CBSE Class 10 Mathematics Quadratic Equations VBQs Set 08

Exercise

Question. The roots of the equation \( \frac{4}{x}x^2 - 2x + \frac{3}{4} = 0 \) are
(a) \( \frac{2}{3}, \frac{3}{2} \)
(b) \( \frac{3}{4}, \frac{3}{4} \)
(c) \( \frac{1}{2}, -\frac{1}{2} \)
(d) None of the options
Answer: (b) \( \frac{3}{4}, \frac{3}{4} \)

Question. The required solution of \( 4x^2 - 25x = 0 \) are
(a) \( x = 0, x = \frac{12}{7} \)
(b) \( x = 0, x = \frac{25}{4} \)
(c) \( x = 1, x = \frac{5}{9} \)
(d) \( x = 1, x = \frac{12}{7} \)
Answer: (b) \( x = 0, x = \frac{25}{4} \)

Question. Assertion (A): When the quadratic equation \( 6x^2 - x - 2 = 0 \) is factorised, we get its roots as \( \frac{2}{3} \) and \( -\frac{1}{2} \).
Reason (R): \( 6x^2 - x - 2 = 0 \Rightarrow 2x(3x - 2) + (3x - 2) = 0 \Rightarrow (3x - 2)(2x + 1) = 0 \Rightarrow x = \frac{2}{3}, -\frac{1}{2} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Question. Find the roots of the quadratic equation by factorisation: \( 3x^2 + 10x + 7\sqrt{3} = 0 \).
Answer: \( -\sqrt{3}, -\frac{7\sqrt{3}}{3} \)

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

Question. Find the roots of the quadratic equation by factorisation: \( 3x^2 - 2ax - a^2 = 0 \).
Answer: \( a, -\frac{a}{3} \)

Question. Find the roots of the quadratic equation by factorisation: \( 3a^2x^2 + 8abx + 4b^2 = 0 \).
Answer: \( -\frac{2b}{a}, -\frac{2b}{3a} \)

Question. Solve for \( x \): \( 4\sqrt{3}x^2 + 5x - 2\sqrt{3} = 0 \).
Answer: \( x = \frac{\sqrt{3}}{4} \text{ and } -\frac{2}{\sqrt{3}} \)

Question. Solve for \( x \): \( x^2 - (\sqrt{2} + 1)x + \sqrt{2} = 0 \).
Answer: \( x = \sqrt{2} \text{ or } x = 1 \)

Question. Solve for \( x \): \( \sqrt{2x + 9} + x = 13 \).
Answer: \( x = 8 \)

Question. Solve for \( x \): \( \sqrt{3}x^2 - 2\sqrt{2}x - 2\sqrt{3} = 0 \).
Answer: \( x = \sqrt{6} \text{ and } -\frac{\sqrt{6}}{3} \)

Question. Solve for \( x \): \( \frac{1}{x - 3} - \frac{1}{x + 5} = \frac{1}{6} \), \( x \neq 3, -5 \).
Answer: \( x = 7 \text{ or } x = -9 \)

Question. Solve for \( x \): \( 3\sqrt{3}x^2 + 14x - 5\sqrt{3} = 0 \).
Answer: \( x = \frac{1}{\sqrt{3}} \text{ or } -5\sqrt{3} \)

Question. Solve for \( x \): \( \frac{x + 1}{x - 1} + \frac{x - 2}{x + 2} = 4 - \frac{2x + 3}{x - 2} \); \( x \neq 1, -2, 2 \).
Answer: \( x = -5 \text{ or } x = \frac{6}{5} \)

Question. The difference of two natural numbers is 3 and the difference of their reciprocals is \( \frac{3}{28} \). Find the numbers.
Answer: 4, 7

Question. Solve the equation for \( x \): \( \frac{1}{x + 1} + \frac{2}{x + 2} = \frac{5}{x + 4} \), \( x \neq -1, -2, -4 \).
Answer: \( x = 2 \text{ or } x = -\frac{3}{2} \)

Question. Some students planned a picnic. The total budget for food was ₹ 2,000. But 5 students failed to attend the picnic and thus the cost of food for each member increased by ₹ 20. How many students attended the picnic and how much did each student pay for the food?
Answer: Number of students = 25, Cost of food for each student = ₹ 80.

Question. A two-digit number is such that the product of its digits is 14. When 45 is added to the number, the digits interchange their places. Find the number.
Answer: 27

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 = 5 years, Asha’s age = 27 years

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 the length and breadth of the park.
Answer: Breadth = 4 m, Length = 7 m

The speed of a motor boat is 20 km/hr. For covering the distance of 15 km the boat took 1 hour more for upstream than downstream.

Question. Let speed of the stream be \( x \) km/hr, then speed of the motorboat in upstream will be
(a) 20 km/hr
(b) \( (20 + x) \) km/hr
(c) \( (20 - x) \) km/hr
(d) 2 km/hr
Answer: (c) \( (20 - x) \) km/hr

Question. What is the relation between speed, distance and time?
(a) speed = \( \frac{\text{distance}}{\text{time}} \)
(b) distance = \( \frac{\text{speed}}{\text{time}} \)
(c) time = speed \( \times \) distance
(d) speed = distance \( \times \) time
Answer: (a) speed = \( \frac{\text{distance}}{\text{time}} \)

Question. Which is the correct quadratic equation for the speed of the stream?
(a) \( x^2 + 30x - 200 = 0 \)
(b) \( x^2 + 20x - 400 = 0 \)
(c) \( x^2 + 30x - 400 = 0 \)
(d) \( x^2 - 20x - 400 = 0 \)
Answer: (c) \( x^2 + 30x - 400 = 0 \)

Question. What is the speed of stream?
(a) 20 km/hour
(b) 10 km/hour
(c) 15 km/hour
(d) 25 km/hour
Answer: (b) 10 km/hour

Question. How much time boat took in downstream?
(a) 90 minutes
(b) 15 minutes
(c) 30 minutes
(d) 45 minutes
Answer: (c) 30 minutes

SOLUTION OF A QUADRATIC EQUATION BY QUADRATIC FORMULA

The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are given by \( \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), if \( b^2 - 4ac \ge 0 \).

This formula for finding the roots of a quadratic equation is often referred to as the quadratic formula. The expression \( b^2 - 4ac \) is called the discriminant of the quadratic equation and generally denoted by D. If \( b^2 - 4ac < 0 \), then the equation will have no real roots. As this formula was given by an ancient Indian mathematician Sridharacharya around AD 1025, it is known as Sridharacharya’s formula.

Question. Find the solution of the quadratic equations by quadratic formula: (i) \( \frac{1}{2}x^2 - \sqrt{11}x + 1 = 0 \)
Answer: \( \sqrt{11} + 3 \) and \( \sqrt{11} - 3 \)

Question. Find the solution of the quadratic equations by quadratic formula: (ii) \( -x^2 + 7x - 10 = 0 \)
Answer: Roots are 2 and 5.

Question. Solve the quadratic equation \( 2x^2 + ax - a^2 = 0 \) for \( x \) using quadratic formula.
Answer: \( \frac{a}{2}, -a \)

Question. Solve the following quadratic equation: \( 9x^2 - 9(a + b)x + [2a^2 + 5ab + 2b^2] = 0 \)
Answer: \( \frac{2a + b}{3} \) or \( \frac{a + 2b}{3} \)

Question. The discriminant of the equation \( 9x^2 + 6x + 1 = 0 \) is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (a) 0

Question. If D is the discriminant of the equation \( x^2 + 2x - 4 \), then 2D is:
(a) 20
(b) 40
(c) 60
(d) 80
Answer: (b) 40

Question. The discriminant of the quadratic equation \( 4x^2 - 6x + 3 = 0 \) is:
(a) 12
(b) 84
(c) 2\(\sqrt{3}\)
(d) -12
Answer: (d) -12

Question. The roots of the quadratic equation \( ax^2 + bx + c = 0 \) are given by \( \frac{-b \pm \sqrt{b^2 - 4ac}}{2ac} \) if \( b^2 - 4ac \)
(a) \( < 0 \)
(b) \( \le 0 \)
(c) \( > 0 \)
(d) \( \ge 0 \)
Answer: (d) \( \ge 0 \)

Question. The quadratic formula was given by an ancient Indian mathematician.
(a) Sridharacharya
(b) Aryabhata
(c) Brahmagupta
(d) None of the options
Answer: (a) Sridharacharya

Question. The quadratic equation whose roots are \( a, \frac{1}{a} \) is :
(a) \( ax^2 - (a^2 + 1)x + a = 0 \)
(b) \( ax^2 - (a^2 - 1)x + a = 0 \)
(c) \( ax^2 - (a^2 - 1)x - a = 0 \)
(d) None of the options
Answer: (a) ax^2 - (a^2 + 1)x + a = 0

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. The speed of the train is :
(a) 30 km/h
(b) 35 km/h
(c) 12 km/h
(d) 40 km/h
Answer: (d) 40 km/h

Question. The roots of the equation \( \sqrt{2x+9} + x = 13 \) are :
(a) 8, - 20
(b) 20, - 8
(c) - 20, - 8
(d) 20, 8
Answer: (d) 20, 8

Question. If \( x = 1 \) is a common root of the equation \( ax^2 + ax + 3 = 0 \) and \( x^2 + x + b = 0 \) then \( ab = \)
(a) 6
(b) 3
(c) - 3
(d) \( \frac{7}{2} \)
Answer: (b) 3

Question. Find the values of \( k \) for which the quadratic equation \( kx(x - 3) + 9 = 0 \) has real equal roots
(a) \( k = 0 \) or \( k = 4 \)
(b) \( k = 1 \) or \( k = 4 \)
(c) \( k = - 3 \) or \( k = 3 \)
(d) \( k = - 4 \) or \( k = 4 \)
Answer: (a) k = 0 or k = 4

Question. If the quadratic equation \( px^2 - 2\sqrt{5}px + 15 = 0 \) has two equal roots then value of \( p \) :
(a) 0, 3
(b) 0, - 3
(c) 3, 4
(d) 5, 4
Answer: (a) 0, 3

Question. The sum of a number and its reciprocal is \( \frac{10}{3} \). Find the number :
(a) 3
(b) \( \frac{1}{3} \)
(c) both (a) and (c)
(d) none of the options
Answer: (c) both (a) and (c)

Question. If one of the roots of a quadratic equation having rational coefficients is \( \sqrt{7} - 4 \), then the quadratic equation is
(a) \( x^2 - 2\sqrt{7}x - 9 = 0 \)
(b) \( x^2 - 8x + 9 = 0 \)
(c) \( x^2 + 8x + 9 = 0 \)
(d) \( x^2 - 2\sqrt{7}x + 9 = 0 \)
Answer: (a) x^2 - 2\sqrt{7}x - 9 = 0
Explanation : Solution: One root of the equation is \( \sqrt{7} - 4 \) so its another roots is \( \sqrt{7} + 4 \).