MODEL EXAM QUESTION PAPER
40022 ENGINEERING MATHEMATICS – II
Time : 3.00 Hours Date: 22-05-2021 Max.Marks: 100
Note:
1. Answer all question in PART A. Each question carries one mark.
2. Answer any ten questions in PART B. Each question carries two marks.
3. Answer all question by selecting either A or B. Each question carries fifteen marks.
4. Clarkes Table and programmable calculators are not permitted.
PART – A (5×1=5)
1. Find the value of ‘p’if the pair of lines px2-5xy + 7y2 = 0 are perpendicular
to each Other.
\[2.\ Find\ \overrightarrow{a}× \overrightarrow{b}if\ \overrightarrow{a}= \overrightarrow{i}+ \overrightarrow{j}+ \overrightarrow{k}\ and \overrightarrow{b}= 2\overrightarrow{i}- \overrightarrow{j}+ 3\overrightarrow{k}\ \hspace{20cm}\]
3. Define vector Differential operator.
\[4.\ Evaluate: \int(x^2 -x -1)\ dx\ \hspace{30cm}\]
\[5.\ Evaluate: \int_1^2 x^2\ dx\ \hspace{30cm}\]
PART – B (10×2=20)
\[6.\ Find\ the\ angle\ between\ the\ lines\ y = \sqrt{3x}\ and\ x- y = 0\ \hspace{30cm}\]
7. Find the equation of the circle passing through the point A (2, – 3) and having its centre at
C ( – 5 , 1).
8. Prove that the circles x2 + y2 – 4x + 6y + 4 = 0 and
x2 + y2 + 2x + 4y + 4 = 0 cut orthogonally.
\[9.\ Find\ the\ projection\ of\ the\ vector\ 3\overrightarrow{i}+ 4\overrightarrow{j}- 5\overrightarrow{k} on\ the\ vector\ \overrightarrow{i}+ 2\overrightarrow{j}+2\overrightarrow{k}\ \hspace{30cm}\]
\[10.\ Find\ the\ value\ of\ m\ if\ the\ vectors\ 2\overrightarrow{i}+ m\overrightarrow{j}- 3\overrightarrow{k} and\ 3\overrightarrow{i}+ 1\overrightarrow{j}+4\overrightarrow{k} are\ perpendicular\ \hspace{30cm}\]
\[11.\ Find\ the\ area\ of\ the\ parellelogram\ whose\ adjacent\ sides\ are\ 3\overrightarrow{i}- \overrightarrow{k} and\ \overrightarrow{i}+ \overrightarrow{j}+\overrightarrow{k}.\ \hspace{30cm}\]
\[12.\ Prove\ that\ the\ vectors\ 2\overrightarrow{i}+ \overrightarrow{j}+ \overrightarrow{k},\ 3\overrightarrow{i} + 4\overrightarrow{j}+ \overrightarrow{k}\ and\ \overrightarrow{i} – 2\overrightarrow{j}+ \overrightarrow{k}\ are\ coplanar.\ \hspace{30cm}\]
13. Find the unit normal to the surface xy +yz + zx= 3 at the point (1, 1, 1).
\[14.\ If\ \overrightarrow{F}=xyz\overrightarrow{i} + 3x^2y\overrightarrow{j} +(x y ^2- zy^3)\overrightarrow{k}\ then\ find\ curl\ \overrightarrow{F}\ \hspace{30cm}\]
\[15.\ Evaluate: \int tan^2 x \ dx\ \hspace{30cm}\]
\[16.\ Evaluate: \int(2 sin x + 7)\ dx\ \hspace{30cm}\]
\[17.\ Evaluate:\ \int \frac{dx}{{\sqrt{9 – x^2}}}\ \hspace{30cm}\]
\[18.\ Evaluate: \int \frac{cos^2 x}{ 1- sin x} \ dx\ \hspace{30cm}\]
\[19.\ Evaluate:\ \int x\ e^x\ dx\ \hspace{30cm}\]
\[20.\ Evaluate: \int_0^\frac{\pi}{2} sin^2 x\ dx\ \hspace{30cm}\]
![Magzter2 [CPS] IN](https://ad.admitad.com/b/hjdpvbg24h109003a688aadae9dc1a/)
PART – C (5×15=75)
21 A). i) Prove that the circles x2 + y2 + 2x – 4y – 3 = 0 and x2 + y2 – 8x + 6y +7 = 0 touch each other. ( 7 )
ii) Find K if 2x2 – 7xy + 3y2 + 5x – 5y + k= 0 represents a pair of straight lines. ( 8 )
(or)
B) i. Find vertex, focus , equation of directrix and latus rectum for ( 7 )
y2 + 8x – 6y + 1 = 0.
ii. The slope of one of the lines ax2 + 2hxy + by2 = 0 is thrice that of the (8) other. Show that 3h2= 4ab
22 A) i. Prove that the points whose position vectors are
\[3\overrightarrow{i}\ – \overrightarrow{j}+ 6\overrightarrow{k}, 5\overrightarrow{i}\ – 2\overrightarrow{j}+ 7\overrightarrow{k} and\ 6\overrightarrow{i}\ -5 \overrightarrow{j}+ 2\overrightarrow{k}\ form\ a\ right\ angled\ triangle\ \hspace{4cm}\ (7)\]
\[\hspace{-1cm}\ ii.\ Find\ the\ projection\ of\ the\ vector\ 3\overrightarrow{i}+ \overrightarrow{j}- 2\overrightarrow{k} on\ 7\overrightarrow{i}+ \overrightarrow{j}+2\overrightarrow{k}. Also\ find\ the\ angle\ between\ them\ \hspace{1cm}\ (8)\]
(or)
B) i. A particle acted on by the forces
\[\hspace{-2cm}\ 3\overrightarrow{i}+ 2\overrightarrow{j}- 3\overrightarrow{k} and\ \overrightarrow{i}+ 7\overrightarrow{j}+7\overrightarrow{k} acting\ on\ the\ particle\ \hspace{11cm}\ (7)\]
\[displaces\ the\ particle\ from\ the\ point\ \overrightarrow{i}+ 2\overrightarrow{j}+ 3\overrightarrow{k} to\ the\ point\ 3\overrightarrow{i}- 5\overrightarrow{j}+4\overrightarrow{k}. Find\ the\ total\ work\ done\ by\ the\ forces.\]
\[\hspace{-1cm}\ ii.\ Find\ the\ moment\ of\ the\ force\ 3\overrightarrow{i}+\overrightarrow{k}\ acting\ through\ the\ point\ \overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k} about\ the\ point\ 2\overrightarrow{i}+ \overrightarrow{j}-2\overrightarrow{k}.\ \hspace{1cm}\ (8)\]
23. A) i. Show that the points whose position vectors
\[4\overrightarrow{i}+ 5\overrightarrow{j}+ \overrightarrow{k},\ – \overrightarrow{j}- \overrightarrow{k},\ 3\overrightarrow{i} + 9\overrightarrow{j}+ 4\overrightarrow{k}\ \hspace{15cm}\ (7)\]\[and\ -4\overrightarrow{i} + 4\overrightarrow{j}+ 4\overrightarrow{k}\ lie\ on\ the\ same\ plane.\ (or)\ Coplanar.\]
ii. Find the acute angle between the surface xy2z = 4 and x2 + y2 + z2 =6 at the point (2, 1, 1). (8)
(or)
\[\hspace{-2cm}\ B)\ i.\ If\ \overrightarrow{F}=( 2x + 2y + 2z)\overrightarrow{i} – (xy + yz + zx)\overrightarrow{j} + 3xyz\overrightarrow{k}\ then\ find\ \nabla\ × \overrightarrow{F}\ and\ \nabla\ × ( \nabla\ × \overrightarrow{F})\ \hspace{5cm}\ (7)\]
\[\hspace{-2cm}\ ii.\ If\ \overrightarrow{a} = 2\overrightarrow{i} – \overrightarrow{j}+ 2\overrightarrow{k},\ \overrightarrow{b} = \overrightarrow{i} +\overrightarrow{j}+ \overrightarrow{k},\ \overrightarrow{c} = \overrightarrow{i}+2\overrightarrow{j} +3\overrightarrow{k}\}\ and\ \overrightarrow{d} = \overrightarrow{i}-\overrightarrow{j} – \overrightarrow{k}\ \hspace{5cm}\ (8)\]\[find\ (\overrightarrow{a} × \overrightarrow{ b} ) . (\overrightarrow{c} × \overrightarrow{ d} )\]
24. A) Evaluate
\[a)\ \hspace{1cm}\ (i)\ \hspace{1cm}\ \int ( sin x + cos x ) ^ 2\ dx\ \hspace{5cm}\ (ii)\ \hspace{1cm}\ \int {\sqrt{9x^2 +16}}\ dx\ \hspace{5cm}\ (7)\]
\[b)\ \hspace{1cm}\ (i)\ \hspace{1cm}\ \int \frac{cos^2 x}{ 1- sin x} \ dx\ \hspace{5cm}\ (ii)\ \hspace{1cm}\ \int \frac{dx}{(3x + 2)^2 + 16}\ dx\ \hspace{5cm}\ (8)\]
(or)
B) Evaluate
\[a)\ \hspace{1cm}\ (i)\ \hspace{1cm}\ \int\ \frac{2ax + b}{{\sqrt{(ax^2 + bx + c)}}}\ dx\ \hspace{5cm}\ (ii)\ \hspace{1cm}\ \int cos^3 x \ dx\ \hspace{5cm}\ (7)\]
\[b)\ \hspace{1cm}\ (i)\ \hspace{1cm}\ \int sin 5x\ cos 2x\ dx\ \hspace{5cm}\ (ii)\ \hspace{1cm}\ \int \frac{dx}{4 + 9 x^2}\ \hspace{5cm}\ (8)\]
25. A) Evaluate
\[a)\ \hspace{1cm}\ (i)\ \hspace{1cm}\ \int x^2\ sin\ 2x\ dx\ \hspace{5cm}\ (ii)\ \hspace{1cm}\ \int x^n\ log\ x\ dx\ \hspace{5cm}\ (7)\]
\[b)\ \hspace{1cm}\ (i)\ \hspace{1cm}\ \int x^2\ cos\ 5x\ dx\ \hspace{5cm}\ (ii)\ \hspace{1cm}\ \int x^3\ e^{2x}\ dx\ \hspace{5cm}\ (8)\]
(or)
B) Evaluate
\[a)\ \hspace{1cm}\ (i)\ \hspace{1cm}\ \int x^3\ log\ x\ dx\ \hspace{5cm}\ (ii)\ \hspace{1cm}\ \int_0^\frac{\pi}{2} sin^3 x\ dx\ \hspace{5cm}\ (7)\]
\[b)\ \hspace{1cm}\ (i)\ \hspace{1cm}\ \int x^3\ e^{-x}\ dx\ \hspace{5cm}\ (ii)\ \hspace{1cm}\ \int_0^\frac{\pi}{2} \frac{sin\ x}{sin\ x + cos\ x}\ dx \hspace{5cm}\ (8)\]
Like this:
Like Loading...
You must log in to post a comment.