Mathematics & Numerical Analysis Practice Questions (Q81-85) - Statistical Sciences

ICAR PG Statistical Sciences 29 Jun 2024Shift 1

MCQ (Single Correct) +4 −1

If $\ln u = \frac{x^3 + y^3}{3x + 4y}$, then the value of $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}$ is given by:

यदि $\ln u = \frac{x^3 + y^3}{3x + 4y}$ हो, तो $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}$ का मान है:

$u \ln\frac{u}{2}$ $u \ln\frac{u}{2}$ $\frac{u}{2}\ln u$ $\frac{u}{2}\ln u$ $2u \ln u$ $2u \ln u$ $\frac{u}{2}\ln\frac{u}{2}$ $\frac{u}{2}\ln\frac{u}{2}$

ICAR PG Statistical Sciences 29 Jun 2024Shift 1

MCQ (Single Correct) +4 −1

If $A$ is any real number and $r$ is positive rational number, then the value of $\lim_{x \to +\infty} \frac{A}{x^r}$ is given by:

यदि $A$ कोई वास्तविक संख्या है और $r$ एक धनात्मक परिमेय संख्या है, तो $\lim_{x \to +\infty} \frac{A}{x^r}$ का मान है:

$e$ $e$ -1 -1 0 0 1 1

ICAR PG Statistical Sciences 29 Jun 2024Shift 1

MCQ (Single Correct) +4 −1

Find constants $a$ and $b$ so that $\lim_{x \to 0} \left(\frac{\sin 2x}{x^3} + \frac{a}{x^2} + b\right) = 1$:

अचर $a$ और $b$ के मान ज्ञात कीजिए ताकि $\lim_{x \to 0} \left(\frac{\sin 2x}{x^3} + \frac{a}{x^2} + b\right) = 1$ हो:

$a = -2, b = \frac{7}{3}$ $a = -2, b = \frac{7}{3}$ $a = 2, b = -\frac{7}{3}$ $a = 2, b = -\frac{7}{3}$ $a = 1, b = -1$ $a = 1, b = -1$ $a = -1, b = 1$ $a = -1, b = 1$

ICAR PG Statistical Sciences 29 Jun 2024Shift 1

MCQ (Single Correct) +4 −1

Find the interval in which the graph of the function given by $f(x) = x^4 - 4x^3 + 10$ is concave up:

वह अंतराल ज्ञात कीजिए जिसमें फलन $f(x) = x^4 - 4x^3 + 10$ का आलेख अवतल ऊपर (concave up / उत्तल) है:

$(-\infty, 1) \cup (3, \infty)$ $(-\infty, 1) \cup (3, \infty)$ $(-\infty, 0) \cup (2, \infty)$ $(-\infty, 0) \cup (2, \infty)$ $(1.5, \infty)$ $(1.5, \infty)$ $(0, 2)$ $(0, 2)$

ICAR PG Statistical Sciences 29 Jun 2024Shift 1

MCQ (Single Correct) +4 −1

The line $y = L$ is a horizontal asymptote of the graph of the function $f$ if:

रेखा $y = L$ फलन $f$ के आलेख की एक क्षैतिज अनंतस्पर्शी (horizontal asymptote) कहलाती है यदि:

$\lim_{x \to +\infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$ $\lim_{x \to +\infty} f(x) = L$ अथवा $\lim_{x \to -\infty} f(x) = L$ $\lim_{x \to L^+} f(x)$ do not exist $\lim_{x \to L^+} f(x)$ का अस्तित्व न हो $\lim_{x \to L^-} f(x)$ do not exist $\lim_{x \to L^-} f(x)$ का अस्तित्व न हो $\lim_{x \to +\infty} f(x)$ do not exist $\lim_{x \to +\infty} f(x)$ का अस्तित्व न हो