Cfe Iat-1 (Question Bank)
Cfe Iat-1 (Question Bank)
Cfe Iat-1 (Question Bank)
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.
11.Find the canonical form of the quadratic form 𝑄 = 2𝑥𝑦 + 2𝑦𝑧 − 2𝑧𝑥.
𝑑𝑦
8. If 𝑦 = √sin 𝑥 + √sin 𝑥 + √sin 𝑥 + ⋯ . Then find
𝑑𝑥
Part B
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
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).
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