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The number of real solutions of the equation \( x\lvert x+5\rvert + 2\lvert x+7\rvert - 2 = 0 \) is \( \_\_\_\_\_\_\_ \).

If the domain of the function \( f(x)=\log_{(10x^{2}-17x+7)}(18x^{2}-11x+1) \) is \( (-\infty,a)\cup(b,c)\cup(d,\infty)-\{e\} \), then \( 90(a+b+c+d+e) \) equals:

If the set of all solutions of \( |x^{2}+x-9| = |x| + |x^{2}-9| \) is \([\alpha,\beta] \cup [\gamma,\infty)\), then \( (\alpha^{2}+\beta^{2}+\gamma^{2}) \) is equal to:

Let \( S \) be the set of the first \( 11 \) natural numbers. Then the number of elements in \( A = \{ B \subseteq S : n(B) \ge 2 \text{ and the product of all elements of } B \text{ is even} \} \) is ________ .

Let \( S = \left\{ x^{3} + a x^{2} + b x + c : a,b,c \in \mathbb{N} \ \text{and} \ a,b,c \leq 20 \right\} \) be a set of polynomials. Then the number of polynomials in \( S \), which are divisible by \( x^{2} + 2 \), is:

Let the number \( (22)^{2022}+(2022)^{22} \) leave the remainder \( \alpha \) when divided by \( 3 \) and \( \beta \) when divided by \( 7 \). Then \( (\alpha^2+\beta^2) \) is equal to:

The sum of squares of all the real solutions of the equation \( \log_{(x+1)}(2x^2+5x+3)=4-\log_{(2x+3)}(x^2+2x+1) \) is equal to \( \_\_\_\_\_ \).

The missing value in the following figure is ______________ .

For a natural number \( n \), let \( \alpha_n = 19^n-12^n \). Then, the value of \( \frac{31\alpha_9-\alpha_{10}}{57\alpha_8} \) is ________.

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JEE Main 2023 10th April Evening Shift

Let the number \( (22)^{2022}+(2022)^{22} \) leave the remainder \( \alpha \) when divided by \( 3 \) and \( \beta \) when divided by \( 7 \). Then \( (\alpha^2+\beta^2) \) is equal to: