JEE Mains JEE Advance BITSAT
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The number of real solutions of the equation, \( x^2 - |x| - 12 = 0 \) is:

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:

If the sum of all the roots of equation \( e^{2x}-11e^{x}-45e^{-x}+\dfrac{81}{2}=0 \) is \( \log_{e} p \), then \( p \) is equal to \( \_\_\_\_\_ \).

Let \( \alpha,\beta \) be roots of \( x^{2}+\sqrt{2}\,x-8=0 \). If \( U_{n}=\alpha_{n}+\beta^{n} \), then \( \frac{U_{10}+\sqrt{2}\,U_{9}}{2U_{8}} \) is equal to ______.

If \( \alpha \), \( \beta \) and \( \gamma \) are the roots of the equation \( 5x^3 - qx - 1 = 0 \), (\( q \in \mathbb{R} \)) then find the value of \( \dfrac{\alpha^2 - 3}{\beta \gamma} + \dfrac{\beta^2 - 3}{\gamma \alpha} + \dfrac{\gamma^2 - 3}{\alpha \beta} \).

Let \( S \) be the set of positive integral values of \( a \) for which \( \dfrac{a x^{2}+2(a+1)x+9a+4}{x^{2}-8x+32}<0 \, \forall x \in \mathbb{R} \). Then, the number of elements in \( S \) is:

The value of \( 3+\cfrac{1}{\,4+\cfrac{1}{\,3+\cfrac{1}{\,4+\cfrac{1}{\,3+\cdots\infty}}}} \) is equal to:

Let \( \alpha,\beta \) be the roots of the equation \( x^{2}-4\lambda x+5=0 \) and \( \alpha,\gamma \) be the roots of the equation \( x^{2}-\left(3\sqrt{2}+2\sqrt{3}\right)x+7+3\lambda\sqrt{3}=0 \), \( \lambda>0 \). If \( \beta+\gamma=3\sqrt{2} \), then \( (\alpha+2\beta+\gamma)^{2} \) is equal to _______.

Let \( f(x) \) be a quadratic polynomial with leading coefficient \( 1 \) such that \( f(0) = p, \; p \neq 0, \) and \( f(1) = \dfrac{1}{3} \). If the equations \( f(x) = 0 \) and \( fof of of(x) = 0 \) have a common real root, then \( f(-3) \) is equal to \( \_ \_ \_ \_ \_ \_ \).

Let \( f(x) \) be a polynomial of degree \( 3 \) such that \( f(k) = -\dfrac{2}{k} \) for \( k = 2, 3, 4, 5 \). Then the value of \( 52 - 10 f(10) \) is equal to:

If for some \( p, q, r \in \mathbb{R} \), not all have same sign, one of the roots of the equation \( (p^{2}+q^{2})x^{2}-2q(p+r)x+q^{2}+r^{2}=0 \) is also a root of the equation \( x^{2}+2x-8=0 \), then \( \dfrac{q^{2}+r^{2}}{p^{2}} \) is equal to __________.

The \( \alpha,\beta \) are the roots of the equation
\( x^{2}-\left(5+3^{\sqrt{\log_{3}5}-5\sqrt{\log_{5}3}}\right)+3\left(3^{(\log_{3}5)^{\dfrac{1}{3}}}-5^{(\log_{5}3)^{\dfrac{2}{3}}} -1\right)=0 \), then the equation, whose roots are \( \alpha+\dfrac{1}{\beta} \) and \( \beta+\dfrac{1}{\alpha} \), is:

\( \alpha=\sin 36^{\circ} \) is a root of which of the following equation?

Let \( \alpha , \beta \) be the roots of the equation \( x^{2}-\sqrt{2}\,x+2=0 \). Then \( \alpha^{14}+\beta^{14} \) is equal to:

Let \( f(x)=ax^{2}+bx+c \) be such that \( f(1)=3 \), \( f(-2)=\lambda \) and \( f(3)=4 \). If \( f(0)+f(1)+f(-2)+f(3)=14 \), then \( \lambda \) is equal to:

If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a-5)x + 15 = 3a \) has no real root, is the interval \( (\alpha,\beta) \), and \( X = \{ x \in \mathbb{Z} : \alpha < x < \beta \} \), then \( \sum\limits_{x \in X} x^2 \) is equal to:

The number of real solution of the equation \( e^{4x} + 4e^{3x} - 58e^{2x} + 4e^{x} + 1 = 0 \) is ______.

If the sum of the squares of the reciprocals of the roots \( \alpha \) and \( \beta \) of the equation \( 3x^{2} + \lambda x - 1 = 0 \) is \( 15 \), then \( 6(\alpha^{3} + \beta^{3})^{2} \) is equal to:

Let \( \alpha, \beta \in \mathbb{N} \) be roots of the equation \( x^2 - 70x + \lambda = 0 \), where \( \frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{N} \). If \( \lambda \) assumes the minimum possible value, then \( \dfrac{\left(\sqrt{\alpha - 1} + \sqrt{\beta - 1}\right)(\lambda
+ 35)}{|\alpha - \beta|} \) is equal to: ______.

Let \( \alpha \) and \( \beta \) be the roots of \( x^2+\sqrt{3}x-16=0 \), and \( \gamma \) and \( \delta \) be the roots of \( x^2+3x-1=0 \). If \( P_n=\alpha^n+\beta^n \) and \( Q_n=\gamma^n+\delta^n \), then \( \dfrac{P_{25}+\sqrt{3}P_{24}}{2P_{23}}+\dfrac{Q_{25}-Q_{23}}{Q_{24}} \) is equal to:

Let \( S = \{ x \in R : (\sqrt{3} + \sqrt{2})^x + (\sqrt{3} - \sqrt{2})^x = 10 \} \). Then the number of elements in \( S \) is:

Let the set of all values of $p \in \mathbb{R}$, for which both the roots of the equation $x^2 - (p + 2)x + (2p + 9) = 0$ are negative real numbers, be the interval $(\alpha, \beta]$. Then $\beta - 2\alpha$ 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 ______.

If the set of all \( a \in \mathbb{R} - \{1\} \), for which the roots of the equation \( (1 - a)x^{2} + 2(a - 3)x + 9 = 0 \) are positive is \( (-\infty, -\alpha] \cup [\beta, \gamma) \), then \( 2\alpha + \beta + \gamma \) 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:

Let \( P_n=\alpha^n+\beta^n,\, n\in \mathbb{N} \). If \( P_{10}=123,\, P_9=76,\, P_8=47 \) and \( P_1=1 \), then the quadratic equation having roots \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) is:

Let \( \alpha \) and \( \beta \) be the roots of the equation \( p x^{2} + qx - r = 0 \) , where \( p \neq 0 \). If \( p \), \( q \) and \( r \) be the consecutive terms of a non constant G.P. and \( \dfrac{1}{\alpha} + \dfrac{1}{\beta} = \dfrac{3}{4} \) , then the value of \( (\alpha - \beta)^{2} \) is:

No. of integral value(s) of \( p \) for which the roots of the equation \( 3x^2 + 2x + p(p-4)(p-6)(p-8) = 0 \) are of opposite sign is/are ________.

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 sum of all real values of \( x \) for which \( \dfrac{3x^2 - 9x + 17}{x^2 + 3x + 10} = \dfrac{5x^2 - 7x + 19}{3x^2 + 5x + 12} \) is equal to __________.

Let the sum of the maximum and the minimum values of the function \( f(x)=\dfrac{2x^{2}-3x+8}{2x^{2}+3x+8} \) be \( \dfrac{m}{n} \), where \( \gcd(m,n)=1 \). Then \( m+n \) is equal to:

The sum of the squares of the roots of \( |x-2|^{2}+|x-2|-2=0 \) and the squares of the roots of \( x^{2}-2|x-3|-5=0 \), is:

If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to ________.

The number of the real solutions of the equation \( x|x+3|+|x-1|-2=0 \) is:

The sum of all integral values of \( k \) (\( k \neq 0 \)) for which the equation \( \frac{2}{x-1} - \frac{1}{x-2} = \frac{2}{k} \) in \( x \) has no real roots, is _______.

Let \( \alpha, \beta \) be two roots of the equation \( x^{2} + (20)^{1/4}x + (5)^{1/2} = 0 \). Then \( \alpha^{8} + \beta^{8} \) is equal to:

Let \( \alpha,\beta \) be the roots of the equation \( x^{2}-\sqrt{2}\,x+\sqrt{6}=0 \) and \( \dfrac{1}{\alpha^{2}}+1,\, \dfrac{1}{\beta^{2}}+1 \) be the roots of the equation \( x^{2}+ax+b=0 \). Then the roots of the equation \( x^{2}-(a+b-2)x+(a+b+2)=0 \) are:

The number of distinct real solutions of the equation \( x\lvert x + 4\rvert + 3\lvert x + 2\rvert + 10 = 0 \) is:

If the quadratic equation \( (\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0, \ \lambda \neq -2, \) has two positive roots, then the number of possible integral values of \( \lambda \) is:

\( cosec 18^\circ \) is a root of the equation:

The number of real roots of the equation \( \sqrt{x^2 - 4x + 3} + \sqrt{x^2 - 9} = \sqrt{4x^2 - 14x + 6} \), is:

If the set of all values of \( a \), for which the equation \( 5x^{3} - 15x - a = 0 \) has three distinct real roots, is the interval \( (\alpha, \beta) \), then \( \beta - 2\alpha \) is equal to ________.

The number of the real roots of the equation \( (x+1)^2 + |x-5| = \frac{27}{4} \) is _______.

The sum of the cubes of all the roots of the equation \( x^{4} - 3x^{3} - 2x^{2} + 3x + 1 = 0 \) 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 the parabola \( y = x^2 + px + q \) passing through the point \( (1, -1) \) be such that the distance between its vertex and the \( x \)-axis is minimum. Then the value of \( p^2 + q^2 \) is:

Let \( a, b \in R \) be such that the equation \( a x^{2} - 2 b x + 15 = 0 \) has a repeated root \( \alpha \). If \( \alpha \) and \( \beta \) are the roots of the equation \( x^{2} - 2 b x + 21 = 0 \), then \( \alpha^{2} + \beta^{2} \) is equal to:

Let the set \( C = \{(x,y)\mid x^2 - 2^y = 2023,\, x,y \in N\} \). Then \(\displaystyle \sum_{(x,y)\in C} (x+y) \) is equal to _______________.

The sum of all the real roots of the equation \( (e^{2x} - 4)(6e^{2x} - 5e^{x} + 1) = 0 \) is:

If \( a + b + c = 1 \), \( ab + bc + ca = 2 \) and \( abc = 3 \), then the value of \( a^4 + b^4 + c^4 \) is equal to ________.

The number of real roots of the equation \( e^{4x}-e^{3x}-4e^{2x}-e^{x}+1=0 \) is equal to ____________.

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:

If \( \alpha ,\beta \) are roots of the equation \( x^{2}+5(\sqrt{2})\,x+10=0 \), \( \alpha>\beta \) and \( P_{n}=\alpha^{n}-\beta^{n} \) for each positive integer \( n \), then the value of \( \left( \dfrac{P_{17}P_{20}+5\sqrt{2}\,P_{17}P_{19}}{P_{18}P_{19}+5\sqrt{2}\,P_{18}^{2}} \right) \) is equal to __________ .

The smallest positive integral value of \( a \), for which all the roots of \( x^4 - a x^2 + 9 = 0 \) are real and distinct, is equal to:

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 ______.

The number of real solutions of the equation \( x\big(x^{2}+3|x|+5|x-1|+6|x-2|\big)=0 \) is __________.

The roots of the quadratic equation \( 3x^2 - px + q = 0 \) are \( 10^{\text{th}} \) and \( 11^{\text{th}} \) terms of an arithmetic progression with common difference \( \dfrac{3}{2} \). If the sum of the first \( 11 \) terms of this arithmetic progression is \( 88 \), then \( q - 2p \) is equal to _________.

Let \( p \) and \( q \) be two real numbers such that \( p+q=3 \) and \( p^{4}+q^{4}=369 \). Then \( \left(\frac{1}{p}+\frac{1}{q}\right)^{-2} \) is equal to ______.

Let \( a,b \in \mathbb{C} \). Let \( \alpha,\beta \) be the roots of the equation \( x^2 + \alpha x + b = 0 \). If \( \beta - \alpha = \sqrt{11} \) and \( \beta^2 - \alpha^2 = 3i\sqrt{11} \), then \( (\beta^3 - \alpha^3)^2 \) is equal to:

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 \( f:[-1,1]\rightarrow \mathbb{R} \) be defined as \( f(x)=ax^{2}+bx+c \) for all \( x\in [-1,1] \), where \( a,b,c\in \mathbb{R} \) such that \( f(-1)=2 \), \( f'(-1)=1 \) and for \( x\in [-1,1] \) the maximum value of \( f''(x) \) is \( \frac{1}{2} \). If \( f(x)\le \alpha,\, x\in[-1,1] \), then the least value of \( \alpha \) is equal to ________.

If \( 2 \) and \( 6 \) are the roots of the equation \( ax^2+bx+1=0 \), then the quadratic equation, whose roots are \( \dfrac{1}{2a+b} \) and \( \dfrac{1}{6a+b} \), is:

Let \( \alpha, \beta \) be the roots of the equation \( x^2 - 3x + r = 0 \), and \( \frac{\alpha}{2}, 2\beta \) be the roots of the equation \( x^2 + 3x + r = 0 \). If the roots of the equation \( x^2 + 6x = m \) are \( 2\alpha + \beta + 2r \) and \( \alpha - 2\beta - \frac{r}{2} \), then \( m \) is equal to:

The number of distinct real roots of the equation \( |x+1|\,|x+3|-4|x+2|+5=0 \), is __________.

The number of solutions, of the equation \( e^{\sin x}-2e^{-\sin x}=2 \), is:

Let \( x_1, x_2, x_3, x_4 \) be the solution of the equation \( 4x^{4} + 8x^{3} - 17x^{2} - 12x + 9 = 0 \) and \( \left(4 + x_{1}^{2}\right) \left(4 + x_{2}^{2}\right) \left(4 + x_{3}^{2}\right) \left(4 + x_{4}^{2}\right) = \dfrac{125}{16}\, m \). Then the value of \( m \) is:

The number of solutions of the equation \( \left( \frac{9}{x} - \frac{9}{\sqrt{x}} + 2 \right)\left( \frac{2}{x} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is:

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 ________.

If \( \alpha, \beta, \gamma \) are the roots of \( x^3 + 3x + 9 = 0 \), find the value of \( \left(\alpha^9 + \beta^9 + \gamma^9\right) \).

If \( \alpha, \beta \) are roots of equation \( x^2 - 2x - 1 = 0 \), then value of \( 5\alpha^4 + 12\beta^3 \) is ________.

The sum of the roots of the equation, \( x + 1 - 2\log_{2}(3 + 2^{x}) + 2\log_{4}(10 - 2^{-x}) = 0 \), is:

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 sum of all the solutions of the equation \( (8)^{2x} - 16\cdot(8)^{x} + 48 = 0 \) is:

Let \( \lambda \neq 0 \) be in \( \mathbb{R} \). If \( \alpha \) and \( \beta \) are roots of the equation \( x^2 - x + 2\lambda = 0 \), and \( \alpha \) and \( \gamma \) are the roots of the equation \( 3x^2 - 10x + 27\lambda = 0 \), then \( \frac{\beta \gamma}{\lambda} \) 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:

If \( \alpha \) and \( \beta \) are the distinct roots of the equation \( x^{2}+(3)^{1/4}x+3^{1/2}=0 \), then the value of \( \alpha^{96}\left(\alpha^{12}-1\right)^{9}+\beta^{96}\left(\beta^{12}-1\right)^{9} \) is equal to;

The sum of all the roots of the equation \( |x^2 - 8x + 15| - 2x + 7 = 0 \) is:

The minimum value of the sum of the squares of the roots of \( x^2 + (3-a)x + 1 = 2a \) 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:

If \( \alpha, \beta \), where \( \alpha < \beta \), are the roots of the equation \( \lambda x^2 - (\lambda + 3)x + 3 = 0 \) such that \( \frac{1}{\alpha} - \frac{1}{\beta} = \frac{1}{3} \), then the sum of all possible values of \( \lambda \) is:

Suppose \( a,b,c \in \mathbb{R} \) and \( b \neq c \). If \( \alpha, \beta \) are roots of \( x^2 + ax + b = 0 \) and \( \gamma, \delta \) are roots of \( x^2 + ax + c = 0 \), then value of \( \dfrac{(\alpha - \gamma)(\alpha - \delta)}{(\beta - \gamma)(\beta - \delta)} \) is independent of:

If \( \alpha,\beta \) are the roots of the equation, \( x^{2}-x-1=0 \) and \( S_{n}=2023\,\alpha^{n}+2024\,\beta^{n} \), then:

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:

The number of solutions of the equation \( \log_{(x+1)}\!\left(2x{^{2}}+7x+5\right)+\log_{(2x+5)}(x+1)^{2}-4=0, \; x>0, \) is ______.

Let one root of the quadratic equation in \( x \):
\( (k^{2}-15k+27)x^{2}+9(k-1)x+18=0 \) be twice the other. Then the length of the latus rectum of the parabola \( y^{2}=6kx \) is equal to:

The sum of all the real solutions of the equation \( \log_{(x+3)}(6x^2 + 28x + 30) = 5 - 2\log_{(6x+10)}(x^2 + 6x + 9) \) is equal to:

The given equation is \( 3x^4 - 2x^3 + 4x^2 - 4x + 12 = 0 \). Now, which of the following options is correct?

Let \( \alpha , \alpha+2 , \alpha \in \mathbb{Z} \), be the roots of the quadratic equation \( x(x+2) + (x+1)(x+3) + (x+2)(x+4) + \ldots + (x+n-1)(x+n+1) = 4n \) for some \( n \in \mathbb{N} \). Then \( n+\alpha \) is equal to:

If \( x \in [5,7] \) the greatest value of expression \( -3x^2 + 54x - 200 \) is ________.

The sum of all the roots of the equation \( (x - 1)^2 - 5|x - 1| + 6 = 0 \), is:

The number of real roots of the equation \( x|x-2|+3|x-3|+1=0 \) is:

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 \( x^2 - (m-3)x + m = 0 \) (\( m \in \mathbb{R} \)) be a quadratic equation. Find the values of \( m \) for which both the roots are negative?

Let \( \alpha, \beta \) be the distinct roots of the equation \( x^2 - (t^2 - 5t + 6)x + 1 = 0, \; t \in \mathbb{R} \) and \( a_n = \alpha^n + \beta^n \). Then the minimum value of \( \dfrac{a_{2023} + a_{2025}}{a_{2024}} \) is:

If \( \alpha \) and \( \beta (\alpha < \beta) \) are the roots of the equation \( (-2 + \sqrt{3})\lvert \sqrt{x} - 3 \rvert + (x - 6\sqrt{x}) + (9 - 2\sqrt{3}) = 0, \ \ x \ge 0 \), then \( \sqrt{\frac{\beta}{\alpha}} + \sqrt{\alpha \beta} \) is equal to:

The value of $4 + \frac{1}{5 + \frac{1}{4 + \frac{1}{5 + \frac{1}{4 + \dots \infty}}}} \text{ is : }$

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:

The set of all values of \( k > -1 \), for which the equation \( (3x^{2} + 4x + 3)^{2} - (k + 1)\,(3x^{2} + 4x + 3)\,(3x^{2} + 4x + 2) + k\,(3x^{2} + 4x + 2)^{2} = 0 \) has real roots, is:

Let the equation \( x(x+2)(12-k)=2 \) have equal roots. Then the distance of the point \( \left(k,\dfrac{k}{2}\right) \) from the line \( 3x+4y+5=0 \) is:

The smallest positive integer \( p \) for which expression \( x^2 - 2px + 3p + 4 \) is negative for atleast one real \( x \) is:

Consider the equation \( x^{2} + 4x - n = 0 \), where \( n \in [20,100] \) is a natural number. Then the number of all distinct values of \( n \), for which the given equation has integral roots, is equal to:

The number of pairs \( (a,b) \) of real numbers, such that whenever \( \alpha \) is a root of the equation \( x^{2} + ax + b = 0 \), \( \alpha^{2} - 2 \) is also a root of this equation is:

The sum, of the squares of all the roots of the equation \( x^{2}+|2x-3|-4=0 \), is:

Let \( f(x) \) and \( g(x) \) be two real polynomials of degree \( 2 \) and \( 1 \) respectively. If \( f(g(x)) = 8x^2 - 2x \), and \( g(f(x)) = 4x^2 + 6x + 1 \), then the value of \( f(2) + g(2) \) is ________.

Let \( \alpha,\beta;\alpha>\beta \) be the roots of the equation \( x^{2}-\sqrt{2}\,x-\sqrt{3}=0 \). Let \( P_{n}=\alpha^{n}-\beta^{n},\, n\in\mathbb{N} \). Then \( \left(11\sqrt{3}-10\sqrt{2}\right)P_{10}+\left(11\sqrt{2}+10\right)P_{11}-11P_{12} \) is equal to

The product of all the rational roots of the equation \( (x^2 - 9x + 11)^2 - (x - 4)(x - 5) = 3 \), is equal to:

Let \( \displaystyle \alpha = \max_{x \in R} \{8^{2 \sin 3x} \cdot 4^{4 \cos 3x}\} \text{ and } \beta = \min_{x \in R} \{8^{2 \sin 3x} \cdot 4^{4 \cos 3x}\} \). If \( 8x^{2} + bx + c = 0 \) is a quadratic equation whose roots are \( \alpha^{1/5} \) and \( \beta^{1/5} \), then the value \( c - b \) is equal to:

Consider the quadratic equation \( (n^2-2n+2)x^2-3x+(n^2-2n+2)^2=0,\, n\in R \). Let \( \alpha \) be the minimum value of the product of its roots and \( \beta \) be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is \( \alpha \) and the common ratio is \( \dfrac{\alpha}{\beta} \), is:

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 \( \dfrac{1}{10}(\alpha^6 + \beta^6) \), is:

Let \( \alpha, \beta (\alpha > \beta) \) be the roots of the quadratic equation \( x^{2}-x-4=0 \). If \( P_{n}=\alpha^{n}-\beta^{n}, \ n \in \mathbb{N} \), then \( \dfrac{P_{15}P_{16}-P_{14}P_{16}-P_{15}^{2}+P_{14}P_{15}}{P_{13}P_{14}} \) is equal to __________.

The maximum value of \( z \) in the following \( z = 6xy + y^2 \), where \( 3x + 4y \leq 100 \) and \( 4x + 3y \leq 75 \) for \( x \geq 0 \) and \( y \geq 0 \) is __________.

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 __________.

Let \( f(x) \) be a quadratic polynomial such that \( f(-2) + f(3) = 0 \). If one of the roots of \( f(x) = 0 \) is \( -1 \), then the sum of the roots of \( f(x) = 0 \) is equal to:

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If \( \alpha, \beta, \gamma \) are the roots of \( x^3 + 3x + 9 = 0 \), find the value of \( \left(\alpha^9 + \beta^9 + \gamma^9\right) \).