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Relations between roots and coefficients, nature of roots Short Tricks for JEE

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Syllabus context

Part of Complex Numbers and Quadratic Equations in JEE Main Mathematics.

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Q1MathsUnit 2: Complex Numbers and Quadratic Equations
If x+1x=4,x+\frac{1}{x}=4, then x4+1x4x^{4}+\frac{1}{x^{4}} is equal to
Q2MathsUnit 2: Complex Numbers and Quadratic Equations
If the product of two numbers is 10 and their sum is 7,7, which is the greatest of the two numbers?
Q3MathsUnit 2: Complex Numbers and Quadratic Equations
The zeros of the polynomial n3+9n2+n^{3}+9 n^{2}+ 23n+1523 n+15 are a−d,aa-d, a and a+d.a+d . What is the value of 'a'?
Q4MathsUnit 2: Complex Numbers and Quadratic Equations
Discriminant of the equation −3x2+-3 x^{2}+ 2x−8=02 x-8=0 is
Q5MathsUnit 2: Complex Numbers and Quadratic Equations
If the sum of the roots of the equation ax2+bx+c=0a x^{2}+b x+c=0 is equal to sum of the squares of their reciprocals, then bc2,ca2,ab2b c^{2}, c a^{2}, a b^{2} are in
Q6MathsUnit 2: Complex Numbers and Quadratic Equations
If x−y=7x-y=7 and x3−y3=133;x^{3}-y^{3}=133 ; find: the value of xyx y
Q7MathsUnit 2: Complex Numbers and Quadratic Equations
Find p∈Rp \in R for x2−px+p+3=0x^{2}-p x+p+3=0 has
Q8MathsUnit 2: Complex Numbers and Quadratic Equations
Assertion Let equations ax2+bx+c=a x^{2}+b x+c= 0(a,b,c∈R)&x2+2x+5=0\mathbf{0}(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c} \in \boldsymbol{R}) \& \boldsymbol{x}^{2}+\mathbf{2} \boldsymbol{x}+\mathbf{5}=\mathbf{0} have common root, then a+cb=13\frac{\boldsymbol{a}+\boldsymbol{c}}{\boldsymbol{b}}=\frac{\mathbf{1}}{\mathbf{3}} Reason If both roots of Ax2+Bx+K1=0&A x^{2}+B x+K_{1}=0 \& A′x2+B′x+K2=0\boldsymbol{A}^{\prime} \boldsymbol{x}^{2}+\boldsymbol{B}^{\prime} \boldsymbol{x}+\boldsymbol{K}_{2}=\mathbf{0} are identical then⁡AA1=BB1=K1K2( where A,B,K1\operatorname{then} \frac{\boldsymbol{A}}{\boldsymbol{A}_{1}}=\frac{\boldsymbol{B}}{\boldsymbol{B}_{1}}=\frac{\boldsymbol{K}_{1}}{\boldsymbol{K}_{2}}\left(\text { where } \boldsymbol{A}, \boldsymbol{B}, \boldsymbol{K}_{1}\right. and A′,B,′K2∈R)\left.A^{\prime}, B,^{\prime} K_{2} \in R\right)

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