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    Question Bank TMAT101L Updated

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    Question Bank TMAT101L Updated

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    of 5

    SCHOOL OF ADVANCED SCIENCES

    DEPARTMENT OF MATHEMATICS
    FALL SEMESTER – 2024 ~ 2025
    TMAT101L – Mathematics
    A1+TA1+TAA1 / A2+TA2+TAA2 / B2+TB2+TBB2 Slot
    Question Bank for Slow Learners (Modulewise)

    Module – 1: Matrices

    Practice Problems:
     1 2  x 
    1. Find the value of x, if  2 x 3    = 0
     −3 0   3 
     4 6 1 0
    2. Solve: X + 2Y =   ; X −Y =  
     −8 10  −2 −2
     −2 1 3  4 7 0
    3. Solve: X + 2Y =  5 −7 3  , X − Y =  −1 2 −6 
       
     4 5 4   −2 8 −5
    4. Solve by matrix inversion method:
    x + y + z = 9, 2x + 5y + 7z = 52, 2x + y – z = 0
    5. Examine the consistency of the system of equations. If it is consistent then
    solve: x + y – z = 1 2x + 2y – 2z = 2, –3x + 3y + 3z = –3
    6. Solve by matrix inversion method 2x – y + 3z = 9, x + y + z = 6, x – y + z = 2
    7. Solve the non-homogeneous equations of three unknowns.
    x + y +2z = 6, 3x + y – z = 2, 4x + 2y + z = 8.
    8. Verify whether the given system of equations is consistent. If it is
    consistent, solve them. 2x - 3y + 7z = 52, x + y + z = 9, 2x + y - z = 0.

    9. Show that equation x + y + z = 6, x + 2y + 3z = 14, x + 4y + 7z = 30 are


    consistent and solve them.

    1
     − 1 2 − 2
    10. For A =  4 − 3 4  , show that A = A-1
     4 − 4 5 
    11. Find the rank of the following matrices:
     3 1 2 0 1 2 − 1 3 
    i) 1 0 − 1 0 ii) 2 4 1 − 2
    2 1 3 0 3 6 3 − 7

     1 −2 3 4  − 2 1 3 4

    iii) − 2 4 − 1 − 3 iv)  0 1 1 2
     − 1 2 7 6   1 3 4 1 
    12. Examine the consistency of the system of equations. If it is consistent then
    solve : x + y + z = 7,
    1 2 3 − 1 
    13. Find the rank of the matrix 2 4 6 − 2
    3 6 9 − 3

    3 1 − 5 − 1
    14. Find the rank of the matrix 1 − 2 1 − 5
    1 5 − 7 2 
    15. Solve the following non-homogeneous equations of three unknowns.
    (each)
    i) 2x + 2y + z = 5; x – y + z = 1; 3x + y + 2z = 4
    ii) x + y + 2z = 4; 2x + 2y + 4z = 8; 3x + 3y + 6z = 10

    Module -2 – Differential Calculus

    Practice Problems:

    1. Obtain the taylor series of f ( x) = Sin x about x = .
    2
    1
    2. Obtain the taylor series of f ( x) = about x = 2 .
    x
    1
    3. Obtain the taylor series of f ( x) = 2 about x = 2 .
    x
    4. Find the local minimum and maximum and absolute minimum values
    of
    f(x) = x4 – 3x3 + 3x2 – x
    5. Find the local minimum and maximum and absolute minimum values of
    y = x3 – 3x + 1

    2
    6. Find the local minimum and maximum and absolute minimum values of
    y = x3 – 3x + 2
    7. Find the local minimum and maximum and absolute minimum values of
    f(x) = 2x3 + 5x2 – 4x

    Module -3 – Integral Calculus

    −1 −1
    esin x
    e tan x
    1. Integrate the following: i ) ii)
    1 − x2 1 + x2
    2. Find the area between the curves y = x2 – x – 2, x – axis and the lines x = -2
    and x = 4

    3. Find the area bounded by the curve y = x3 and the line y = x.


    4. Find the area of the region enclosed by y2 = x and y = x -2
    5. Find the area of the region bounded by the curve y = 3 x2 – x and the x –
    axis between x = -1 and x = 1.
     
    2 2
    6. Evaluate i ).  log(tan x) dx ii ).  log(cot x) dx
    0 0

    7. Evaluate the following problems using properties of intergration.


     
    3 3
    dx dx
    i)  1 + tan x (ii) 
     1+ cot x
    6 6

    Module - 4 – Linear Ordinary Differential Equations

    d2y dy
    1. Solve: 2
    -3 + 2y = 0 when x = 0, y = 0 and when x = 0, y’ = 1.
    dx dx
    2. Solve: (D2 – 6D + 9) y = 0
    3. Solve: (D2 – l) y = 0
    4. Find the general solution to the differential equation:
    2
    d y dy
    2
    − 5 + 6 y = 0 and then find the particular solution that satisfies the
    dx dx
    initial conditions y (0) = 0 & y(0) = 1 .
    dy y y
    5. Solve : = + tan
    dx x x

    3
    dy
    6. Solve : + y cot x = 2 cos x
    dx
    7. Solve:
    dy
    8. Solve : + 2 y tan x = sin x
    dx

    Module – 5 and 6 – Analytical Geometry & Vector Algebra

    1. Find the shortest distance between the parallel lines


                
    r = (i − j ) + t (2i − j + k ) and r = (2i + j + k ) + s(2i − j + k )
              
    2. Show that the two lines r = (i − j ) + t (2i + k ) and r = (2i − j ) + s(i + j − k ) are
    skew lines and find the distance between them.
    3. Find the vector & Cartesian equations of the plane passing through the
    points
    (-1, 1, 1) and (1, -1, 1) and perpendicular to the plane x + 2y + 2z = 5.
    4. Find the vector and Cartesian equations of the plane passing through the
    (2, 2, -1), (3, 4, 2) and (7, 0, 6).
    x −1 y +1 z x − 2 y −1 − z −1
    5. Show that the lines = = and = = are intersect
    1 −1 3 1 2 1
    and find their point of intersection.
    6. Find the vector and Cartesian equation of the plane through the point (1,
    x +1 y + 2 z + 3 x − 2 y +1 z + 2
    3, 2) and parallel to the lines. = = and = = .
    2 −1 3 1 2 2
    Module - 7 – Logics and Probability

    1. Construct the truth table for (p  q)  (~ r)


    2. Construct the truth table for (p  q)  r
    3. Construct the truth table for (p  q)  r
    4. Show that (pq)→(pq) is a tautology
    5. Verify whether the given statement is a tautology or not.
    (( P → Q)  (Q → R)) → ( P → R )
    6. Two coins are tossed simultaneously, what is the probability of getting
    (i) exactly one head
    (ii) at least one head
    (iii) . at most one head
    7. The probability that a girl will get an admission in IIT is 0.16, the
    probability that she will get an admission in Government Medical College is

    4
    0.24, and the probability that she will get both is 0.11. Find the probability
    that (i) She will get at least one of the two seats (ii) She will get only one of
    the two seats.

    8. The probability that a new ship will get an award for its design is 0.25, the
    probability that it will get an award for the efficient use of materials is 0.35,
    and that it will get both awards is 0.15. What is the probability, that (i) it will
    get atleast one of the two awards (ii) it will get only one of the awards.

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