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Engineering
116 Views   |

The largest interval lying in $\left ( \frac{-\pi }{2} ,\frac{\pi }{2}\right )$ for which the function,

$f\left ( x \right )= 4^{-x^{2}}+\cos ^{-1}\left ( \frac{x}{2} -1\right )+\log \left ( \cos x \right ),$ is defined, is

• Option 1)

• Option 2)

• Option 3)

$\left [ 0,\pi \right ]$

• Option 4)

$\left ( -\frac{\pi}{2} ,\frac{\pi}{2}\right )$

As we learnt in  Domains and Ranges of Inverse Trigonometric Functions - For  Domain  Range  -     Now  always differed   is  differed if   log    is difred if  Combining  Option 1) Incorrect Option 2) Correct Option 3) Incorrect Option 4) Incorrect
Engineering
106 Views   |

If $\sin ^{-1}\left ( \frac{x}{5} \right )+cosec^{-1}\left ( \frac{5}{4} \right )= \frac{\pi }{2},$ then the values of $x$ is

• Option 1)

$4$

• Option 2)

$5$

• Option 3)

$1$

• Option 4)

$3$

As we learnt in  Important Results of Inverse Trigonometric Functions - - wherein When          taking cos on both sides,        Option 1) Incorrect Option 2) Incorrect Option 3) Incorrect Option 4) Correct
Engineering
126 Views   |

The differential equation of all circles passing through the origin and having their centres on the $\dpi{100} x$-axis is

• Option 1)

$y^{2}=x^{2}+2xy\frac{dy}{dx}$

• Option 2)

$y^{2}=x^{2}-2xy\frac{dy}{dx}$

• Option 3)

$x^{2}=y^{2}+xy\frac{dy}{dx}$

• Option 4)

$x^{2}=y^{2}+3xy\frac{dy}{dx}$

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 circle is   where (a,0) is center and a is radius.     Option 1) Correct option Option 2) Incorrect option Option 3) Incorrect option Option 4) Incorrect option
Engineering
106 Views   |

If (2, 3, 5) is one end of a diameter of the sphere $\dpi{100} x^{2}+y^{2}+z^{2}-6x-12y-2z+20=0,$ then the coordinates of the other end of the diameter are

• Option 1)

(4, 3, 5)

• Option 2)

(4, 3, –3)

• Option 3)

(4, 9, –3)

• Option 4)

(4, –3, 3)

As we learnt in  Section Formula - 1)    Internal Division 2)    External Division 3)    Mid Point Formula - wherein    Coordinate of centre of sphere= (3,6,1) Let other end be (x,y,z) Option 1) (4, 3, 5) Incorrect Option   Option 2) (4, 3, –3) Incorrect Option   Option 3) (4, 9, –3) Correct Option   Option 4) (4, –3, 3) Incorrect Option
Engineering
150 Views   |

Let $\dpi{100} \vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+2\hat{k}\; \; and\; \;\vec{c}=x\hat{i}+(x-2)\hat{j}-\hat{k}.$ If the vectors $\dpi{100} \vec{c}$ lies in the plane of $\dpi{100} \vec{a}$ and $\dpi{100} \vec{b}$ , then $\dpi{100} x$ equals

• Option 1)

– 4

• Option 2)

–2

• Option 3)

0

• Option 4)

1

As we learnt in  Plane passing through a point and parallel to two given vectors (Cartesian form) - Let the plane passes through and parallel to vectors having DR's , then the plane is given by   - wherein    Normal vector of plane with   is     Option 1) – 4 Incorrect Option   Option 2) –2 Correct Option   Option 3) 0 Incorrect Option   Option 4) 1 Incorrect Option
Engineering
146 Views   |

Let $\dpi{100} L$ be the line of intersection of the planes $\dpi{100} 2x+3y+z=1\; \; and\; \; x+3y+2z=2.$ If $\dpi{100} L$ makes an angle $\dpi{100} \alpha$ with the positive $\dpi{100} x$-axis,then $\dpi{100} \cos \alpha$ equals

• Option 1)

1

• Option 2)

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

• Option 3)

$\frac{1}{\sqrt{3}}$

• Option 4)

$\frac{1}{2}$

As we learnt in  Direction Cosines - i)    ii)    If OP =r then the co-ordinates of P will be (lr,mr,nr) iii)    Direction cosines of X-axis are (1,0,0) iv)    Direction cosines of Y-axis are (0,1,0) v)    Direction cosines of Z-axis are (0,0,1) -    Let DCs of L be (l,m,n) then 2l+3m+2n=0 and l + 3m + 2n = 0  On solving Option 1) 1 Incorrect Option   Option 2) Incorrect Option   Option...
Engineering
135 Views   |

Let   If $\left | A^{2} \right |=25,then\; \left | \alpha \right |\; equals$

• Option 1)

1/5

• Option 2)

5

• Option 3)

52

• Option 4)

1

As we learnt in

Multiplication of matrices -

-

$A=\begin{vmatrix} 5 &5\alpha &\alpha \\ 0&\alpha &5\alpha \\ 0&0 & 5\end{vmatrix}$

$|A^{2}|= 25$

$A^{2}=\begin{vmatrix} 5 &5\alpha &\alpha \\ 0&\alpha &5\alpha \\ 0&0 & 5\end{vmatrix}\begin{vmatrix} 5 &5\alpha &\alpha \\ 0&\alpha &5\alpha \\ 0&0 & 5\end{vmatrix}$

$A^{2}=\begin{vmatrix} 25 &25\alpha+5\alpha^{2} &10\alpha+25\alpha^{2} \\ 0&\alpha^{2} &5\alpha^{2}+25\alpha\\ 0&0 & 25\end{vmatrix}$

$|A^{2}|=25\begin{vmatrix} 25 &25\alpha+5\alpha^{2} \\ 0&\alpha^{2} \end{vmatrix}= 25\, \, \, \, \, given$

$\therefore 25\alpha ^{2}= 1 ,\, \, \, \alpha ^{2}= \frac{1}{25},\, \, \, \alpha= \frac{1}{5}$

Option 1)

1/5

Correct Option

Option 2)

5

Incorrect Option

Option 3)

52

Incorrect Option

Option 4)

1

Incorrect Option

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Engineering
121 Views   |

If

• Option 1)

divisible by $x$ but not $y$

• Option 2)

divisible by $y$ but not $x$

• Option 3)

divisible by neither $x$ nor  $y$

• Option 4)

divisible by both $x$ and   $y$

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

$D=\begin{vmatrix} 1& 1&1\\ 1 & 1+x &1\\ 1&1 &1+y \end{vmatrix}$

$=1\left ( 1+x+y+xy-1 \right )-1\left ( 1+y-1 \right )+1\left ( 1-1-x \right )= xy$

= x + y + xy - y - x

= xy

which is divisible by both x and y

Option 1)

divisible by $x$ but not $y$

Incorrect Option

Option 2)

divisible by $y$ but not $x$

Incorrect Option

Option 3)

divisible by neither $x$ nor  $y$

Incorrect Option

Option 4)

divisible by both $x$ and   $y$

Correct Option

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Engineering
129 Views   |

If $p \: and \: q$ are positive real numbers such that $p^{2} +q^{2}=1$, then the maximum value of $\left ( p +q \right )$ is

• Option 1)

$\frac{1}{2}$

• Option 2)

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

• Option 3)

$\sqrt{2}$

• 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) This option is incorrect. Option 2) This option is incorrect. Option 3) This option is correct. Option...
Engineering
137 Views   |

The function $f:R/\left \{ 0 \right \}\rightarrow R$ given by

$f\left ( x \right )= \frac{1}{x}-\frac{2}{e^{2x}-1}$ can be made continuous at  $x= 0$ by defining $f \left ( 0 \right )$ as

• Option 1)

$0$

• Option 2)

$1$

• Option 3)

$2$

• Option 4)

$-1$

As we learnt in  Evaluation of Exponential Limits - - wherein     We know that Put  Option 1) Incorrect Option 2) Correct Option 3) Incorrect Option 4) Incorrect
Engineering
115 Views   |

Let $f:R\rightarrow R$ be a function defined by $f\left ( x \right )= min\left \{ x+1,\left | x \right |+1 \right \}$

Then which of the following is true ?

• Option 1)

$f\left ( x \right )$ is differentiable everywhere

• Option 2)

$f\left ( x \right )$ is not differentiable at $x= 0$

• Option 3)

$f\left ( x \right )$ $\geq 1\: for \: all\: x\in R$

• Option 4)

$f\left ( x \right )$ is not differentiable at $x= 1$

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 -     So f(x) is differentiable every where. Option 1) is differentiable everywhere Correct Option 2) is not differentiable at Incorrect Option 3) Incorrect Option 4) is not differentiable at Incorrect
Engineering
101 Views   |

A  value of $\large c$ for which conclusion of Mean Value Theorem holds for the function

$f\left ( x \right )= \log _{e}x$  on the interval $\left [ 1,3 \right ]$ is

• Option 1)

$\log_{3}e$

• Option 2)

$\log_{e}3$

• Option 3)

$2\log_{3}e$

• Option 4)

$\frac{1}{2}\log_{e}3$

As we learnt in Lagrange's mean value theorem - If  a  function  f(x)  1.   is continuous in the closed interval [a,b] and  2.   is differentiable in the open interval [a, b] then  -       Option 1) Incorrect Option 2) Incorrect Option 3) Correct Option 4) Incorrect
Engineering
138 Views   |

For the Hyperbola   $\dpi{100} \frac{x^{2}}{\cos ^{2}\alpha }-\frac{y^{2}}{\sin ^{2}\alpha }=1,$  which of the following remains constant when $\dpi{100} \alpha$ varies ?

• Option 1)

abscissae of vertices

• Option 2)

abscissae of foci

• Option 3)

eccentricity

• Option 4)

directrix

As we learnt in  Coordinates of foci - - wherein For the ellipse       Here  Coordnita of focus = ae Option 1) abscissae of vertices Incorrect Option 2) abscissae of foci Correct Option 3) eccentricity Incorrect Option 4) directrix Incorrect
Engineering
117 Views   |

The equation of a tangent to the parabola $\dpi{100} y^{2}=8x\; is\; y=x+2.$ The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

• Option 1)

(2, 4)

• Option 2)

(–2, 0)

• Option 3)

(–1, 1)

• Option 4)

(0, 2)

As we learnt in    Equation of tangent - - wherein Tengent to is slope form.       Tangent is  Slope of tangent,  When two lines are perpendicular to each other, Let the other tangent be  Now we have    touching  So,  Point is (-2, 0)   Option 1) (2, 4) This option is incorrect. Option 2) (–2, 0) This option is correct. Option 3) (–1, 1) This option is incorrect. Option 4) (0, 2) This...
Engineering
108 Views   |

Let $\dpi{100} A(h,k),B(1,1)\; and\; C(2,1)$ be the vertices of a right angled triangle with $\dpi{100} AC$ as its hypotenuse. If the area of the triangle is 1 square unit, then the set of values which $\dpi{100} ' k'$ can take is given by

• Option 1)

{–1, 3}

• Option 2)

{–3, –2}

• Option 3)

{1, 3}

• Option 4)

{0, 2}

As we learnt in    Distance formula - The distance between the point is - wherein   Also,   Option 1) {–1, 3} This option is correct. Option 2) {–3, –2} This option is incorrect. Option 3) {1, 3} This option is incorrect. Option 4) {0, 2} This option is incorrect.
Engineering
120 Views   |

Let $\dpi{100} P=(-1,0),Q=(0,0)\; and\; R=(3,3\sqrt{3})$ be three points. The equation of the bisector of the angle $\dpi{100} PQR$  is

• Option 1)

$\frac{\sqrt{3}}{2}x+y=0\;$

• Option 2)

$\; x+\sqrt{3y}=0\;$

• Option 3)

$\; \sqrt{3}x+y=0\;$

• Option 4)

$\; x+\frac{\sqrt{3}}{2}y=0$

As learnt in Slope of a line - - wherein Slope of line joining A(x1,y1) and  B(x2,y2) .     Slope of line  is  Equation is     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
92 Views   |

If one of the lines of  $\dpi{100} my^{2}+(1-m^{2})xy-mx^{2}=0$ is a bisector of the angle between the lines $\dpi{100} xy=0,\; \; then \; \; m\; \; is$

• Option 1)

1

• Option 2)

2

• Option 3)

–1/2

• Option 4)

–2

As we learnt in    Slope of a line - If  is the angle at which a straight line is inclined to a positive direction of x-axis, then the slope is defined by . - wherein     So slope of line equally inclined is: Option 1) 1 This option is correct. Option 2) 2 This option is incorrect. Option 3) –1/2 This option is incorrect. Option 4) –2 This option is incorrect.
Engineering
157 Views   |

Consider a family of circles which are passing through the point (–1, 1) and are tangent to $\dpi{100} x-$axis. If  $\dpi{100} (h,k)$ are the coordinate of the centre of the circles, then the set of values of $\dpi{100} k$ is given by the interval

• Option 1)

$-\frac{1}{2}\leq k\leq \frac{1}{2}\;$

• Option 2)

$\; k\leq \frac{1}{2}\;$

• Option 3)

$\; 0\leq k\leq \frac{1}{2}\;$

• Option 4)

$\; k\geq \frac{1}{2}$

As we learnt in General form of a circle -   - wherein centre = radius =  Equation of circle with center  is: Radius of circle is   We get,   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
114 Views   |

The normal to a curve at $\dpi{100} P(x,y)$ meets the $\dpi{100} x-$axis at $\dpi{100} G$ . If the distance of  $\dpi{100} G$ from the origin is twice the abscissa of $\dpi{100} P$ , then the curve is a

• Option 1)

circle

• Option 2)

hyperbola

• Option 3)

ellipse

• Option 4)

parabola

As we learnt in Slope of a line - - wherein Slope of line joining A(x1,y1) and  B(x2,y2) .    If slope of tangent =  We get      as equation of normal. If normal meets x-axis again, we get: Distance of QG = 2 (abscissa of P) i.e.,     Generalising       which is a Hyperbola. Option 1) circle This option is incorrect. Option 2) hyperbola This option is correct. Option 3) ellipse This...
Engineering
121 Views   |

Which one of the following is the strongest base in aqueous solution?

• Option 1)

Methylamine

• Option 2)

Trimethylamine

• Option 3)

Aniline

• Option 4)

Dimethylamine

4

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