Mathematics II Past Paper PDF - Silver Oak University - Summer 2024

## SILVER OAK UNIVERSITY **Programme:** Diploma Engineering **Branch:** ALL **SEMESTER:** II (REG/REM) **EXAMINATION-SUMMER-2024** **Subject Code:** 1010272102 **Subject Name:** MATHEMATICS II **Time:** 10:30 AM to 12:45 PM **Date:** 05/06/2024 **Total Marks:** 60 **Instructions:** 1. Attempt al...

## SILVER OAK UNIVERSITY **Programme:** Diploma Engineering **Branch:** ALL **SEMESTER:** II (REG/REM) **EXAMINATION-SUMMER-2024** **Subject Code:** 1010272102 **Subject Name:** MATHEMATICS II **Time:** 10:30 AM to 12:45 PM **Date:** 05/06/2024 **Total Marks:** 60 **Instructions:** 1. Attempt all questions. 2. Make suitable assumptions wherever necessary. 3. Figures to the right indicate full marks. 4. Only normal calculator is allowed. **Q-1** Fill in the blanks (Each of 1 mark) (i) If $\frac{4x}{[2 -3]}$ = 0, then x = (ii) If A = $\begin{bmatrix} 3 & 2 & 1 \\ -6 & 5 & 4 \\ -1 & 3 & -2 \end{bmatrix}$, then A^T = (iii) If A = $\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$ then A^2 = (iv) li-j-k = (v) $\int \frac{1}{x+5}dx$ = (vi) $\int cotx dx$ = (vii) $\int \frac{1}{x^2+16}dx$ = (viii) $\int sin x dx$ = (ix) The slope of the line ax - by - c = 0 is (x) The distance between the points (2,-3) & (3,-2) is (xi) The equation of the line passing through the points (1,3) & (2,5) is (xii) If x = (3,-1) and y = (-5,4) then 2x - y = (xiii) The value of î.î + j.j + k.k = (xiv) If x = (1,2,3) & y = (4,5,6) then x.ỹ = (xv) The condition of slope of two parallel lines is **Q-2** (A) $\int (x^5-2x^4-2x + 3x + cosx - e^{-5x})dx$ = **OR** (A) If the lines 2x - py = 4 and 5x + 3y = 5 are (i)parallel (ii) mutually perpendicular then find the value of p (B) Find the adjoint of the matrix A = $\begin{bmatrix} 1 & -3 & 2 \\ 3 & 1 & -4 \\ -3 & 2 & 5 \end{bmatrix}$ **OR** (B) Evaluate: $\begin{bmatrix} -1 & -3 & 2 \\ 3 & -2 & 1 \\ -4 & 1 & 2 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ 3 & -2 \\ -1 & 5 \end{bmatrix}$ (C) Using matrix method solve the equations: x-y+2z = 3, 3x + y-2z = 9, 2x - y + z = 5 **OR** (C) Using matrix method solve the equations: x-y+z=2, 2x+y-3z = 5, x+y-4z = 1 **Q-3** (A) $\int x sin x dx $ (B) $\int cosx sinx dx$ (C) (i) $\int \frac{2x+5}{(x-1)(x-2)(x-3)} dx$ (ii) $\int (x + 1)^3 dx$ **OR** (A) Find the equation of circle with center (3,5) and radius 2. (B) Prove that the points (0, -1), (6,7), (-2,3) and (8,3) are the vetices of rectangle. (C) (i) Find the equations of the tangent and normal at (3,-1) to the circle x² + y² + 6x + 4y + 3 = 0 (ii) Find slope, x - intercept, y - intercept of line 5x + 3y - 2 = 0. **Q-4** (A) Find unit perpendicular vector to given vectors x = (5,-3,1) & y = (1,5,-2) (B) Find the angle between the vectors i - j&j-k (C) (i) Prove that angle between vectors i + 2j & i + j + 3k is $sin^{-1}\frac{46}{55}$ (ii) Find direction cosines of vector (3,2,1). **OR** (A) Find the order and degree of the $\frac{d^3y}{dx^3} = \frac{dy}{dx}^5 + 1$ (B) For which value of p the vectors (p, -4, 3) and (2,-3, 5) are perpendicular. (C) (i) Show that y = ae^(2x) + be^(-2x) is the general solution of differential equation y^2 - 4y = 0. (ii) Solve: y(1 + e^x) dy - (1 + y) e^xdx = 0. ## SILVER OAK UNIVERSITY **Mid Semester Examination (April 2024)** **Semester:** II **Programme:** Diploma Engineering **Subject Code:** 1010272102 **Subject Name:** Mathematics-II **Date:** 05/04/2024 **Time:** 10:00 AM to 11:45 AM **Duration:** 1 hr. 45 minutes **Total Marks:** 50 **Instructions:** 1. Attempt all questions. 2. Make suitable assumptions wherever necessary. 3. Figures to the right indicate full marks. 4. No scientific or programmable calculator is allowed. (Only normal calculator) **Q-1** Attempt all questions. (14 MARKS) 1. If A= $\begin{bmatrix} 2 & 7 \\ 5 & 3 \\ 1 & 4 \end{bmatrix}$. then A^T = 2. If A= $\begin{bmatrix} -5 & 3 \\ 1 & -2 \\ 2 & 1 \end{bmatrix}$ then adj A = 3. If A= $\begin{bmatrix} -1 & 2 \\ 2 & 4 \\ 5 & 7 \end{bmatrix}$ then Co-factor of 7 = 4. If $\int \frac{2}{|x|} dx$ = 2, then x = 5. $\int cotx dx$ = 6. $\int \frac{1}{x} dx$ = 7. $\int x dx$ 8. $\int u v' dx$ = 9. If a=i+j+k&b=3i-6j+2k, then |2a+3b|= 10. If a = (-5,2,4) & b = (-3,1,2), then (a + b). (a - b) = 11. If a = (2,x, 1) & b = (5,2,3), and if a ⊥ b, then x = 12. Find the distance between the points A(1,5) & B(3,8). 13. General equation of line is given by. 14. The triangle having two sides equal is said to be **Q-2** (a) If A= $\begin{bmatrix} 5 & 2 & 3 \\ 1 & 8 & 6 \\ -1 & 2 & 2 \end{bmatrix}$ , B= $\begin{bmatrix} 3 \\ 6 \\ 1 \end{bmatrix}$ , C= $\begin{bmatrix} 2 & 3 & 2 \end{bmatrix}$ , then find 3A + 2B - 4C, AC + AB **OR** a) Find Adjoint of matrix if A = $\begin{bmatrix} 1 & 2 & 1 \\ -4 & 1 & 3 \\ 3 & -1 & 6 \end{bmatrix}$ b) Find $\int [4x^3+3x^2 - sinx + 3x+5]dx$ **OR** b) Find $\int x cosx dx$ and $\int x sinx dx$ c) Show that the points (-2,-1), (5, -4), (-1,-18) & (-8,-15) form a rectangle. Show that the angle between the vector i + j-k & 2i-2j+k is $sin^{-1}(\frac{\sqrt{27}}{3})$ **OR** c) If (-1,3), (-1,x) and (4,3) are the vertices of right angled triangle, Find x. Find the angle between two vectors (1,0,1) & (1,1,0) **Q-3** a) Solve: 3x +7y=13 & x-3y=-1 using matrices. b) Find $\int \frac{3x+2}{(x-2)(x-4)}$ dx c) Find Modulus, unit vector, direction cosines of a = 5î-3j+k Show that the points (1,4), (3,-2) & (-3,16) are colliear. **OR** a) If A = $\begin{bmatrix} 5 & 4 & 3 \\ 1 & 2 & 1 \end{bmatrix}$ & B= $\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$ ,prove that (A+B)^* = B* + A^* b) Find $\int \frac{1}{x-1} dx$ c) Find unit vector perpendicular to the given vectors (2,-3,4) & (1,5,-3) Find the equation of lines from the points A(2,5) & B (1,6). Also find slope,x - intercept, y - intercept. **Q-4** Answer any 4 out of 6. (12 MARKS) a) Evaluate: $\begin{bmatrix} 1 & 2 & 3 \\ -5 & 6 & 1 \\ 1 & -3 & 7 \end{bmatrix} \begin{bmatrix} 3 & -2 \\ 5 & 2 \\ -2 & 14 \end{bmatrix}$ b) Find A^-1, A = $\begin{bmatrix} 1 & 2 & 3 \\ 4 & -3 & 5 \\ 1 & 1 & 3 \end{bmatrix}$ c) $\int (x + 1)^3 dx$ d) Find $\int \frac{cosx}{(1-sinx)(2-sinx)} dx$ e) Find p if the lines 5x + py-3=0&2x-y+1 = 0 are parallel and perpendicular. f) $\int (x^2+2x+5) dx$