JEE Main Quadratic Equation and Inequalities Practice Questions with Solutions
The Quadratic Equations and Inequalities are two fundamental chapters of Algebra in Mathematics. Mastering these two topics is essential for scoring well in the Quantitative Aptitude section of the exam. The JEE Main Quadratic Equation and Inequalities Practice Questions with Solutions contain a variety of questions developed for students to practise and improve their problem-solving abilities. You may find about 2 to 3 questions from the Quadratic Equations and Inequalities on the JEE Main question paper. A strong understanding of this topic and its basic concepts will be beneficial for students in the JEE Main entrance exam. The JEE Main Quadratic Equation and Inequalities Practice Test is likely to contain 15 to 20 questions on these chapters, along with their detailed solutions. Students must practise these questions and refer to the solution to understand the methods of algebra used in solving the equations and improve their conceptual knowledge in this area.
The important topics from Quadratic Equation and Inequalities include Solving Quadratic Equations, Relations Between Roots and Coefficients, Making Equations from Root Relationships, Discriminant-based Reasoning, Linear Inequalities, Counting Integer Solutions, Combined Inequalities, Modulus, Quadratic Inequalities, Formation of Quadratic Equations, Graph of Quadratic Functions, Maximum Value of Quadratic Expressions, Minimum Value of Quadratic Expressions, Graph of Quadratic Functions, Word Problems and Applications. Practising JEE Main Quadratic Equation and Inequalities Practice Questions with Solutions regularly will improve the students’ ability to understand complex mathematical problems with ease. They can solve about 10 to 15 problems on the Quadratic Equation and Inequalities chapter every week to be able to tackle questions from this area well in the entrance test. It will also help the JEE Main aspirants to gain speed and develop confidence over time, which is necessary for solving problems in the JEE Main’s Quantitative Aptitude section and performing well on the test.
JEE Main Mathematics Quadratic Equation and Inequalities Practice Questions
Let \alpha , \beta ; \alpha > \beta, be the roots of the equation x^{2} - \sqrt{2} x - \sqrt{3} = 0. Let P_{n} = \alpha^{n} - \beta^{n} , n \in N. Then \left(\right. 11 \sqrt{3} - 10 \sqrt{2} \left.\right) P_{10} + \left(\right. 11 \sqrt{2} + 10 \left.\right) P_{11} - 11 P_{12} is equal to
Question 2.
Let \alpha , \beta be the roots of the equation x^{2} + 2 \sqrt{2} x - 1 = 0. The quadratic equation, whose roots are \alpha^{4} + \beta^{4} and \frac{1}{10} \left(\right. \alpha^{6} + \beta^{6} \left.\right), is:
Question 3.
The sum of all the solutions of the equation \left(\right. 8 \left.\right)^{2 x} - 16 \cdot \left(\right. 8 \left.\right)^{x} + 48 = 0 is :
Question 4.
Let \alpha , \beta be the distinct roots of the equation x^{2} - \left(\right. t^{2} - 5 t + 6 \left.\right) x + 1 = 0 , t \in \mathbb{R} and a_{n} = \alpha^{n} + \beta^{n}. Then the minimum value of \frac{a_{2023} + a_{2025}}{a_{2024}} is
Question 5.
If 2 and 6 are the roots of the equation a x^{2} + b x + 1 = 0, then the quadratic equation, whose roots are \frac{1}{2 a + b} and \frac{1}{6 a + b}, is :
Question 6.
Let \alpha and \beta be the roots of the equation p x^{2} + q x - r = 0, where p \neq 0. If p , q and r be the consecutive terms of a non constant G.P. and \frac{1}{\alpha} + \frac{1}{\beta} = \frac{3}{4}, then the value of \left(\right. \alpha - \beta \left.\right)^{2} is :
Question 7.
Let \mathbf{S} = \left{\right. x \in \mathbf{R} : \left(\right. \sqrt{3} + \sqrt{2} \left.\right)^{x} + \left(\right. \sqrt{3} - \sqrt{2} \left.\right)^{x} = 10 \left.\right}. Then the number of elements in S is :
Question 8.
Let S be the set of positive integral values of a for which \frac{a x^{2} + 2 \left(\right. a + 1 \left.\right) x + 9 a + 4}{x^{2} - 8 x + 32} < 0 , \forall x \in \mathbb{R}. Then, the number of elements in S is :
Question 9.
If \alpha , \beta are the roots of the equation, x^{2} - x - 1 = 0 and S_{n} = 2023 \alpha^{n} + 2024 \beta^{n}, then :
Question 10.
The number of real roots of the equation x \left|\right. x \left|\right. - 5 \left|\right. x + 2 \left|\right. + 6 = 0, is :
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Question 1.
Let \alpha , \beta be the roots of the equation x^{2} - \sqrt{2} x + 2 = 0. Then \alpha^{14} + \beta^{14} is equal to
Question 2.
The set of all a \in \mathbb{R} for which the equation x \left|\right. x - 1 \left|\right. + \left|\right. x + 2 \left|\right. + a = 0 has exactly one real root, is :
Question 3.
Let \alpha , \beta be the roots of the quadratic equation x^{2} + \sqrt{6} x + 3 = 0. Then \frac{\alpha^{23} + \beta^{23} + \alpha^{14} + \beta^{14}}{\alpha^{15} + \beta^{15} + \alpha^{10} + \beta^{10}} is equal to :
Question 4.
Let \alpha , \beta , \gamma be the three roots of the equation x^{3} + b x + c = 0. If \beta \gamma = 1 = - \alpha, then b^{3} + 2 c^{3} - 3 \alpha^{3} - 6 \beta^{3} - 8 \gamma^{3} is equal to :
Question 5.
Let A = \left{\right. x \in R : \left[\right. x + 3 \left]\right. + \left[\right. x + 4 \left]\right. \leq 3 \left.\right} ,
B = \left{\right. x \in R : 3^{x} \left(\right. \sum_{r = 1}^{\infty} \frac{3}{10^{r}} \left.\right)^{x - 3} < 3^{- 3 x} \left.\right} , where [t] denotes greatest integer function. Then,
Question 6.
The sum of all the roots of the equation \left|\right. x^{2} - 8 x + 15 \left|\right. - 2 x + 7 = 0 is :
Question 7.
The number of integral values of k, for which one root of the equation 2 x^{2} - 8 x + k = 0 lies in the interval (1, 2) and its other root lies in the interval (2, 3), is :
Question 8.
Let S = \left{\right. x : x \in \mathbb{R} and \left(\left(\right. \sqrt{3} + \sqrt{2} \left.\right)\right)^{x^{2} - 4} + \left(\left(\right. \sqrt{3} - \sqrt{2} \left.\right)\right)^{x^{2} - 4} = 10 \left.\right}. Then n \left(\right. S \left.\right) is equal to
Question 9.
The equation e^{4 x} + 8 e^{3 x} + 13 e^{2 x} - 8 e^{x} + 1 = 0 , x \in \mathbb{R} has :
Question 10.
The number of real roots of the equation \sqrt{x^{2} - 4 x + 3} + \sqrt{x^{2} - 9} = \sqrt{4 x^{2} - 14 x + 6}, is :
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