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Engineering
121 Views   |

If in a triangle $\dpi{100} ABC$  ,$\dpi{100} a\cos ^{2}\left ( \frac{C}{2} \right )+c\cos ^{2}\left ( \frac{A}{2} \right )= \frac{3b}{2},$  then the sides  $\dpi{100} a,b \: and\: c$

• Option 1)

$are\: in \: G.P.$

• Option 2)

$are\: in \: H.P.$

• Option 3)

$satisf\! y\: \: a+b=c$

• Option 4)

$are\: in \: A.P.$

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. -     now by sine law , we have  Thus , using it , we have using sine law, we have     Option 1) This is incorrect option Option 2) This is incorrect option Option 3) This is incorrect option Option...
Engineering
90 Views   |

The trigonometric equation $sin^{-1}x=2sin^{-1}a,$ has a solution for

• Option 1)

all real values

• Option 2)

$\left | a \right |< \frac{1}{2}\; \;$

• Option 3)

$\; \left | a \right |\geqslant \frac{1}{\sqrt{2}}\; \; \;$

• Option 4)

$\; \frac{1}{2}< \left | a \right |< \frac{1}{\sqrt{2}}$

As we learnt in  Domains and Ranges of Inverse Trigonometric Functions - For  Domain  Range  -     For (1) to have solution,  Thus  Also, for  to be defined .  Thus solution is  None of the solution is correct.  Option 1) all real values Option 2) Option 3) Option 4)
Engineering
136 Views   |

The sum of the radii of inscribed and circumscribed circles for an $n$ sides regular polygon of side $a$ , is

• Option 1)

$\frac{a}{2}cot\left ( \frac{\pi }{2n} \right )\; \;$

• Option 2)

$\; a\, cot\left ( \frac{\pi }{2n} \right )\; \;$

• Option 3)

$\; \frac{a}{4}cot\left ( \frac{\pi }{2n} \right )\; \;$

• Option 4)

$\; a\, cot\left ( \frac{\pi }{n} \right )$

As we learnt in  Trigonometric Ratios of Functions - - wherein   let AB represent any side of polygon with O as center  AB=a OC = in radius, OA=OB = circum radius = R Now,  Similarly,        Option 1) This is correct option Option 2) This is incorrect option Option 3) This is incorrect option Option 4) This is incorrect option
Engineering
117 Views   |

The upper 3/4th portion of a vertical pole subtends an angle $\dpi{100} tan^{-1}(3/5)$   at a point in the horizontal plane through its foot and at a distance 40 m from the foot. A possible height of the vertical pole is

• Option 1)

40 m

• Option 2)

60 m

• Option 3)

80 m

• Option 4)

20 m

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. -   Now In                 -----(ii) Similarly,    using (i) & (ii) solving,   Option 1) 40 m This is correct option Option 2) 60 m This is incorrect option Option 3) 80 m This is incorrect option Option 4) 20...
Engineering
106 Views   |

The vectors  $\dpi{100} \overrightarrow{AB}=3\hat{i}+4\hat{k}\; \; and\; \;\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}$  are the sides of a triangle $\dpi{100} ABC$.  The length of the median through  $\dpi{100} A$ is

• Option 1)

$\sqrt{72}\;$

• Option 2)

$\; \sqrt{33}\;$

• Option 3)

$\; \sqrt{288}\;$

• Option 4)

$\; \sqrt{18}\; \;$

As we learnt in Mid point formula - - wherein If  and  , position vector of mid-point of AB  Median is  Length of Mediam  Option 1) This option is incorrect. Option 2) This option is correct. Option 3) This option is incorrect. Option 4) This option is incorrect.
Engineering
138 Views   |

Let $\dpi{100} \vec{u}=\hat{i}+\hat{j},\; \; \vec{v}=\hat{i}-\hat{j}\; and\; \vec{w}=\hat{i}+2\hat{j}+3\hat{k}.$ If $\dpi{100} \hat{n}$ is a unit vector such that $\dpi{100} \vec{u}\cdot \hat{n}=0$ and $\dpi{100} \vec{v}\cdot \hat{n}=0,$ then $\dpi{100} \left |\vec{w}\cdot \vec{n} \right |$  is equal to

• Option 1)

1

• Option 2)

2

• Option 3)

3

• Option 4)

0

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      Scalar Product of two vectors (dot product) - - wherein  is the angle between the vectors    Since  is perpendicular to    Unit vector of  Now,  Option 1) 1 This option is incorrect. Option 2) 2 This option is...
Engineering
105 Views   |

$\dpi{100} \vec{a},\vec{b},\vec{c}$  are three vectors, such that $\dpi{100} \vec{a}+\vec{b}+\vec{c}=0,\; \; \left | \vec{b} \right |=2,\; \left | \vec{c} \right |=3,$  then $\dpi{100} \vec{a}\cdot \vec{b}\; +\;\vec{b}\cdot \vec{c}\; +\vec{c}\cdot \vec{a}\;$  is equal to

• Option 1)

–7

• Option 2)

7

• Option 3)

1

• Option 4)

0

As we learnt in Scalar Product of two vectors (dot product) - - wherein  is the angle between the vectors       Option 1) –7 This option is correct. Option 2) 7 This option is incorrect. Option 3) 1 This option is incorrect. Option 4) 0 This option is incorrect.
Engineering
110 Views   |

If $\dpi{100} \vec{u},\vec{v}\; and\; \vec{w}$ are three non-­coplanar vectors, then $\dpi{100} (\vec{u}+\vec{v}-\vec{w})\cdot (\vec{u}-\vec{v})\times (\vec{v}-\vec{w})$  equals

• Option 1)

$\vec{u}\cdot \vec{v}\times \vec{w}$

• Option 2)

$\vec{u}\cdot \vec{w}\times\vec{v}$

• Option 3)

$3u\cdot \vec{u}\times\vec{w}$

• Option 4)

0

As we learnt in  Scalar Triple Product - - wherein Scalar Triple Product of three vectors .         Option 1) This option is correct. Option 2) This option is incorrect. Option 3) This option is incorrect. Option 4) 0 This option is incorrect.
Engineering
152 Views   |

The degree and order of the differential equation of the family of all parabolas whose axis is $x$-axis, are respectively

• Option 1)

1, 2

• Option 2)

3, 2

• Option 3)

2, 3

• Option 4)

2, 1

As we learnt in  Formation of Differential Equations - A differential equation can be derived from its equation by the process of differentiation and other algebraical process of elimination -     Let the equation of Parabola be: Order is 2 Degree is 1   Option 1) 1, 2 This option is correct. Option 2) 3, 2 This option is incorrect. Option 3) 2, 3 This option is incorrect. Option 4) 2,...
Engineering
104 Views   |

The solution of the differential equation  $\dpi{100} (1+y^{2})+(x-e^{\tan^{ -1}y})\frac{dy}{dx}=0$   is

• Option 1)

$2xe^{\tan^{ -1}\, y}=e^{2\tan^{-1} \, y}+k$

• Option 2)

$xe^{\tan^{-1} \, y}=\tan ^{ -1}y+k$

• Option 3)

$xe^{2\tan^{-1} \, y}=e^{\tan^{-1} \, y}+k$

• Option 4)

$(x-2)=ke^{-\tan ^{-1}y}$

As we learnt in  Bernoulli's Equation - - wherein     Now put   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
139 Views   |

The lines $\dpi{100} \frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}\; and\; \frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}\;$   are coplanar if

• Option 1)

$k=1\; or-1$

• Option 2)

$k=0\; or-3$

• Option 3)

$k=3\; or-3$

• Option 4)

$k=0\; or-1$

As we leant in Condition for lines to be intersecting (cartesian form) - Their shortest distance should be 0 Also the condition for coplanar lines -    Condition for coplanarity =  k=0   or   -3   Option 1) Incorrect option Option 2) Correct option Option 3) Incorrect option Option 4) Incorrect option
Engineering
124 Views   |

The two lines $\dpi{100} x=ay+b,z=cy+d\; and\; x=a'y+b',z=c'y+d$  will be perpendicular, if and only if

• Option 1)

$aa'+bb'+cc'=0\; \;$

• Option 2)

$(a+a')(b+b')+(c+c')=0$

• Option 3)

$aa'+cc'+1=0$

• Option 4)

$aa'+bb'+cc'+1=0$

As we learnt in Condition of perpendicularity - Two lines will be perpendicular if  or  .     -    Equation of lines :   ;    Option 1) Incorrect option Option 2) Incorrect option Option 3) Correct option Option 4) Incorrect option
Engineering
119 Views   |

A tetrahedron has vertices at $\dpi{100} O$ (0, 0, 0) , $\dpi{100} A$ (1,2,1), $\dpi{100} B$ (2,1,3) and $\dpi{100} C$ (-1,1,2). Then the angle between the faces $\dpi{100} OAB$ and $\dpi{100} ABC$ will be

• Option 1)

$\cos ^{-1}(17/31)\;$

• Option 2)

$\; 30^{\circ}\;$

• Option 3)

$\; 90^{\circ}\;$

• Option 4)

$\; \cos ^{-1}(19/35)\;$

As we learnt in Ange between two lines in terms of direction cosines and direction ratios - If two lines having direction ratios a1,b1,c1 and a2,b2,c2 then the angle between them is given by If two lines have direction ratios as l1,m1,n1 and l2,m2,n2 then the angle betwee them is given by     -    Vector perpendicular to face OAB=  Similarily, vector perpendicular to face ABC=  Angle...
Engineering
106 Views   |

The radius of the circle in which the sphere $\dpi{100} x^{2}+y^{2}+z^{2}+2x-2y-4z-19=0$  is cut by the plane $\dpi{100} x+2y+2z+7=0$  is

• Option 1)

2

• Option 2)

3

• Option 3)

4

• Option 4)

1

As we learnt in Distance of a point from plane (Cartesian form) - The length of perpendicular from to the plane is given by    -  Center of sphere = (-1,1,2) radius=  Perpendicular distance =  Hence,  r= 3   Option 1) 2 Incorrect option Option 2) 3 Correct option Option 3) 4 Incorrect option Option 4) 1 Incorrect option
Engineering
144 Views   |

Two systems of rectangular axes have the same origin. If a plane cuts them at distance $\dpi{100} a,b,c\; \; and\; \; a',b',c'$ from the origin, then

• Option 1)

$\frac{1}{a^{2}}+\frac{1}{b^{2}}-\frac{1}{c^{2}}+\frac{1}{a'^{2}}+\frac{1}{b'^{2}}-\frac{1}{c'^{2}}=0$

• Option 2)

$\frac{1}{a^{2}}-\frac{1}{b^{2}}-\frac{1}{c^{2}}+\frac{1}{a'^{2}}-\frac{1}{b'^{2}}-\frac{1}{c'^{2}}=0$

• Option 3)

$\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}-\frac{1}{a'^{2}}-\frac{1}{b'^{2}}-\frac{1}{c'^{2}}=0$

• Option 4)

$\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{a'^{2}}+\frac{1}{b'^{2}}+\frac{1}{c'^{2}}=0$

As we learnt in Distance of a point from plane (Cartesian form) - The length of perpendicular from to the plane is given by    -    Clearly with reference to both rectangular axis, the distance of plane from origin doesn't change.    Option 1) Incorrect option Option 2) Incorrect option Option 3) Correct option Option 4) Incorrect option
Engineering
131 Views   |

The shortest distance from the plane $\dpi{100} 12x+4y+3z=327$  to the sphere  $\dpi{100} x^{2}+y^{2}+z^{2}+4x-2y-6z=155$  is

• Option 1)

$11\frac{3}{4}\;$

• Option 2)

$\; 13\;$

• Option 3)

$\; 39\;$

• Option 4)

$\; 26$

As we learnt in Distance of a point from plane (Cartesian form) - The length of perpendicular from to the plane is given by    -    Center of sphere = C(-2,1,3) and r= 13 Distance of center from  plane =  Hence shortest distance is 26-13= 13 units.   Option 1) Incorrect option Option 2) Correct option Option 3) Incorrect option Option 4) Incorrect option
Engineering
120 Views   |

if then

• Option 1)

$\alpha =a^{2}+b^{2},\beta =2ab$

• Option 2)

$\alpha =a^{2}+b^{2},\beta =a^{2}-b^{2}$

• Option 3)

$\alpha =2ab,\beta =a^{2}+b^{2}$

• Option 4)

$\alpha =a^{2}+b^{2},\beta =ab$

As we learnt in

Multiplication of matrices -

-

$A= \begin{bmatrix} a & b\\ b& a \end{bmatrix}$

$A^{2}= \begin{bmatrix} \alpha & \beta\\ \beta & \alpha \end{bmatrix}$

$A^{2}= \begin{bmatrix} a & b\\ b & a \end{bmatrix} \begin{bmatrix} a & b\\ b & a \end{bmatrix} = \begin{bmatrix} a^{2}+ b^{2} &2ab\\ 2ab & b^{2}+ a^{2} \end{bmatrix}$

so that $\alpha= a^{2}+ b^{2}$

$\beta= 2ab$

Option 1)

$\alpha =a^{2}+b^{2},\beta =2ab$

Correct

Option 2)

$\alpha =a^{2}+b^{2},\beta =a^{2}-b^{2}$

Incorrect

Option 3)

$\alpha =2ab,\beta =a^{2}+b^{2}$

Incorrect

Option 4)

$\alpha =a^{2}+b^{2},\beta =ab$

Incorrect

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Engineering
128 Views   |

If $\dpi{100} 1,\omega ,\omega ^{2}$ are the cube roots of unity, then

• Option 1)

1

• Option 2)

$\omega$

• Option 3)

$\omega ^{2}$

• Option 4)

0

As we learnt in

Value of determinants of order 3 -

-

$\Delta =\begin{vmatrix} 1 &\omega^{^{n}} &\omega^{2n} \\ \omega^{n} &\omega^{2n} &1 \\ \omega^{2n}&1 &\omega^{n} \end{vmatrix}$

$C_1 \rightarrow C_1+C_2+C_3$

$\Delta =\begin{vmatrix} 1+\omega^{^{n}} +\omega^{^{2n}} &\omega^{^{n}} &\omega^{2n} \\ 1+\omega^{^{n}} +\omega^{^{2n}} &\omega^{2n} &1 \\ 1+\omega^{^{n}} +\omega^{^{2n}}&1 &\omega^{n} \end{vmatrix}$

for  $n\epsilon N , 1+\omega^{^{n}} +\omega^{^{2n}}=0$

Option 1)

1

Incorrect option

Option 2)

$\omega$

Incorrect option

Option 3)

$\omega ^{2}$

Incorrect option

Option 4)

0

Correct option

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Engineering
110 Views   |

Let$\dpi{100} f\left ( a \right )= g\left ( a \right )= k$ and there $\dpi{100} n^{th}$ derivatives

$\dpi{100} f^{n}\left ( a \right ),g^{n}\left ( a \right )$ exist and are not equal for some $\dpi{100} n$. Further if

$\dpi{100} \lim_{x\rightarrow a}\frac{f\left ( a \right )g\left ( x \right )-f\left ( a \right )-g\left ( a \right )f\left ( x \right )+g\left ( a \right )}{g\left ( x \right )-f\left ( x \right )}= 4$

then the value $\dpi{100} k$ is

• Option 1)

$2$

• Option 2)

$1$

• Option 3)

$0$

• Option 4)

$4$

As we learnt L - Hospital Rule - - wherein     Option 1) Incorrect Option 2) Incorrect Option 3) Incorrect Option 4) Correct
Engineering
125 Views   |

The real number $\dpi{100} x$ when added to its inverse gives the minimum value of the sum at $\dpi{100} x$ equal to

• Option 1)

1

• Option 2)

-1

• Option 3)

-2

• Option 4)

2

As we learnt in  Rate Measurement - Rate of any of variable with respect to time is rate of measurement. Means according to small change in time how much other factors change is rate measurement: - wherein Where dR / dt  means Rate of change of radius.     Option 1) 1 This option is correct. Option 2) -1 This option is incorrect. Option 3) -2 This option is incorrect. Option...
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