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Engineering
108 Views   |

If $\dpi{100} A=\sin ^{2}x+\cos ^{4}x,$ then for all real $\dpi{100} x$

• Option 1)

$1\leq A\leq 2\;$

• Option 2)

$\; \frac{3}{4}\leq A\leq \frac{13}{16}\;$

• Option 3)

$\; \frac{3}{4}\leq A\leq 1\;$

• Option 4)

$\; \frac{13}{16}\leq A\leq 1$

As we learnt in Trigonometric Identities - - wherein They are true for all real values of      Range of A depends on  . Minimum value occurs at   Maximum value occurs at  For all other values of  , A varies between these two values[ function is continues] Option 1) This option is incorrect. Option 2) This option is incorrect. Option 3) This option is correct. Option 4) This option is...
Engineering
99 Views   |

If   $\dpi{100} \vec{a}=\frac{1}{\sqrt{10}}\left ( 3\hat{i}+\hat{k} \right )\; and\;\; \vec{b}=\frac{1}{7}\left ( 2\hat{i}+3\hat{j}-6\hat{k} \right ),$   then the value of $\dpi{100} (2\vec{a}-\vec{b})\cdot \left [ \left (\vec{a}\times \vec{b} \right )\times \left ( \vec{a}+2\vec{b} \right ) \right ]\; \; is$

• Option 1)

5

• Option 2)

3

• Option 3)

-5

• Option 4)

-3

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 Triple Product - - wherein Scalar Triple Product of three vectors .     STP We saw So value = -5 Option 1) 5 This option is incorrect. Option 2) 3 This option is incorrect. Option 3) -5 This...
Engineering
124 Views   |

The vectors $\dpi{100} \vec{a}\; and \; \vec{b}$ are not perpendicular and $\dpi{100} \vec{c}\; and \; \vec{d}$ are two vectors satisfying : $\dpi{100} \vec{b}\; \times \; \vec{c}=\vec{b}\; \times \; \vec{d}$and $\dpi{100} \vec{a}\cdot \vec{d}=0$. Then the vector $\dpi{100} \vec{d}$ is equal to

• Option 1)

$\vec{b}+\left ( \frac{\vec{b}\cdot \vec{c}}{\vec{a}\cdot \vec{b}} \right )\vec{c}$

• Option 2)

$\vec{c}-\left ( \frac{\vec{a}\cdot \vec{c}}{\vec{a}\cdot \vec{b}} \right )\vec{b}$

• Option 3)

$\vec{b}-\left ( \frac{\vec{b}\cdot \vec{c}}{\vec{a}\cdot \vec{b}} \right )\vec{c}$

• Option 4)

$\vec{c}+\left ( \frac{\vec{a}\cdot \vec{c}}{\vec{a}\cdot \vec{b}} \right )\vec{b}$

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

Let $\dpi{100} I$ be the purchase value of an equipment and $\dpi{100} V(t)$ be the value after it has been used for $\dpi{100} t$ years. The value $\dpi{100} V(t)$ depreciates at a rate given by differential equation $\dpi{100} \frac{dV(t)}{dt}=-k(T-t);$  where $\dpi{100} k> 0$ is a constant and $\dpi{100} T$ is the total life in years of the equipment. Then the scrap value $\dpi{100} V(T)$ of the equipment is

• Option 1)

$I-\frac{k(T-t)^{2}}{2}\;$

• Option 2)

$\; \; e^{-kT}\;$

• Option 3)

$\; T^{2}-\frac{I}{k}\;$

• Option 4)

$\; I-\frac{kT^{2}}{2}$

As we learnt in  Temperature Problems - - wherein K is Proportionality constant T = Temperature of body   Temperature of Surrounding     Option 1) Incorrect Option 2) Incorrect Option 3) Incorrect Option 4) Correct
Engineering
111 Views   |

If    $\dpi{100} \frac{dy}{dx}=y+3> 0\; and\; y(0)=2,then\; y(1n\: 2)$  is equal to

• Option 1)

13

• Option 2)

– 2

• Option 3)

7

• Option 4)

5

As we learnt in  Solution of Differential Equation - put       - wherein Equation with convert to           Option 1) 13 This option is incorrect. Option 2) – 2 This option is incorrect. Option 3) 7 This option is correct. Option 4) 5 This option is incorrect.
Engineering
161 Views   |

Statement-1 : The point $\dpi{100} A$ (1,0,7) is the mirror image of the point $\dpi{100} B$ (1,6,3) in the line $\dpi{100} \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$

Statement-2 : The line : $\dpi{100} \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ bisects  the line segment joining $\dpi{100} A$ (1,0,7) and $\dpi{100} B$ (1,6,3).

• 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.

As we learnt in  Image of a point - Let be the image of point in the plane will be given by the formula  -    Image of B(1,6,3)is A(1,0,7) But it is not necessary that a plane will always bisect the line AB. So,it is not the correct explanation of statement 1.   Option 1) Statement-1 is true, Statement-2 is false. Incorrect option Option 2) Statement-1 is false, Statement-2 is...
Engineering
111 Views   |

If the angle between the line $\dpi{100} x=\frac{y-1}{2}=\frac{z-3}{\lambda }$   and  the plane $\dpi{100} x+2y+3z=4\; is\; \cos ^{-1}\left ( \sqrt{\frac{5}{14}} \right ),then\; \lambda \; equals$

• Option 1)

2/5

• Option 2)

5/3

• Option 3)

2/3

• Option 4)

3/2

As learnt in Angle between two lines (Vector form ) - Let the two lines be  .The angle between two lines will be equal to angle between their parallel vectors . -     Projection of a line segment on a line - Projection of line segment joining the points P(x1,y1,z1) and Q(x2,y2,z2) on a line having direction cosines (l,m,n) is - wherein   Then,        Option 1) 2/5 Incorrect...
Engineering
173 Views   |

The number of values of $\dpi{100} k$ for which the linear equations

$\dpi{100} 4x+ky+2z=0$

$\dpi{100} kx+4y+z=0$

$\dpi{100} 2x+2y+z=0$

possess a non-zero solution is

• Option 1)

1

• Option 2)

zero

• Option 3)

3

• Option 4)

2

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

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

$4x+ky+2z=0$

$kx+4y+z=0$

$2x+2y+z=0$

$\therefore \begin{vmatrix}4&k&2\\ k&4&1 \\ 2&2&1\end{vmatrix}= 0$

$4\left ( 2 \right )-k\left ( k-2 \right )+2\left ( 2k-8 \right )= 0$

$= > 8-k^{2}+2k+4k-16=0$

$= > 6k-k^{2}-8=0$

$= > k^{2}-6k+8=0$

$= >\left ( k-4 \right )\left ( k-2 \right )= 0$

$\therefore k=2\, \, and \, \,k=4$

$\therefore k=2$

Option 1)

1

Incorrect Option

Option 2)

zero

Incorrect Option

Option 3)

3

Incorrect Option

Option 4)

2

Correct Option

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Engineering
145 Views   |

Let A and B be two symmetric matrices of order 3 .

Statement -1 : $A(BA)$ and $(AB)A$  are symmetric matrices.

Statement -2 : $AB$  is symmetric matrix if matrix multiplication of A and B is commutative.

• 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.

As we learnt in

Property of Transpose Conjugate -

$\left ( AB \right )^{\Theta }=B^{\Theta } A^{\Theta }$

- wherein

$A^{\Theta }$ is the conjugate matrix of $A$

Given that $A^{T}=A\, \, \, and\, \, \,B^{T}=B$

Then $\left [ A\left ( BA\right ) \right ]^{T}= \left ( BA \right )^{T}A^{T}= A^{T}B^{T}A^{T}= ABA$

Similarly $\left [\left ( AB\right )A \right ]^{T}=ABA$

AB is symmmetric matrix.

If AB is commulative.

Option 1)

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

Incorrect Option

Option 2)

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

Incorrect Option

Option 3)

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

Incorrect Option

Option 4)

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

Correct Option

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Engineering
132 Views   |

$\dpi{100} \lim_{x\rightarrow 2}\left ( \frac{\sqrt{1-\cos \left \{ 2(x-2) \right \}}}{x-2} \right )$

• Option 1)

$equals-\sqrt{2}\;$

• Option 2)

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

• Option 3)

does not exist

• Option 4)

$\; equals\sqrt{2}\;$

As we learnt in Evalution of Trigonometric limit - -     For x-2    it is For x-2    it is So LHL PHL So limit does not exist. Option 1) This option is incorrect. Option 2) This option is incorrect. Option 3) does not exist This option is correct. Option 4) This option is incorrect.
Engineering
118 Views   |

For $\dpi{100} x\in \left ( 0,\frac{5\pi }{2} \right ),$   define  $\dpi{100} f(x)=\int_{0}^{x}\sqrt{t}\sin t\, dt\; \; Then\; f\; has$

• Option 1)

local minimum at $\pi$ and local maximum at $2\pi$

• Option 2)

local maximum at $\pi$ and local minimum at $2\pi$

• Option 3)

local maximum at $\pi$ and $2\pi$

• Option 4)

local minimum at $\pi$ and $2\pi$

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.     & x=0 at   at      It is   Option 1) local minimum at and local maximum at This option is incorrect. Option 2) local...
Engineering
118 Views   |

$\dpi{100} \frac{d^{2}x}{dy^{2}}$  equals  to

• Option 1)

$\left ( \frac{d^{2}y}{dx^{2}} \right )\left ( \frac{dy}{dx} \right )^{-2}\;$

• Option 2)

$\; \; -\left ( \frac{d^{2}y}{dx^{2}} \right )\left ( \frac{dy}{dx} \right )^{-3}\;$

• Option 3)

$\; \; \left ( \frac{d^{2}y}{dx^{2}} \right )^{-1}\;$

• Option 4)

$\; -\left ( \frac{d^{2}y}{dx^{2}} \right )^{-1}\left ( \frac{dy}{dx} \right )^{-3}$

As we learnt in Second order derivative for parametric function - When we find  -     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
109 Views   |

The values of $\dpi{100} p\; and\; q$ for which the function

$\dpi{100} f(x)=\left\{\begin{matrix} \frac{\sin (p+1)x+\sin x}{x} &,x< 0 \\ q&,x=0 \\ \frac{\frac{q}{\sqrt{x+x^{2}-\sqrt{x}}}}{x^{3/2}}&,x> 0 \end{matrix}\right.$

is continuous for all $\dpi{100} x$ in R,are

• Option 1)

$p=-\frac{3}{2},q=\frac{1}{2}\;$

• Option 2)

$\; \; p=\frac{1}{2},q=\frac{3}{2}\;$

• Option 3)

$\; p=\frac{1}{2},q=-\frac{3}{2}\;$

• Option 4)

$\; \; p=\frac{5}{2},q=\frac{1}{2}$

As we learnt in Continuity - If the function is continuous, Its graph does not break but for discontinuous functions there is a break in the graph. - wherein     Since f(x) is continuous so that = P+2 = q = P = q = Option 1) This solution is correct. Option 2) This solution is incorrect  Option 3) This solution is incorrect  Option 4) This solution is incorrect
Engineering
128 Views   |

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point (–3, 1) and has eccentricity $\dpi{100} \sqrt{\frac{2}{5}}$   is

• Option 1)

$3x^{2}+5y^{2}-15=0\;$

• Option 2)

$\; 5x^{2}+3y^{2}-32=0\;$

• Option 3)

$\; 3x^{2}+5y^{2}-32=0\; \;$

• Option 4)

$\; 5x^{2}+3y^{2}-48=0$

As we learnt in  Eccentricity - - wherein For the ellipse      and   Standard equation -   - wherein Semi major axis Semi minor axis     Let equation of ellipse be   It passes through (-3,1) we get               On solving,  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   |

The shortest distance between line $\dpi{100} y-x=1\; and\; curve\; x=y^{2}\; is$

• Option 1)

$\frac{8}{3\sqrt{2}}\;$

• Option 2)

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

• Option 3)

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

• Option 4)

$\; \frac{3\sqrt{2}}{8}$

As we learnt in Perpendicular distance of a point from a line -     - wherein   is the distance from the line .     Let (a2, a) be a point of x=y2 Distance between (a2, a) and y-x-1=0 is This is minima when   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
96 Views   |

The lines $\dpi{100} L_{1}:y-x=0\; and\; L_{2}:2x+y=0$  intersect the line $\dpi{100} L_{3}:y+2=0$ at $\dpi{100} P\; and\; Q$  respectively. The bisector of the acute angle between $\dpi{100} L_{1}\; and\; L_{2}\;\; intersects\; L_{3}\;at\; R.$

Statement-1 : The ratio $PR:RQ\; equals\; 2\sqrt{2}:\sqrt{5}.$

Statement-2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.

• 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.

As we learnt in  Angle Bisector - A line segment that bisects an angle of a triangle. - wherein    For point P: y-x=0 and y+2=0 So P is (2,-2)  For point Q: 2x+y=0 and y+2=0 So Q is (-1, -2) But statement 2 is false Option 1) Statement-1 is true, Statement-2 is false. This option is correct Option 2) Statement-1 is false, Statement-2 is true. This option is incorrect Option 3) Statement-1...
Engineering
107 Views   |

The two circles $\dpi{100} x^{2}+y^{2}=ax\; and\; x^{2}+y^{2}=c^{2}(c> 0)$   touch each other if

• Option 1)

$a=2c\;$

• Option 2)

$\; \left | a \right |=2c\;$

• Option 3)

$\; 2\left | a \right |=c\;$

• Option 4)

$\; \left | a \right |=c$

As we learnt in  Common tangents of two circles - When two circles touch  each other externally, there are three common tangents, two of them are direct.   - wherein     and   Common tangents of two circles - When two circles touch each other internally, there is only one common tangent. - wherein     They can touch internally or externally Internally,         We get, Externally,  ...
Engineering
113 Views   |

The area of the region enclosed by the curves $\dpi{100} y=x,x=e,y=1/x$  and the positive $\dpi{100} x$-axis is

• Option 1)

3/2 square units

• Option 2)

5/2 square units

• Option 3)

1/2 square units

• Option 4)

1 square units

As we learnt in  Introduction of area under the curve - The area between the curve axis and two ordinates at the point  is given by - wherein   Option 1) 3/2 square units This is correct option Option 2) 5/2 square units This is incorrect option Option 3) 1/2 square units This is incorrect option Option 4) 1 square units This is incorrect option
Engineering
130 Views   |

If C and D are two events such that $\dpi{100} C\subset D$ and $\dpi{100} P(D)\neq 0,$ then the correct statement among the following is :

• Option 1)

$P(C/D)< P(C)\;$

• Option 2)

$\; P(C/D)=\frac{P(D)}{P(C)}\;$

• Option 3)

$\; P(C/D)=P(C)\;$

• Option 4)

$\; P(C/D)\geqslant P(C)$

As we learnt in Conditional Probability -   and   - wherein where  probability of A when B already happened.     If C and d are independent events other wise if     Option 1) Incorrect option     Option 2) Incorrect option     Option 3) Incorrect option     Option 4) Correct option
Engineering
138 Views   |

Consider 5 independent Bernoulli's trials each with probability of success $\dpi{100} p$ . If the probability of at least one failure is greater than or equal to $\dpi{100} \frac{31}{32}$ , then $\dpi{100} p$ lies in the interval

• Option 1)

$\left [ 0,\frac{1}{2} \right ]\;$

• Option 2)

$\; \left ( \frac{11}{12},1 \right ]\; \;$

• Option 3)

$\; \; \left ( \frac{1}{2},\frac{3}{4} \right ]\; \;$

• Option 4)

$\; \; \left ( \frac{3}{4},\frac{11}{12} \right ]\; \; \;$

As we learnt in Binomial Distribution - Let E be an event and p+q = 1 then X :          0                         1                            2         ..................     n P(x):      qn                                          pn -    Probability of atleast one failure   Probability of no failure    Option 1) Correct Option 2) Incorrect Option 3) Incorrect Option 4) Incorrect
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