25 questions from 2023
Let \( \alpha, \beta \) be the roots of the quadratic equation \( x^2 + \sqrt{6}x + 3 = 0 \). Then \( \dfrac{\alpha^{23} + \beta^{23} + \alpha^{14} + \beta^{14}}{\alpha^{15} + \beta^{15} + \alpha^{10} + \beta^{10}} \) is equal to:
Let \( \alpha , \beta \) be the roots of the equation \( x^{2}-\sqrt{2}\,x+2=0 \). Then \( \alpha^{14}+\beta^{14} \) is equal to:
If \( a \) and \( b \) are the roots of the equation \( x^{2}-7x-1=0 \), then the value of \( \dfrac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}} \) is equal to ______.
Let \( \alpha, \beta, \gamma \) be the three roots of the equation \( x^{3} + bx + c = 0 \). If \( \beta\gamma = 1 - \alpha \), then \( b^{3} + 2c^{3} - 3\alpha^{3} - 6\beta^{3} - 8\gamma^{3} \) is equal to:
The equation \( x^{2}-4x+[x]+3=x[x] \), where \( [x] \) denotes the greatest integer function, has:
The set of all \( a \in \mathbb{R} \) for which the equation \( x|x-1| + |x+2| + a = 0 \) has exactly one real root, is:
The number of real roots of the equation \( \sqrt{x^2 - 4x + 3} + \sqrt{x^2 - 9} = \sqrt{4x^2 - 14x + 6} \), is:
Let \(\displaystyle S=\left\{ x \in \left( -\frac{\pi}{2},\, \frac{\pi}{2} \right) : 9^{\,1-\tan^{2}x} + 9^{\,\tan^{2}x} = 10 \right\} \) and \( \beta =\displaystyle \sum\limits_{x\in S} \tan^{2}\!\left( \frac{x}{3} \right) \), then \( \dfrac{1}{6} (\beta - 14)^{2} \) is equal to:
Let \( a \in \mathbb{R} \) and let \( \alpha, \beta \) be the roots of the equation \( x^2 + 60^{\frac{1}{4}}x + a = 0 \). If \( \alpha^4 + \beta^4 = -30 \), then the product of all possible values of \( a \) is _____.
Let \( S=\left\{\alpha:\log_{2}\!\left(9^{\,2\alpha-4}+13\right)-\log_{2}\!\left(\frac{5}{2}\cdot 3^{\,2\alpha-4}+1\right)=2\right\}. \) Then the maximum value of \( \beta \) for which the equation \( x^{2}-2\!\left(\ \displaystyle \sum_{a\in S} \alpha\right)^{2}x+\displaystyle \sum_{a\in S}(\alpha+1)^{2}\beta=0 \) has real roots, is _______.
The domain of \( f(x) = \dfrac{\log_{(x+1)}(x-2)}{e^{2\log_e x} - (2x + 3)} , \ x \in \mathbb{R} \) is:
Let \( \lambda \in \mathbb{R} \) and let the equation \( E \) be \( |x|^{2} - 2|x| + |\lambda - 3| = 0 \). Then the largest element in the set \( S = \{x + \lambda : x \text{ is an integer solution of } E\} \) is ______.
Let \( m \) and \( n \) be the numbers of real roots of the quadratic equations \( x^{2}-12x+[ x ] +31=0 \) and \( x^{2}-5|x+2|-4=0 \) respectively, where \( [ x] \) denotes the greatest integer \( \le x \). Then \( m^{2}+mn+n^{2} \) is equal to __________.
Let \( a, b, c \) be three distinct positive real numbers such that \( (2a)^{\log_e a} = (bc)^{\log_e b} \) and \( b^{\log_e 2} = a^{\log_e c} \). Then \( 6a + 5bc \) is equal to ________.
Let \( \lambda \neq 0 \) be a real number. Let \( \alpha, \beta \) be the roots of the equation \( 14x^2 - 31x + 3\lambda = 0 \) and \( \alpha, \gamma \) be the roots of the equation \( 35x^2 - 53x + 4\lambda = 0 \). Then \( \dfrac{3\alpha}{\beta} \) and \( \dfrac{4\alpha}{\gamma} \) are the roots of the equation:
The number of points, where the curve \( f(x) = e^{8x} - 6^{6x} - 3e^{4x} - e^{2x} + 1, \ x \in \mathbb{R} \) cuts x-axis is equal to _______.
The number of real solutions of the equation \( 3\left( x^2 + \frac{1}{x^2} \right) - 2\left( x + \frac{1}{x} \right) + 5 = 0 \), is:
If \( \alpha > \beta > 0 \) are the roots of the equation \( a x^{2}+bx+1=0 \), and \(\displaystyle \lim_{x\to \dfrac{1}{\alpha}} \left(\frac{1-\cos(x^{2}+bx+a)}{2(1-\alpha x)^{2}}\right)^{\dfrac{1}{2}}=\frac{1}{k}\left(\frac{1}{\beta}-\frac{1}{\alpha}\right) \), then \( k \) is equal to:
The sum of all the roots of the equation \( |x^2 - 8x + 15| - 2x + 7 = 0 \) is:
The number of integral values of \( k \), for which one root of the equation \( 2x^{2}-8x+k=0 \) lies in the interval \( (1,2) \) and its other root lies in the interval \( (2,3) \), is:
Let \( S = \{ x : x \in \mathbb{R} \text{ and } (\sqrt{3}+\sqrt{2})^{\,x^2-4} + (\sqrt{3}-\sqrt{2})^{\,x^2-4} = 10 \} .\) Then \( n(S) \) is equal to:
Let \( \displaystyle \sum_{n=0}^{\infty} \dfrac{n^{3}\,(2n)! + (2n-1)(n!)}{(n!)((2n)!)} = a e + \dfrac{b}{e} + c \), where \( a, b, c \in \mathbb{Z} \) and \( e = \displaystyle \sum_{n=0}^{\infty} \dfrac{1}{n!} \). Then \( a^{2} - b + c \) is equal to ________.
Let \( \alpha_1, \alpha_2, \ldots, \alpha_7 \) be the roots of the equation \( x^7 + 3x^5 - 13x^3 - 15x = 0 \) and \( |\alpha_1| \ge |\alpha_2| \ge \ldots \ge |\alpha_7| \). Then \( \alpha_1 \alpha_2 - \alpha_3 \alpha_4 + \alpha_5 \alpha_6 \) is equal to ________.
The parabolas: \( ax^{2}+2bx+cy=0 \) and \( dx^{2}+2ex+fy=0 \) intersect on the line \( y=1 \). If \( a,b,c,d,e,f \) are positive real numbers and \( a,b,c \) are in G.P., then:
If the value of real number \( a>0 \) for which \( x^{2}-5ax+1=0 \) and \( x^{2}-ax-5=0 \) have a common real root is \( \dfrac{3}{\sqrt{2\beta}} \) then \( \beta \) is equal to __________.
