The integral \(\int_{0}^{1}\frac{dx}{\sqrt{(1-x)}}\) is equal to ________
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Solution
\(\int_{0}^{1}\frac{dx}{\sqrt{(1-x)}}=\int_{0}^{1}(1-x)^{-1/2}\, dx\)
=\(\left [ \frac{(1-x)^{-\frac{1}{2}+1}}{-\frac{1}{2}+1} \right ]_{0}^{1}=[-2\sqrt{1-x}]\)
=\([-2\sqrt{1-1}-\sqrt{1-0}]=2\)
The probability of getting a “head” in a single toss of a biased coin is 0.3. The coin is tossed repeatedly till a “head”is obtained. If the tosses are independent, then the probability of getting “head” for the first time in the fifth toss is __________
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Solution
P(H)=0.3 (Given)
∴P(T)=1 – P(H) = 0.7
Required probability=P(T) P(T) P(T) P(T) P(H)
= (0.7)4. (0.3)
= 0.07203
How many distinct values of x satisfy the equation sin(x) = x/2,where x is in radians ?
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Solution
Let y = sin(x) = x/2
⇒y =x/2 and y =sin(x)
The curves y= sin x and y=x⁄2 touch each other at 3 points(as shown in the figure).
Consider the plot of f (x) versus x as shown below.
Suppose F(x) =\(\int_{-5}^{x}\)f(y)dy Which one of the following is a graph of F (x) ?
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Solution
Given function is a combination of ramp and step function.
Given : F(x) =\(\int_{-5}^{x}\) f(y)dy as integration of a ramp function is parabolic function and integration of a step function is a ramp function.
Which one of the following is aproperty of the solutions to the Laplace equation: ∇2 f = 0?
The bearing length of a seat angle in the beam to column connection is given by
A precast concrete pile of size 40 cm × 40 cm is driven into stiff clay (Cu = 80 KN/m2). Determine the length of pile required to carry a safeload of 300 KW with a factor of safety 3. Take adhesion factor as 0.5.
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Solution
Qu= Qup+ Qf
= C Nc Ab+ αCAs
= (80 ×9 × .4 × 0.4) +(0.5 × 80 × 0.4 × l× 4)
= 115.2 + 64 l
Qu= 300 × 3900
= 115.2 + 64 l
l= 12.26 m
Match (List I) and (List II) and select the correct answe using codes given below:
List I List II
A.Steel pipe 1.House plumbing
B.Cast Iron pipe 2.Hot water carrying
C.GI pipe 3.Distribution main
D.PVC pipe 4.Pumping main ABCD
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Solution
Steel pipe -used in pressure sopumping main .
PVC pipes are used in house plumbing.
Cast Iron pipes are black in colour, large in diameter and used in distribution mains.
Galvanised Iron pipes are for carrying hot water.
If the turbine having a flow of 1.2m3/sec the net head is 150m . If the efficiency of the turbine is 80% the shaft power developed in KW is
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Solution
Power developed = ηγQH
= 0.8 × 9.81 × 1.2 × 150
= 1412.64 KW
A frustum of cone of base radius R = 100 cm, top radius
r = 30 cm,and height H = 3 m.
Volume of the frustum is ________m3.
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Solution
Volume of frustum =\(\frac{\pi 4(R^{2}+r^{2}+rR)}{3}\)
=\(\frac{\pi (300)(100^{2}+30^{2}+100\times 30)}{3}\)
= 4366813.788 cm3.
= 4.36m3.