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Engineering
92 Views   |  

If    \cos \alpha \: +\: \cos \beta \: =\frac{3}{2} \: and\: \sin \alpha +\sin \beta =\frac{1}{2}\; and\; \theta     is the arithmetic mean of \alpha and \beta , then 

\sin2\theta +\cos 2\theta  is  equal to :

  • Option 1)

    \frac{3}{5}

  • Option 2)

    \frac{4}{5}

  • Option 3)

    \frac{7}{5}

  • Option 4)

    \frac{8}{5}

 
As we learnt in  Trigonometric Ratios of Submultiples of an Angle - - wherein This shows the formulae for half angles and their doubles.            dividing  (2) by (1), we get Thus,    Option 1) This option is incorrect. Option 2) This option is incorrect. Option 3) This option is correct. Option 4) This option is incorrect.
Engineering
127 Views   |  

if f(x)=2\tan ^{-1}x+\sin ^{-1}\left ( \frac{2x}{1+x^{2}} \right ),x>1then f(5) is equal to :

  • Option 1)

    \frac{\pi }{2}

  • Option 2)

    \pi

  • Option 3)

    4\tan ^{-1}(5)

  • Option 4)

    \tan ^{-1}\left ( \frac{65}{156} \right )

 
As we learnt in Inverse Trigonometric Functions - The functions  and  are the inverse trigonometric functions. - wherein If  then,                   now,                 .......................... ( 1 ) Now,              ............................( 2 )                                                                                 [From ( 1 ) and ( 2 )]         Option 1) This option is...
Engineering
193 Views   |  

Let 10 vertical poles standing at equal distances on a straight line, subtend the same angle of elevation \alpha at a point O on this line and all the poles are on the same side of O. If the height of the longest pole is ‘h’ and the distance of the foot of the smallest pole from O is ‘a’ ; then the distance between two consecutive poles, is :

  • Option 1)

    \frac{h\sin \alpha +a\cos \alpha }{9\sin \alpha }

  • Option 2)

    \frac{h\\cos \alpha -a\\sin \alpha }{9\cos \alpha }

  • Option 3)

    \frac{h\\cos \alpha -a\\sin \alpha }{9\sin \alpha }

  • Option 4)

    \frac{h\sin \alpha +a\cos \alpha }{9\cos \alpha }

 
As we learnt in Height and Distances - The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios. -   For all poles to subtend the same angle, the longest pole must be farthest, smallest pole must be nearest, while all other poles are arranged in ascending order of height. Let distance between each pole be x....
Engineering
485 Views   |  

 If the angles of elevation of the top of a  tower from three collinear points A, B and C, on a line leading to the foot of the

tower, are 300, 450 and 600 respectively,then the ratio, AB : BC, is :

  • Option 1)

    \sqrt{3}:1

  • Option 2)

    \sqrt{3}:\sqrt{2}

  • Option 3)

    1:\sqrt{3}

  • Option 4)

    2:3

 
As we learnt in Height and Distances - The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios. - In                                   ...................... ( 1 ) In                                           .................... ( 2 ) In                                      ..................... ( 3...
Engineering
133 Views   |  

Let  \tan ^{-1}y=\tan ^{-1}x\, +\, \tan ^{-1}\left ( \frac{2x}{1-x^{2}} \right ),    where \left | x \right |< \frac{1}{\sqrt{3}}.  

Then a value of y  is :

  • Option 1)

    \frac{3x-x^{3}}{1-3x^{2}}

  • Option 2)

    \frac{3x+x^{3}}{1-3x^{2}}

  • Option 3)

    \frac{3x-x^{3}}{1+3x^{2}}

  • Option 4)

    \frac{3x+x^{3}}{1+3x^{2}}

 
As we learned in  Formulae of Inverse Trigonometric Functions - - wherein                                                                                                      Option 1) This option is correct Option 2) This option is incorrect Option 3) This option is incorrect Option 4) This option is incorrect
Engineering
101 Views   |  

Let   \overrightarrow{a},\overrightarrow{b}and\, \overrightarrow{c}   be three non-zero vectors such that no two of them are collinear and

(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}=\frac{1}{3}\left | \overrightarrow{b} \right |\, \left | \overrightarrow{c} \right |\,\overrightarrow{a}.   if  \theta is the angle between vectors \overrightarrow{b}\, and \, \overrightarrow{c},

then a value of \sin \theta  is:

  • Option 1)

    \frac{2\sqrt{2}}{3}

  • Option 2)

    \frac{-\sqrt{2}}{3}

  • Option 3)

    \frac{2}{3}

  • Option 4)

    \frac{-2\sqrt{3}}{3}

 
As we learnt in  Vector Triple Product (VTP) - - wherein are three vectors.   and   Scalar Product of two vectors (dot product) - - wherein  is the angle between the vectors     Thus     Option 1) This option is correct. Option 2) This option is incorrect. Option 3) This option is incorrect. Option 4) This option is incorrect.
Engineering
109 Views   |  

In    a    parallelogram ABCD ,   \left | \overrightarrow{AB} \right |=a,\: \left | \overrightarrow{AD} \right |=b\; and\: \left | \overrightarrow{AC} \right |=c,\; then\: \overrightarrow{DB}\cdot \overrightarrow{AB}\; has\: the\; value:

  • Option 1)

    \frac{1}{2}\left ( a^{2}-b^{2}+c^{2} \right )

  • Option 2)

    \frac{1}{4}\left ( a^{2}+b^{2}-c^{2} \right )

  • Option 3)

    \frac{1}{3}\left ( b^{2}+c^{2}-a^{2} \right )

  • Option 4)

    \frac{1}{2}\left ( a^{2}+b^{2}+c^{2} \right )

 
As we learnt in Scalar Product of two vectors (dot product) - - wherein  is the angle between the vectors Thus let A be position  vector of B is  , D is                                          ..............(i) Also here, Put in (i)     Option 1) This option is correct. Option 2) This option is incorrect. Option 3) This option is incorrect. Option 4) This option is incorrect.
Engineering
172 Views   |  

Let \vec{a}\: and \: \vec{b} be two unit vectors such that \left | \vec{a} +\vec{b}\right |=\sqrt{3}

if\vec{c}= \vec{a}+2\vec{b}+3\left ( \vec{a} \times \vec{b}\right )    then   2\left | \vec{c} \right |    is equal to:

  • Option 1)

    \sqrt{55}

  • Option 2)

    \sqrt{51}

  • Option 3)

    \sqrt{43}

  • Option 4)

    \sqrt{37}

 
As we learnt in  Vector Product of two vectors(cross product) - If  and  are two vectors and  is the angle between them , then  - wherein  is unit vector perpendicular to both    and    Scalar Product of two vectors (dot product) - - wherein  is the angle between the vectors   Squaring both sides            Option 1) This option is correct. Option 2) This option is...
Engineering
101 Views   |  

If  y(x) is the solution of the differential equation (x+2)\frac{dy}{dx}=x^{2}+4x-9,x\neq -2\: and\, \, y(0)=0,\; then\; y(-4)\; is\; equal\; to\, :

  • Option 1)

    0

  • Option 2)

    1

  • Option 3)

    -1

  • Option 4)

    2

 
As we learnt in Solution of Differential Equation - put       - wherein Equation with convert to                           =8 - 8 - 1 log | 2 | + 13 log 2 = 0    Option 1) 0 This option is correct  Option 2) 1 This option is incorrect  Option 3) -1 This option is incorrect  Option 4) 2 This option is incorrect 
Engineering
107 Views   |  

The solution of the differential equation  ydx-(x+2y^{2})dy=0\, is\,\, x=f(y).\,  if f(-1)=1,\: then\: f(1)

 is equal to :

 

  • Option 1)

    4

  • Option 2)

    3

  • Option 3)

    2

  • Option 4)

    1

 
As we learnt in  Linear Differential Equation - - wherein P, Q are functions of x alone.     Put           & Q = 2y Its solution is 2+1=x   x=3 Option 1) 4 This option is incorrect. Option 2) 3 This option is correct. Option 3) 2 This option is incorrect. Option 4) 1 This option is incorrect.
Engineering
101 Views   |  

Let y(x) be the solution of the differential equation

(x\, \log \, x)\frac{dy}{dx}+y=2x\, \log x,\left ( x\geq 1 \right ). 

Then y(e) is equal to :

  • Option 1)

    e

  • Option 2)

    0

  • Option 3)

    2

  • Option 4)

    2e

 
As we learnt in  Linear Differential Equation - - wherein P, Q are functions of x alone.     I.F.   Put x=1 0=c-2          = c=2 at x=e Option 1) e This option is incorrect. Option 2) 0 This option is incorrect. Option 3) 2 This option is correct. Option 4) 2e This option is incorrect.
Engineering
101 Views   |  

The equation of the plane containing the line 2x-5y+z=3;x+y+4z=5,

and parallel to the plane, x+3y+6z=1,\, is:

  • Option 1)

    2x+6y+12z=13

  • Option 2)

    x+3y+6z=-7

  • Option 3)

    x+3y+6z=7

  • Option 4)

    2x+6y+12z=-13

 
As we learnt in Equation of any plane passing through the line of intersection of two planes (Cartesian form ) -   The equation of any plane passing through the line of intersection of two planes and is given by      -    Equation of a plane be             ..........(i) Since a plane is parallel to  x+3y+6z-1 After substituting  Hence requred plane will be    Option 1) This option is...
Engineering
116 Views   |  

The distance of the point (1, 0, 2) from the line point of intersection of the line

\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}      and the plane  x-y+z=16,  is:

  • Option 1)

    2\sqrt{14}

  • Option 2)

    8

  • Option 3)

    3\sqrt{21}

  • Option 4)

    13

 
As we learnt in  Intersection of line and plane - Let the line plane intersect at P to find P assume general point on line as now put it in plane to find , -    Point of intesection of line x=3k+2, y=4k-1,z=12k+2 x-y+z=16 3k+2-4k+1+12k+2=16 11k-11    k=1 Point is (5,3,14) Distance between (5,3,14) and (1,0,2) is Option 1) This is incorrect option Option 2) This is incorrect...
Engineering
107 Views   |  

A plane containing the point (3, 2, 0) and the line  \frac{x-1}{1}=\frac{y-2}{5}=\frac{z-3}{4}  also contains the point :

  • Option 1)

    (0, -3, 1)

  • Option 2)

    (0, 7, 10)

  • Option 3)

    (0, 7, -10)

  • Option 4)

    (0, 3, 1)

 
As we learnt in  Normal form (cartesian form ) - where d is the distance from origin.   - wherein putting in  We get -      Normal vector of Plane is   15x-11y+10z=23 It passes through (0,7,10) Option 1) (0, -3, 1) This option is incorrect Option 2) (0, 7, 10) This option is correct Option 3) (0, 7, -10) This option is incorrect Option 4) (0, 3, 1) This option is incorrect
Engineering
93 Views   |  

The shortest distance between the z-axis and the line

x+y+2z-3=0=2x+3y+4z-4,\; is:

 

  • Option 1)

    1

  • Option 2)

    2

  • Option 3)

    3

  • Option 4)

    4

 
As we learnt in  Shortest distance between two skew lines (Cartesian form) - Shortest distance between and is given by Where    -    Normal vector of line is Put z=0 We get x+y=3 2x+3y=4 y=-2, x=5 (5,-2,0) is the point Equation of line will be Equation of z-axis is Shortest distance = Option 1) 1 This is incorrect option Option 2) 2 This is correct option Option 3) 3 This is...
Engineering
84 Views   |  

 If the shortest distance between the lines\frac{x-1}{\alpha }=\frac{y+1}{-1}= \frac{z}{1},\left ( \alpha \neq -1 \right )and

x+y+z+1=0=2x-y+z+3 \: is\: \frac{1}{\sqrt{3}},then
a value of  \alpha  is :

 

  • Option 1)

    -\frac{16}{19}

  • Option 2)

    -\frac{19}{16}

  • Option 3)

    \frac{32}{19}

  • Option 4)

    \frac{19}{32}

 
As we learnt in  Shortest distance between two skew lines (Cartesian form) - Shortest distance between and is given by Where    -    Normal vector of line is One point on is (0,1,-2) and on is (1,-1,0) Distance = Option 1) This option is incorrect Option 2) This option is incorrect Option 3) This option is correct Option 4) This option is incorrect
Engineering
111 Views   |  

If the points (1, 1, \lambda) and (-3, 0, 1) are equidistant from the plane, 3x+4y-12z+13=0, then \lambdasatisfies the equation :

  • Option 1)

    3x^{2}-10x+7=0

  • Option 2)

    3x^{2}+10x+7=0

  • Option 3)

    3x^{2}+10x-13=0

  • Option 4)

    3x^{2}-10x+21=0

 
As we learnt in  Distance of a point from plane (Cartesian form) - The length of perpendicular from to the plane is given by    -    Distance from plane are         or                             or             These are roots of equation Option 1) This option is correct Option 2) This option is incorrect Option 3) This option is incorrect Option 4) This option is incorrect
Engineering
142 Views   |  

The set of all values of \lambda for which the system of linear equations :

2x_{1}-2x_{2}+x_{3}=\lambda x_{1}

2x_{1}-3x_{2}+2x_{3}=\lambda x_{2}

-x_{1}+2x_{2}\; \; \; =\lambda x_{3}

has a non-trivial solution,

  • Option 1)

    is an empty set.

  • Option 2)

    is a singleton.

  • Option 3)

    contains two elements.

  • Option 4)

    contains more than two elements.

 

As we learnt in 

Cramer's rule for solving system of linear equations -

When \Delta =0  and \Delta _{1}=\Delta _{2}=\Delta _{3}=0 ,

then  the system of equations has infinite solutions.

- wherein

a_{1}x+b_{1}y+c_{1}z=d_{1}

a_{2}x+b_{2}y+c_{2}z=d_{2}

a_{3}x+b_{3}y+c_{3}z=d_{3}

and 

\Delta =\begin{vmatrix} a_{1} &b_{1} &c_{1} \\ a_{2} & b_{2} &c_{2} \\ a_{3}&b _{3} & c_{3} \end{vmatrix}

\Delta _{1},\Delta _{2},\Delta _{3} are obtained by replacing column 1,2,3 of \Delta by \left ( d_{1},d_{2},d_{3} \right )  column

 

\begin{vmatrix} 2-\lambda & -2 & 1\\ 2 & -3-\lambda & 2\\ -1 & 2 & -\lambda \end{vmatrix}=0

\therefore \left ( 2-\lambda \right )\left ( \lambda ^{2}+3\lambda -4 \right )+4\left ( 1-\lambda \right )+\left ( 1-\lambda \right )=0

\therefore \left ( \lambda -1 \right )\left ( \lambda ^{2}+2\lambda -3 \right )=0

\left ( \lambda +3 \right )\left ( \lambda -1 \right )^{2}=0

\therefore \lambda =1\:and\:\lambda =-3 


Option 1)

is an empty set.

This option is incorrect.

Option 2)

is a singleton.

This option is incorrect.

Option 3)

contains two elements.

This option is correct.

Option 4)

contains more than two elements.

This option is incorrect.

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Engineering
324 Views   |  

if              is a matrix satisfying the equation AAT=9I, where I is 3x3 identity matrix, then the ordered pair

(a, b) is equal to :

  • Option 1)

    (2,-1)

  • Option 2)

    (-2,1)

  • Option 3)

    (2,1)

  • Option 4)

    (-2,-1)

 
Using Multiplication of matrices - -                    by given condition Option 1) This option is incorrect. Option 2) This option is incorrect. Option 3) This option is incorrect. Option 4) This option is correct.
Engineering
98 Views   |  

If

then ‘a’ is equal to :

 

  • Option 1)

    12

  • Option 2)

    24

  • Option 3)

    -12

  • Option 4)

    -24

 

As we learnt in 

Value of determinants of order 3 -

-

 

 \begin{vmatrix} x^{2}+x & x+1 & x-2\\ 2x^{2}+3x-1 & 3x & 3x-3\\ x^{2}+2x+3& 2x-1 & 2x-1 \end{vmatrix} =ax-12

R_{2}\rightarrow R_{2}-\left ( R_{1}+R_{3} \right )

\begin{vmatrix} x^{2}+x & x+1 & x-2\\ -4 & 0 & 0\\ x^{2}+2x+3 & 2x-1 & 2x-1 \end{vmatrix}

\Rightarrow 4\left ( 6x-3 \right )=ax-12

\therefore 24x-12=ax-12

\Rightarrow a=24


Option 1)

12

This option is incorrect.

Option 2)

24

This option is correct.

Option 3)

-12

This option is incorrect.

Option 4)

-24

This option is incorrect.

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