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Engineering
109 Views   |

If $0< x< \pi$ and $\cos x+\sin x= 1/2$ then $\tan x$ is

• Option 1)

$\frac{\left ( 1-\sqrt{7} \right )}{4}$

• Option 2)

$\frac{\left ( 4-\sqrt{7} \right )}{3}$

• Option 3)

$-\frac{\left ( 4+\sqrt{7} \right )}{3}$

• Option 4)

$\frac{\left ( 1+\sqrt{7} \right )}{4}$

As we learnt in  Trigonometric Equations - The equations involving trigonometric function of unknown angles are known as trigonometric equations. - wherein e.g.        Square both side we get       Now ,    or     (1)     if              But  Not Possible. (2)       if   Here    Option 1) Incorrect Option 2) Incorrect Option 3) Correct Option 4) Incorrect
Engineering
138 Views   |

The number of value of $x$ in the interval $\left [ 0,3\pi \right ]$  satisfying the equation $2\sin ^{2}x+5\sin x-3= 0$is

• Option 1)

$4$

• Option 2)

$6$

• Option 3)

$1$

• Option 4)

$2$

As we learnt in  Trigonometric Equations - The equations involving trigonometric function of unknown angles are known as trigonometric equations. - wherein e.g.                                   Now, in rage    is satisfied by  4 solution Option 1) Correct Option 2) Incorrect Option 3) Incorrect Option 4) Incorrect
Engineering
98 Views   |

The differential equation whose solution is $\dpi{100} Ax^{2}+By^{2}=1,\; where\; A\; and\; B$ are arbitrary constants is of

• Option 1)

second order and second degree

• Option 2)

first order and second degree

• Option 3)

first order and first degree

• Option 4)

second order and first degree

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 -     By eliminating A and B,      Option 1) second order and second degree Incorrect option Option 2) first order and second degree Incorrect option Option 3) first order and first degree Incorrect...
Engineering
98 Views   |

The two lines $\dpi{100} x=ay+b,z=cy+d\;$ and $\dpi{100} x=a'y+b',z=c'y+d'\;$ are perpendicular to each other if

• Option 1)

$aa'+cc'=-1\;$

• Option 2)

$\; aa'+cc'=1\;$

• Option 3)

$\; \frac{a}{a'}+\frac{c}{c'}=-1\;$

• Option 4)

$\; \frac{a}{a'}+\frac{c}{c'}=1$

As we learnt in  Condition for perpendicularity - or -     Option 1) Incorrect Option   Option 2) Correct Option   Option 3) Incorrect Option   Option 4) Incorrect Option
Engineering
111 Views   |

The image of the point (–1, 3, 4) in the 3 plane $\dpi{100} x-2y=0$  is

• Option 1)

$\left ( -\frac{17}{3},-\frac{19}{3},4 \right )\;$

• Option 2)

$(15,11,4)$

• Option 3)

$\left ( -\frac{17}{3},-\frac{19}{3},1 \right )\; \;$

• Option 4)

$\left ( \frac{9}{5},-\frac{13}{5},4 \right )\; \;$

As we learnt in  Image of a point - Let be the image of point in the plane will be given by the formula  -    To find image we take Option 1) Incorrect Option   Option 2) Incorrect Option   Option 3) Incorrect Option   Option 4) Correct Option
Engineering
118 Views   |

Let   Then

• Option 1)

there cannot exist any B such that AB = BA

• Option 2)

there exist more than one but finite number B's such that AB = BA

• Option 3)

there exists exactly one B such that AB = BA

• Option 4)

there exist infinitely many B's such that AB = BA

As we learnt in

Multiplication of matrices -

-

$A= \begin{bmatrix} 1 &2 \\ 3& 4 \end{bmatrix}$

$B= \begin{bmatrix} a &0\\ 0& b \end{bmatrix}$

$AB= \begin{bmatrix} 1 &2\\ 3& 4 \end{bmatrix}\begin{bmatrix} a &0\\ 0& b \end{bmatrix}= \begin{bmatrix} a &2b\\ 3a& 4b \end{bmatrix}$

$BA= \begin{bmatrix} a &0\\ 0& b \end{bmatrix}\begin{bmatrix} 1 &2\\ 3& 4 \end{bmatrix}= \begin{bmatrix} a &2a\\ 3b& 4b \end{bmatrix}$

$AB= BA$

$\begin{bmatrix} a &2b\\ 3a& 4b \end{bmatrix}= \begin{bmatrix} a &2a\\ 3b& 4b \end{bmatrix}$

$\Rightarrow$    2b = 2a

3a  = 3b

$a=b$

$B=a \begin{bmatrix} 1 &0\\ 0& 1 \end{bmatrix}= aI \; \: \: \: \: \:where A\leftarrow R$

Option 1)

there cannot exist any B such that AB = BA

Incorrect Option

Option 2)

there exist more than one but finite number B's such that AB = BA

Incorrect Option

Option 3)

there exists exactly one B such that AB = BA

Incorrect Option

Option 4)

there exist infinitely many B's such that AB = BA

Correct Option

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Engineering
120 Views   |

If A and B are square matrices of size n x n such that $\dpi{100} A^{2}-B^{2}=(A-B)(A+B),$ then which of the following will be always true?

• Option 1)

A = B

• Option 2)

AB = BA

• Option 3)

either A or B is a zero matrix

• Option 4)

either A or B is an identity matrix

As we learn tin

Multiplication of matrices -

-

$A^{2}-B^{2}= (A-B)(A+B)$

$= AA+ AB- BA -BB$

$\therefore AB=BA$

Option 1)

A = B

Incorrect option

Option 2)

AB = BA

Correct option

Option 3)

either A or B is a zero matrix

Incorrect option

Option 4)

either A or B is an identity matrix

Incorrect option

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Engineering
107 Views   |

If $\dpi{100} x^{m}\cdot y^{n}= \left ( x+y \right )^{m+n},$ then $\dpi{100} dy/dx \: is$

• Option 1)

$\frac{y}{x}$

• Option 2)

$\frac{x+y}{xy}$

• Option 3)

$xy$

• Option 4)

$\frac{x}{y}$

As we learnt in  Chain Rule for differentiation (indirect) - Let  y = f(x)  is not in standard form then    - wherein     Option 1) Correct Option 2) Incorrect Option 3) Incorrect Option 4) Incorrect
Engineering
128 Views   |

The set of points where  $\dpi{100} f\left ( x \right )= \frac{x}{1+\left | x \right |}$ is differentiable, is

• Option 1)

$\left ( -\infty ,0 \right )\cup \left ( 0,\infty \right )$

• Option 2)

$\left ( -\infty ,-1 \right )\cup \left ( -1,\infty \right )$

• Option 3)

$\left ( -\infty ,-\infty \right )$

• Option 4)

$\left ( 0 ,\infty \right )$

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 for each                       Option 1) Incorrect Option 2) Incorrect Option 3) Correct Option 4) Incorrect
Engineering
123 Views   |

The function $\dpi{100} g\left ( x \right )= \frac{x}{2}+\frac{2}{x}$ has a local minimum at

• Option 1)

$x=2$

• Option 2)

$x=-2$

• Option 3)

$x=0$

• Option 4)

$x=1$

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 is correct option Option 2) This is incorrect option Option 3) This is incorrect option Option 4) This is...
Engineering
116 Views   |

If $\dpi{100} x$ is real, the maximum value of  $\dpi{100} \frac{3x^{2}+9x+17}{3x^{2}+9x+7}$ is

• Option 1)

$1/4$

• Option 2)

$41$

• Option 3)

$1$

• Option 4)

$17/7$

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.    let For real x        maximum value =41 Option 1) This option is incorrect Option 2) This option is correct Option...
Engineering
122 Views   |

A straight line through the point $\dpi{100} A(3,4)$ is such that its intercept between the axes is bisected at $\dpi{100} A$ . Its equation is,

• Option 1)

$x+y=7$

• Option 2)

$3x-4y+7=0$

• Option 3)

$4x+3y=24$

• Option 4)

$3x+4y=25$

As we learnt in Intercept form of a straight line -   - wherein and are the -intercept and -intercept respectively.                a = 6         b = 8 Option 1) This is incorrect option Option 2) This is incorrect option Option 3) This is correct option Option 4) This is incorrect option
Engineering
141 Views   |

The locus of the vertices of the family of parabolas $\dpi{100} y=\frac{a^{3}x^{2}}{3}+\frac{a^{2}x}{2}-2a\; \; is$

• Option 1)

$xy=\frac{105}{64}$

• Option 2)

$xy=\frac{3}{4}$

• Option 3)

$xy=\frac{35}{16}$

• Option 4)

$xy=\frac{64}{105}$

As we learnt in  Rectangular Hyperbola - - wherein    and Standard equation of parabola - - wherein    Family of parabolas For quadratic, A2x2+Bx + C vertex is  so, Eliminating a from h and k 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
155 Views   |

In an ellipse, the distance between its focii is 6 and minor axis is 8. Then its eccentricity is,

• Option 1)

3/5

• Option 2)

1/2

• Option 3)

4/5

• Option 4)

1/$\dpi{80} \sqrt{5}$

As we learnt in  Coordinates of foci - - wherein For the ellipse       Length of major axis - - wherein Semi minor axis     Eccentricity - - wherein For the ellipse   Distance between its focii is 6.  2ae = 6; ae = 3 minor axis, 2b = 8; b=4 b2 = a2(1-e2) b2 = a2- a2e2 a2=25  a=5   Option 1) 3/5 this is correct option Option 2) 1/2 this is incorrect option Option 3) 4/5 this is...
Engineering
104 Views   |

If the lines $\dpi{100} 3x-4y-7=0\; and\; 2x-3y-5=0\;$  are two diameters of a circle of area $\dpi{100} 49\pi$  square units, then the equation of the circle is

• Option 1)

$x^{2}+y^{2}+2x-2y-47=0$

• Option 2)

$x^{2}+y^{2}+2x-2y-62=0$

• Option 3)

$x^{2}+y^{2}-2x+2y-62=0$

• Option 4)

$x^{2}+y^{2}-2x+2y-47=0$

As we learnt in  General form of a circle -   - wherein centre = radius =    Point of intersection of 3x - 4y - 7= 0 and 2x - 2y - 5 = 0 is (1, -1) which is the center and r = 7 Equation is x2 + y2 - 2x + 2y - 47 = 0. Option 1) this is incorrect Option 2) this is incorrect Option 3) this is incorrect Option 4) this is correct
Engineering
119 Views   |

Let $\dpi{100} C$ be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the mid points of chord of the circle $\dpi{100} C$        that subtend an angle of  $\dpi{100} 2\pi /3$  at its centre is

• Option 1)

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

• Option 2)

$x^{2}+y^{2}=1$

• Option 3)

$x^{2}+y^{2}=\frac{27}{4}$

• Option 4)

$x^{2}+y^{2}=\frac{9}{4}$

As we learnt in  Equation of a circle - - wherein Circle with centre and radius .  In h2 + k2 = 9/4 Option 1) This is incorrect option Option 2) This is incorrect option Option 3) This is incorrect option Option 4) This is correct option
Engineering
116 Views   |

If $\dpi{100} (a,a^{2})$   falls inside the angle made by the lines $\dpi{100} y=\frac{x}{2},x> 0\; and \, y=3x,x> 0,\; then\; a$ belongs to

• Option 1)

$\left ( 0,\; \frac{1}{2} \right )\;$

• Option 2)

$\; (3,\infty )\;$

• Option 3)

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

• Option 4)

$\; \left ( -3,\; -\frac{1}{2} \right )\;$

As we learnt in  Slope of a line - - wherein Slope of line joining A(x1,y1) and  B(x2,y2) . Slope of now  slop of    is    Slop of y=3x is m=3 Thus        Option 1) Incorrect option Option 2) Incorrect option Option 3) Correct option Option 4) Incorrect option
Engineering
119 Views   |

The value of    $\dpi{100} \int_{1}^{a}[x]f'(x)dx,\; a> 1,where\; [x]$  denotes the greatest integer not exceeding $\dpi{100} x$ is

• Option 1)

$af(a)- \left \{f(1)+f(2)+...+f([a]) \right \}\;$

• Option 2)

$\; \; [a]f(a)-\left \{f(1)+f(2)+...+f([a]) \right \}$

• Option 3)

$[a]f([a])-\left \{f(1)+f(2)+...+f(a) \right \}$

• Option 4)

$af([a])-\left \{f(1)+f(2)+...+f(a) \right \}$

As learnt in concept Fundamental Properties of Definite integration - If the function is continuous in (a, b ) then integration of a function a to b will be same as the sum of integrals of the same function from a to c and c to b.   - wherein         has to be split into integral limits. = = Terms start cancelling out, We get, = Option 1) This option is incorrect Option 2) This option...
Engineering
92 Views   |

$\dpi{100} \int_{-3\pi /2}^{-\pi /2}\left [ (x+\pi )^{3}+cos^{2}(x+3\pi ) \right ]dx$    is equal to

• Option 1)

$\frac{\pi ^{4}}{32}\;$

• Option 2)

$\; \; \frac{\pi ^{4}}{32}+\frac{\pi }{2}\;$

• Option 3)

$\; \frac{\pi }{2}\;$

• Option 4)

$\; \frac{\pi }{2}-1$

As learnt in concept Properties of definite integration - If is an EVEN function of x: then integral of the function from - a to a is the same as twice the integral of the same function from o to a.   - wherein Check even function and symmetrical about y axis.     Put = = Option 1) This is incorrect option Option 2) This is incorrect option Option 3) This correct option Option...
Engineering
123 Views   |

$\dpi{100} \int_{0}^{\pi}x\; f(sin\: x)\: dx\;$   is equal to

• Option 1)

$\pi \int_{0}^{\pi }f(cos\, x)dx\;$

• Option 2)

$\; \; \pi \int_{0}^{\pi }f(sin\, x)dx\;$

• Option 3)

$\; \frac{\pi }{2}\int_{0}^{\pi /2}\! \! f(sin\, x)dx\; \;$

• Option 4)

$\; \pi \int_{0}^{\pi /2}\! \! f(cos\, x)dx$

As learnt in concept Properties of Definite Integration - -     Properties of Definite integration - When   - wherein Put the at the place of x in     Also Option 1) This option is incorrect Option 2) This option is incorrect Option 3) This option is incorrect Option 4) This option is correct
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