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Engineering
181 Views   |  

 Which one of the following complexes shows optical isomerism ?

 

  • Option 1)

     [Co(NH3)3Cl3]

     

  • Option 2)

     cis[Co(en)2Cl2]Cl

     

  • Option 3)

    trans[Co(en)2Cl2]Cl

     

  • Option 4)

     [Co(NH3)4Cl2]Cl

     

     

     

    (en=ethylenediamine)

     

 
As discussed in   These molecules are optically active having absence of plane of symmetry.  This molecule has absence of plane of symmetry and hence option 2 is correct. Option 1)  [Co(NH3)3Cl3]   This option is incorrect. Option 2)  cis[Co(en)2Cl2]Cl   This option is correct. Option 3) trans[Co(en)2Cl2]Cl   This option is incorrect. Option...
Engineering
779 Views   |  

 If  0 ≤ x < 2 π, then the number of real values of x, which satisfy the equation

cos x + cos 2x + cos 3x + cos 4x = 0, is :

  • Option 1)

    3

  • Option 2)

    5

  • Option 3)

    7

  • Option 4)

    9

 
As we learnt in in General Solution of Trigonometric Ratios - - wherein  is the given angle     Thus,                           Hence, there are a total number of 7 solutions.   Option 1) 3 This option is incorrect. Option 2) 5 This option is incorrect. Option 3) 7 This option is correct. Option 4) 9 This option is incorrect.
Engineering
295 Views   |  

A man is walking towards a vertical pillar in a straight path, at a uniform speed.  At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 300.  After walking for 10 minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is 600. Then the time taken (in minutes) by him, from B to reach the pillar, is :

  • Option 1)

    6

  • Option 2)

    10

  • Option 3)

    20

  • Option 4)

    5

 
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. -  Let PQ represent pillar. Say height = h In  ,   In      Now, the person is walking at a uniform speed, hence time taken will be proportional to distance travelled. Thus,  Hence, time required is 5...
Engineering
102 Views   |  

 Let\vec{a},\vec{b},and\: \vec{c}  be three unit vectors such that

\vec{a}\times \left ( \vec{b}\: \: \times\: \: \vec{c} \right )= \frac{\sqrt{3}}{2}\left ( \vec{b} +\vec{c}\right )\: if \: \vec{b}

is not parallel  to  \vec{c} then the angle between \vec{a}\: and\: \vec{b}\: is

  • Option 1)

    \frac{3\pi }{4}

  • Option 2)

    \frac{\pi }{2}

  • Option 3)

    \frac{2\pi }{3}

  • Option 4)

    \frac{5\pi }{6}

 
As we learnt in  Vector Triple Product (VTP) - - wherein are three vectors.     Hence,    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
151 Views   |  

In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are respectively 3\hat{i}+\hat{j}-\hat{k},-\hat{i}+3\hat{j}+p\hat{k} \: and\: 5\hat{i}+q\, \hat{j}-4\hat{k},

 then the point (p, q) lies on a line :

  • Option 1)

    parallel to x-axis.

  • Option 2)

    parallel to y-axis.

  • Option 3)

    making an acute angle with the positive direction of x-axis.

  • Option 4)

    making an obtuse angle with the positive direction of x-axis

 
As we learnt in Scalar Product of two vectors - - wherein   is the angle between the vectors    Replace (p,q) with (x,y) 3x-2y+13=0 Acute angle with +x-axis   Option 1) parallel to x-axis. This option is incorrect. Option 2) parallel to y-axis. This option is incorrect. Option 3) making an acute angle with the positive direction of x-axis. This option is correct. Option 4) making an...
Engineering
92 Views   |  

If a curve y=f(x)  passes through the point  \left ( 1,-1 \right ) and satisfies the differential
equation,  y(1+xy)dx=x\: dx  then     f\left ( -\frac{1}{2} \right )

is equal to :

 

  • Option 1)

    -\frac{2}{5}

  • Option 2)

    -\frac{4}{5}

  • Option 3)

    \frac{2}{5}

  • Option 4)

    \frac{4}{5}

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

 If the line, \frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3} lies in


the plane,lx+my-z=9,then l^{2}+m^{2}  is equal to:

  • Option 1)

    26

  • Option 2)

    18

  • Option 3)

    5

  • Option 4)

    2

 
As we learnt in  Plane passing through a point and a line (vector form) - Let the plane passes through and a line then the plane is given by   - wherein    line lies in plane lx+my-z=9 So (3,-2,-4) statisfies lx+my-z=9 3l-2m+4=9 3l-2m=5                ----------(i) and 2l-m=3            -------------(ii) l=1, m=-1 Option 1) 26 This option is incorrect Option 2) 18 This option is...
Engineering
100 Views   |  

The distance of the point (1, −5, 9) from the plane x − y + z = 5 measured along the

line x = y = z is :

  • Option 1)

    3\sqrt{10}

  • Option 2)

    10\sqrt{3}

  • Option 3)

    \frac{10}{\sqrt{3}}

  • Option 4)

    \frac{20}{3}

 
As we learnt in  Distance formula - The distance between two points is =   - wherein     = -           Cartesian eqution of a line - The equation of a line passing through two points and parallel to vector having direction ratios as is given by The equation of a line passing through two points  is given by - wherein    Coordinates of point A Line AP is x-1=y+5=z-9=k Thus, x=k+1,...
Engineering
98 Views   |  

 The number of distinct real values of λ for which the lines

\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{\lambda ^{2}} and

\frac{x-3}{1}=\frac{y-2}{\lambda ^{2}}=\frac{z-1}{2}

are coplanar is :

  • Option 1)

    4

  • Option 2)

    1

  • Option 3)

    2

  • Option 4)

    3

 
As we learnt in  Condition for lines to be intersecting (cartesian form) - Their shortest distance should be 0 Also the condition for coplanar lines -   For two lines to be coplanar But Two values of are possible Option 1) 4 This option is incorrect Option 2) 1 This option is incorrect Option 3) 2 This option is correct Option 4) 3 This option is incorrect
Engineering
677 Views   |  

ABC is a triangle in a plane with vertices A(2, 3, 5), B(−1, 3, 2) and C(λ, 5, µ).  If the median through A is equally inclined to the coordinate axes, then the value of (λ33+5) is :

  • Option 1)

    1130

  • Option 2)

     1348

  • Option 3)

    676

  • Option 4)

    1077

 
As learnt in concept Direction Ratios - (i)     if a,b,c are direction ratios then direction cosines will be (ii)        Direction ratios of line joining two given points is given by          (iii)    If  be a vector with direction cosines l, m, n then -  DRs of median So, So, = Option 1) 1130 This option is incorrect Option 2)  1348 This option is correct Option 3) 676 This option...
Engineering
178 Views   |  

 The distance of the point (1, −2, 4) from the plane passing through the point (1, 2, 2) and perpendicular to the planes x − y + 2z = 3 and  2x −2y + z + 12=0, is :

 

  • Option 1)

    2\sqrt{2}

  • Option 2)

    2

  • Option 3)

    \sqrt{2}

  • Option 4)

    \frac{1}{\sqrt{2}}

 
As we learnt in  Distance of a point from plane (Cartesian form) - The length of perpendicular from to the plane is given by    -    Normal vector of plane is So equation of plane is passes through (1,2,2) 3+6=C 3x+3y=9    =>    x+y=3 Distance 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
86 Views   |  

The shortest distance between the lines

\frac{x}{2}= \frac{y}{2}= \frac{z}{1}  and  \frac{x+2}{-1}= \frac{y-4}{8}= \frac{z-5}{4}   

lies in the interval :

  • Option 1)

     [0, 1)

  • Option 2)

     [1, 2)

  • Option 3)

    (2, 3]

  • Option 4)

    (3, 4]

 
As we learnt in  Shortest distance between two skew lines (Cartesian form) - Shortest distance between and is given by Where    -    Cross product of DRS is So shortest distance = It lies in interval (2,3) Option 1)  [0, 1) This option is incorrect Option 2)  [1, 2) This option is incorrect Option 3) (2, 3] This option is correct Option 4) (3, 4] This option is incorrect
Engineering
117 Views   |  

The system of linear equations

x +λy −z = 0
λx − y − z = 0
x + y − λz = 0

has a non-trivial solution for :

 

  • Option 1)

    infinitely many values of λ.

     

  • Option 2)

    exactly one value of λ.

     

  • Option 3)

    exactly two values of λ.

     

  • Option 4)

    exactly three values of λ.

     

 

Solution Video

Engineering
230 Views   |  

If and A adj A=A AT, then

5 a + b is equal to :

 

  • Option 1)

    -1

  • Option 2)

    5

  • Option 3)

    4

  • Option 4)

    13

 

As we learnt in 

Adjoint of a square matrix -

Transpose of the matrix of co-factors of elements of A is called the adjoint of A

- wherein

 

 

Transpose of a Matrix -

The matrix obtained from any given matrix A, by interchanging its rows and columns.

- wherein

 

 

A = \begin{bmatrix} 5a & -b\\ 3 & 2 \end{bmatrix}

\Rightarrow adj (A)= \begin{bmatrix} 2 & b\\ -3 & 5a \end{bmatrix}

\\ \Rightarrow A adj (A)= \begin{bmatrix} 5a & -b\\ 3 & 2 \end{bmatrix} \begin{bmatrix} 2 & b \\ -3 & 5a \end{bmatrix} = \begin{bmatrix} 10a + 3b & 0\\ 0 & 10a +3b \end{bmatrix}

\\ \Rightarrow A A ^{T}= \begin{bmatrix} 5a & -b\\ 3 & 2 \end{bmatrix} \begin{bmatrix} 5a & 3\\ -b & 2 \end{bmatrix}\\ = \begin{bmatrix} 25a^{2} + b^{2}& 15a-2b\\ 15a-2b & 13 \end{bmatrix}

so that 15a - 2b = 0

and 10a + 3b =13

\therefore a = \frac{2}{5} \ and \ b =3

\therefore 5a +b = 2+3 = 5

 


Option 1)

-1

This option is incorrect.

Option 2)

5

This option is correct.

Option 3)

4

This option is incorrect.

Option 4)

13

This option is incorrect.

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Engineering
98 Views   |  

The number of distinct real roots of the

equation,in the

interval \left [ -\frac{\pi }{4} ,\frac{\pi }{4}\right ] is

 

  • Option 1)

    4

  • Option 2)

    3

  • Option 3)

    2

  • Option 4)

    1

 
As we leant in Cramer's rule for solving system of linear equations - When   and  , then  the system of equations has infinite solutions. - wherein and   are obtained by replacing column 1,2,3 of  by   column     C1 C1+C2+C3   Option 1) 4 This option is incorrect. Option 2) 3 This option is incorrect. Option 3) 2 This option is correct. Option 4) 1 This option is incorrect.
Engineering
102 Views   |  

Let p= \lim_{x\rightarrow 0+}\left ( 1+\tan ^{2} \sqrt{x}\right )^{\frac{1}{2x}}then  log p
is equal to :

  • Option 1)

    2

  • Option 2)

    1

  • Option 3)

    \frac{1}{2}

  • Option 4)

    \frac{1}{4}

 
As we learnt in  1 to the power of infinity Form - -   It is  form   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
91 Views   |  

If m and M are the minimum and the maximum values of

4+\frac{1}{2}\sin ^{2}2x-2\cos ^{4}x\, ,x\, \epsilon \, R,

 then Mm is equal to :

 

 

  • Option 1)

    \frac{15}{4}

  • Option 2)

    \frac{9}{4}

  • Option 3)

    \frac{7}{4}

  • Option 4)

    \frac{1}{4}

 
As we learnt in  Method for maxima or minima - First and second derivative method :       - wherein   Let,          So that                                                  m = 4 and    Correct option is 4.   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
109 Views   |  

If f(x) is a differentiable function in the interval (0, ∞) such that f(1) = 1 and  \lim_{t\rightarrow x} \frac{t^{2}f(x)-x^{2}f(t)}{t-x} = 1,  for each x> 0 , then f(3/2) is equal to :

  • Option 1)

    \frac{13}{6}

  • Option 2)

    \frac{23}{18}

  • Option 3)

    \frac{25}{9}

  • Option 4)

    \frac{31}{18}

 
As we learnt in L - Hospital Rule - - wherein   Now, let y=f(x)  If  Put, x=1, y=1 Put,      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
113 Views   |  

If the function   is differentiable at x = 1, then
\frac{a}{b}  is equal  to :

  • Option 1)

    \frac{\pi - 2}{2}

  • Option 2)

    \frac{-\pi - 2}{2}

  • Option 3)

    \frac{\pi + 2}{2}

  • Option 4)

    -1-\cos ^{-1}(2)

 
As we learnt in Rule for continuous - A function is continuous at  x = a if and only if           L.H.L    R.H.L   value at  x = a. - wherein Where          1+b=0        b=-1 Now f(x) will be continuous at x=1     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
103 Views   |  

   If   \lim_{x\rightarrow \infty }\left ( 1 + \frac{a}{x} - \frac{4}{x^{2}} \right )^{2x} = e^{3}  , then  'a' is equal to :

  • Option 1)

     2

  • Option 2)

    \frac{3}{2}

  • Option 3)

    \frac{2}{3}

  • Option 4)

    \frac{1}{2}

 

Option 2

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