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
