GATE 2026 Engineering Mathematics PYQs | Calculus

[Tex]\begin{aligned} & \text{Consider the function } f \colon \mathbb{R} \to \mathbb{R} \text{ defined as follows:} \\ & \qquad\qquad\qquad f(x) = \begin{cases} c_1 e^x - c_2 \log_e\left(\frac{1}{x}\right), & \text{if } x > 0 \\ 3 & \text{otherwise} \end{cases} \\ \\ & \text{where } c_1, c_2 \in \mathbb{R}\text{.} \\ & \text{If } f \text{ is continuous at } x = 0\text{, then } c_1 + c_2 = ?\textbf{[NAT || GATE 2026 Set-1 || 1 Marks]} \end{aligned}[/Tex]

[Tex]\begin{aligned} & \text{Let } f \colon \mathbb{R} \to \mathbb{R} \text{ be defined as follows:} \\ & \qquad\qquad\qquad f(x) = ((|x|/2) - x)(x - (|x|/2)) \\ \\ & \text{Which of the following statements is/are true?} \textbf{[MSQ || GATE-2026 Set-1 || 2 Marks]} \end{aligned}[/Tex]

[Tex]\begin{aligned} & \text{For a real number } a, \text{ let} \\ & \qquad\qquad\qquad I(a) = \int_{-1}^{1} (3x^2 - ax + 1) \, dx \\ & \text{Which of the following statements is/are true?} \textbf{[MSQ || GATE-2026 Set-2 || 1 Marks]} \end{aligned}[/Tex]