5) The sum of the ages of a father and his son is 50 years. Five years ago, the father's age was four times the son's age. Find their present ages.

Father's Age = years and Son's Age = years

Perfect! Father = 37 years, Son = 13 years.

Long Answer Questions (4 Marks Each)

(1) A fraction becomes 12 when 1 is subtracted from the numerator and 2 is added to the denominator. If 2 is added to the numerator and 3 is subtracted from the denominator, it becomes 23. Find the fraction.

The fraction is 2036 = 59.

(2) A motorboat travels 30 km downstream and 18 km upstream in 3 hours. It can travel 21 km downstream and return in 3 hours 15 minutes. Find the speed of the boat in still water and the speed of the stream.

Boat speed in still water: kmh

Stream speed: kmh

The exact algebraic solution involves solving a complex system, but checking shows these values are very close to satisfying both conditions.

(3) The difference between two numbers is 26. Four times the smaller number added to three times the larger number is 178. Find the numbers.

Smaller number:

Larger number:

Perfect! The numbers are 1007 and 2827 (or 14 27 and 40 27).

(4) Raju purchased 4 pens and 3 pencils for ₹50. Ravi purchased 2 pens and 5 pencils for ₹42. Find the cost of one pen and one pencil.

Cost of one pen: ₹

Cost of one pencil: ₹

Good solution! The costs are ₹ 627 per pen and ₹ 347 per pencil.

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) Which of the following methods is not used to solve a pair of linear equations?

(a) Elimination (b) Substitution (c) Trial and error (d) Cross multiplication

Correct! Trial and error is not a systematic method for solving linear equations.

(2) The pair of equations x + 2y = 7 and 2x + 4y = 15 is:

(a) Consistent with unique solution (b) Inconsistent (c) Consistent with infinite solutions (d) None

Correct! The ratios are a1a2 = 12, b1b2 = 12, c1c2 = 715. Since a1a2 = b1b2 ≠ c1c2, the system is inconsistent.

(3) The solution of the pair x + y = 10, x - y = 2 is:

(a) (6, 4) (b) (4, 6) (c) (5, 5) (d) (2, 8)

Correct! Adding the equations: 2x = 12, so x = 6. Then y = 4.

(4) In cross multiplication method, the denominator used to find the value of x is:

(a) a1b2 - a2b1 (b) b1c2 - b2c1 (c) c1a2 - c2a1 (d) a1c2 - a2c1

Correct! In cross multiplication: xb1c2−b2c1 = yc1a2−c2a1 = 1a1b2−a2b1.

(5) If the ratio a1a2 = b1b2 ≠ c1c2, the pair of equations is:

(a) Consistent (b) Dependent (c) Inconsistent (d) Homogeneous

Correct! This condition means parallel lines (no solution), so the system is inconsistent.

(6) Which of the following equations does NOT represent a straight line?

(a) 3x + 4y = 12 (b) x + y = 0 (c) x2+y=5 (d) 5x - 2y = 1

Correct! x2+y=5 is quadratic in x, so it represents a parabola, not a straight line.

(7) Which method involves multiplying the equations to eliminate one variable?

(a) Substitution (b) Graphical (c) Elimination (d) None of the above

Correct! Elimination method involves multiplying equations by suitable numbers to eliminate variables.

(8) If two equations represent the same line, then the system is:

(a) Independent (b) Inconsistent (c) Consistent and dependent (d) None

Correct! Same line means infinite solutions, so the system is consistent and dependent.

(9) In the pair 4x - y = 5, 2x + 3y = 12, the value of y when x = 2 is:

(a) 1 (b) 2 (c) 3 (d) 4

Correct! From 4x - y = 5 with x = 2: 8 - y = 5, so y = 3.

(10) The pair of equations x + y = 4, x + y = 6 is:

(a) Consistent (b) Inconsistent (c) Dependent (d) Both a and c

Correct! These are parallel lines (same slope, different intercepts), so inconsistent.

Linear Equations Challenge

Determine whether these statements about pairs of linear equations are True or False:

Linear Equations Quiz