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Cfe Iat-1 (Question Bank)

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KNOWLEDGE INSTITUTE OF TECHNOLOGY (AUTONOMOUS),SALEM

DEPARTMENT OF MATHEMATICS
BE23MA201 / CALCULUS FOR ENGINEERS (Regulation 2023)
IAT-I (QUESTION BANK)
UNIT I – MATRICES
Part – A
1 1 3
1. Find the sum and product of all the Eigenvalues of 𝐴 = [ 1 5 1].
3 1 1
2. If the sum of two Eigenvalues and trace of 3x3 matrix A are equal , find the value of
|𝐴|.
8 −6 2
3. Two eigenvalues of the matrix 𝐴 = [−6 7 −4] are 3 and 0. What is the third
2 −4 3
Eigenvalue?
6 −2 2
4. The product of two Eigenvalues of the matrix 𝐴 = [−2 3 −1] is 16. Find the
2 −1 3
third eigenvalue of A .
2 0 1
5. If 2, 3 are the Eigenvalues of [ 0 2 0] then find the value of b.
𝑏 0 2
𝑎 4
6. Find the constants 𝑎 and 𝑏 such that the matrix [ ] has 3 and -2 as its Eigen
1 𝑏
values.
7. State Cayley- Hamilton theorem and write its uses.
1 2
8. Verify that [ ] satisfies its own characteristic equation.
2 −1
𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 0
9. Check whether the matrix B is orthogonal? Justify. 𝐵 = [−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 0].
0 0 1
2 2 2
10.Find the Index and Signature of the Q.F 𝑥 +𝑦 -3𝑧 .
11.Write down the matrix of the quadratic form 2𝑥 2 + 8𝑧 2 + 4𝑥𝑦 + 10𝑥𝑧 − 2𝑦𝑧.
2 2 2
12.Prove that the quadratic form 𝑥 + 2𝑦 + 3𝑧 + 2𝑥𝑦 + 2𝑦𝑧 − 2𝑧𝑥 is indefinite.

PART-B
2 −2 2
1. Find the Eigenvalues and Eigenvectors of the matrix 𝐴 = [1 1 1 ].
1 3 −1
1 0 0
2. Find the Eigenvalues and Eigenvectors of the matrix 𝐴 = [0 3 −1 ].
0 −1 3
2 −1 2
3. Show that the matrix [−1 2 −1] satisfies its own characteristic equation and
1 −1 2
hence find its inverse.

4. Using Cayley Hamilton theorem evaluate 𝐴8 -5𝐴7 +7𝐴6 -3𝐴5 +𝐴4 -5𝐴3 -8𝐴2 +2A-I
2 1 1
for [0 1 0].
1 1 2
2 −1 1
5. Find the characteristic equation of the matrix A given 𝐴 = [−1 2 −1]
1 −1 2
−1 4
.Hence find 𝐴 and 𝐴 .

6. Use Cayley – Hamilton theorem to find the value of the matrix given
by 𝐴8 − 5𝐴7 + 7𝐴6 − 3𝐴5 + 𝐴4 − 5𝐴3 + 8𝐴2 − 2𝐴 + 𝐼 , if the matrix 𝐴 =
2 1 1
[0 1 0].
1 1 2
8 −6 2
7. If the Eigen values of [−6 7 −4] are 0, 3, 15. Find the Eigen vectors of A
2 −4 3
and diagonalize the matrix A.
8. Reduce the quadratic form 2𝑥 2 + 6𝑦 2 + 2𝑧 2 + 8𝑥𝑧 into a canonical form by an
orthogonal transformation and discuss its nature, Rank , Index and Signature.

9. Reduce the quadratic form 6𝑥 2 + 3𝑦 2 + 3𝑧 2 − 4𝑥𝑦 − 2𝑦𝑧 + 4𝑥𝑧 into a canonical


form by an orthogonal transformation and discuss its Rank , Nature, Index and
Signature.

10.Reduce the quadratic form 3𝑥 2 + 5𝑦 2 + 3𝑧 2 − 2𝑥𝑦 + 2𝑦𝑧 − 2𝑧𝑥 into a canonical


form by an orthogonal reduction. Hence find its Rank and Nature.

11.Find the canonical form of the quadratic form 𝑄 = 2𝑥𝑦 + 2𝑦𝑧 − 2𝑧𝑥.

UNIT – II (DIFFERENTIAL CALCULUS)


PART A
𝑥 2 −1
1.Find lim
𝑥→1 𝑥−1
2. Find lim [3𝑥 2 + 2𝑥 2 − 𝑥 + 1]
𝑥→−2
3
3. Find the domain and range of 𝑓(𝑥) =
7−𝑥
4. Find the domain and range of 𝑓(𝑥) = √5𝑥 + 10
5. Show that the function 𝑓(𝑥) = 1 − √1 − 𝑥 2 is continuous on the interval [-1,1]
6. Check whether the function is even or odd
𝑓(𝑥) = 2𝑥 + 𝑥 2
𝑑
7. Find [𝑥 2 sin 𝑥]
𝑑𝑥

𝑑𝑦
8. If 𝑦 = √sin 𝑥 + √sin 𝑥 + √sin 𝑥 + ⋯ . Then find
𝑑𝑥

9. Find the critical values of the function 𝑓(𝑥) = 5𝑥 3 − 6𝑥


2
10. Find the critical values of the function 𝑓(𝑥) = 𝑥 2 +
𝑥
1
11. Find the absolute maximum and minimum of 𝑓(𝑥) = (3𝑥 − 1)
2
1
12. Find the absolute maximum and minimum of 𝑓(𝑥) =
𝑥

Part B

1. Find the values of a and b that make f continuous on ( −,  ) .


 x3 − 8
 ,
if x  2
 x − 2

f ( x) = ax 2 − bx + 3, if 2  x  3
 2 x − a + b, if x  3


2. For what value of the constant “c” is the function “f” continuous on
cx 2 + 2 x; x  2
( −,  ) , f ( x) =  .
 x − cx ; x  2

3

3. Find the local maximum and minimum values of the function


( i ) f ( x ) = 2 x3 + 5x 2 − 4 x ( ii ) f ( x ) = x 4 − 3x3 + 3x 2 − x ( iii ) f ( x ) = x5 − 5x 4 + 5x3 + 10

4. For the function f ( x ) = 2 + 2 x 2 − x 4 , find the intervals of increase or decrease, local


maximum and minimum values, the intervals of concavity and the inflection points.

5. If f ( x ) = 2 x3 + 3x 2 − 36 x , find the intervals on which it is increasing or decreasing, the


local maximum and local minimum values of f ( x ) .

𝑑 sec 𝑥
6.Find [ ]
𝑑𝑥 1+tan 𝑥

𝑑
7. Find [𝑥 2 𝑒 2𝑥 (𝑥 2 + 1)4 ]
𝑑𝑥
8.Find the equation of the tangent line and the normal line to the curve 𝑦 = 2𝑥𝑒 𝑥 at
origin.

9.Find the equation of the tangent line and normal line to the curve 𝑦 = 3𝑥 2 − 𝑥 3 at
(1,2).

10.Discuss the curve 𝑦 = 𝑥 4 − 4𝑥 3 with respective to concavity points of inflection


and local maxima and minima.

11. Find the Maxima and Minima of 𝑓(𝑥) = 𝑥 4 + 2𝑥 3 − 3𝑥 2 − 4𝑥 + 4


by using first derivative test and second derivative test.

12. Discuss the curve 𝑦 = 2𝑥 3 + 3𝑥 2 − 36𝑥 with respective to concavity points of


inflection and local maxima and minima.

UNIT-III (ORDINARY DIFFERENTIAL EQUATION)

PART A
1. Solve (𝐷2 + 𝐷 + 1) 𝑦 = 0
2. Solve [−3𝐷2 + 4𝐷 − 2]𝑦 = 0
3. Solve (𝐷3 + 3𝐷2 + 3𝐷 + 1)𝑦 = 0
4. Find the particular integral of (𝐷2 + 6𝐷 + 9)𝑦 = 5𝑒 3𝑥
5. Find the particular integral of (𝐷2 + 4)𝑦 = 𝑠𝑖𝑛2𝑥
6. Find the particular integral of (𝐷2 − 4𝐷 + 4)𝑦 = 2𝑥
7. Find the particular integral of (𝐷2 + 2𝐷 + 1)𝑦 = 𝑒 −𝑥 𝑥 2
8. Find the particular integral of (𝐷2 − 4𝐷 + 4)𝑦 = 8sin 2𝑥.
PART B
1. Solve (𝐷2 + 2𝐷 + 2)𝑦 = 𝑒 −2𝑥
+ 𝑐𝑜𝑠2𝑥
2. Solve (𝐷2 − 3𝐷 + 2)𝑦 = 2𝑥 2 + 1
3. Solve (𝐷2 − 1)𝑦 = 𝑥
4. Solve (𝐷2 − 4𝐷 + 3)𝑦 = 𝑒 𝑥 𝑐𝑜𝑠 2𝑥
5. Solve (𝐷2 − 2𝐷 + 5)𝑦 = 𝑒 2𝑥 𝑠𝑖𝑛𝑥
6. Solve (𝐷2 + 4𝐷 + 3)𝑦 = 𝑒 3𝑥 𝑥
7. Solve (𝐷2 − 3𝐷 + 2)𝑦 = (1 + 𝑥)𝑒 𝑥
8. Solve (𝐷2 − 1)𝑦 = 𝑐𝑜𝑠𝑥 𝑐𝑜𝑠ℎ𝑥

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