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

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

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:

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

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

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 __________ .

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 ________.

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 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 ______________.

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 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 ______.

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

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:

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:

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:

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 __________.

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JEE Main 2021 25th July Evening Shift

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 ________.