Model Question Bank | ENGINEERING MATHEMATICS – I | Diploma 2020

Model Question Bank | ENGINEERING MATHEMATICS – I | Diploma 2020



UNIT-1: MATRICES AND DETERMINANTS

 

3 MARK QUESTIONS

1.  

]

 
IfA = [ 3           −9 , find A + A.

−4         7

5     −2

2.   If A = [2         −1      3] and B = [3              1 ] , find AB matrix.

2       4

2     −1       3

3.   If matrix A=[5

1

0] is a singular matrix, then find the value of x.

1

0

x


4.     Find the adjoint of the matrix


4     −5 .


A = [                 ]


 

5.     If


3     −1


3      −2

find the characteristic equation.


A = [                 ]

0     −2

 

 

5  MARK QUESTIONS

 

1.  


Solve the equations x + y = 3, 2x + 3y = 8 by Cramer’s rule.

2.     Solve for x, if


3.   Verify Cayley-Hamilton theorem if  A = [1           3  ].

2     −4


4.     VerifyA(AdjA) =  |A|. I. if


5     −2 .


A = [                 ]

3       1

 

3

−1

2

5. Find the adjoint of the matrix A = [2

−3

1]

0

4

2

6  MARK QUESTIONS

 

1.    Solve for x &y from the equations 4x + y = 7, 3y + 4z = 5, 5x + 3z = 2by Cramer’s rule.

1

2

2

2. Find the inverse of the matrix [−1

3

0]

0

−2

1

3.    Prove that adj(AB)=(adjB).(adjA) if A = [−1            0                                                                                             3                                                                                             5

5          ] and B = [        ]

3              2              4


4.    Find the characteristic roots of a matrix[ 1         1].

−6       −2

5.    Solve the equations by Gauss elimination method 3x y + z = 0, x + 2y 2z =

3, 3x + z = 4. UNIT-2: VECTORS 3 MARK QUESTIONS

1. Find the magnitude of vector 2i +3j – 6k

2.   If a = i + 2j 3k, b¯ = 3i 5j + 2k find magnitude of ¯3¯¯¯a ¯2¯¯¯b

3.   Show that cos 8i sin 8j is unit vector

4.   Show that the vectors 2i + 5j 6k,and 7i + 2j + 4k orthogonal vectors.

5.   If  a = 5i + 2j 4k, and  b¯ = 2i 5j + 3kfind aXb¯ .

 

5 MARK QUESTIONS

 

1.                             Find cosine of the angle between the vectors 4i 2j 3kand 2i 3j + 4k.

2.                             Find the projection of b¯ on a if a = 5i + 2j 4kand b¯ = 2i 5j + 6k.

3.                             If a = 3i + 2j 4k and b¯ = i 2j + 5k are two sides of a triangle, find its area. 4.   Simplify (a + b¯). (a b¯)and (a + b¯)X(a b¯).

5. Find the magnitude of moment of force 4i 2j + 5k about (2,5,-7) acting at (4,7,0) 6 MARK QUESTIONS

1.           If A=(2,5,7), B=(3,9,4) and C=(-2,5,7) are three vertices of parallelogram find its  area.

2.           If a force 4i + 6j + 2k acting on a body displaces it from (2,7,-8) to (3,9,4). Find the work done by the force.

3.           Find the sine of the angle between the vectors 4i 2j 3kand 2i 3j + 4k.

4.           Find the unit vector in the direction perpendicular to both vector 2i 5j + kand 5i + j + 7k.

5.           Show that the points whose position vectors are i 3j 5k, 2i j + k and 3i 4j 4k form a right angled triangle.

 

 

UNIT-3: PROBABILITY AND LOGARITHMS

 

3 MARK QUESTIONS

 

1.       Define equally likely events, Independent event, and mutually exclusive event.

2.       Define probability of an event.

3.       A coin is tossed twice. What is the probability that at least one head occurs.

4.       A die is thrown once, what is the probability an odd number appears.

5.       If E and F are events such that P(E)=0.6, P(F)=0.3 and P(E∩F)= 0.2. Find P(E/F).


5  MARK QUESTIONS

1.        Prove that         1

1+logc ab


+         1

1+loga bc


+         1            = 1

1+logb ca


2.        If x = logc ab , y = logb bc , z = loga ca , Prove that xyz = x + y + z + 2

3.        If x = log2a a , y = log3a 2a , z = log4a 3a , prove that xyz + 1 = 2yz


4.        If a2 + b2 = 7ab, prove that log (a+b) = 1

3              2


(log a + log b)


5.        Solve for x given that (log2 x)2 + (log2 x) 20 = 0

 

6  MARK QUESTIONS

 

1.        An integer is chosen at random from the numbers ranging from 1 to 50 . What is the probability that the integer chosen is a multiple of 3 or 10 ?

2.        Two unbiased dice are thrown once . Find the probability of getting the sum of the numbers obtained on the two dice is neither a multiple of 2 nor a multiple of 4 .

3.        One card is drawn from a well shuffled pack of 52 cards. If E is the event “the card drawn is a king or an ace” and F is the event “ the card drawn is an ace or a jack “  then find the conditional probability of the event E, when the event F has already occurred .

4.        A pair of dice is thrown once. If the two numbers appearing on them are different, find the probability that the sum of the numbers is 6.

5.        A family has two children. What is the probability that both the children are boys given that (i) the youngest is a boy. (ii) at least one is a boy ?

 

 

UNIT-4: ALLIED ANGLES AND COMPOUND ANGLES

 

ALLIED ANGLES

3 MARKS QUESTIONS

 

1.     Find the value of

 

2.     Find the value of

 

3.     3.If sin           and                   , find cos

 

4.     4. If A+B+C =1800  Prove that     cot

5.     5.find the value of tan


5 MARKS QUESTIONS

 

 

1.     Prove that                                                                     =1

 

2.      If secx = 13/5 and 2700                    , Find the value of

3.     Find the value of

 

4.     Evaluate

 

5.     Show that tan2250xcot4050+tan7650xcot6750+cosec1350xsec3150 = 0 6 MARK QUESTIONS

1 .Evaluate tan3150xcot4050+tan7650xcot6750+cosec1350xsec3150

2.     Find x if

3.     If sin   , find the value of

 

4.     Evaluate

 

 

5.     Show that

 

COMPOUND ANGLES

 

3 MARKS QUESTIONS

 

1.               Find the value of sin150

 

2.               Show that

 

 

 

3.               Prove that

4.                Using tan(A+B), prove that cot(A+B)=

 

5.                Prove that


5 MARKS QUESTIONS

 

1.   Prove that cos(A-B) cos(A+B)= cos2A-sin2B

2.   Show that

3.   If sinA=

 

4.   Prove that tan3

5.   If A+B =

 

 

TRASFORMATION FORMULAE

 

3 MARKS QUESTIONS

 

1             P.T

2             P.T

3             Show that

4             Show that

5              Show that

 

MARKS QUESTIONS

 

1             P.T

2             In and triangle ABC prove that tanA + tanB +tanC = tanA tanB tanC

3             Show that

4             Prove that

5              Prove that

MARKS QUESTIONS

 

1         Prove that cos200xcos400xcos800xcos600= 1/16

2         In any triangle ABC prove that sinA + sinB + sinC=4Cos(A/2)cos(B/2)cos(C/2)

 

3         Show that

4         If A+B+C = 1800 prove that


5         If A+B+C = 1800 prove that sin2A-sin2B+sin2C=4cosAcosCsinB

 

 

UNIT-5: COMPLEX NUMBERS

 

3 MARK QUESTIONS

 

1.                              Evaluate i–999

2.                              Find the complex conjugate of (1 + 2i)(3i 4)

3.                              Express (3 + 4i)–1 in the form a+ib

4.                              Find the real part and imaginary part of 1

2 + i


5.                              ifx + iy = cos 8 + i sin 8 show that


x + 1

s


= 2 cos 8


 

5 MARK QUESTIONS

 

1 25 2

1.  Evaluate (i19 + ( )              )


i

2.  Find the modulus and amplitude of(1  i3)

3.  Express in a + ib form:      (2+3i)

(1+3i).(2+i)

4. 


Express the complex number 1 + i in the polar form.

5.  Find the amplitude of 3 + i and represent in Argand diagram.

 

UNIT-6: INTRODUCTION TO CALCULUS

 

3 MARK QUESTIONS

 


1.                            Evaluate: lim


s2–9


 

2.                            Evaluate:


s→–3 s+3

tan mθ

 


lim (

8→0


)

sin n8


3.                            Evaluate:


n+1 n

lim ( ) .


 

4.                            Evaluate:


n→        n

3s2–2s+1

 


lim (        2                    )


5.                          Evaluate:


s→


2s +5s–1

1–cos 2s)

 


lim (

s→0             x2

 

5  MARK QUESTIONS

 

1.   Evaluate:lim s2+s–2.


s→1


s2–1

 

                       


2.   Evaluate:  lim (a+sa–s)

s→0                3s

3.   Evaluate: lim (xm–1)


x→1


xn–1


4.   Evaluate:


1–cosx+ tan2 x

lim (                             )


8→0


scins


5.   Evaluate: lim (eax–ebx).

s→0              x

 

6  MARK QUESTIONS

1.   Prove that lim sin8 = 1, if θ is in "radian".

8→0 8

2.   Evaluate: lim (sin πx)


s→0


s–1


3.   Evaluate: lim (          (5–n2)(n–2)              ).


n→

4.   Evaluate: lim


(2n–3)(n+3)(5–n)

s2–5s+4 .


s→1


s2–12s+11


5.   Evaluate: lim (            s2–4            )

s2    s+23s–2

 



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