CBSE Class 10 Maths HOTs Quadratic Equations Set F

Multiple Choice Questions

Question. Which of the following is a quadratic equation? 
(a) \(x^2 + 2x + 1 = (4 - x)^2 + 3\)
(b) \(-2x^2 = (5 - x)(2x - \frac{2}{5})\)
(c) \((k + 1)x^2 + \frac{3}{2}x = 7\), where \(k = -1\)
(d) \(x^3 - x^2 = (x - 1)^3\)
Answer: (d)

Question. Which of the following is not a quadratic equation? 
(a) \(2(x - 1)^2 = 4x^2 - 2x + 1\)
(b) \(2x - x^2 = x^2 + 5\)
(c) \((\sqrt{2}x + \sqrt{3})^2 = 3x^2 - 5x\)
(d) \((x^2 + 2x)^2 = x^4 + 3 + 4x^2\)
Answer: (d)

Question. If a number \(x\) is added to twice its square, then the resultant is 21. Then the quadratic representation of this statement is
(a) \(2x^2 - x + 21 = 0\)
(b) \(2x^2 + x - 21 = 0\)
(c) \(2x^2 - x - 20 = 0\)
(d) None of these
Answer: (b)

Question. Which of the following equations has 2 as a root?
(a) \(x^2 - 4x + 5 = 0\)
(b) \(x^2 + 3x - 12 = 0\)
(c) \(2x^2 - 7x + 6 = 0\)
(d) \(3x^2 - 6x - 2 = 0\)
Answer: (c)

Question. If \(\frac{1}{2}\) is a root of the equation \(x^2 + kx - \frac{5}{4} = 0\), then the value of \(k\) is 
(a) 2
(b) \(-2\)
(c) \(\frac{1}{4}\)
(d) \(\frac{1}{2}\)
Answer: (a)

Question. Which of the following equation has root as 3?
(a) \(x^2 - 5x + 6 = 0\)
(b) \(-x^2 + 3x - 3 = 0\)
(c) \(\sqrt{2}x^2 - \frac{3}{\sqrt{2}}x + 1 = 0\)
(d) \(3x^2 - 3x + 3 = 0\)
Answer: (a)

Question. 0.2 is a root of the equation \(x^2 - 0.4 = 0\)? 
(a) True
(b) False
(c) Can't determined
(d) None of these
Answer: (b)

Question. A quadratic equation with integral coefficient has integral roots.
(a) True
(b) False
(c) Can't determined
(d) None of these
Answer: (b)

Question. If \(b = 0\), \(c < 0\), then the roots of \(x^2 + bx + c = 0\) are numerically equal and opposite in sign. 
(a) True
(b) False
(c) Can't determined
(d) None of these
Answer: (a)

Question. The roots of the quadratic equation \(x^2 - 8x - 20 = 0\) are
(a) \(5, -4\)
(b) \(-4, 5\)
(c) \(10, -2\)
(d) \(-10, 2\)
Answer: (c)

Question. Which constant must be added and subtracted to solve the quadratic equation \(9x^2 + \frac{3}{4}x - \sqrt{2} = 0\).
(a) \(\frac{1}{8}\)
(b) \(\frac{1}{64}\)
(c) \(\frac{1}{4}\)
(d) \(\frac{9}{64}\)
Answer: (b)

Question. Solve \(12x^2 + 5x - 3 = 0\).
(a) \(\frac{1}{3}, \frac{4}{3}\)
(b) \(\frac{1}{2}, \frac{3}{4}\)
(c) \(-\frac{1}{3}, \frac{3}{4}\)
(d) \(\frac{1}{3}, -\frac{3}{4}\)
Answer: (d)

Question. The discriminant of the quadratic equation \(x^2 - 4x + 1 = 0\) is 
(a) \(2\sqrt{3}\)
(b) 4
(c) 12
(d) 16
Answer: (c)

Question. If the discriminant of the equation \(6x^2 - bx + 2 = 0\) is 1, then the value of \(b\) is 
(a) 7
(b) \(-7\)
(c) Both (a) and (b)
(d) None of these
Answer: (c)

Question. Value(s) of \(k\) for which the quadratic equation \(2x^2 - kx + k = 0\) has equal roots is/are 
(a) 0
(b) 4
(c) 8
(d) 0, 8
Answer: (d)

Question. The quadratic equation \(2x^2 - \sqrt{5}x + 1 = 0\) has
(a) two distinct real roots
(b) two equal real roots
(c) no real roots
(d) more than 2 real roots
Answer: (c)

Question. If the discriminant of the equation \(kx^2 - 3\sqrt{2}x + 4\sqrt{2} = 0\) is 14, then the value of \(k\) is
(a) \(\sqrt{2}\)
(b) \(\frac{1}{3\sqrt{2}}\)
(c) \(\frac{1}{\sqrt{2}}\)
(d) \(\frac{1}{4\sqrt{2}}\)
Answer: (d)

Question. Which of the following equations has two distinct real roots? 
(a) \(2x^2 - 3\sqrt{2}x + \frac{9}{4} = 0\)
(b) \(x^2 + x - 5 = 0\)
(c) \(x^2 + 3x + 2\sqrt{2} = 0\)
(d) \(5x^2 - 3x + 1 = 0\)
Answer: (b)

Question. Which of the following equations has no real roots?
(a) \(x^2 - 4x + 3\sqrt{2} = 0\)
(b) \(x^2 + 4x - 3\sqrt{2} = 0\)
(c) \(x^2 - 4x - 3\sqrt{2} = 0\)
(d) \(3x^2 + 4\sqrt{3}x + 4 = 0\)
Answer: (a)

Question. \((x^2 + 1)^2 - x^2 = 0\) has 
(a) four real roots
(b) two real roots
(c) no real roots
(d) one real root
Answer: (c)

Question. The sum of the squares of three consecutive integers is 110, then the smallest positive integer is
(a) 6
(b) 5
(c) 7
(d) 4
Answer: (b)

Question. A line segment AB is 8 cm in length. AB is produced to P such that \(BP^2 = AB \cdot AP\). Then, the length of BP is 
(a) \(5(\sqrt{5} + 1)\)
(b) \(\sqrt{5} + 1\)
(c) \(4(\sqrt{5} + 1)\)
(d) \(\sqrt{3} + 1\)
Answer: (c)

Question. One year ago, a man was 8 times as old as his son. Now, his age is equal to the square of his son's age. Present age of man is
(a) 49 yr
(b) 37 yr
(c) 59 yr
(d) 39 yr
Answer: (a)

Case Based MCQs

Raj and Ajay are very close friends. Both the families decide to go to Ranikhet by their own cars. Raj's car travels at a speed of \(x\) km/h while Ajay's car travels 5 km/h faster than Raj's car. Raj took 4 h more than Ajay to complete the journey of 400 km. 

Question. What will be the distance covered by Ajay's car in two hours?
(a) \(2(x + 5)\) km
(b) \((x - 5)\) km
(c) \(2(x + 10)\) km
(d) \((2x + 5)\) km
Answer: (a)

Question. Which of the following quadratic equation describe the speed of Raj's car?
(a) \(x^2 - x - 500 = 0\)
(b) \(x^2 + 4x - 400 = 0\)
(c) \(x^2 + 5x - 500 = 0\)
(d) \(x^2 - 4x + 400 = 0\)
Answer: (c)

Question. What is the speed of Raj's car?
(a) 20 km/h
(b) 15 km/h
(c) 25 km/h
(d) 10 km/h
Answer: (a)

Question. How much time took Ajay to travel 400 km?
(a) 20 h
(b) 40 h
(c) 25 h
(d) 16 h
Answer: (d)

Question. How much time took Raj to travel 400 km?
(a) 15 h
(b) 20 h
(c) 18 h
(d) 22 h
Answer: (b)

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

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

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

Question. Which is the correct quadratic equation for the speed of the current?
(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)

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

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

By quadratic formula, the roots of the quadratic equation \(ax^2 + bx + c = 0\), \(a \neq 0\) are given by
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) or \(x = \frac{-b \pm \sqrt{D}}{2a}\)
where, \(D = b^2 - 4ac\) is called discriminant.

Question. The roots of the quadratic equation \(8x^2 - 22x - 21 = 0\) are
(a) \(-\frac{7}{2}, -\frac{3}{4}\)
(b) \(\frac{7}{2}, \frac{3}{4}\)
(c) \(\frac{7}{2}, -\frac{3}{4}\)
(d) \(-\frac{7}{2}, \frac{3}{4}\)
Answer: (c)

Question. The discriminant of \(x^2 + x + 7 = 0\) is
(a) 27
(b) \(-27\)
(c) \(\sqrt{27}\)
(d) \(-\sqrt{27}\)
Answer: (b)

Question. Roots of \(4x^2 - 2x - 3 = 0\) are
(a) Real and distinct
(b) Real and equal
(c) Imaginary
(d) More than two real roots
Answer: (a)

Question. The value of \(k\) for which \(4x^2 + kx + 9 = 0\) has real and equal roots is
(a) 12
(b) \(-12\)
(c) Both (a) and (b)
(d) None of these
Answer: (c)

Question. The least positive value of \(k\) for which \(x^2 + kx + 16 = 0\) has real roots, is
(a) 18
(b) 4
(c) 2
(d) 8
Answer: (d)

Seven years ago, Varun’s age was five times the square of Swati’s age. Three years hence, Swati’s age will be two-fifth of Varun’s age.

Question. If seven years ago, Swati’s age be \(x\) yr, then Varun’s age is
(a) \((5x - 7)^2\) yr
(b) \(5x^2\) yr
(c) \((5x^2 + 7)\) yr
(d) \((5x^2 - 7)\) yr
Answer: (b)

Question. After three years, Swati’s age is
(a) \((x + 3)\) yr
(b) \((x - 3)\) yr
(c) \((x + 7)\) yr
(d) \((x + 10)\) yr
Answer: (d)

Question. The quadratic equation related to the given problem is
(a) \(2x^2 - x - 6 = 0\)
(b) \(5x^2 - x + 6 = 0\)
(c) \(3x^2 - 2x + 5 = 0\)
(d) \(7x^2 - 3x + 1 = 0\)
Answer: (a)

Question. Present age of Varun’s is
(a) 27 yr
(b) 20 yr
(c) 30 yr
(d) 37 yr
Answer: (a)

Question. If Swati’s present age 10 yr, then present age of Varun’s is
(a) 40 yr
(b) 47 yr
(c) 45 yr
(d) 52 yr
Answer: (d)