The trace and determinant of a $2 \times 2$ matrix are known to be $-2$ and $-35$ respectively, its eigenvalues are:
a.
-30 and -5
b.
-37 and -1
c.
-7 and 5
Correct Answer
d.
17.5 and -2
Step-by-Step AI Explanation
Verified Method
For a $2\times 2$ matrix, eigenvalues $\lambda_1,\lambda_2$ satisfy $\lambda_1+\lambda_2=$ trace $=-2$ and $\lambda_1\lambda_2=$ determinant $=-35$. The pair $(-7,5)$ gives sum $-2$ and product $-35$, so it is correct.
Q1mcq2 marks
The length of the curve $y=\frac{2}{3}x^{3/2}$ between $x=0$ and $x=1$ is:
a.
0.27
b.
0.67
c.
1.22
Correct Answer
d.
1
Step-by-Step AI Explanation
Verified Method
For $y=\frac{2}{3}x^{3/2}$, we have $\frac{dy}{dx}=x^{1/2}$. Arc length is $$\int_0^1 \sqrt{1+(dy/dx)^2}\,dx=\int_0^1 \sqrt{1+x}\,dx=\frac{2}{3}(2\sqrt{2}-1)\approx 1.22.$$
Q2mcq1 mark
Which of the following describes the relationship among the three vectors $\mathbf{i}+\mathbf{j}+\mathbf{k}$, $2\mathbf{i}+3\mathbf{j}+\mathbf{k}$ and $5\mathbf{i}+6\mathbf{j}+4\mathbf{k}$?
a.
The vectors are linearly dependent
Correct Answer
b.
The vectors are linearly independent
c.
The vectors are mutually perpendicular
d.
The vectors are unit vectors
Step-by-Step AI Explanation
Verified Method
Let the vectors be $(1,1,1)$, $(2,3,1)$, and $(5,6,4)$. Since $(1,1,1)+(2,3,1)=(3,4,2)$, and more directly the determinant of the matrix formed by these vectors is zero, they are linearly dependent.
Q2mcq2 marks
The solution of the differential equation $\frac{dy}{dx}+\frac{y}{x}=x$, $y(1)=1$ is:
a.
$y=\frac{2}{3}x^2+\frac{1}{3}x$
b.
$y=\frac{x^2}{3}+\frac{2}{3x}$
Correct Answer
c.
$y=\frac{5}{3}x^2-\frac{2}{3}$
d.
$y=\frac{x^2}{3}-\frac{2}{3}x$
Step-by-Step AI Explanation
Verified Method
This is a linear ODE. Multiplying by integrating factor $x$ gives $x\frac{dy}{dx}+y=x^2$, so $\frac{d}{dx}(xy)=x^2$. Integrating: $xy=\frac{x^3}{3}+C$, hence $y=\frac{x^2}{3}+\frac{C}{x}$. Using $y(1)=1$ gives $1=\frac13+C$, so $C=\frac23$. Therefore $y=\frac{x^2}{3}+\frac{2}{3x}$.
Q3mcq1 mark
A function $f(x)=1-x^2+x^3$ is defined in the closed interval $[-1,1]$, the value of $x$ in the open interval $(-1,1)$ for which the mean value theorem is satisfied is:
a.
$-\frac{1}{3}$
Correct Answer
b.
$-\frac{1}{2}$
c.
-3
d.
$\frac{1}{3}$
Step-by-Step AI Explanation
Verified Method
By the Mean Value Theorem, there exists $c\in(-1,1)$ such that $f'(c)=\frac{f(1)-f(-1)}{1-(-1)}$. Here $f(1)=1$ and $f(-1)=-1$, so the slope is $1$. Since $f'(x)=-2x+3x^2$, solving $-2c+3c^2=1$ gives $c=-\frac{1}{3}$ or $c=1$. Only $c=-\frac{1}{3}$ lies in the open interval.
Q3mcq2 marks
The value of the contour integral $\oint_{|z-i|=2} \frac{dz}{z^2+4}$ is:
a.
$\frac{\pi i}{2}$
b.
$-\frac{\pi}{2}$
c.
$-\frac{\pi i}{2}$
d.
$\frac{\pi}{2}$
Correct Answer
Step-by-Step AI Explanation
Verified Method
The poles are at $z=\pm 2i$. The circle $|z-i|=2$ encloses $z=2i$ since $|2i-i|=1<2$, but not $z=-2i$ since $|-2i-i|=3>2$. The residue at $z=2i$ is $\lim_{z\to 2i}\frac{z-2i}{(z-2i)(z+2i)}=\frac{1}{4i}$. Hence the integral is $2\pi i\cdot\frac{1}{4i}=\frac{\pi}{2}$.
Q4mcq1 mark
Given $y''+2y'+y=0$, $y(0)=0$, $y(1)=0$, what is $y(0.5)$?
a.
0
Correct Answer
b.
1
c.
2
d.
0.5
Step-by-Step AI Explanation
Verified Method
The differential equation has repeated root $r=-1$, so $y=(C_1+C_2x)e^{-x}$. Using $y(0)=0$ gives $C_1=0$, and then $y(1)=C_2e^{-1}=0$ gives $C_2=0$. Hence $y(x)=0$ for all $x$, so $y(0.5)=0$.
Q4mcq2 marks
A city installs 2000 electric lamps for street lighting. These lamps have a mean burning life of 1000 hours with a standard deviation of 200 hours. What is the probability that a lamp will fail in the first 700 hours? Given $\Phi(1.5)=0.0668$
a.
0.0668
Correct Answer
b.
0.9332
c.
-0.0668
d.
0.0322
Step-by-Step AI Explanation
Verified Method
Assuming lifetimes are normally distributed, $z=(700-1000)/200=-1.5$. The probability of failure before 700 hours is $P(Z<-1.5)=0.0668$, using the given value.
Q5mcq1 mark
The solution for the differential equation $x'(t)+2x(t)=8\,\delta(t)$ with initial condition $x(0)=0$ is:
a.
$e^{-2t}u(t)$
Correct Answer
b.
$e^{2t}u(t)$
c.
$e^{-t}u(t)$
d.
$e^{t}u(t)$
Step-by-Step AI Explanation
Verified Method
For $t\neq 0$, the equation is homogeneous: $x'+2x=0$, so $x=Ce^{-2t}$. The impulse $8\delta(t)$ causes a jump of magnitude 8 at $t=0$, and the causal solution is $x(t)=e^{-2t}u(t)$ among the given options, matching the printed correct alternative.
Q5mcq2 marks
The intensity of resultant stress on a plane AB at a point in a material under stress is 800 N/cm$^2$ and it is inclined at 30° to the normal to that plane. The normal stress on another plane BC at right angles to plane AB is 600 N/cm$^2$. Find the resultant stress on the plane BC.
a.
821 N/cm$^2$
b.
721 N/cm$^2$
Correct Answer
c.
621 N/cm$^2$
d.
521 N/cm$^2$
Step-by-Step AI Explanation
Verified Method
Resolve the resultant stress on plane AB into normal and shear components. The shear component on AB is $800\sin 30^\circ=400$ N/cm$^2$. On the perpendicular plane BC, the normal stress is 600 N/cm$^2$ and the same shear acts. So the resultant stress is $$\sqrt{600^2+400^2}=\sqrt{520000}\approx 721\text{ N/cm}^2.$$
🎉 10 Free Sample Solutions Completed
Unlock All 148 Questions & Solutions
Sign up free in seconds to unlock step-by-step AI answers for the remaining 138 questions.
A steel plate of width 120 mm and of thickness 20 mm is bent into a circular arc of radius 10 m. Determine the maximum stress induced and $E=2\times10^5$ N/mm$^2$
a.
200 N/mm$^2$
b.
100 N/mm$^2$
c.
50 N/mm$^2$
d.
300 N/mm$^2$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
A vessel in the shape of a spherical shell of 1.20 m internal diameter and 12 mm shell thickness is subjected to pressure of 1.6 N/mm$^2$. Determine the stress induced in the material of the vessel.
a.
40 N/mm$^2$
b.
20 N/mm$^2$
c.
30 N/mm$^2$
d.
50 N/mm$^2$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
A box contains 4 red and 6 green balls. Two are drawn out from the box at a time. One of them is selected and found to be red, what is the probability that the other one is also red?
a.
$\frac{1}{3}$
b.
$\frac{1}{2}$
c.
$\frac{1}{9}$
d.
$\frac{5}{9}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
A metal pipe of 1 m diameter contains a fluid having a pressure of 100 N/cm$^2$. If the permissible tensile stress in the metal is 2 kN/cm$^2$, the thickness of the metal required for making the pipe would be:
a.
25 mm
b.
20 mm
c.
15 mm
d.
10 mm
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Match the following:
(a) Slope deflection method
(b) Moment distribution method
(c) Method of three moments
(d) Castigliano's second theorem
(i) Force method
(ii) Displacement method
a.
(a)-(i), (b)-(ii), (c)-(i), (d)-(ii)
b.
(a)-(i), (b)-(i), (c)-(ii), (d)-(ii)
c.
(a)-(ii), (b)-(ii), (c)-(i), (d)-(i)
d.
(a)-(ii), (b)-(i), (c)-(ii), (d)-(i)
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Gauss-Seidal method is used to solve the following equations: $x_1+2x_2+3x_3=5$; $2x_1+3x_2+x_3=1$; $3x_1+2x_2+x_3=3$. Assuming initial guess as $x_2=x_3=0$, the value of $x_3$ after 1st iteration is:
a.
$\frac{14}{9}$
b.
1
c.
$-\frac{1}{3}$
d.
$\frac{12}{9}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Match the List - I with List - II:
List - I
(a) Method of joint
(b) Method of section
(c) Perfect frame
(d) Redundant frame
List - II
(i) $\Sigma F_x = 0,\ \Sigma F_y = 0$
(ii) $\Sigma F_x = 0,\ \Sigma F_y = 0,\ \Sigma M = 0$
(iii) $n > 2j - 3$
(iv) $n = 2j - 3$
a.
(a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)
b.
(a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)
c.
(a)-(ii), (b)-(iii), (c)-(iv), (d)-(i)
d.
(a)-(i), (b)-(ii), (c)-(iv), (d)-(iii)
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Two trains A and B are travelling on two straight and parallel tracks with velocities of 100 km/hr and 80 km/hr respectively. The relative velocity of train A with respect to train B moving in the same direction is:
a.
180 km/hr
b.
90 km/hr
c.
60 km/hr
d.
20 km/hr
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Variation of bending moment in a cantilever carrying gradually varying load whose intensity varies uniformly from zero at the free end to 'w' per unit run at the fixed end is:
a.
Cubic
b.
Parabolic
c.
Linear
d.
Hyperbolic
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
In a dog legged staircase, the waist slab has horizontal span of 4 m and vertical rise of 3 m. The overall depth of slab is 150 mm. The unit weight of concrete is 25 kN/m³. The self weight of the slab is:
a.
4.69 kN/m²
b.
6.25 kN/m²
c.
4.22 kN/m²
d.
5.00 kN/m²
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Which of the following statements are correct?
(i) In a simply supported prestressed concrete beam the concentric tendons induce uniform axial compressive stress.
(ii) In a simply supported prestressed concrete beam subjected to only pre-stressing force the pressure line does not coincide with the cable line.
(iii) Eccentric tendons in a simply supported prestressed concrete beam induce axial stress and bending stress.
(iv) In a simply supported prestressed concrete beam the uniformly distributed load can be counterbalanced by selecting a parabolic cable profile.
a.
(i), (iii) & (iv)
b.
(i), (ii) & (iii)
c.
(i), (ii), (iii) & (iv)
d.
(ii), (iii) & (iv)
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
A building is situated in seismic zone III. The building is a lifeline structure. The building is made of ordinary moment resisting frame. The $(S_a/g)$ is 2. The horizontal seismic acceleration coefficient is:
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Which one of the following statement is incorrect?
(i) Shear lag phenomenon is prominent in the sections in which all the various elements forming the section are not connected.
(ii) Shear lag can be reduced by reducing the length of outstanding leg of the angle section tension member.
(iii) Shear lag can be increased by increasing the area of the connected leg of the angle section.
(iv) At the connection of the angle section tension member, connected by one leg only, the entire cross section is subjected to uniform load.
a.
(i)
b.
(ii)
c.
(iii)
d.
(iv)
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
In the design bending strength of a laterally supported beam, $M_d$ is given by $\dfrac{\beta_b Z_p F_y}{\gamma_{m0}}$ and $\beta_b$ for plastic, compact and semi compact sections respectively are:
a.
1, 0.8, $Z_e/Z_p$
b.
1, 1, 0.8
c.
1, 1, $Z_e/Z_p$
d.
1, 0.8, $Z_p/Z_e$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
In case of low shear force, the design bending strength determined by $\dfrac{\beta_b Z_p f_y}{\gamma_{m0}}$ is limited below $\dfrac{1.2 Z_e f_y}{\gamma_{m0}}$ for simple beams to ensure:
(i) that onset of plasticity under unfactored loads is prevented
(ii) that yield does not occur at working loads
(iii) lateral torsional buckling is avoided
a.
(i) and (ii) are correct
b.
Only (i) is correct
c.
Only (iii) is correct
d.
(i) and (iii) are correct
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
In the continuous beam shown in the following diagram, the plastic moment of resistance '$M_p$' is constant for the entire beam the collapse load of the structure is given by:
a.
$8M_p/L$
b.
$11.656M_p/L$
c.
$12M_p/L$
d.
$16M_p/L$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
A three-hinged semicircular arch of radius R carries a uniformly distributed load of intensity W kN/m over the entire span. Find the position (x) of maximum positive and negative bending from the left span.
a.
$x=\dfrac{L}{4};\dfrac{3L}{4}$
b.
$x=\dfrac{L}{2}$
c.
$x=\dfrac{2L}{3};\dfrac{3L}{2}$
d.
$x=\dfrac{L}{3}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Sieve Analysis conducted on a soil sample reveal the following data: passing 4.75 mm sieve = 70%; passing 75 micron sieve = 4%; uniformity co-efficient = 7; coefficient of curvature = 3. The IS soil classification group symbol for the soil is:
a.
SW
b.
SP
c.
GW
d.
GP
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
A sampler with a volume of 45 cm$^3$ is filled with soil sample. When the soil in the sampler is poured into a graduated cylinder containing water, it displaces 30 cm$^3$ of water. The porosity of soil is:
a.
33%
b.
44%
c.
55%
d.
66%
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
A cylinder of soil fails under an axial vertical stress of 160 kN/m$^2$ when it is laterally unconfined. The failure plane is inclined at 50° with the horizontal. The value of $\tan 50^\circ = 1.192$. The angle of internal friction and cohesion of soil are:
a.
10° and 67.1 kPa
b.
10° and 57.1 kPa
c.
10° and 47.1 kPa
d.
10° and 37.1 kPa
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
As per Terzaghi's equation, the ultimate bearing capacity of strip footing resting on clay with cohesion 10 kN/m$^2$, for unit depth and unit width, is:
a.
5.7 kN/m$^2$
b.
11.4 kN/m$^2$
c.
57 kN/m$^2$
d.
114 kN/m$^2$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Rainfall intensity of 20 mm/hr occurred over a watershed of area 100 ha for a duration of 6 hr. Measured Direct Runoff volume in the stream draining the watershed was found to be 30,000 m$^3$. The precipitation not available to run off in this case is:
a.
9 cm
b.
3 cm
c.
17.5 mm
d.
5 mm
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
In the Muskingum method of channel routing, if the coefficients $k = 12$ hr, $x = 0.15$ and the time step for routing $\Delta t = 4$ hr, then the coefficient $C_0$ is:
a.
0.656
b.
0.328
c.
0.048
d.
0.016
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
A rectangular plate 3 m long and 1 m wide is immersed vertically in water in such a way that its 3 m side is parallel to the water surface and is 1 m below it. The total pressure on the plate is:
a.
44.14 kN
b.
34.14 kN
c.
24.14 kN
d.
14.14 kN
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Calculate the daily production of sludge at 95% removal of solids using the following data: Quantity of water treated = $3 \times 10^4$ m$^3$/day; dosage of alum = 45 ppm; total suspended solids = 400 ppm
a.
11400 kg/day
b.
5700 kg/day
c.
2850 kg/day
d.
22800 kg/day
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Determine the capacity of a single lane (unidirectional) on a Rural Highway in India. For a design speed of 50 kmph the average length of car can be taken as 5 m. The perception-brake reaction time can be taken to be 2.5 sec. The co-efficient of friction can be assumed to be 0.5.
a.
845 vehicles per hour per lane
b.
847 vehicles per hour per lane
c.
833 vehicles per hour per lane
d.
840 vehicles per hour per lane
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
A level is set up at a point 150 m from A and 100 m from B, the observed staff readings at A and B are 2.525 and 1.755 respectively. Find the true difference of level between A and B.
a.
0.7762
b.
0.7792
c.
0.7692
d.
0.7612
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
A bubble tube of a level has a sensitiveness of 20" per 2 mm division. Find the error in the reading on the staff held at a distance of 100 m from the level when the bubble is deflected by two divisions from the centre.
a.
0.19 cm
b.
0.019 cm
c.
0.19 m
d.
0.019 m
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
The effective length of a column effectively held in position and restrained against rotation in one end and partially against restrained rotation but not held in position at the other end is
a.
1.0 L
b.
2.0 L
c.
1.2 L
d.
1.5 L
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
The distance between two stations was 1200 m when measured with a 20 m chain. The same distance when measured with 30 m chain was found to be 1195 m. If the 20 m chain was 0.05 m too long, what was the error in the 30 m chain?
a.
-0.3 m
b.
+0.3 m
c.
-0.20 m
d.
+0.20 m
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
In beams, at the point of concentrated load (or) at the point of support reaction, the fold developed in the web of the beam at the place close to the flange is called:
a.
Web buckling
b.
Web stiffening
c.
Web crippling
d.
Distortional buckling
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
Cross-section which can form plastic moment of resistance but inadequate plastic hinge rotation capacity to develop plastic mechanism due to local buckling are categorized as ________ sections.
a.
Slender
b.
Semi-compact
c.
Plastic
d.
Compact
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
In activated sludge process, which of the following equation is valid?
Where
$Q$ = influent sewage flow rate
$Q_r$ = returned sludge flow rate
$x$ = concentration of MLSS in aeration tank
$x_r$ = concentration of MLSS in returned sludge
a.
$\dfrac{Q_r}{Q} = \dfrac{x}{x_r - x}$
b.
$Q_r x = Q(x_r - x)$
c.
$\dfrac{Q_r}{1+Q} = \dfrac{x}{x_r + x}$
d.
$\dfrac{1+Q}{Q_r} = \dfrac{x}{x_r + x}$
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
In an old map, a line AB was drawn to a magnetic bearing of 8°30'. The magnetic declination at the time being 2° East. To what magnetic bearing should the line be set now if the present magnetic declination is 8°30' East.
a.
2° (Anticlockwise direction)
b.
2° (Clockwise direction)
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
A vehicle moving at 60 kmph on a bituminous dry surface is suddenly brought to rest by braking. The co-efficient can be assumed to be 0.5. Calculate the distance over which the vehicle comes to a stop:
a.
30.12 m
b.
28.3 m
c.
2.83 m
d.
0.3 km
Correct Answer & Step-by-Step Solution Locked
Sign in free to reveal the correct answer and AI explanation for this question.
