Engineering Mathematics Notes pdf in English 2023

Engineering Mathematics Notes pdf in English 2023

Engineering Mathematics Notes pdf in English 2023

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Engineering Mathematics Notes pdf in English 2023 :-  We all know that math subject is the most important subject for our all government job exams so today, we are sharing easy and important the Engineering Mathematics Notes pdf in English 2023 Free for your math exams score improvement. This Engineering Mathematics Notes pdf in English 2023 for competitive exams pdf free download Download for the preparation of upcoming government exams like SSC CGL, BANK, RAILWAYS, RRB NTPC, LIC AAO, and many other exams. Math Book pdf Download For Competitive Exams 2022-23.

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Engineering Mathematics Questions and Answers 

Q 1. The pth derivative of a qth degree monic polynomial, where p, q are positive integers and 2p4 + 3pq3⁄2 = 3q3⁄2 + 2qp3 is given by?
a) Cannot be generally determined
b) (q – 1)!
c) (q)!
d) (q – 1)! * pq
Answer: c
Explanation: First consider the equation
2p4 + 3pq3⁄2 = 3q3⁄2 + 2qp3
After simplification, we get
(2p3 + 3q1⁄2) (p – q) = 0
This gives us two possibilities
2p3 = – 3q1⁄2
OR
p = q
The first possibility can’t be true as we are dealing with positive integers
Hence, we get
p = q
Thus the pth derivative of any monic polynomial of degree p(p = q) is
p! =q!.
Q 2. The first and second derivatives of a quadratic Polynomial at x = 1 are 1 and 2 respectively. Then the value of f(1) – f(0) Is given by?
a) 3⁄2
b) 1⁄2
c) 1
d) 0
Answer: d
Explanation: Let the quadratic polynomial be
f(x) = ax2 + bx + c
The first derivative at x = 1 is given by
2a + b = 1
Now consider the second derivative at x = 1 which is given by
2a = 2
Solving for the coefficients using equations, we get the values as a = 1 and b = -1
Putting these values back in the polynomial yields
f(x) = x2 – x + c
Now the required value can be computed as
f(1) – f(0) = (12 – 1 + c) – (02 – 0 + c)
= (0 + c) – (0 + c) = 0.
Q 3. Let f(x) = sin(x)x–54, then the value of f(100)(54) is given by?
a) Undefined
b) 100
c) 10
d) 0
Answer: a
Explanation: The key here is to expand the numerator into a taylor series centered at 54
Doing this gives us the following
Sin(x)=Sin(54)+(x−54)×Cos(54)1!−(x−54)2×Sin(54)2!…∞
Now the function transforms into f(x)=Sin(54)(x−54)+Cos(54)1!−(x−54)×Sin(54)2!…∞
Q 4. Let f(x) = ln(x2 + 5x + 6) then the value of f(30)(1) is given by?
a) (29!)(1330+1430)
b) (-29!)(1330+1430)
c) (30!)(1330+1430)
d) (-30!)(1330+1430)
Answer: b
Explanation: Given function
f(x) = ln(x2 + 5x + 6)
Factorising the inner polynomial we get
f(x) = ln((x + 3) (x + 2))
Now using the rule of logarithms ln (m * n) = ln(m) + ln(n)
we get
f(x) = ln(x + 3) + ln(x + 2)
Now using the nth derivative of logarithmic function dn(ln(x+a))dxn=(−1)n+1×(n−1)!(x+a)n
We have
dn(f(x))dxn=dn(ln(x+3))dxn+dn(ln(x+2))dxn
Which simplifies to dn(f(x))dxn=(−1)n+1×(n−1)!(x+3)n+(−1)n+1×(n−1)!(x+2)n
Substituting x=1 and n=30
Gives us the answer as (-29!)(1330+1430)

Engineering Mathematics Questions and Answers 

Q 5. f(x) = ∫π/20sin(ax)da then the value of f(100)(0) is?
a) a(100) sin(a)
b) -a(100) sin(a)
c) a(100) cos(a)
d) 0
Answer: d
Explanation: First solve the integral f(x)=∫π/20sin(ax)da
Which gives =[−cos(ax)x]π/20=1−cos(πx2)x
Now expanding into Taylor series yields
=((πx2)2×2!−(πx2)4×4!….∞)
Observe that every term in the expansion is odd powered
Hence even derivative at x = 0 has to be 0.
Check this: Numerical Methods MCQ | Engineering Mathematics Books
Q 6. Let f(x) = x9 ex then the ninth derivative of f(x) at x = 0 is given by?
a) 9!
b) 9! * e9
c) 10!
d) 21!
Answer: a
Explanation: The key here is to expand ex as a mclaurin series and then multiply it by x9
We have ex=x00!+x11!+x22!…∞
Now multiplying it by x9 we get
f(x)=x90!+x101!+x112!…∞
This simplifies into polynomial differentiation and the ninth derivative is as follows
f(9)(x)=9!+10!×x1!+p119×x22!…∞
Substituting x = 0 gives us the result
f(9)(0) = 9!.
Q 7. The following moves are performed on g(x).
(i) Pick (x0, y0) on g(x) and travel toward the left/right to reach the y = x line. Now travel above/below to reach g(x). Call this point on g(x) as (x1, y1)
(ii) Let the new position of (x0, y0)be (x0, y1)
This is performed for all points on g(x) and a new function is got. Again these steps may be repeated on new function and another function is obtained. It is observed that, of all the functions got, at a certain point (i.e. after finite number of moves) the nth derivatives of the intermediate function are constant, and the curve passes through the origin. Then which of the following functions could be g(x)
a) y = 1–x2−−−−√
b) xy3⁄2 + y = constant
c) x9 y3⁄2 + y6 x3⁄2 = constant
d) x7 y 8 + 4y = constant
Answer: d
Explanation: The phrasing is a bit convoluted. It is asking us to convert a given function f(x) into its own composition f(f(x))
Now the set of finite number of moves asks us to find the composition of composition i.e. f(f(x)) and f(f(x)) and so on.
It also says that the intermediate composition function has the property of all nth derivatives being constant. This is possible only if the intermediate function is of the form k(x) = ax + b (Linear Function)
Given another condition that it has to pass through origin leads to the conclusion that b = 0
So the intermediate function has the form
k(x) = ax
The first possibility gives rise to
k(x) = x
Now one has to select from the options as to which curve would give k(x) = f(f(….(x)…)) = x when composed with itself a finite number of times.
This can be done by simply interchanging the position of y and x
in the options and check whether it preserves its structure.
For y = 1–x2−−−−√ we have after changing the position
This has the same structure.

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Engineering Mathematics Questions and Answers 

8. The first, second and third derivatives of a cubic polynomial f(x) at x = 1, are 13, 23 and 33 respectively. Then the value of f(0) + f(1) – 2f(-1) is?
a) 76
b) 86
c) 126
d) 41.5
Answer: d
Explanation: Assume the polynomial to be of the form f(x) = ax3 + bx2 + cx + d
Now the first derivative at x = 1 yields the following equation
13 = 1 = 3a + 2b + c
The second derivative at x = 1 yields the following expression
23 = 8 = 6a + 2b
The third derivative at x = 1 yields the following equation
33 = 27 = 6a
Solving for a, b and c simultaneously yields
(a, b, c) = (9⁄2, -19⁄2, 13⁄2)
Hence the assumed polynomial is f(x) = 9×3 – 19×2 + 13x ⁄ 2 + d
Now the given expression can be evaluated as
f(0) + f(1) – 2f(-1) = (d) + (3⁄2 + d) – 2(-20 + d)
= 40 + 3⁄2
= 41.5.
Q 9. Let g(x) = ln(x) ⁄ x – 1 Then the hundredth derivative at x = 1 is?
a) 100!⁄101
b) 99!⁄101
c) 101⁄100!
d) 1⁄99!
Answer: a
Explanation: The key here is to again expand the numerator as a Taylor series centered at x = 1
Hence we have the Taylor series as
ln(x)=(x−1)1−(x−1)22+(x−1)33…..∞
Hence our function g(x) transforms into
g(x)=1−(x−1)2−(x−1)23+(x−1)34…..∞
This now simplifies into polynomial differentiation and the hundredth derivative can be written as
g(100)(x)=100!101−101!×(x−1)102…..∞
Substituting x = 1 yields
100!⁄101.

Engineering Mathematics Questions and Answers 

Q 10. Let f(x) = ln(x3 – 3×2 – 16x – 12) , then the 1729th derivative at x = 234 is?
a) (1728!)×(12281729+12361729+12351729)
b) (-1728!)×(12281729+12361729+12351729)
c) (1728!)×(12281729+12361728+12351729)
d) (-1729!)×(12281729+12361729+12351729)
Answer: a
Explanation: The function can be written as f(x) = ln((x – 6)(x + 1)(x + 2))
Using property of logarithms we have
f(x) = ln(x + 1) + ln(x + 2) + ln(x – 6)
Using the nth derivative of logarithmic functions
dn(ln(x+a))dxn=(−1)n+1×(n−1)!(x+a)n
We have the 1729th derivative of f(x) as
f(1729)(x)=1728!(x−6)1729+1728!(x+2)1729+1728!(x+1)1729
Now substituting x=234 we get our answer as
= (1728!)×(12281729+12361729+12351729).
Q 11. Find the value of S=∑∞n=1(−1)n+1×(2n−1)3(2n−1)! using nth derivatives.
a) – 2 * sin(1)
b) 3 * sin(1)
c) 3 * cos(1)
d) – 3 * cos(1)
Answer: a
Explanation: We have to consider the function f(x) = sin(ex) in order to get the series in some way.
Expanding the given function into a Taylor series we have
f(x)=ex1!−e3x3!+e5x5!…∞
Now observe that our series in question doesn’t have the exponential function, this gives us the hint that some derivative of this function has to be taken at x = 0
Observe that the term (2n – 1)3 has exponent equal to 3
Hence we have to take the third derivative of the function to get the required series
Now taking the third derivative yields
f(x)=ex1!−33e3x3!+55e5x5!…∞
Now substituting x=0 we get
131!−333!+535!…∞=∑∞n=1(−1)n+1×(2n−1)3(2n−1)!
To find the value of this series we need to take the third derivative of original function at the required point, this is as follows
f(3)(x) = -2exsin(ex)
Substituting x = 0 we get
f(3)(0) = -2sin(1).
Q 12. Let f(x)=ln(1−x)ex. Find the third derivative at x = 0.
a) 4
b) 1⁄3
c) Undefined
d) 1⁄4
Answer: b
Explanation: Again the key here is to expand the given function into appropriate Taylor series.
Rewriting the function as f(x) = e-x(ln(1 – x)) and then expanding into Taylor series we have
f(x)=(1−x1!+x22!−x33!…∞)×(x1+x22+x33…∞)
Now the question asks us to find the third derivative at x = 0. Thus, it is enough for us to find the coefficient of the x3 term in the infinite polynomial product above
The third degree terms can be grouped apart as follows
= x3⁄3 – x3⁄2 + x3⁄2
Hence the third derivative at x = 0 is simply the coefficient of the third degree term, which is
coefficient(x3⁄3) = 1⁄3.

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