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Engineering
103 Views   |

The domain of $\dpi{100} \sin ^{-1}\left [ \log _{3}\left ( \frac{x}{3} \right ) \right ]$ is

• Option 1)

$\left [ 1,9 \right ]$

• Option 2)

$\left [ -1,9 \right ]$

• Option 3)

$\left [ -9,1 \right ]$

• Option 4)

$\left [ -9,-1 \right ]$

As we learnt in  Domains and Ranges of Inverse Trigonometric Functions - For  Domain  Range  -    For expression to be defined,     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
132 Views   |

The number of solution of $tanx+secx=2cosx\; in\; \left [ 0,2\pi \right )\; is$

• Option 1)

2

• Option 2)

3

• Option 3)

0

• Option 4)

1

As we learnt in  Trigonometric Equations - The equations involving trigonometric function of unknown angles are known as trigonometric equations. - wherein e.g.      multiplying both sides by , we get now   given from solution now L.H.S of equ.(i) when   Thus Option 1) 2 This is incorrect option Option 2) 3 This is correct option Option 3) 0 This is incorrect option Option 4) 1 This...
Engineering
108 Views   |

$cot^{-1}\left [ (cos\, \alpha )^{\frac{1}{2}} \right ]+tan^{-1}\left [ (cos\, \alpha )^{\frac{1}{2}} \right ]=x\; \; then\; sinx=$

• Option 1)

1

• Option 2)

$\dpi{100} cot^{2}(\alpha /2)\;$

• Option 3)

$\dpi{100} \; \; tan\alpha \;$

• Option 4)

$\dpi{100} \; \; cot(\alpha /2)$

As we learnt in  Important Results of Inverse Trigonometric Functions - - wherein When      let Then [identity] Thus, Option 1) 1 This is correct option Option 2) This is incorrect option Option 3) This is incorrect option Option 4) This is incorrect option
Engineering
128 Views   |

If $\dpi{100} \left | \vec{a} \right |=4,\left | \vec{b} \right |=2,$  and the angle between $\dpi{100} \vec{a}\; \; and \; \; \vec{b} \; \; is \; \; \frac{\pi }{6}\; then\; \; (\vec{a}\times \vec{b} )^{2}$ is equal to

• Option 1)

48

• Option 2)

16

• Option 3)

$\vec{a}$

• Option 4)

none of these

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      Option 1) 48 This option is incorrect. Option 2) 16 This option is correct. Option 3) This option is incorrect. Option 4) none of these This option is incorrect.
Engineering
103 Views   |

If   $\dpi{100} \vec{a},\vec{b},\vec{c}$  are vectors show that $\dpi{100} \vec{a}+\vec{b}+\vec{c}=0$ and $\dpi{100} \left | \vec{a} \right |=7,\left | \vec{b} \right |=5,\left | \vec{c} \right |=3,$ then angle between vector $\dpi{100} \vec{b}\; \; and \; \; \vec{c}$ is

• Option 1)

$60^{\circ}\;$

• Option 2)

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

• Option 3)

$\; 45^{\circ}\;$

• Option 4)

$\; 90^{\circ}$

As we learnt in  Scalar Product of two vectors (dot product) - - wherein  is the angle between the vectors     We need to find angle between and Squaring both sides 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
126 Views   |

$\dpi{100} \vec{a}=3\hat{i}-5\hat{j}\;\; and\;\; \vec{b}=6\hat{i}+3\hat{j}$  are two vectors and $\dpi{100} \vec{c}$ is a vector such that $\dpi{100} \vec{c}=\vec{a}\times \vec{b}$ then $\dpi{100} \left | \vec{a} \right |:\left |\vec{b} \right |:\left | \vec{c} \right |=$

• Option 1)

$\sqrt{34}:\sqrt{45}:\sqrt{39}$

• Option 2)

$\sqrt{34}:\sqrt{45}:39$

• Option 3)

$34:39:45$

• Option 4)

$39:35:34$

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      Ratio =   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
98 Views   |

If   $\dpi{100} y=(x+\sqrt{1+x^{2}})^{n},\; then\; (1+x^{2})\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}$     is

• Option 1)

$n^{2}y\;$

• Option 2)

$\; -n^{2}y\;$

• Option 3)

$\; -y\;$

• Option 4)

$\; 2x^{2}y$

As we learnt in  Differential Equations - An equation involving independent variable (x), dependent variable (y) and derivative of dependent variable with respect to independent variable  - wherein eg:        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
102 Views   |

The solution of the equation  $\dpi{100} \frac{d^{2}y}{dx^{2}}=e^{-2x}$

• Option 1)

$\frac{1}{4}e^{-2x}\;$

• Option 2)

$\; \frac{1}{4}e^{-2x}+cx+d\;$

• Option 3)

$\; \; \frac{1}{4}e^{-2x}+cx^{2}+d\;$

• Option 4)

$\; \; \frac{1}{4}e^{-2x}+c+d\;$

As we learnt in  Solution of differential equations - A function y =f(x) is a solution of differential equation, if the substitution of f(x) and its derivative (s) in differential equation reduces it to an identity. -   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
123 Views   |

The order and degree of the differential equation  $\dpi{100} \left ( 1+3\frac{dy}{dx} \right )^{\frac{2}{3}}=4\frac{d^{3}y}{dx^{3}}$   are

• Option 1)

$1,\frac{2}{3}\;$

• Option 2)

$\; \; 3,1\;$

• Option 3)

$\; 3,3\;$

• Option 4)

$\; 1,2$

As we learnt in  Order of a Differential Equation - The order of a differential equation is order of highest order occuring in differential equation - wherein order of    is 2.      and   Degree of a Differential Equation - Degree of Highest order differential coefficient appearing in it, provided it can be expressed as a polynomial equation in derivatives - wherein Degree =...
Engineering
115 Views   |

The d.r. of normal to the plane through (1, 0, 0), (0, 1, 0) which makes an angle $\dpi{100} \pi /4$ with plane $\dpi{100} x+y=3$  are

• Option 1)

$1,\sqrt{2},1$

• Option 2)

$1,1,\sqrt{2}$

• Option 3)

$1,1,2$

• Option 4)

$\sqrt{2},1,1$

As we learnt in  Direction Ratios - If l,m,n are direction cosines then three numbers a,b,c propotional to l,m,n are called direction cosines.    -    Plane through (1,0,0) is  It passes though (0,1,0)  We get,  DRs of normal are    Option 1) Incorrect option Option 2) Correct option Option 3) Incorrect option Option 4) Incorrect option
Engineering
119 Views   |

If $\dpi{100} l,m,n\; are\; the\; p^{th},q^{th}and\; r^{th}\; term\; of\; a\; GP,all$

positive, then   equals

• Option 1)

–1

• Option 2)

2

• Option 3)

1

• Option 4)

0

As we learnt in

Property of determinant -

If each element in a row ( or column ) of a determinant is written as the sum of two or more terms then the determinant can be written as the sum of two or more determinants

- wherein

Given that

$l= a\times R^{p-1}$

$m= a\times R^{q-1}$

$n= a\times R^{r-1}$

$\therefore \log l= \log a +(p-1) \log R$

$\log m= \log a +(q-1) \log R$

$\log n= \log a +(r-1) \log R$

so that $\begin{vmatrix} \log a & p &1 \\ \log a& q & 1\\ \log a & r & 1 \end{vmatrix} + \log R \begin{vmatrix} p-1 &p &1 \\ q-1&q &1 \\ r-1&r &1 \end{vmatrix}$

$\log a\begin{vmatrix} \1 & p &1 \\ \1& q & 1\\ \1 & r & 1 \end{vmatrix} + \log a \begin{vmatrix} p-1 &p-1 &1 \\ q-1&q-1 &1 \\ r-1&r-1 &1 \end{vmatrix}$

0+0 = 0

Option 1)

–1

Incorrect option

Option 2)

2

Incorrect option

Option 3)

1

Incorrect option

Option 4)

0

Correct option

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Engineering
101 Views   |

If a>0 and discriminant of $\dpi{100} ax^{2}+2bx+c$ is -ve, then

is

• Option 1)

+ve

• Option 2)

$(ac-b^{2})(ax^{2}+2bx+c)$

• Option 3)

-ve

• Option 4)

0

As we learnt in  Value of determinants of order 3 - -   So that it is -ve.  Option 1) +ve This option is incorrect. Option 2) This option is incorrect. Option 3) -ve This option is correct. Option 4) 0 This option is incorrect.
Engineering
114 Views   |

If $\dpi{100} 2a+3b+6c=0\left ( a,b,c\: \in R\right )$

then the quadratic equation $\dpi{100} ax^{2}+bx+c= 0$ has

• Option 1)

At least one in $\left ( 0.1 \right )$

• Option 2)

At least one root in  $\left [ 2,3 \right ]$

• Option 3)

At least one root in $\left [ 4,5 \right ]$

• Option 4)

none of these

As we learnt in  Rolle's Theorems - Let f(x) be a function of x subject to the following conditions. 1.  f(x) is continuous function of     2.  f'(x) is exists for every point :   3.   -               So at least one root between (0,1) Option 1) At least one in   This option is correct Option 2) At least one root in  This option is incorrect Option 3) At least one root in This option is...
Engineering
146 Views   |

Let$\dpi{100} f\left ( 2 \right )= 4\: and \: {f}'\left ( 2 \right )= 4$ then

$\dpi{100} \lim_{x\rightarrow 2}\frac{xf\left ( 2 \right )-2f\left ( x \right )}{x-2}$ equals

• Option 1)

$2$

• Option 2)

$-2$

• Option 3)

$-4$

• Option 4)

$3$

As we learnt in  L - Hospital Rule - - wherein             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
102 Views   |

$\dpi{100} \lim_{x\rightarrow \infty }\left ( \frac{x^{2}+5x+3}{x^{2}+x+3} \right )^{\frac{1}{x}}$

• Option 1)

$e^{4}$

• Option 2)

$e^{2}$

• Option 3)

$e^{3}$

• Option 4)

$1$

As we learnt in  1 to the power of infinity Form - -     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
116 Views   |

If $\dpi{100} f\left ( x+y \right )=f\left ( x \right ).f\left ( y \right )\forall\: x,y\: and\: f\left ( 5 \right )= 2,f{}'\left ( 0 \right )= 3,$

Then $\dpi{100} {f}'\left ( 5 \right )$ is

• Option 1)

$0$

• Option 2)

$1$

• Option 3)

$6$

• Option 4)

$2$

As we learnt in  Condition for differentiable - A function  f(x) is said to be differentiable at    if      both exist and are equal otherwise non differentiable -     Now,                                            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   |

The maximum distance from origin of a point on the curve   $\dpi{100} x=a\sin t-b\sin \left ( \frac{at}{b} \right )$

$\dpi{100} y=a\cos t-b\cos \left ( \frac{at}{b} \right ),both\; a,b> 0\; is$

• Option 1)

$a-b\; \;$

• Option 2)

$\; \; a+b\;$

• Option 3)

$\; \sqrt{a^{2}+b^{2}}\;$

• Option 4)

$\; \; \; \sqrt{a^{2}-b^{2}}$

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.        for min  Option 1) Correct Option 2) Incorrect Option 3) Incorrect Option 4) Incorrect
Engineering
141 Views   |

$\dpi{100} \lim_{x\rightarrow 0}\frac{\sqrt{1-\cos 2x}}{\sqrt{2}x}$ is

• Option 1)

$1$

• Option 2)

$-1$

• Option 3)

$0$

• Option 4)

does not exist.

As we learnt in  Evalution of Trigonometric limit - -   for  it is 1 for  it is -1 So limit does not exist. Option 1) Correct Option 2) Incorrect Option 3) Incorrect Option 4) does not exist. Incorrect
Engineering
128 Views   |

If $\dpi{100} f(1)=1,f'(1)=2\; then\;$      is

• Option 1)

2

• Option 2)

4

• Option 3)

1

• Option 4)

1/2

As we learnt in  L - Hospital Rule - - wherein    If   Then     Option 1) 2 Correct Option 2) 4 Incorrect Option 3) 1 Incorrect Option 4) 1/2 Incorrect
Engineering
95 Views   |

$\dpi{100} f$ is defined in [ -5 , 5 ] as

• Option 1)

$\dpi{100} f(x)\; is \; continuous\; at\; every\; x,except\; x=0$

• Option 2)

$\dpi{100} f(x)\; is \; discontinuous\; at\; every\; x,except\; x=0$

• Option 3)

$\dpi{100} f(x)\; is \; continuous\; everywhere$

• Option 4)

$\dpi{100} f(x)\; is \; discontinuous\; everywhere$

As we learnt in  Condition for discontinuity - limit of function at x = a does not exist. limit exist but not equal to  x = a - ab  both  are 0 so thas  continuous at     Option 1) Incorrect   Option 2) Correct Option 3) Incorrect   Option 4) Incorrect
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