Situational Problems (Quadratic) MCQs Quiz | Class 10

This quiz on Situational Problems (Quadratic) for Class X Mathematics (Code 041), Unit II: Algebra, focuses on applying quadratic equations to solve problems from day-to-day contexts, emphasizing the skills of forming and solving such equations. After attempting all 10 multiple-choice questions, submit your quiz to view your score and download a detailed answer PDF for revision.

Understanding Situational Problems involving Quadratic Equations

Situational problems involving quadratic equations require us to translate real-world scenarios into mathematical models that can be solved using quadratic techniques. These problems often appear in various day-to-day contexts such as age, speed, distance, area, perimeter, and numbers.

Key Steps to Solve Quadratic Situational Problems:

  1. Read and Understand: Carefully read the problem to identify what is given and what needs to be found.
  2. Define Variables: Assign variables (usually ‘x’) to the unknown quantities. If there are multiple unknowns, try to express them in terms of a single variable.
  3. Form the Quadratic Equation: Translate the problem statement into a mathematical equation. Look for keywords that suggest relationships like product, sum of squares, difference, or area formulas. Ensure the equation is in the standard quadratic form: ax^2 + bx + c = 0.
  4. Solve the Equation: Use appropriate methods to solve the quadratic equation. Common methods include:
    • Factorization: Splitting the middle term and factoring.
    • Quadratic Formula: x = [-b ± sqrt(b^2 - 4ac)] / 2a
    • Completing the Square: Transforming the equation into a perfect square trinomial.
  5. Check and Interpret: Verify if the solutions obtained make sense in the context of the problem. Sometimes, a mathematical solution might not be valid in a real-world context (e.g., negative length, fractional number of people).

Common Contexts and Equation Formation:

Context Keywords/Concept Example Variable Representation
Numbers Consecutive integers, product, sum of squares x, x+1 (consecutive); x, x+2 (consecutive even/odd)
Area/Perimeter Rectangles, squares, triangles Length l, Width w. Area lw, Perimeter 2(l+w)
Age Problems Past age, future age, product of ages Present age x. Age ‘n’ years ago x-n. Age ‘n’ years later x+n.
Speed, Distance, Time Upstream/downstream, increased/decreased speed Distance D, Speed S, Time T. D = S * T
Work Problems Work done per unit time Time taken x. Work rate 1/x

Quick Revision Points:

  • A quadratic equation is of the form ax^2 + bx + c = 0 where a is not equal to 0.
  • The degree of a quadratic equation is 2.
  • Roots of a quadratic equation are the values of ‘x’ that satisfy the equation.
  • The discriminant D = b^2 - 4ac determines the nature of the roots:
    • If D > 0, two distinct real roots.
    • If D = 0, two equal real roots.
    • If D < 0, no real roots.
  • Always check for extraneous solutions that do not fit the problem's context.

Extra Practice Questions:

  1. The sum of the areas of two squares is 468 square meters. If the difference of their perimeters is 24 meters, find the sides of the two squares.
  2. A takes 6 days less than B to finish a piece of work. If both A and B together can finish the work in 4 days, find the time taken by B alone to finish the work.
  3. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.
  4. Is it possible to design a rectangular park of perimeter 80 meters and area 400 square meters? If so, find its length and breadth.
  5. The hypotenuse of a right triangle is 25 cm. The difference between the other two sides is 5 cm. Find the lengths of the other two sides.