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Previous Year Topic Wise Questions With Solutions: Quadratic Equations | CSAT Preparation - UPSC PDF Download
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Q1. The total cost of 4 oranges, 6 mangoes and 8 apples is equal to twice the total cost of 1 orange, 2 mangoes and 5 apples. Consider the following statements:
1. The total cost of 3 oranges, 5 mangoes and 9 apples is equal to the total cost of 4 oranges, 6 mangoes and 8 apples.
2. The total cost of one orange and one mango is equal to the cost of one apple.
Which of the statements given above is/are correct? (2024)
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
View Answer 
Ans: (c)
Let the costs of oranges, mangoes and apples be Q, M and A. 4Q + 6M + 8A = 2(Q + 2M + 5A)
Or 4Q + 6M + 8A = 2Q + 4M + 10A Or 2Q + 2M = 2A
Or Q + M = A
Statement 1: 3Q + 5M + 9A = 4Q + 6M + 8A Or A = Q + M
It is true.
Statement 2: Q + M = A
It is true.
Hence, both statements 1 & 2 are true.
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FAQs on Previous Year Topic Wise Questions With Solutions: Quadratic Equations - CSAT Preparation - UPSC
| 1. What are quadratic equations, and how are they typically represented? |  |
Ans. Quadratic equations are polynomial equations of degree two, which can be represented in the standard form as ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to these equations, known as the roots, can be found using various methods including factoring, completing the square, or applying the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
| 2. What are the different methods to solve quadratic equations? | |
Ans. There are several methods to solve quadratic equations:
1. <b>Factoring</b>: This involves expressing the equation in a product form, if possible.
2. <b>Completing the Square</b>: This method transforms the equation into a perfect square trinomial.
3. <b>Quadratic Formula</b>: This universal method uses the formula x = (-b ± √(b² - 4ac)) / (2a) to find the roots directly.
4. <b>Graphing</b>: Plotting the equation on a graph can visually show the points where it intersects the x-axis, representing the roots.
| 3. What is the significance of the discriminant in a quadratic equation? | |
Ans. The discriminant, denoted as D, is the value calculated from the coefficients of the quadratic equation using the formula D = b² - 4ac. It determines the nature of the roots:
- If D > 0, there are two distinct real roots.
- If D = 0, there is exactly one real root (a repeated root).
- If D < 0, there are no real roots, but two complex roots.
| 4. How can quadratic equations be applied in real-life situations? | |
Ans. Quadratic equations have numerous applications in real life, including:
1. <b>Physics</b>: They describe projectile motion, where the height of an object can be modeled as a quadratic function of time.
2. <b>Finance</b>: They are used in calculating profit maximization and break-even points in business.
3. <b>Engineering</b>: Quadratics model various design elements, such as the shape of arches and bridges.
4. <b>Biology</b>: They can represent population growth models under certain conditions.
| 5. Can quadratic equations have complex roots, and if so, how are they represented? | |
Ans. Yes, quadratic equations can have complex roots when the discriminant is negative (D < 0). In such cases, the roots are represented in the form of complex numbers: x = (-b ± i√(4ac - b²)) / (2a), where i is the imaginary unit. This shows that the roots consist of a real part and an imaginary part, indicating that they do not intersect the x-axis on a graph.
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