JEE Main 2026 · 5 Apr · Shift 1 · Mathematics Q12

MathematicsSingle correct+4 / -1Question 12 of 75

The sum of all the integral values of $p$ such that the equation $3\sin^2x+12\cos x-3=p$ , $x\in\mathbb R$ , has at least one solution, is:

-54

-60

-75

-84

Answer and worked solution

Official answer: C

In terms of $c=\cos x$ .
$p=3(1-c^2)+12c-3=12-3(c-2)^2,\qquad c\in[-1,1].$

Range. This is increasing on $[-1,1]$ : from $-15$ at $c=-1$ to $9$ at $c=1$ .

Integers $-15,\dots,9$ : the terms from $-9$ to $9$ cancel, leaving $-15-14-13-12-11-10=-75$ .

(C)

Topics:Trigonometric equations Quadratic expressions

JEE Main 2026 · 5 Apr · Shift 1, Mathematics question 12