TS EAPCET (Engineering) Limits and Continuity — practice questions
59 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice TS EAPCET (Engineering) Limits and Continuity in the app →Evaluate $\displaystyle\lim_{x \to 0} \dfrac{ in 3x}{x}$.Evaluate $\displaystyle\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2}$.Evaluate $\displaystyle\lim_{x \to 0} \dfrac{1 - \cos 2x}{x^2}$.Evaluate $\displaystyle\lim_{x \to \infty} \left(1 + \dfrac{2}{x}\right)^x$.Consider $f(x) = |x|$ at $x = 0$. Which statement is correct?$ in(180° - \theta)$ equals:$ in^2\theta + \cos^2\theta$ equals:$ in(A + B) = $A **random variable** is best defined as:A **discrete** random variable is one whose possible values:For a continuous random variable $X$, the probability that $X$ equals any specific value is:The **cumulative distribution function (c.d.f.)** $F(x)$ of a discrete random variable is:When two fair dice are rolled and $X$ denotes the sum of the upper faces, the range of $X$ is:If a fair coin is tossed twice and $X$ = number of heads, then $P[X = 1]$ equals:If two cards are drawn with replacement from a deck of 52, $P[X = 2]$ where $X$ = number of aces equals:A fair die is thrown. Let $X$ = number of factors of the number on the upper face. $P[X = 2]$ equals:$f(x) = 3x^2$ for $0 < x < 1$ is the p.d.f. of $X$. The probability $P[1/2 < X < 1]$ is:If $f(x) = kx^2(1 - x)$ for $0 < x < 1$ is the p.d.f. of $X$, the value of $k$ is:For p.d.f. $f(x) = x/8$, $0 < x < 4$, the probability $P[x < 1.5]$ is:Find $k$ if $f(x) = kx$ for $0 < x < 2$ and 0 otherwise is a valid p.d.f.The c.d.f. $F(x)$ of the p.d.f. $f(x) = 3x^2$ for $0 < x < 1$ is:Two cards are drawn with replacement. $X$ = number of aces. The probability $P[X \geq 1]$ equals:For p.d.f. $f(x) = x^2/3$ for $-1 < x < 2$, the probability $P[0 < X \leq 1]$ equals:For p.d.f. $f(x) = k(4 - x^2)$ for $-2 \leq x \leq 2$ (and 0 elsewhere), the value of $k$ is:For the c.d.f. $F(x) = 3x - 2x^3$ (from p.d.f. $f(x) = 3(1 - 2x^2)$ on $[0, 1]$), $F(0.5)$ equals:If a fair coin is tossed $n$ times, the probability of any exact pattern (like all heads) is:If a die is thrown and 'getting an odd number' is success, then $p$ and $q$ are:A bag has balls marked 0–9. Four balls drawn with replacement. Probability that **none** is marked '0' is:Probability of bomb hitting target = 0.8. Out of 10 bombs dropped, probability that **exactly 2 miss** equals sin(0°) equals:cos(90°) equals:sin²θ + cos²θ equals:tan(45°) equals:In which quadrant is sin θ positive and cos θ negative?lim_(x→2) (x² - 4)/(x - 2) =lim_(x→0) sin(x)/x =For limit to EXIST at x = a, left-hand and right-hand limits must:A function f is continuous at x = a if:d/dx (xⁿ) = ?lim_(x→0) (1 - cos x)/x² =lim_(x→∞) (1 + 1/x)^x =lim_(x→0) (e^x - 1)/x =lim_(x→0) (sin 3x)/(sin 5x) =d/dx (sin x) =d/dx (eˣ) =d/dx (ln x) =Find d/dx (x² sin x) by product rule:By chain rule, d/dx [sin(x²)] =lim_(x→∞) (3x² - 5)/(7x² + 2) =l'Hôpital's rule applies to limits of form:Find dy/dx by implicit differentiation for x² + y² = 25:d/dx (xˣ) where x > 0:For f(x) = sin(1/x) for x ≠ 0, what is the limit as x → 0?Differentiation by chain rule for d/dx [ln(sin x)]:lim_(x→0) [(1 + x)^(1/x)] =Higher-order derivative: d²/dx² (x⁴) =f(x) is discontinuous at x = a if which condition fails?For y = x³ - 6x² + 9x, find critical points (where y' = 0):By Taylor expansion around x = 0: e^x = 1 + x + ___: