CBSE Class 10 Quadratic Equations Sure Shot Questions Set 18

SA TYPE – I

Question. Represent the following situation in the form of quadratic equation: The product of two consecutive positive integers is 306. We need to find the integers.
Answer: \( x^2 + x - 306 = 0 \), where x is the smaller integer.

Question. Find the roots of the quadratic equation: \( 6x^2 - x - 2 = 0 \).
Answer: -1/2 and 2/3

Question. Find the nature of the roots of the quadratic equation \( 2x^2 - 3x + 5 = 0 \).
Answer: no real roots

Question. Solve: \( 6x^2 + 40 = 31x \).
Answer: 8/3 and 5/2

Question. If the discriminant of the equation \( 6x^2 - bx + 2 = 0 \) is 1, then find the value of b.
Answer: 7, -7

Question. Find the values of k for each of the quadratic equation \( kx (x - 2) + 6 = 0 \), so that they have two equal roots.
Answer: k=6

Question. Represent the situation in the form of Quadratic equation: “The product of Rohan’s age (in years) 5 years ago with his age 9 years later is 15”
Answer: 6 yrs.

Question. The product of two consecutive odd numbers is 483. Find the numbers.
Answer: 21, 23

Question. Solve: \( x - \frac{1}{x} = 3 (x \neq 0) \).
Answer: \( x = \frac{3 \pm \sqrt{13}}{2} \)

Question. Find the sum and product of the roots of the quadratic equation \( 2x^2 + 7x - 4 = 0 \).
Answer: Sum = -7/2, Product = -2

SA TYPE – I 

Question. The total cost of a certain length of a piece of wire is ₹200. If the piece was 5 metres longer and each metre of wire costs ₹2 less, the cost of the piece would have remained unchanged. How long is the piece and what is its original rate per metre?
Answer: ₹ 10 per m.

Question. If the quadratic equation \( (1 + m^2)x^2 + 2mcx + c^2 - a^2 = 0 \) has equal roots, prove that \( c^2 = a^2(1 + m^2) \).
Answer: \( x = \frac{3 \pm \sqrt{10}}{4} \)

Question. If a and b are real and \( a \neq b \) then show that the roots of the equation \( (a - b)x^2 + 5(a + b)x - 2(a - b) = 0 \) are real and unequal.
Answer: 40 km/h

Question. Find the roots of quadratic equation \( 16x^2 - 24x - 1 = 0 \) by using the quadratic formula.
Answer: c/a, -b/a

Question. 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.
Answer: 1, 2

Question. Using quadratic formula, solve for x: \( abx^2 + (b^2 - ac)x - bc = 0 \).
Answer: \( X = 20, 8 \) but x=20 does not satisfy the equation. (34) x= a+2b, a-2b

Question. Solve the equation: \( \frac{1}{x+4} - \frac{1}{x-7} = \frac{11}{30}, x \neq -4, 7 \).
Answer: Marks in mathematics = 12, marks in English = 18; or, Marks in mathematics = 13, marks in English = 17

Question. Solve for x: \( \sqrt{2x + 9} + x = 13 \).
Answer: 42 km/h.

Question. Solve the equation: \( x^2 - 2ax - (4b^2 - a^2) = 0 \).
Answer: Length = 7m and breadth = 4m.

Question. In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.
Answer: 5 km/h. (40) -2,7

LA TYPE – I 

Question. A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km/h more than its original speed. If it takes 3 hours to complete the total journey, what is its original average speed?
Answer: 42 km/h.

Question. A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 square metres more than the area of a park that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular park and of altitude 12 m. Find its length and breadth.
Answer: Length = 7m and breadth = 4m.

Question. If the roots of the equation \( (a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0 \) are equal, prove that \( a/b = c/d \).
Answer: \( a/b = c/d \)

Question. A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
Answer: 5 km/h.

Question. Solve for x: \( \frac{1}{(x-1)(x-2)} + \frac{1}{(x-2)(x-3)} + \frac{1}{(x-3)(x-4)} = 1/6 \).
Answer: (40) -2, 7

CASE STUDY BASED QUESTIONS

In the picture given below, one can see a rectangular in-ground swimming pool installed by a family in their backyard. There is a concrete sidewalk around the pool of width x m. The outside edges of the sidewalk measure 7 m and 12 m. The area of the pool is 36 sq. m.
(a) Based on the information given above, form a quadratic equation in terms of x.
(b) Find the width of the sidewalk around the pool.
Answer: (a) \( 2x^2 - 19x + 24 = 0 \) (ii) 3/2 metre

The tradition of pottery making in India is very old. In fact, it is older than Indus Valley Civilization. The shaping and baking of clay articles has continued through the ages. The picture of a potter is shown below:
A potter makes a certain number of pottery articles in a day. It was observed on a particular day the cost of production of each article(in RS.) was one more than twice the number of articles produced on that day. The total cost of production on that day was Rs.210
(a) Taking number of articles produced on that day as x, form a quadratic equation in x.
(b) Find the number of articles produced and the cost of each article.
Answer: (a) \( 2x^2 + x - 210 = 0 \) (b) NO. Of articles=10 and cost of each articles =Rs.21

Riya has a field with a flowerbed and grass land. The grass land is in the shape of rectangle while flowerbed is in the shape of square. The length of the grassland is found to be 3 m more than twice the length of the flowerbed. Total area of the whole land is 1260m². (a) If the length of the square is x m then find the total length of the field i (b) What will be the perimeter of the whole figure in terms of x? (c) Find the value of x if the area of total field is 1260 m². (d) Find area of grassland and the flowerbed separately.
Answer: (a) (3x+3) m (b) (8x+3) m (c) 20m (d) 860m²

Jackson throws a ball with a speed of 14 m/s which follows the curve \( h = -5t^2 + 14t + 3 \). where “h” represents height in meters and time “t” in seconds
(i) What is the height of the ball initially?
(a) 12m (b) 13 m (c) 3m (d) -3 m
(ii) What is the height of the ball after 3 sec.?
(a) 3m (b) 12 m (c) 13 m (d) 0 m
(iii) Find the possible values of ’t’ when the ball touches the ground.
(iv) Find the maximum height attained by the ball.
Answer: (i) 3 m (ii) 0m (iii) t=-0.2 0r t=3 (iv) 12.5m to 12.8 m

The angry Arjun carried some arrows for fighting with Bheeshm. With the half of the arrows, he cut down the arrows thrown by Bheeshm on him and with six other arrows, he killed the charioteer of Bheeshm. With one arrow each, he knocked down respectively the chariot, flag and the bow of Bheeshma. Finally, with one more than four times the square root of arrows, he laid Bheeshm unconscious on an arrow bed.
(a) If Arjun had x arrows then by how many arrows he cut down arrows thrown by Bheeshm?
(b) If Arjun had x arrows then by how many arrows he laid Bheeshm unconscious on arrow bed?
(c) Find the total number of arrows Arjun had.
Answer: (a) x/2 arrows (b) \( 4\sqrt{x} + 1 \) (c) x=100