Grade 10 : Mathematics - 3. Linear Equations (Set 4)

Question 1

Taxi Company	Fixed Fee	Rate per Kilometre
City Cab (Service A)	₹50	₹12 / km
Metro Cab (Service B)	₹80	₹10 / km

Based on the pricing table above, at what travel distance $x$ (in km) will both taxi companies charge exactly the same total fare?

Accepted Answer

C) 15 km

Answer

A) 10 km

Answer

B) 12 km

Answer

D) 18 km

Question 2

A rectangular community park has its length $L$ defined as 4 metres more than twice its width $W$. The perimeter of the park is measured to be 68 metres.

What is the length of the rectangular community park?

Accepted Answer

B) 24 m

Answer

A) 20 m

Answer

C) 10 m

Answer

D) 28 m

Question 3

System of linear equations:
Equation 1: $2x + 3y = 7$
Equation 2: $6x + ky = 21$

For what constant value of $k$ does this system possess infinitely many solutions?

Accepted Answer

B) 9

Answer

A) 6

Answer

C) 12

Answer

D) 3

Question 4

System of linear equations:
Equation 1: $x + 2y = 5$
Equation 2: $3x + ky = 15$

Which condition must $k$ satisfy for the system to have a unique intersection point?

Accepted Answer

B) $k 
eq 6$

Answer

A) $k = 6$

Answer

C) $k = 3$

Answer

D) $k 
eq 3$

Question 5

System of linear equations:
Equation 1: $2x + 3y - 5 = 0$
Equation 2: $4x + py - 8 = 0$

Find the value of $p$ for which the pair of linear equations has NO solution (parallel lines).

Accepted Answer

B) 6

Answer

A) 5

Answer

C) 8

Answer

D) 10

Question 6

Taxi Company	Fixed Fee	Rate per Kilometre
City Cab (Service A)	₹50	₹12 / km
Metro Cab (Service B)	₹80	₹10 / km

From the purchases recorded above, what is the combined cost of purchasing exactly 1 notebook and 1 pen?

Accepted Answer

B) ₹40

Answer

A) ₹35

Answer

C) ₹45

Answer

D) ₹50

Question 7

Two cyclists ride along a circular track of circumference 400 metres. When riding in opposite directions, they pass each other every 20 seconds. When riding in the same direction, the faster cyclist laps the slower one every 100 seconds.

What is the speed of the faster cyclist?

Accepted Answer

B) 12 m/s

Answer

A) 10 m/s

Answer

C) 15 m/s

Answer

D) 8 m/s

Question 8

A motorboat travels 30 km upstream and 44 km downstream in 10 hours. On another trip, it travels 40 km upstream and 55 km downstream in 13 hours.

Find the speed of the boat in still water.

Accepted Answer

B) 8 km/h

Answer

A) 6 km/h

Answer

C) 10 km/h

Answer

D) 12 km/h

Question 9

A fraction reduces to $\frac{1}{3}$ when 1 is subtracted from its numerator. The same fraction becomes $\frac{1}{4}$ when 8 is added to its denominator.

Determine the original fraction.

Accepted Answer

B) $\frac{5}{12}$

Answer

A) $\frac{4}{11}$

Answer

C) $\frac{7}{15}$

Answer

D) $\frac{3}{8}$

Question 10

Five years ago, Nuri was three times as old as Sonu. Ten years from now, Nuri will be twice as old as Sonu.

How old is Nuri currently?

Accepted Answer

C) 50 years

Answer

A) 40 years

Answer

B) 45 years

Answer

D) 60 years

Question 11

The sum of the digits of a two-digit positive integer is 9. Nine times this number is equal to twice the number obtained by reversing the order of its digits.

Find the original two-digit number.

Accepted Answer

B) 18

Answer

A) 27

Answer

C) 36

Answer

D) 45

Question 12

In triangle $ riangle ABC$, the interior angle measures satisfy the relation: $\angle C = 3\angle B = 2(\angle A + \angle B)$.

Calculate the measure of angle $\angle B$.

Accepted Answer

B) $40^\circ$

Answer

A) $20^\circ$

Answer

C) $60^\circ$

Answer

D) $120^\circ$

Question 13

Taxi Company	Fixed Fee	Rate per Kilometre
City Cab (Service A)	₹50	₹12 / km
Metro Cab (Service B)	₹80	₹10 / km

How many litres of Solution A and Solution B must be mixed to create 100 litres of a 22% acid mixture?

Accepted Answer

B) 40 L of A, 60 L of B

Answer

A) 50 L of A, 50 L of B

Answer

C) 60 L of A, 40 L of B

Answer

D) 30 L of A, 70 L of B

Question 14

A library charges a fixed rate for borrowing a book for the first 3 days and an additional per-day charge thereafter. Saritha paid ₹27 for keeping a book for 7 days, while Susy paid ₹21 for 5 days.

What is the fixed charge for the first 3 days?

Accepted Answer

B) ₹15

Answer

A) ₹12

Answer

C) ₹18

Answer

D) ₹9

Question 15

A rectangular plot's area decreases by 9 sq units if length $x$ is decreased by 5 units and breadth $y$ is increased by 3 units. However, if length is increased by 3 units and breadth by 2 units, the area increases by 67 sq units.

Find the dimensions (length $x$ and breadth $y$) of the rectangle.

Accepted Answer

B) 17 units, 9 units

Answer

A) 15 units, 8 units

Answer

C) 19 units, 10 units

Answer

D) 12 units, 7 units

Question 16

The straight line represented by $4x + 3y = 24$ intersects the $x$-axis at point $A$ and the $y$-axis at point $B$, forming a right-angled triangle OAB with the origin $O(0,0)$.

Calculate the area of $ riangle OAB$.

Accepted Answer

B) 24 sq units

Answer

A) 12 sq units

Answer

C) 48 sq units

Answer

D) 18 sq units

Question 17

Linear system:
Equation 1: $x + 2y = 3$
Equation 2: $5x + ky + 7 = 0$

For what numerical value of $k$ will the system have no solution?

Accepted Answer

B) 10

Answer

A) 5

Answer

C) -10

Answer

D) 15

Question 18

Towns A and B are 100 km apart along a straight highway. Two cars start simultaneously: one from A and one from B. If they drive in the same direction, they meet in 5 hours. If they drive towards each other, they meet in 1 hour.

What are the speeds of the two cars?

Accepted Answer

B) 60 km/h and 40 km/h

Answer

A) 50 km/h and 30 km/h

Answer

C) 70 km/h and 30 km/h

Answer

D) 55 km/h and 45 km/h

Question 19

Taxi Company	Fixed Fee	Rate per Kilometre
City Cab (Service A)	₹50	₹12 / km
Metro Cab (Service B)	₹80	₹10 / km

How many days would it take for 1 woman working alone to finish the entire task?

Accepted Answer

B) 18 days

Answer

A) 12 days

Answer

C) 24 days

Answer

D) 36 days

Question 20

Pair of non-linear equations reducible to linear form:
$\frac{2}{x} + \frac{3}{y} = 13$
$\frac{5}{x} - \frac{4}{y} = -2$

Find the solution set $(x, y)$.

Accepted Answer

B) $x = \frac{1}{2}, y = \frac{1}{3}$

Answer

A) $x = 2, y = 3$

Answer

C) $x = \frac{1}{3}, y = \frac{1}{2}$

Answer

D) $x = 3, y = 2$

Grade 10 : Mathematics - 3. Linear Equations (Set 4)