CBSE Class 10 Mathematics Quadratic Equations Worksheet Set 09

Objective Type Questions: 

Question. Choose and write the correct option in each of the following questions.

Question. A sum of Rs. 4000 was divided among \( x \) persons. Had there been 10 more persons, each would have got Rs. 80 less. Which of the following represents the above situation?
(a) \( x^2 + 10x + 500 = 0 \)
(b) \( x^2 + 10x - 400 = 0 \)
(c) \( x^2 + 10x - 500 = 0 \)
(d) \( x^2 + 10x + 400 = 0 \)
Answer: (c) \( x^2 + 10x - 500 = 0 \)

Question. What is the smallest positive integer value of \( k \) such that the roots of the equation \( x^2 - 9x + 18 + k = 0 \) can be calculated by factorising the equation?
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (a) 2

Question. If the roots of \( ax^2 + bx + c = 0 \) are equal, then the value of \( c \) is
(a) \( \frac{-b}{2a} \)
(b) \( \frac{b}{2a} \)
(c) \( \frac{-b^2}{4a} \)
(d) \( \frac{b^2}{4a} \)
Answer: (d) \( \frac{b^2}{4a} \)

Question. The product of three consecutive integers is equal to 6 times the sum of three integers. If the smallest integer is \( x \), which of the following equations represent the above situation?
(a) \( 2x^2 + x - 9 = 0 \)
(b) \( 2x^2 - x + 9 = 0 \)
(c) \( x^2 + 2x + 18 = 0 \)
(d) \( x^2 + 2x - 18 = 0 \)
Answer: (d) \( x^2 + 2x - 18 = 0 \)

Question. For what value of \( k \), the roots of the quadratic equation \( 3x^2 + 2kx + 27 = 0 \) are real and equal?
(a) \( k = \pm 4 \)
(b) \( k = \pm 3 \)
(c) \( k = \pm 6 \)
(d) \( k = \pm 9 \)
Answer: (d) \( k = \pm 9 \)

Question. If the discriminant of a quadratic equation is less than zero then it has
(a) equal roots
(b) real roots
(c) no real roots
(d) can't be determined
Answer: (c) no real roots

Very Short Answer Questions:

Question. If one root of the quadratic equation \( 6x^2 - x - k = 0 \) is \( \frac{2}{3} \), then find the value of \( k \).
Answer: \( k = 2 \)

Question. If one root of \( 5x^2 + 13x + k = 0 \) is the reciprocal of the other root, then find value of \( k \). 
Answer: \( k = 5 \)

Question. Find the nature of roots of the quadratic equation \( 2x^2 - 4x + 3 = 0 \).
Answer: No real root

Question. For what values of 'a' the quadratic equation \( 9x^2 - 3ax + 1 = 0 \) has equal roots? 
Answer: \( a = \pm 2 \)

Question. Find the value(s) of \( k \) for which the quadratic equation \( 3x^2 + kx + 3 = 0 \) has real and equal roots. 
Answer: \( k = \pm 6 \)

Question. For what values of \( k \) does the quadratic equation \( 4x^2 - 12x - k = 0 \) have no real roots? 
Answer: \( k < -9 \)

Short Answer Questions-I: 

Question. State whether the equation \( (x + 1)(x - 2) + x = 0 \) has two distinct real roots or not. Justify your answer.
Answer: Yes, it has two distinct real roots

Question. Is 0.3 a root of the equation \( x^2 - 0.9 = 0 \)? Justify.
Answer: No

Question. Find the value of \( k \), for which \( x = 2 \) is a solution of the equation \( kx^2 + 2x - 3 = 0 \). 
Answer: \( k = -\frac{1}{4} \)

Question. If \(-5\) is a root of the quadratic equation \( 2x^2 + px - 15 = 0 \) and the quadratic equation \( p(x^2 + x) + k = 0 \) has equal roots, then find the value of \( k \).  
Answer: \( k = \frac{7}{4} \)

Question. Find the roots of the quadratic equation: \( x^2 - 3\sqrt{5}x + 10 = 0 \) by factorisation method.
Answer: \( \sqrt{5}, 2\sqrt{5} \)

Question. Find the roots of the quadratic equation: \( 2x^2 + \frac{5}{3}x - 2 = 0 \) using quadratic method.
Answer: \( x = \frac{-3}{2}, \frac{2}{3} \)

Question. If \( (x^2 + y^2)(a^2 + b^2) = (ax + by)^2 \), prove that \( \frac{x}{a} = \frac{y}{b} \).
Answer: Hence proved.

Short Answer Questions-II: 

Question. Find the value of \( p \), for which one root of the quadratic equation \( px^2 - 14x + 8 = 0 \) is 6 times the other. 
Answer: \( p = 3 \)

Question. Solve for \( x: \frac{1}{(x-1)(x-2)} + \frac{1}{(x-2)(x-3)} = \frac{2}{3}, x \neq 1, 2, 3 \).
Answer: \( x = 0, 4 \)

Question. Solve for \( x: 2\left(\frac{2x-1}{x+3}\right) - 3\left(\frac{x+3}{2x-1}\right) = 5; x \neq -3, \frac{1}{2} \). 
Answer: \( x = -\frac{1}{5} \) or \(-10\)

Question. Solve for \( x: \frac{14}{x+3} - 1 = \frac{5}{x+1}; x \neq -3, -1 \).
Answer: \( x = 1 \) and 4

Question. Solve for \( x: 4x^2 - 4a^2x + (a^4 - b^4) = 0 \). 
Answer: \( x = \frac{a^2+b^2}{2}, \frac{a^2-b^2}{2} \)

Question. Sum of the areas of two squares is \( 157\text{ m}^2 \). If the sum of their perimeters is 68 m, find the sides of the two squares. 
Answer: 6 m and 11 m

Question. Find the dimensions of a rectangular park whose perimeter is 60 m and area \( 200\text{ m}^2 \). 
Answer: Length = 20 m and Breadth = 10 m

Question. Find that non-zero value of \( k \), for which the quadratic equation \( kx^2 + 1 - 2(k - 1)x + x^2 = 0 \) has equal roots. Hence find the roots of the equation. 
Answer: \( k = 3, x = \frac{1}{2}, \frac{1}{2} \)

Question. Find the value of \( k \) for which the quadratic equation \( (k + 1)x^2 - 6(k + 1)x + 3(k + 9) = 0, k \neq -1 \) has equal roots. 
Answer: \( k = 3 \)

Question. If 2 is a root of the quadratic equation \( 3x^2 + px - 8 = 0 \) and the quadratic equation \( 4x^2 - 2px + k = 0 \) has equal roots, find the value of \( k \).
Answer: \( k = 1 \)

Long Answer Questions:

Question. Solve for \( x: \frac{x-3}{x-4} + \frac{x-5}{x-6} = \frac{10}{3}; x \neq 4, 6 \). 
Answer: \( \frac{9}{2} \) and 7

Question. Find \( x \) in terms of \( a, b \) and \( c \): \( \frac{a}{x-a} + \frac{b}{x-b} = \frac{2c}{x-c}, x \neq a, b, c \). 
Answer: \( x = \frac{2ab - ac - bc}{a + b - 2c}, 0 \)

Question. Solve for \( x: \frac{1}{x+1} + \frac{3}{5x+1} = \frac{5}{x+4}, x \neq -1, -\frac{1}{5}, -4 \). 
Answer: \( \frac{-11}{17}, 1 \)

Question. \( \frac{1}{2a + b + 2x} = \frac{1}{2a} + \frac{1}{b} + \frac{1}{2x}; x \neq 0, x \neq \frac{-2a-b}{2}, a, b \neq 0 \). 
Answer: \( x = -a \) or \( -\frac{b}{2} \)

Question. Solve the following equation for \( x \): \( \frac{1}{x+1} + \frac{2}{x+2} = \frac{7}{x+5}, x \neq -1, -2, -5 \). 
Answer: \( x = 1, x = \frac{-3}{2} \)

Question. The sum of the squares of two consecutive even numbers is 340. Find the numbers. 
Answer: 12 and 14

Question. Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number.
Answer: 12

Question. At present Asha's age (in years) is 2 more than the square of her daughter Nisha's age. When Nisha grows to her mother's present age, Asha's age would be one year less than 10 times the present age of Nisha. Find the present ages of both Asha and Nisha. 
Answer: Nisha's age is 5 years, Asha's age is 27 years

Question. There is a square field whose side is 44 m. A square flower bed is prepared in its centre leaving a gravel path all round the flower bed. The total cost of laying the flower bed and the gravel path at Rs. 2.75 and Rs. 1.50 per \( m^2 \) respectively, is Rs. 4904. Find the width of gravel path.
Answer: 2 m

Question. A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the journey. Find the original speed of the train.
Answer: 45 km/h

Question. In a class test, the sum of the marks obtained by Puneet in Mathematics and Science is 28. Had he got 3 marks more in Mathematics and 4 marks less in Science, the product of their marks, would have been 180. Find his marks in two subjects. 
Answer: Mathematics: 9 and Science: 19 or Mathematics: 12 and Science: 16

Question. A faster train takes one hour less than a slower train for a journey of 200 km. If the speed of slower train is 10 km/h less than that of faster train, find the speeds of two trains. 
Answer: Speed of faster train = 50 km/h and speed of slower train = 40 km/h

Question. Two water taps together can fill a tank in \( 1\frac{7}{8} \) hours. The tap with longer diameter takes 2 hours less than the tap with smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately. 
Answer: 5 hours, 3 hours

Question. The total cost of a certain length of a piece of cloth is Rs. 200. If the piece was 5 m longer and each metre of cloth costs Rs. 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: \( l = 20\text{ m} \), Rs. 10 per metre