1
A school is organizing a field trip for its students. The cost of the bus is a fixed amount of ₹5000. Additionally, each student needs to pay ₹250 for entry tickets and food. If 'x' is the number of students going on the trip, write a linear expression representing the total cost. If the school has allocated ₹20,000 for the trip and 40 students are going, calculate the total cost and determine if the allocation is sufficient. 5marks
2
Assertion: A linear polynomial is always represented by a straight line when plotted on a graph.Reason: A linear polynomial is an equation of degree one, and equations of degree one in two variables, when plotted, always result in a straight line.Select the correct option:(a) Both assertion and reason are true, and reason is the correct explanation of assertion.(b) Both assertion and reason are true, but reason is not the correct explanation of assertion.(c) Assertion is true, but reason is false.(d) Assertion is false, but reason is true. 1mark
3
A farmer uses a rectangular plot of land to grow two types of crops, A and B. The plot has a length of 'l' meters and a width of 'w' meters. The cost of seeds for crop A is ₹50 per square meter and for crop B is ₹70 per square meter. If the farmer decides to use half of the plot for crop A and the other half for crop B, write an algebraic expression for the total cost of seeds. If the plot is 20m by 15m, and crop A is planted in one half, what is the cost of seeds for crop A? 3marks
4
The number of mangoes a vendor has decreases each day. On day 1, he has 150 mangoes. Each day, he sells 15 mangoes.(i) Write a linear expression for the number of mangoes left on day 'n'.(ii) How many mangoes will be left on day 7?(iii) On which day will he have 45 mangoes left? 5marks
5
Consider the cost of a taxi ride: a fixed charge of ₹30 for the first kilometer and ₹15 for each subsequent kilometer. If 'd' is the total distance in kilometers, write an expression for the total fare for a journey of 'd' km, where d > 1. Calculate the fare for a journey of 8 km. 3marks
6
A student is solving a quadratic equation problem and mistakenly uses a linear polynomial expression
p(x) = 3x + 5 instead of the correct quadratic one. For what value of x will the student's incorrect linear polynomial give the same value as the correct quadratic polynomial q(x) = x² + 2x + 3? 5marks7
Two students, Anjali and Rohan, are calculating the perimeter of a rectangle. Anjali uses the formula P = 2(l+w) and gets a value. Rohan uses a simplified linear expression
P = 4s assuming the rectangle is a square with side 's'. If the actual rectangle has length 12 cm and width 8 cm, and Rohan mistakenly calculates the perimeter as if it were a square with side length equal to the average of length and width, compare their results and identify the error in Rohan's approach in terms of linear relationships. 5marks8
Assertion: The graph of the equation y = 5x is a straight line passing through the origin.Reason: In the equation y = ax + b, if b = 0, the line passes through the origin.Select the correct option:(a) Both assertion and reason are true, and reason is the correct explanation of assertion.(b) Both assertion and reason are true, but reason is not the correct explanation of assertion.(c) Assertion is true, but reason is false.(d) Assertion is false, but reason is true. 1mark
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A company manufactures mobile phones. The cost of producing 'n' mobile phones is given by the linear expression
C(n) = 50000 + 250n, where ₹50000 is the fixed cost and ₹250 is the cost per phone.(i) If the company produces 100 phones, what is the total cost?(ii) If the company wants to keep the total cost below ₹100000, what is the maximum number of phones they can produce? 5marks10
A train ticket costs
50 for the first 100 km and 10 for every additional 50 km. If 'd' is the distance in km, write a piecewise linear function to represent the cost. Calculate the cost for a journey of 350 km. 5marks11
Consider the equation of a line
y = 3x - 2.(i) Find the y-coordinate when x = 0.(ii) Find the x-coordinate when y = 7.(iii) Plot the graph of this line using the points found in (i) and (ii). 3marks12
Two linear relationships are given:Relationship 1:
y = 2x + 5Relationship 2: y = -x + 2Find the point of intersection of these two lines. 5marks13
A rectangular garden has a length of
(2x + 3) meters and a width of (x + 1) meters.(i) Write an expression for the perimeter of the garden.(ii) If the perimeter is 46 meters, find the value of x and the dimensions of the garden. 5marks14
Assertion: The expression
5x³ + 2x² - x + 8 is a linear polynomial.Reason: A linear polynomial is an algebraic expression where the highest power of the variable is 1.Select the correct option:(a) Both assertion and reason are true, and reason is the correct explanation of assertion.(b) Both assertion and reason are true, but reason is not the correct explanation of assertion.(c) Assertion is true, but reason is false.(d) Assertion is false, and reason is true. 1mark15
A small company’s profit
P (in thousands of rupees) depends on the number of units sold x. The relationship is linear: P(x) = 3x - 15.(i) What is the profit if 10 units are sold?(ii) How many units must be sold to achieve a profit of ₹45,000? 3marks16
A gardener is planting trees in a row. The first tree is planted at a distance of 2 meters from the start. Each subsequent tree is planted 3 meters away from the previous one.(i) Write a linear expression for the distance of the nth tree from the start.(ii) Find the distance of the 6th tree from the start. 3marks
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A survey was conducted on the average monthly electricity consumption of households in a locality. It was found that the consumption increases linearly with the number of rooms in a house. A house with 2 rooms consumes 120 units, and a house with 5 rooms consumes 210 units.(i) Find the linear relationship between the number of rooms (
x) and the electricity consumption (y). Let the relationship be y = ax + b.(ii) What is the consumption for a house with 8 rooms? 5marks18
Identify the degree of the following polynomials:(i)
7x - 3(ii) 10(iii) x² + 5x - 1 2marks19
A water tank is being filled. The height of the water increases by 0.5 meters every hour. If the initial height of the water was 1 meter, represent the height of the water
h after t hours using a linear expression. What will be the height after 5 hours? 3marks20
Assertion: The graphs of
y = 2x + 3 and y = 2x - 1 are parallel lines.Reason: Lines with the same slope are parallel.Select the correct option:(a) Both assertion and reason are true, and reason is the correct explanation of assertion.(b) Both assertion and reason are true, but reason is not the correct explanation of assertion.(c) Assertion is true, but reason is false.(d) Assertion is false, but reason is true. 1markAnswers
1
The linear expression for the total cost is
C(x) = 5000x + 250x.If 40 students are going, the total cost is C(40) = 5000 + 250 * 40 = 5000 + 10000 = ₹15000.Since ₹15000 is less than ₹20000, the allocation is sufficient. 5marks2
(a) Both assertion and reason are true, and reason is the correct explanation of assertion. 1mark
3
The area of the rectangular plot is
l * w square meters.The area for crop A is (l * w) / 2.The cost of seeds for crop A is (l * w) / 2 * 50 = 25lw.If the plot is 20m by 15m, then l = 20 and w = 15.The cost of seeds for crop A is 25 * 20 * 15 = 25 * 300 = ₹750. 3marks4
(i) Let
M(n) be the number of mangoes left on day n.M(n) = 150 - 15(n-1) or M(n) = 165 - 15n.(ii) Number of mangoes left on day 7: M(7) = 165 - 15 * 7 = 165 - 105 = 60 mangoes.(iii) To find the day when 45 mangoes are left:45 = 165 - 15n15n = 165 - 4515n = 120n = 120 / 15 = 8. So, on day 8. 5marks5
For
d > 1, the cost for the subsequent (d-1) kilometers is 15 * (d-1).The total fare is Fare(d) = 30 + 15(d-1) = 30 + 15d - 15 = 15d + 15.For a journey of 8 km: Fare(8) = 15 * 8 + 15 = 120 + 15 = ₹135. 3marks6
The student's linear polynomial is
p(x) = 3x + 5.The correct quadratic polynomial is q(x) = x² + 2x + 3.We need to find x such that p(x) = q(x).3x + 5 = x² + 2x + 3x² + 2x - 3x + 3 - 5 = 0x² - x - 2 = 0Factoring the quadratic equation: (x - 2)(x + 1) = 0.So, x = 2 or x = -1. 5marks7
Actual rectangle:
l = 12 cm, w = 8 cm.Anjali's perimeter: P_Anjali = 2(12 + 8) = 2(20) = 40 cm.Rohan's approach: Average of length and width = (12 + 8) / 2 = 20 / 2 = 10 cm.Rohan's perimeter (assuming a square): P_Rohan = 4 * 10 = 40 cm.In this specific case, Rohan's result matches Anjali's. However, Rohan's approach is flawed because he assumes a square shape with a side equal to the average, which is not generally correct for a rectangle. A linear relationship between length and width for a rectangle's perimeter would require both dimensions. Rohan's formula P = 4s represents a linear relationship only when s is the side of a square. 5marks8
(a) Both assertion and reason are true, and reason is the correct explanation of assertion. 1mark
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C(n) = 50000 + 250n(i) Cost for 100 phones: C(100) = 50000 + 250 * 100 = 50000 + 25000 = ₹75000.(ii) To keep the cost below ₹100000:50000 + 250n < 100000250n < 100000 - 50000250n < 50000n < 50000 / 250n < 200.The maximum number of phones they can produce is 199. 5marks10
Let
C(d) be the cost for a journey of d km.For d ≤ 100, C(d) = 50.For d > 100, the additional distance is d - 100 km. The number of 50 km segments in the additional distance is (d - 100) / 50.The cost for the additional distance is 10 * ((d - 100) / 50).So, for d > 100, C(d) = 50 + 10 * ((d - 100) / 50) = 50 + (d - 100) / 5 = 50 + d/5 - 20 = d/5 + 30.Piecewise function:C(d) = { 50, if d ≤ 100{ d/5 + 30, if d > 100Cost for a journey of 350 km: Since 350 > 100, use the second part.C(350) = 350/5 + 30 = 70 + 30 = ₹100. 5marks11
The equation of the line is
y = 3x - 2.(i) When x = 0, y = 3(0) - 2 = -2. So the point is (0, -2).(ii) When y = 7, 7 = 3x - 2. 3x = 9, x = 3. So the point is (3, 7).(iii) Plot the points (0, -2) and (3, 7) on a coordinate plane and draw a line passing through them. 3marks12
Relationship 1:
y = 2x + 5Relationship 2: y = -x + 2To find the point of intersection, we set the two expressions for y equal:2x + 5 = -x + 22x + x = 2 - 53x = -3x = -1Now, substitute x = -1 into either equation to find y. Using the second equation:y = -(-1) + 2 = 1 + 2 = 3.The point of intersection is (-1, 3). 5marks13
Length
l = (2x + 3) meters, Width w = (x + 1) meters.(i) Perimeter P = 2(l + w) = 2((2x + 3) + (x + 1)) = 2(3x + 4) = 6x + 8 meters.(ii) If the perimeter is 46 meters:6x + 8 = 466x = 46 - 86x = 38x = 38 / 6 = 19 / 3.Dimensions:Length l = 2(19/3) + 3 = 38/3 + 9/3 = 47/3 meters.Width w = (19/3) + 1 = 19/3 + 3/3 = 22/3 meters. 5marks14
(d) Assertion is false, and reason is true. 1mark
15
P(x) = 3x - 15 (where P is in thousands of rupees)(i) Profit if 10 units are sold: P(10) = 3(10) - 15 = 30 - 15 = 15 thousand rupees, which is ₹15,000.(ii) To achieve a profit of ₹45,000 (which is 45 thousand rupees):3x - 15 = 453x = 45 + 153x = 60x = 20.20 units must be sold. 3marks16
(i) Let
D(n) be the distance of the nth tree from the start.D(n) = 2 + 3(n-1) or D(n) = 3n - 1.(ii) Distance of the 6th tree from the start:D(6) = 3(6) - 1 = 18 - 1 = 17 meters. 3marks17
Let
x be the number of rooms and y be the electricity consumption in units.The relationship is y = ax + b.Given:For 2 rooms, y = 120: 120 = 2a + b (Equation 1)For 5 rooms, y = 210: 210 = 5a + b (Equation 2)Subtract Equation 1 from Equation 2:(210 - 120) = (5a - 2a) + (b - b)90 = 3aa = 30Substitute a = 30 into Equation 1:120 = 2(30) + b120 = 60 + bb = 60(i) The linear relationship is y = 30x + 60.(ii) Consumption for a house with 8 rooms:y = 30(8) + 60 = 240 + 60 = 300 units. 5marks18
(i) The degree of
7x - 3 is 1.(ii) The degree of 10 (a constant) is 0.(iii) The degree of x² + 5x - 1 is 2. 2marks19
Let
h be the height of the water and t be the time in hours.The linear expression for the height is h(t) = 1 + 0.5t.After 5 hours: h(5) = 1 + 0.5 * 5 = 1 + 2.5 = 3.5 meters. 3marks20
(a) Both assertion and reason are true, and reason is the correct explanation of assertion. 1mark
