N-SCHEME-ENGINEERING MATHEMATICS-II MODEL EXAM QUESTION PAPER/2021

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.

\[4.\ Evaluate: \int(x^2 -x -1)\ dx\ \hspace{30cm}\]
\[5.\ Evaluate: \int_1^2 x^2\ dx\ \hspace{30cm}\]
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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).

\[14.\ If\ \overrightarrow{F}=xyz\overrightarrow{i} + 3x^2y\overrightarrow{j} +(x y ^2- zy^3)\overrightarrow{k}\ then\ find\ curl\ \overrightarrow{F}\ \hspace{30cm}\]
\[15.\ Evaluate: \int tan^2 x \ dx\ \hspace{30cm}\]
\[16.\ Evaluate: \int(2 sin x + 7)\ dx\ \hspace{30cm}\]
\[17.\ Evaluate:\ \int \frac{dx}{{\sqrt{9 – x^2}}}\ \hspace{30cm}\]
\[18.\ Evaluate: \int \frac{cos^2 x}{ 1- sin x} \ dx\ \hspace{30cm}\]
\[19.\ Evaluate:\ \int x\ e^x\ dx\ \hspace{30cm}\]
\[20.\ Evaluate: \int_0^\frac{\pi}{2} sin^2 x\ dx\ \hspace{30cm}\]
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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

\[a)\ \hspace{1cm}\ (i)\ \hspace{1cm}\ \int x^3\ log\ x\ dx\ \hspace{5cm}\ (ii)\ \hspace{1cm}\ \int_0^\frac{\pi}{2} sin^3 x\ dx\ \hspace{5cm}\ (7)\]
\[b)\ \hspace{1cm}\ (i)\ \hspace{1cm}\ \int x^3\ e^{-x}\ dx\ \hspace{5cm}\ (ii)\ \hspace{1cm}\ \int_0^\frac{\pi}{2} \frac{sin\ x}{sin\ x + cos\ x}\ dx \hspace{5cm}\ (8)\]

                 

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