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Engineering
1070 Views   |

The speed of sound in oxygen  (O2) at a certain temperature is  460 ms-1 The speed of sound in helium (He) at the same temperature will be (assume both gases to be ideal)

• Option 1)

• Option 2)

• Option 3)

• Option 4)

Laplace formula (correct value )

$= \sqrt{\frac{\gamma P}{\rho }}$

-

$\nu = \sqrt{\frac{\gamma P}{\rho }}= \sqrt{\frac{\gamma RT}{M}}$

$\gamma \: for\: O_{2}= 1+\frac{2}{5}=\frac{7}{5}$

$\gamma \: for\: H\! e= 1+\frac{2}{3}=\frac{5}{3}$

$\frac{\nu _{2}}{\nu _{1}}= \sqrt{\frac{\gamma _{He}}{4} \times \frac{32}{\gamma _{O_{2}}}}\times 460$$= \sqrt{\frac{5}{3}\times \frac{1}{4}\times \frac{32\times 5}{7}} \times 460=1420m/s$

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Engineering
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AB is a vertical pole with  B at the ground level and A at the top. A man finds that the angle of elevation of the point ,A  from a certain point C   on the ground is $60^{\circ}$. He moves away from the pole along the line BC  to the point such that  CD=7m. From D The angle of elevation of the point A   is 45°. Then the height of the pole is

• Option 1)

$\frac{7\sqrt{3}}{2}\frac{1}{\sqrt{3}+1}m$

• Option 2)

$\frac{7\sqrt{3}}{2}\frac{1}{\sqrt{3}-1}m$

• Option 3)

$\frac{7\sqrt{3}}{2}\left ( \sqrt{3}+1 \right )m$

• Option 4)

$\frac{7\sqrt{3}}{2}\left ( \sqrt{3}-1 \right )m$

As we leant 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 height be h   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
100 Views   |

The value of is $\cot \left (cosec^{-1}\frac{5}{3} +\tan ^{-1}\frac{2}{3}\right )$

• Option 1)

$\frac{5}{17}$

• Option 2)

$\frac{6}{17}$

• Option 3)

$\frac{3}{17}$

• Option 4)

$\frac{4}{17}$

As we learnt in  Results of Compound Angles - - wherein Where A and B are two angles.      Now                     Thus expression becomes                                   Option 1) Incorrect Option 2) Correct Option 3) Incorrect Option 4) Incorrect
Engineering
98 Views   |

The differential equation of the family of circles with fixed radius 5 units and centre on the line $\dpi{100} y=2$ is

• Option 1)

$(x-2)^{2}y'^{2}=25-(y-2)^{2}$

• Option 2)

$(x-2)y'^{2}=25-(y-2)^{2}$

• Option 3)

$(y-2)y'^{2}=25-(y-2)^{2}$

• Option 4)

$(y-2)^{2}y'^{2}=25-(y-2)^{2}$

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 center is (a,2) and radius is 5.   Option 1) Incorrect option Option 2) Incorrect option Option 3) Incorrect option Option 4) Correct option
Engineering
99 Views   |

The solution of the differential equation   $\dpi{100} \frac{dy}{dx}=\frac{x+y}{x}$    satisfying the condition $\dpi{100} y(1)=1$  is

• Option 1)

$y=x\; \ln \; x+x$

• Option 2)

$y= \ln \; x+x$

• Option 3)

$y=x\; \ln \; x+x^{2}$

• Option 4)

$y=x\, e^{(x-1)}$

As we learnt in    Linear Differential Equation - - wherein P, Q are functions of x alone.     Put       Solution is  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
99 Views   |

The line passing through the points $\dpi{100} (5,1,a)\; and\; (3,b,1)\;$ crosses the $\dpi{100} yz$-plane at the point $\dpi{100} \left ( 0,\frac{17}{2},\frac{-13}{2} \right ).$  Then

• Option 1)

$a=8,b=2$

• Option 2)

$a=2,b=8$

• Option 3)

$a=4,b=6$

• Option 4)

$a=6,b=4$

As we learnt in  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    Equation of line is If it crosses y-z plane, x=0 Option 1) Incorrect Option Option 2) Incorrect Option Option 3) Incorrect Option Option 4) Correct Option
Engineering
147 Views   |

If the straight lines $\dpi{100} \frac{x-1}{k}=\frac{y-2}{2}=\frac{z-3}{3}\; and\; \frac{x-2}{3}=\frac{y-3}{k}=\frac{z-1}{2}\;$  intersect at a point, then the integer $\dpi{100} k$ is equal to

• Option 1)

– 2

• Option 2)

– 5

• Option 3)

5

• Option 4)

2

As we learnt in  Condition for lines to be intersecting (cartesian form) - Their shortest distance should be 0 Also the condition for coplanar lines -    If two lines intersect Option 1) – 2 Incorrect Option   Option 2) – 5 Correct Option   Option 3) 5 Incorrect Option   Option 4) 2 Incorrect Option
Engineering
230 Views   |

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

• Option 1)

If $det\; A=\pm 1,then\; A^{-1}$ need not exist

• Option 2)

If $det\; A=\pm 1,then\; A^{-1}$  exists  but all its entries are not necessarily integers

• Option 3)

If $det\; A\neq \pm 1,then\; A^{-1}$ exists and all its entries are non­integers

• Option 4)

If $det\; A=\pm 1,then\; A^{-1}$ exists and all its entries are integers

As we learnt in  Inverse of a matrix -   -     exist and all its entries are non integers. Option 1) If need not exist Incorrect Option   Option 2) If   exists  but all its entries are not necessarily integers Incorrect Option   Option 3) If exists and all its entries are non­integers Incorrect Option   Option 4) If exists and all its entries are integers Correct Option
Engineering
141 Views   |

Let a, b, c, be any real numbers. Suppose that there are real numbers x, y, z not all zero such that $x=cy+bz,y=az+cx\; and\; z=bx+ay.$ $Then\; a^{2}+b^{2}+c^{2}+2abc$ is equal to

• Option 1)

1

• Option 2)

2

• Option 3)

-1

• Option 4)

0

As we learnt in

Cramer's rule for solving system of linear equations -

When $\Delta =0$ and atleast one of   $\Delta_{1},\Delta _{2} and \Delta _{3}$  is non-zero , system of equations has no solution

- 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

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

$x-cy-bz=0$

$-cx+y-az=0$

$-bx-ay+z=0$

$= > D=\begin{vmatrix} 1 &-c &-b\\ -c &1&-a \\ -b&-a& 1\end{vmatrix}= 0$

$= >1\left (1-a^{2} \right )+c \left (-c-ab\right )-b\left (ac+b\right )= 0$

$= 1-a^{2}-c^{2}-abc-abc-b^{2}= 0$

$= a^{2}+b^{2}+c^{2}+2abc= 1$

Option 1)

1

Correct Option

Option 2)

2

Incorrect Option

Option 3)

-1

Incorrect Option

Option 4)

0

Incorrect Option

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Engineering
87 Views   |

Directions : Question is Assertion - ­Reason type. This question contains two statements : Statement - 1 (Assertion) and Statement­-2 (Reason). Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice.

Question: Let A be a 2 × 2 matrix with real entries. Let $I$ be the 2 × 2 identity matrix. Denote by $tr(A)$ , the sum of diagonal entries of $A$ . Assume that $A^{2}=I$.

Statement - 1 :  If  $A\neq I\; and\; A\neq -I,then\; det\; A=-1.$

Statement - 2 : If $A\neq I\; and\; A\neq -I,then\; tr(A)\neq 0$

• Option 1)

Statement -1 is true, Statement­-2 is false

• Option 2)

Statement-­1 is false, Statement­-2 is true ;

• Option 3)

Statement­-1 is true, Statement­-2 is true Statement­-2 is a correct explanation for Statement­-1

• Option 4)

Statement­-1 is true, Statement­-2 is true Statement-­2 is not a correct explanation for Statement­-1

Engineering
117 Views   |

Suppose the cubic $x^{3}-px+q$ has three distinct real roots

where $p> 0\: and \: q> 0$. Then which one of the following holds?

• Option 1)

$The\: cubic\: has \: maxima \: at \: both\sqrt{\frac{p}{3}}\: and -\sqrt{\frac{p}{3}}$

• Option 2)

$The\: cubic\: has \:minima \: at \: \sqrt{\frac{p}{3}}\: and \: maxima\: at-\sqrt{\frac{p}{3}}$

• Option 3)

$The\: cubic\: has \:minima \: at \: -\sqrt{\frac{p}{3}}\: and \: maxima\: at\sqrt{\frac{p}{3}}$

• Option 4)

$The\: cubic\: has \: minima \: at \: both\sqrt{\frac{p}{3}}\: and -\sqrt{\frac{p}{3}}$

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.       at min   max Option 1) Incorrect option Option 2) Correct option Option 3) Incorrect option Option 4) Incorrect option
Engineering
136 Views   |

Let

Then which one of the following is true?

• Option 1)

$f\: is \: di\! f\! \! ferentiable\: at\: x= 1\: but \: not\: at\: x= 0$

• Option 2)

$f\: is\: neither \: di\! f\! \! ferentiable\: at\: x= 0\: nor\: at\: x= 1$

• Option 3)

$f\: is \: di\! f\! \! ferentiable\: at\: x= 0\: \: and\: at\: x= 1$

• Option 4)

$f\: is \: di\! f\! \! ferentiable\: at\: x= 0\: \:but\:not\: at\: x= 1$

As we learnt in  Differentiability - Let  f(x) be a real valued function defined on an open interval (a, b) and   (a, b).Then  the function  f(x) is said to be differentiable at      if -   There fore f is not differentiable at x = 1 similarly   Hence f is also not differentiable at x= 0  Option 1) this is incorrect Option 2) this is correct Option 3) this is incorrect Option...
Engineering
123 Views   |

The perpendicular bisector of the line segment joining $\dpi{100} P(1,4)\; and\; Q(k,3)$ has $\dpi{100} y-$ intercept – 4 . Then a possible value of $\dpi{100} k$ is

• Option 1)

– 4

• Option 2)

1

• Option 3)

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    and,   Mid-point formula -   - wherein If the point P(x,y) is the mid point of line joining A(x1,y1) and B(x2,y2) .    As well as,   Slope – point from of a straight line - - wherein slope point through which line...
Engineering
204 Views   |

A parabola has the origin as its focus and the line $\dpi{100} x=2$ as the directrix. Then the vertex of the parabola is at

• Option 1)

(2, 0)

• Option 2)

(0, 2)

• Option 3)

(1, 0)

• Option 4)

(0, 1)

As we learnt in Standard equation of parabola - - wherein   Vertex is mid-point of focus and foot of directrix. V is (1,0) Option 1) (2, 0) This option is incorrect. Option 2) (0, 2) This option is incorrect. Option 3) (1, 0) This option is correct. Option 4) (0, 1) This option is incorrect.
Engineering
100 Views   |

A focus of an ellipse is at the origin. The directrix is the line $\dpi{100} x=4$  and the eccentricity is $\dpi{100} \frac{1}{2}$ . Then the length of the semi­major axis is

• Option 1)

$\frac{5}{3}$

• Option 2)

$\frac{8}{3}$

• Option 3)

$\frac{2}{3}$

• Option 4)

$\frac{4}{3}$

As we learnt in Equation of directrices - - wherein For the ellipse      and,     Coordinates of foci - - wherein For the ellipse    Distance between focus and directrix:     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
111 Views   |

The point diametrically opposite to the point $\dpi{100} P(1,0)$   on the  circle $\dpi{100} x^{2}+y^{2}+2x+4y-3=0$  is

• Option 1)

(3, 4)

• Option 2)

(3, – 4)

• Option 3)

(– 3, 4)

• Option 4)

(– 3, – 4)

As we learnt in General form of a circle -   - wherein centre = radius =     Mid-point formula -   - wherein If the point P(x,y) is the mid point of line joining A(x1,y1) and B(x2,y2) .   Centre is (-1, -2)  Option 1) (3, 4) This option is incorrect. Option 2) (3, – 4) This option is incorrect. Option 3) (– 3, 4) This option is incorrect. Option 4) (– 3, – 4) This option is correct.
Engineering
73 Views   |

Directions : Question is Assertion­Reason type. This question contains two statements : Statement­ 1 (Assertion) and Statement­2 (Reason). Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice.

Question: let $\dpi{100} p$ be the statement "$\dpi{100} x$ is an irrational number ", $\dpi{100} q$  be the statemen "$\dpi{100} y$ is a transcendental number ", and $\dpi{100} r$ be the statement "$\dpi{100} x$ is a rational number iff $\dpi{100} y$ is a transcendental number ".

Statement­-1: $\dpi{100} r$ is equivalent to either $\dpi{100} q$ or $\dpi{100} p$.

Statement­-2: $\dpi{100} r$ is equivalent to $\dpi{100} \sim (p\leftrightarrow \sim q).$

• Option 1)

Statement-­1 is true, Statement­-2 is false

• Option 2)

Statement-­1 is false, Statement-­2 is true

• Option 3)

Statement­-1 is true, Statement­-2 is true Statement­-2 is a correct explanation for Statement­-1

• Option 4)

Statement­-1 is true, Statement­-2 is true Statement­-2 is not a correct explanation for Statement­-1

Engineering
132 Views   |

The statement $\dpi{100} p\rightarrow (q\rightarrow p)$ is equivalent to

• Option 1)

$p\rightarrow (p\leftrightarrow q)\;$

• Option 2)

$\; p\rightarrow (p\rightarrow q)\;$

• Option 3)

$\; p\rightarrow (p\vee q)\;$

• Option 4)

$\; p\rightarrow (p\wedge q)$

As we learnt in Truth Table of "if-then" - -     Truth table of 'OR' operator - -       T T T T T T F T T T F T F T T F F T T T Both are tautologies. Option 1) Incorrect option Option 2) Incorrect option Option 3) Correct option Option 4) Incorrect option
Engineering
85 Views   |

Directions : Question is Assertion­-Reason type. This question contains two statements :

Statement­-1 (Assertion) and Statement­-2 (Reason). Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice .

Statement­-1: For every natural number $n\geq 2,\; \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+....+\frac{1}{\sqrt{n}}> \sqrt{n}.$

Statement­-2: For every natural number  $n\geq 2,\; \sqrt{n(n+1)}< n+1$ .

• Option 1)

Statement­-1 is true, Statement-­2 is false

• Option 2)

Statemen­-1 is false, Statement-­2 is true

• Option 3)

Statement­-1 is true, Statement­-2 is true Statement­- 2 is a correct explanation for Statement­-1

• Option 4)

Statement­-1 is true, Statement­-2 is true Statement­- 2 is not a correct explanation for Statement­-1

Engineering
1585 Views   |

The correct decreasing order of priority for the functional groups of organic compounds in the IUPAC system of nomenclature is

• Option 1)

– CONH2 , – CHO, – SO3H , – COOH

• Option 2)

– COOH , – SO3H , – CONH2 , – CHO

• Option 3)

– SO3H , – COOH , – CONH2 , – CHO

• Option 4)

– CHO , – COOH ,– SO3H , – CONH2

Option 2 is correct

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