### 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

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 