Testbook Logo
Exams SuperCoaching Test Series Skill Academy
More

Question

Download Solution PDF

A vector \(\vec{P}\) is given by \(\vec{P}\) = \(x^3 \vec{a_x}-x^2 y^2 \vec{a_y}-x^2 y z \vec{a_z}\). Which one of the following statements is correct?

This question was previously asked in
UPSC ESE (Prelims) Electronics and Telecommunication Engineering 19 Feb 2023 Official Paper
Download PDF Attempt Online
View all UPSC IES Papers >
  1. \(​\vec{P}\) is solenoidal, but not irrotational.
  2. \(​\vec{P}\) is irrotational, but not solenoidal.
  3. \(​\vec{P}\) is neither solenoidal nor irrotational.
  4. \(​\vec{P}\) is both solenoidal and irrotational.

Answer (Detailed Solution Below)

Option 3 : \(​\vec{P}\) is neither solenoidal nor irrotational.
Free Tests
View all Free tests >
Free
ST 1: UPSC ESE (IES) Civil - Building Materials
6.2 K Users
20 Questions 40 Marks 24 Mins
Start Now

Detailed Solution

Download Solution PDF

Given that,

\(\vec P = X^3 a_x - X^2y^2\vec A_y - X^2 YZ\vec a_z\)

Div\(\vec P = \frac{\partial} {\partial X} (x^3) - \frac{\partial}{\partial y}(x^2y^2) - \frac{\partial }{\partial z}(x^2yz)\)

= \(3x^2 -2X^2 - x^2y\)

=\(a_x (-X^2Z -0)- a_y (-2XYZ-0)+ a_z(-2xy^2-0)\)

\(\ne 0\)

\(\vec P\)  is neither solenoidal nor irrotational.

Here, option 3 is correct.

Download Solution PDF
Latest UPSC IES Updates

Last updated on May 28, 2025

->  UPSC ESE admit card 2025 for the prelims exam has been released. 

-> The UPSC IES Prelims 2025 will be held on 8th June 2025.

-> The selection process includes a Prelims and a Mains Examination, followed by a Personality Test/Interview.

-> Candidates should attempt the UPSC IES mock tests to increase their efficiency. The UPSC IES previous year papers can be downloaded here.

‹‹ Previous Ques
Next Ques ››

More Vector Calculus Questions

Q1.Consider a cube having the dimension x, y, z ∈ [1, 3]. If a B̅ = \(2 x^{2} y \hat{a}_{x}+3 x^{2} y^{2} \hat{a}_{y}\). Divergence of B̅ at the centre of cube is :
Q2.Three vectors \(\overrightarrow{\mathrm{p}}, \overrightarrow{\mathrm{q}}\) and \(\overrightarrow{\mathrm{r}}\) are given as \(\overrightarrow{\mathrm{p}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}} \) \(\overrightarrow{\mathrm{q}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \) \(\overrightarrow{\mathrm{r}}=2 \hat{i}+3 \hat{j}+4 \hat{k}\) Which of the following is/are CORRECT?
Q3.A vector \(\vec{P}\) is given by \(\vec{P}\) = \(x^3 \vec{a_x}-x^2 y^2 \vec{a_y}-x^2 y z \vec{a_z}\). Which one of the following statements is correct?
Q4.The work done by two forces \(\vec{P}=3 \hat{i}-2 \hat{j}+\hat{k}\) and \(\vec{Q}=\hat{i}+3 \hat{j}-5 \hat{k}\), in displacing a particle from point A, whose position vector is \(-2 \hat{i}+5 \hat{k}\), to point B, whose position vector is \(3 \hat{i}-7 \hat{j}+2 \hat{k}\), is
Q5.Let 𝜙 be a scalar field, and 𝒖 be a vector field. Which of the following identities is true for div(𝜙𝒖)?
Q6.Which of the following options gives the correct correlation between the variables of rectangular and cylindrical coordinate systems, if x, y, z are variables in a rectangular coordinate system and p, ϕ, z are the variables in a cylindrical system?
Q7.Vector \(\frac{\overrightarrow{r}}{\left|\overrightarrow{r}\right|^3}\) is
Q8.The point of intersection of vector lines \((7\widehat{i}+10\widehat{j}+13\widehat{k})+k_1(2\widehat{i}+3\widehat{j}+4\widehat{k})\) and \((3\widehat{i}+5\widehat{j}+7\widehat{k})+k_2(\widehat{i}+2\widehat{j}+3\widehat{k})\) is given by
Q9.If \(\overrightarrow{p}=i-2j+3k\) and \(\overrightarrow{q}=3i+3j+k\), then \(\left(\overrightarrow{p}-\overrightarrow{q}\right)^2 \) is equal to
Q10.\( \overrightarrow{a}\times[\overrightarrow{b}\times(\overrightarrow{c}\times\overrightarrow{a})]=\)

More Calculus Questions

Q1.Consider a cube having the dimension x, y, z ∈ [1, 3]. If a B̅ = \(2 x^{2} y \hat{a}_{x}+3 x^{2} y^{2} \hat{a}_{y}\). Divergence of B̅ at the centre of cube is :
Q2.Three vectors \(\overrightarrow{\mathrm{p}}, \overrightarrow{\mathrm{q}}\) and \(\overrightarrow{\mathrm{r}}\) are given as \(\overrightarrow{\mathrm{p}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}} \) \(\overrightarrow{\mathrm{q}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \) \(\overrightarrow{\mathrm{r}}=2 \hat{i}+3 \hat{j}+4 \hat{k}\) Which of the following is/are CORRECT?
Q3.Let 𝑓, 𝑔 ∶ ℝ2 → ℝ be defined by 𝑓(𝑥, 𝑦) = \(\rm x^2-\frac{3}{2}xy\) and 𝑔(𝑥, 𝑦) = 4𝑥4 − 5𝑥2𝑦 + 𝑦2  for all (𝑥, 𝑦) ∈ ℝ2 . Consider the following statements: 𝑃: 𝑓 has a saddle point at (0,0). 𝑄: 𝑔 has a saddle point at (0,0). Then
Q4.A vector \(\vec{P}\) is given by \(\vec{P}\) = \(x^3 \vec{a_x}-x^2 y^2 \vec{a_y}-x^2 y z \vec{a_z}\). Which one of the following statements is correct?
Q5.The work done by two forces \(\vec{P}=3 \hat{i}-2 \hat{j}+\hat{k}\) and \(\vec{Q}=\hat{i}+3 \hat{j}-5 \hat{k}\), in displacing a particle from point A, whose position vector is \(-2 \hat{i}+5 \hat{k}\), to point B, whose position vector is \(3 \hat{i}-7 \hat{j}+2 \hat{k}\), is
Q6.\(\int x^{n} d x=\frac{x^{n+1}}{n+1}+c\) is not true, if n is equal to
Q7.The gradient of the normal to the curve y = 2x2 + 3sin x at x = 0 is
Q8.Writing Lagrange’s mean value theorem as f(b) - f(a) = (b - a)f'(c), a < c < b the value of c, if f(x) = x(x - 1), a = 0, \(b = \frac{1}{2}\), is
Q9.Evaluate the definite integration \(\int_1^2\) x . dx and choose the right answer:
Q10.Evaluate the definite integration \(\int_a^b\) ex ⋅ dx:
Suggested Test Series
View All >
UPSC ESE (IES) Civil Mock Test Series
116 Total Tests with 1 Free Tests
Start Free Test
UPSC ESE (IES) Prelims Paper 1 Mock Test Series
54 Total Tests with   1 Free Tests
Start Free Test

More Engineering Mathematics Questions

Q1.If L = {1, 2, 3, 4, 6, 9, 36} is the lattice find the number of complements 9 is having in the below given Hasse diagram?
Q2.Which of the following statements is/are TRUE for undirected graphs?
Q3.\(\iint \frac{x y}{\sqrt{1-y^{2}}} d x d y\) Over the positive quadrant of the circle x2 + y2 = 1 is __________.
Q4.The real root of the equation x3 − x − 5 = 0 lying between 1 and 2 after first iteration by Newton-Raphson method is ________, if initial approximation is taken as x0 = 2 ∈ [1, 2] : 
Q5.Let X be uniform random variable on [0, 4] and Y be uniform random variable on [0, 1]. If X and Y are independent, then P(max {X, Y} > 3) is equal to : 
Q6.If R and D are the radius and diameter of the graph K4,7, then the ordered pair (R, D) is equal to : 
Q7.The largest number of faces in a simple connected maximal planar graph with 100 vertices is :
Q8.If for the matrix A, A3 = I then A−1 = __________. 
Q9.Energy of power signal is : 
Q10.With usual notations, the properties of maxima and minima under various conditions are __________. I II (P) Maxima (i) r t− s2 = 0 (Q) Minima (ii) rt − s2 < 0 (R) Saddle Point (iii) rt − s2 > 0, r > 0 (S)  Case of failure (iv) rt − s2 > 0, r < 0
Important Exams
Super Coaching
Exams
Test Series
Previous Year Papers
Objective Questions
Testbook USA SAT Exam
Testbook USA ACT Exam
US College Admissions Guide
Testbook Edu Solutions Pvt. Ltd.
1st & 2nd Floor, Zion Building,
Plot No. 273, Sector 10, Kharghar,
Navi Mumbai - 410210
[email protected]
Toll Free:1800 833 0800
Office Hours: 10 AM to 7 PM (all 7 days)
Company
About usCareers We are hiringTeach Online on TestbookPartnersMediaSitemap
Products
Test SeriesTestbook PassOnline CoursesOnline VideosPracticeBlogRefer & EarnBooks
Our Apps
Follow us on
Copyright © 2014-2022 Testbook Edu Solutions Pvt. Ltd.: All rights reserved
User PolicyTermsPrivacy
Get Free Access Now
Hot Links： teen patti master 2023 all teen patti teen patti comfun card online