NIMCET 2022 — Mathematics PYQ

1. Relation Between Roots and Coefficients

For a quadratic equation $a{x}^2+bx+c=0$, the sum of roots is $-b/a$ and the product of roots is $c/a$.

Given $x^2+px+q=0$, let the roots be $\alpha =\tan 30^{\circ }$ and $\beta =\tan 15^{\circ }$:

• Sum of roots: $\alpha +\beta =\tan 30^{\circ }+\tan 15^{\circ }=-p$

• Product of roots: $\alpha \beta =\tan 30^{\circ }\cdot \tan 15^{\circ }=q$

2. Apply the Tangent Addition Formula

We know that:

Let $A=30^{\circ }$ and $B=15^{\circ }$. Then $A+B=45^{\circ }$:

Substitute the values in terms of $p$ and $q$:

Since $\tan 45^{\circ }=1$:

3. Calculate the Final Value

Rearrange the equation $1-q=-p$ to find the relationship between $p$ and $q$:

The question asks for the value of $2+q-p$:

Final Answer:

The value of $2+q-p$ is 3. The correct option is A.