KCG Mathematics Question Bank 2023 PDF

Department of Science and Humanities (Mathematics) 23MA101- MATRICES AND CALCULUS First Semester B.E./B.Tech. (Common to ALL BRANCHES) QUESTION BANK (Regulation 2023) UNIT I...

Department of Science and Humanities (Mathematics) 23MA101- MATRICES AND CALCULUS First Semester B.E./B.Tech. (Common to ALL BRANCHES) QUESTION BANK (Regulation 2023) UNIT I MATRICES Q.No Part - A CO’s Bloom’ s Level 1. If  is an eigenvalues of A , then prove that 2 is an Eigen value of A 2. CO1 K2  8 6 2  CO1 K2   Two eigenvalues of the matrix   6 7  4  are 3 and 15. What is the third 2.  2 4 3    eigenvalue? What is the product of eigenvalues of A ? If the sum of the two eigenvalues and trace of 3 3 matrix A are equal, find the CO1 K2 3. value of A. 4. State Cayley Hamilton theorem and what are its uses? CO1 K1 7 √6 CO1 K2 5. Given the elastic deformation Y=AX, 𝐴 = ( ) , Find the scalars of extension √6 2 or contraction If 2,-1,-3 are eigenvalues of A, find the eigenvalues of CO1 K2 6. If the eigenvalues of a matrix A of order 3 3 are 1,2& 3, Then find the eigenvalues CO1 K2 7. of adjoint of A.  5 4 CO1 K2 8. Find the eigenvalues of 3 A +2I where A =    0 2 Find the matrix corresponding to the quadratic form 2 x1  5 x 2  4 x1 x 2  2 x3 x1 CO1 K2 2 2 9. CO1 K2 10. Write the quadratic form corresponding to the matrix form 11. Find the nature of the quadratic form x 2  y 2  z 2 in four variables? CO1 K2 3 1 4 CO1 K2 12. Find the eigenvalues of the inverse of the matrix 𝐴 = (0 2 6) 0 0 5  1 0 0  CO1 K2   13. Find the nature of the quadratic form whose matrix is  0  1 0   0 0  2    a 4 CO1 K2 14. Find the constants a & b such that the matrix   has 3 & -2 as its eigenvalues  1 b   1 4 CO1 K2 15. If A =   then find 2A2 -8A-10I where I is the unit matrix.  2 3 ,  Part - B 1 1 3 (i)Find the eigenvalues and eigenvectors of the matrix (1 5 1) (8 M) CO1 K3 3 1 1 1. 2 −1 1 (ii) Using Cayley – Hamilton theorem find A1 of (−1 2 −1) (8 M) 1 −1 2 2 0 1   (i)Find the eigenvalues and eigenvectors of the matrix  0 2 0  (8 M) CO1 K3  1 0 2 2.   (ii) Find the symmetric matrix A whose eigenvalues are 2,3,6 and eigenvectors are respectively (8 M) −2 2 −3 (i) Find eigenvalues and eigenvectors of ( 2 1 −6) (8 M) CO1 K3 −1 −2 0 3. (ii) Verify Cayley Hamilton theorem for A= and hence find (8 M) 2 1 −1 4. Diagonalise the matrix 𝐴 = ( 1 1 −2) by orthogonal transformation.(16 M) CO1 K3 −1 −2 1 3 1 1 5. Diagonalise the matrix 𝐴 = (1 CO1 K3 3 −1) by orthogonal transformation. (16 M) 1 −1 3 Reduce the quadratic form 6 x1  3x 2  3x3  4 x1 x 2  2 x 2 x3  4 x3 x1 to canonical 2 2 2 6. form by an orthogonal reduction. Hence find its rank, index, signature and nature. CO1 K3 (16 M) Reduce the quadratic form to canonical form by an orthogonal 7. reduction and hence find its rank, index, signature and nature. (16 M) CO1 K3 Reduce the quadratic form 8. to canonical form by an orthogonal reduction and find its nature, rank, index and CO1 K3 signature. (16 M)  UNIT II DIFFERENTIAL CALCULUS CO’ Bloom’ Q.No Part - A s Level s 2𝑥 3 −5 CO2 K2 1. Find the domain of 𝑓(𝑥) = 𝑥 2 +𝑥−6 CO2 K2 2. Find the domain and range of the function 𝑓(𝑥) = √𝑥. 1+𝑥 CO2 K2 3. Determine the domain of the function 𝑓(𝑥) = log 𝑒 (1−𝑥) 𝑥 2 −9 CO2 K1 4. Give the value of lim 𝑥→3 𝑥−3 Find the value of lim √𝑢4 + 3𝑢 + 6. CO2 K2 5. 𝑢→−2 𝑒 𝑎𝑥 −𝑒 𝑏𝑥 CO2 K2 6. Evaluate lim. 𝑥→0 𝑥 Test whether the function |𝑥| is differentiable at 𝑥 = 0. CO2 K2 7. Determine whether the given function 𝑓(𝑥) = 𝑥 5 + 𝑥 is even or odd. CO2 K2 8. Write the condition for a function 𝑓(𝑥) to be increasing or decreasing on an interval 𝐼 CO2 K1 9. 𝑒 𝑥 , 𝑖𝑓 𝑥 < 0 CO2 K2 10. Test the continuity of 𝑓(𝑥) = { 2 𝑥 , 𝑖𝑓 𝑥 ≥ 0 Find the point of inflection of 𝑓(𝑥) = 𝑥 3 − 9𝑥 2 + 7𝑥 − 6 CO2 K2 11. 𝑑𝑦 1+sin 𝑥 CO2 K2 12. Find for 𝑦 =. 𝑑𝑥 𝑥+cos 𝑥 13. CO2 K2 Identify the points where the curve has horizontal tangents. The motion of a particle is given by S = 2t 3 − 5𝑡 2 + 3𝑡 + 4. Find the acceleration of CO2 K2 14. the particle after 2 seconds. CO2 K2 15. Find Part - B (i) Evaluate lim𝜋(𝑠𝑒𝑐 𝜃 − 𝑡𝑎𝑛 𝜃) (8 M) 𝜃→ 2 𝑥 2 −4 ,𝑥 < 2 CO2 K3 1. 𝑥−2 ( ) (ii) If 𝑓 𝑥 = {𝑎𝑥 2 − 𝑏𝑥 + 3, 2 ≤ 𝑥 < 3 is continuous for all real 𝑥, find 𝑎 and 𝑏. 2𝑥 − 𝑎 + 𝑏, 𝑥 ≥ 3 (8M)  2𝑥 − 2 𝑖𝑓 𝑥 < −1 (i) If 𝑓(𝑥) = {𝑎𝑥 + 𝑏 𝑖𝑓 − 1 ≤ 𝑥 < 1 is continuous for all real 𝑥 , the find the 5𝑥 + 7 𝑖𝑓 𝑥 ≥ 1 2. values of 𝑎 and 𝑏 (8M) CO2 K3 (ii) Find 𝑦" if 𝑥 4 + 𝑦 4 = 16 (8M) 𝑑𝑦 (i)If 𝑦 = √𝑆𝑖𝑛 𝑥 + √𝑆𝑖𝑛 𝑥 + √𝑆𝑖𝑛 𝑥 + ⋯ then find (8M) 𝑑𝑥 CO2 K3 3. 𝑑 𝑑 𝑥3 (ii) Evaluate (𝑎) (3𝑥 5 log 𝑥 ) (𝑏) ( ) (8M) 𝑑𝑥 𝑑𝑥 3𝑥−2 𝑑𝑦 Find if 𝑥 3 + 𝑦 3 = 6𝑥𝑦.Find the tangent to the equation 𝑥 3 + 𝑦 3 = 6𝑥𝑦 at the 𝑑𝑥 4. point(3,3). At what point in the first quadrant is the tangent line horizontal? (16M) CO2 K3 4 (i)Find the local maximum and minimum values of 𝑓 (𝑥) = √𝑥 − √𝑥 (8M) CO2 K3 5. 𝑒𝑥 𝑒 (ii) Find the equation of tangent line to 𝑦 = 1+𝑥 2 at (1, ) (8M) 2 𝑦 𝑥 𝑑𝑦 𝑦(𝑦−𝑥 𝑙𝑜𝑔𝑦) (i) If 𝑥 = 𝑦 , prove that 𝑑𝑥 = 𝑥(𝑥−𝑦𝑙𝑜𝑔 𝑥) using implicit differentiation (8M) (ii) Find the absolute maximum and minimum values of the CO2 K3 6. 1 function 𝑥 3 − 3𝑥 2 + 1 in the interval [− , 4] (8M) 2 For the function𝑓 (𝑥) = 3𝑥 4 − 4𝑥 3 − 12𝑥 2 + 5, find the intervals of increase or 7. decrease, local maximum and minimum values, the intervals of concavity and the CO2 K3 inflection points. (16M) For the functionf(x) = 2x 3 +3𝑥 2 − 36𝑥, find the intervals of increase or decrease, 8. local maximum and minimum values, the intervals of concavity and the inflection CO2 K3 points. (16M)