Class 10 - Chapter 4: Quadratic Equations

Section A (1 Mark each)

Q1. The product of two consecutive positive integers is 306. We need to find the integers. The quadratic equation representing this situation is:

(a) $x^2 + x + 306 = 0$ (b) $x^2 + x - 306 = 0$ (c) $x^2 - x - 306 = 0$ (d) $2x^2 + x - 306 = 0$

Q2. The sum of a number and its reciprocal is $\frac{10}{3}$. The number is:

(a) 3 or -3 (b) 1/3 or -1/3 (c) 3 or 1/3 (d) -3 or -1/3

Q3. John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. If John had $x$ marbles initially, the equation is:

(a) $x^2 - 45x + 324 = 0$ (b) $x^2 + 45x - 324 = 0$ (c) $x^2 - 45x - 324 = 0$ (d) $x^2 + 45x + 324 = 0$

Q4. Assertion (A): The equation representing "The sum of the squares of two consecutive odd numbers is 290" is $2x^2 + 4x - 288 = 0$ (where $x$ is the smaller odd number).
Reason (R): Consecutive odd integers can be represented as $2k+1$ and $2k+3$ or simply $x$ and $x+2$.

(a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.

Section B (2 Marks each)

Q5. Find two numbers whose sum is 27 and product is 182.

Solve on white paper.

Q6. The sum of the reciprocals of Rehman's ages, (in years) 3 years ago and 5 years from now is $\frac{1}{3}$. Find his present age.

Solve on white paper.

Q7. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was ₹ 90, find the number of articles produced.

Solve on white paper.

Section C (3 Marks each)

Q8. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

Solve on white paper.

Q9. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Solve on white paper.

Section D (5 Marks)

Q10. Two water taps together can fill a tank in $9 \frac{3}{8}$ hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Solve on white paper.

Section E (4 Marks)

Q11. Case Study: Rectangular Park

A municipal corporation wants to design a rectangular park. They have 80 m of wire for fencing the perimeter and want the area of the park to be 400 sq. m.

(i) Form a quadratic equation in terms of length $l$ (or breadth $b$) representing this situation.

(ii) Determine if it is possible to design such a park by checking the discriminant.

(iii) If possible, find the length and breadth of the park.

Solve on white paper.