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Let and   be the roots of equation

is equal to:

• Option 1)

6

• Option 2)

-6

• Option 3)

3

• Option 4)

-3

3

(1+$\omega^{2}$-$\omega$)(1-$\omega^{2}$+$\omega$) is

1. 4

2. $\omega$

3. 2

4. Zero

With the help of 2 properties of cube roots of unity, we can solve this question. Property 1,  Property 2,  Now we have to find the value of (1+-)(1-+) Using property 2,  Now use property 1, Option (1) is correct
Engineering
30 Views   |

Let where       Then   equals :

• Option 1)

• Option 2)

• Option 3)

• Option 4)

Option 2

Engineering
56 Views   |

If (10)9 + 2(11)1   (10)8 + 3(11)2  (10)7 +......  +10 (11)9 = k (10)9, then k is equal to :

• Option 1)

100

• Option 2)

110

• Option 3)

• Option 4)

Use

Sum of n terms of a GP -

$S_{n}= \left\{\begin{matrix} a\frac{\left ( r^{n}-1 \right )}{r-1}, &if \: r\neq 1 \\ n\, a, & if \, r= 1 \end{matrix}\right.$

- wherein

$a\rightarrow$ first term

$r\rightarrow$ common ratio

$n\rightarrow$ number of terms

and

(10)9 + 2(11)1 (10)8 + 3(11)2 (10)7 +......  +10 (11)9 = k(10)9

Take common 109

$10^{9}\left [ 1+2\times \frac{11}{10}+3\times \left ( \frac{11}{10} \right )^{2}+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot 10\times \left ( \frac{11}{10} \right ) ^{9}\right ]= k\left ( 10 \right )^{9}$

$\therefore k= 1+2x+3x^{2}+\cdot \cdot \cdot \cdot \cdot \cdot \cdot 10x^{9} \: where \:\:x=\frac{11}{10}$

$kx= x+2x^{2}+3x^{3}+\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot 10x^{10}$

Subtract

$k-kx= 1+x+x^{2}+x^{3}+x^{4}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot (-10x^{10})$

$k(1-x)= \frac{1(x^{10}-1)}{x-1}-10x^{10}$

$k\left ( 1-\frac{11}{10} \right )= \frac{\left ( \frac{11}{10} \right )^{10}-1}{\frac{11}{10}-1}-10\times \left ( \frac{11}{10} \right )^{10}$

$-\frac{k}{10}= \frac{\left ( \frac{11}{10} \right )^{10}}{\frac{1}{10}}-10-\frac{11^{10}}{10^{9}}$

$\therefore k=100$

Option 1)

100

Option 2)

110

Option 3)

Option 4)

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Engineering
102 Views   |

The sum of all real values of x satisfying the equation

is

• Option 1)

3

• Option 2)

-4

• Option 3)

6

• Option 4)

5

As we have learned Roots of Quadratic Equation with real Coefficients - are roots if is satisfied by   - wherein       Also there is a case  (x-3)(x-2)= 0  = x = 2,3  For x = 3 :  So sum 4+1+6+(-10)+2= 3              Option 1) 3 Option 2) -4 Option 3) 6 Option 4) 5
Engineering
319 Views   |

If the coefficients of x3 and x4 in the expansion of (1 + ax + bx2) (1 - 2x)18 in powers of x are both zero, then (a, b) is equal to :

• Option 1)

• Option 2)

• Option 3)

• Option 4)

As we have learned Expression of Binomial Theorem -   - wherein for n  +ve integral .     coeff of   coeff of    and  and   a = 16  b = 272/3        Option 1) Option 2) Option 3) Option 4)
Engineering
282 Views   |

The sum of first 20 terms of the sequence 0.7 ,0.77,0.777,.........,is :

• Option 1)

• Option 2)

• Option 3)

• Option 4)

As we learnt

Sum of infinite terms of a GP -

$a+ar+ar^{2}+- - - - -= \frac{a}{1-r}\\here \left | r \right |<1$

- wherein

$a\rightarrow$ first term

$r\rightarrow$ common ratio

$S=0.7+0.77+0.777...upto\: \: 20\: \: terms$

$S=\frac{7}{9}(0.9+0.99+0.999...)$

$S=\frac{7}{9}(1-0.1+1-0.01+1-0.001...)$

$S=\frac{7}{9}(20-(\frac{1}{10}+\frac{1}{100}+...upto\: \: 20\: \: terms))$

$S=\frac{7}{9}(20-\frac{\frac{1}{10}(1-\frac{1}{10^{20}})}{(1-\frac{1}{10})})$

$S=\frac{7}{9}(20-\frac{1-10^{-20}}{9})$

$S=\frac{7}{81}(179+10^{-20})$

Option 1)

Option 2)

Option 3)

Option 4)

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Engineering
134 Views   |

The graph of the function  $\dpi{100} y=f\left ( x \right )$  is symmetrical about the line $\dpi{100} x=2$ ,then

• Option 1)

$f\left ( x \right )= f\left ( -x \right )$

• Option 2)

$f\left (2+ x \right )= f\left ( 2-x \right )$

• Option 3)

$f\left ( x +2\right )= f\left ( x-2 \right )$

• Option 4)

$f\left ( x \right )=- f\left ( -x \right )$

As we learnt in

Even Function -

$f(-x)= f(x)$

- wherein

Since a graph symmetric about y-axis

means  x = 0 then it is even function and f(-x) = f(x)

$\therefore$    f(0 - x) = f(0 + x)     (b < z it is symmetric about v = 0 )

But in question it is symmetric about x = 2

then f(x - 2) = f(x + 2)

Correct option is 3.

Option 1)

$f\left ( x \right )= f\left ( -x \right )$

Option 2)

$f\left (2+ x \right )= f\left ( 2-x \right )$

Option 3)

$f\left ( x +2\right )= f\left ( x-2 \right )$

Option 4)

$f\left ( x \right )=- f\left ( -x \right )$

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Engineering
133 Views   |

if     and

• Option 1)

is an empty set.

• Option 2)

contains exactly one element.

• Option 3)

contains exactly two elements.

• Option 4)

contains more than two elements.

As we learnt in

FUNCTIONS -

A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.

-

$f(x)+2f \left(\frac{1}{x} \right )=3x$

Put $\frac{1}{x}$ at the place of

$f\left(\frac{1}{x} \right )+2f(x)=\frac{3}{x}$                                                    (i)

$2f\left(\frac{1}{x} \right )+f(x)=3x$                                                (ii)

Multiplying (i) by 2

$2f\left(\frac{1}{x} \right )+4f(x)=\frac{6}{x}$

$\underline{2f\left(\frac{1}{x} \right )+f(x)=3x}$

$3f(x)=\frac{6}{x}-3x$

$f(x)=\frac{2}{x}-3x$

and             $f(-x)=\frac{2}{-x}+x$

$\therefore\ \; \frac{2}{x}-x=-\frac{2}{x}+x$

$\Rightarrow\ \; \frac{4}{x}-2x=0$

$\Rightarrow\ \; \frac{4-2x^{2}}{x}=0$

$\Rightarrow\ \; 4=2x^{2}$

$\Rightarrow\ \; x^{2}=2$

$x=\pm \sqrt{2}, \; x \neq 0$

Correct option is 3.

Option 1)

is an empty set.

Option 2)

contains exactly one element.

Option 3)

contains exactly two elements.

Option 4)

contains more than two elements.

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Engineering
124 Views   |

The real number for which the equation, $2x^2+3x+k = 0$  has two distinct real roots in

• Option 1)

does not exist.

• Option 2)

lies between 1 and 2 .

• Option 3)

lies between 2 and 3 .

• Option 4)

lies between -1 and 0 .

As we have learned

Quadratic Expression Graph when a> 0 & D > 0 -

Real and distinct roots of

$f\left ( x \right )= ax^{2}+bx+c$

& $D= b^{2}-4ac$

- wherein

$\frac{-b}{2a}=-3/4$    is the abscissa of vertex

and , it should lie in(0,1 ) but it's not true

S, no value of 'k' exists

Option 1)

does not exist.

Option 2)

lies between 1 and 2 .

Option 3)

lies between 2 and 3 .

Option 4)

lies between -1 and 0 .

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Engineering
124 Views   |

Let and be the roots of equation are in A.P. and   then the value of is ?

• Option 1)

• Option 2)

• Option 3)

• Option 4)

As we have learned

Sum of Roots in Quadratic Equation -

$\alpha +\beta = \frac{-b}{a}$

- wherein

$\alpha \: and\beta$ are root of quadratic equation

$ax^{2}+bx+c=0$

$a,b,c\in C$

Product of Roots in Quadratic Equation -

$\alpha \beta = \frac{c}{a}$

- wherein

$\alpha \: and\ \beta$ are roots of quadratic equation:

$ax^{2}+bx+c=0$

$a,b,c\in C$

@1449

$|\alpha -\beta | = \left | \frac{\sqrt{q^2}-4pr}{p} \right |$

$\left ( \because \left | \frac{\sqrt{D}}a{} \right | \right )$

Also $\frac{\alpha +\beta }{\alpha \beta }= 4$

$\Rightarrow \frac{-q}{r}= 4$

$\Rightarrow q = -4r ....(1)$

$= \sqrt{16(\frac{r}{p})^2-(4\frac{r}{p})}$

Also p+r =2q

$\Rightarrow p+r = -8r \Rightarrow r/p = -1/9$

$\therefore \frac{\left | \alpha -\beta \right |}{16\times 1/81+4/9}= \sqrt{\frac{52}{81}}=\frac{2\sqrt{13}}{9}$

Option 1)

Option 2)

Option 3)

Option 4)

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Engineering
133 Views   |

If z is a complex number of unit modulus and argument  ,then arg    equals:

• Option 1)

• Option 2)

• Option 3)

• Option 4)

As we have learned

Euler's Form of a Complex number -

$z=re^{i\theta}$

- wherein

r denotes modulus of z and $\theta$ denotes argument of z.

Polar Form of a Complex Number -

$z=r(cos\theta+isin\theta)$

- wherein

r= modulus of z and $\theta$ is the argument of z

$|z| = 1$

Arg (z)= $\theta$

$\Rightarrow z = e^{i\theta }= \cos \theta + i \sin \theta$

So, $\frac{1+z}{1+z}= \frac{1+\cos \theta +i \sin \theta }{1+\cos \theta -i\sin \theta }$

$\frac{2 \cos^2\theta h+2 i\sin \theta h\cos \theta /2}{2\cos ^{2}\theta h-2i\sin \theta h\cos \theta }$

$=\frac{\cos \theta h+i\sin \theta h}{\cos \theta h-i\sin \theta h}$

$=\frac{e^{i\theta h}}{e^{-i\theta h}}= e^{i\theta }$

$\left ( \frac{1+z}{1+\bar{z}} \right )= \theta$

Option 1)

Option 2)

Option 3)

Option 4)

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Engineering
135 Views   |

If the equations have a common root, then a : b : c is :

• Option 1)

3 : 1 : 2

• Option 2)

1 : 2 : 3

• Option 3)

3 : 2 : 1

• Option 4)

1 : 3 : 2

As we have learned

Quadratic Expression Graph when a > 0 & D < 0 -

No Real and Equal root of

$f\left ( x \right )= ax^{2}+bx+c$

& $D= b^{2}-4ac$

- wherein

Condition for both roots common -

$\frac{a}{{a}'}=\frac{b}{{b}'}=\frac{c}{{c}'}$

- wherein

$ax^{2}+bx+c=0$ &

$a'x^{2}+b'x+c'=0$

are the 2 equations

For $x^2+2x+3=0$

Discriminant = 4-12 = -8 < 0

Both the roots are common as complex roots occur in conjugate $\therefore a:b:c= 1:2:3$

Option 1)

3 : 1 : 2

Option 2)

1 : 2 : 3

Option 3)

3 : 2 : 1

Option 4)

1 : 3 : 2

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Engineering
88 Views   |

If z is a complex number such that    then the minimum value of

• Option 1)

is strictly greater than

• Option 2)

is strictly greater than   but less than

• Option 3)

is equal to

• Option 4)

lies in the interval (1, 2)

As we have learned

Triangle Law of Inequality in Complex Numbers -

$|z_{1}-z_{2}|\geq \left || z_{1} \right |-| z_{2} \right |||$

- wherein

|.| denotes modulus of z in complex numbers

$\left | z+\frac{1}{z} \right |= \left | z-(-\frac{1}{z}) \right |$

$\geq \left | |z|- (-\frac{1}{z}) \right |$

$= |z| - 1/z (\because |z|\geq 2)$

$\geq 2-1/2 = 3/2$

$\left | z+1/z \right |\geq 3/2$

3/2 lies in the interval (1,2)

Option 1)

is strictly greater than

Option 2)

is strictly greater than   but less than

Option 3)

is equal to

Option 4)

lies in the interval (1, 2)

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Engineering
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A  complex  number  z  is  said  to  be   unimodular if  . Suppose z1 and z2 are complex numbers such that

is unimodular and z2 is not unimodular.Then the point 1 lies on a :

• Option 1)

straight line parallel to x-axis.

• Option 2)

straight line parallel to y-axis.

• Option 3)

• Option 4)

As we have learned

Property of conjugate of complex number -

$z\bar{z}=\left |z \right |^{2}$

- wherein

$z=x+iy$$\bar{z}=conjugate \: of\: z$

$\left |z \right |=\sqrt{x^{2}+y^{2}}$

Given , $\left | \frac{z_1-2z_2}{2-z_1\bar{z}_2} \right |= 1$

$\Rightarrow |(z_1-2z_2)|^2= |(z-z_1\bar{z}_2)|^2$

$\Rightarrow (z_1-2z_2)(\bar{z}_1-2\bar{z} _2)= (2-z_1\bar{z}_2)(2-\bar{z}_1z_2)$

$\Rightarrow (z_1)^2 (1-|z_2|^2)= 4 (1-|z_2|^2)$

$\Rightarrow |z_1|^2 = 4 (\because |z|\neq 1)$

$\Rightarrow |z_1|= 2$

Option 1)

straight line parallel to x-axis.

Option 2)

straight line parallel to y-axis.

Option 3)

Option 4)

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Engineering
105 Views   |

If    is the adjoint of a 3 x 3  matrix A and = 4,then is equal to :

• Option 1)

0

• Option 2)

4

• Option 3)

11

• Option 4)

5

As we have learned

Property of adjoint of A -

$\left | adj A \right |=\left | A \right |^{n-1}$

- wherein

$adj A$ denotes adjoint of $A$ and  $\left |A \right |$  denotes determinant  of $A$ and $n$ is the order of the matrix

$|adj \; \; A| = |A|^{3-1}$

$\Rightarrow |adj \; \; A| =4^2=16$

$\Rightarrow 0-\alpha (-2)+3(-2)=16$

$\Rightarrow 2\alpha -6 = 16$

$\Rightarrow \alpha = 11$

Option 1)

0

Option 2)

4

Option 3)

11

Option 4)

5

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Engineering
139 Views   |

Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of AB having 3 or more elements is :

• Option 1)

211

• Option 2)

256

• Option 3)

220

• Option 4)

219

n(A) = 4, n(B) = 2

$n(A\times B)=8$

Number of sbsets having atlest 3 elements

$=2^{8}-\left(1+^{8}C_{1}+^{8}C_{2} \right )=219$

Option 1)

211

Incorrect

Option 2)

256

Incorrect

Option 3)

220

Incorrect

Option 4)

219

Correct

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Engineering
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In how many differnt ways can 3 different rings be worn in 5 fingers of a hand?

• Option 1)

3! x 5!

• Option 2)

35

• Option 3)

210

• Option 4)

180

As learnt in Number of Permutations without repetition - Arrange n objects taken r at a time equivalent to filling r places from n things.   - wherein Where      First ring can be worn in 7 ways Second in 6 ways Third in 5 ways Option 1) 3! x 5! This option is incorrect. Option 2) 35 This option is incorrect. Option 3) 210 This option is correct. Option 4) 180 This option is incorrect.
Engineering
115 Views   |

if a, b and c from G.P with common ration r ,the sum of the y cordinates of the points of intersection of the line ax+by+c=0 and the curve $x+2y^{2}=0$ is

• Option 1)

$-\frac{r}{4}$

• Option 2)

$-\frac{r}{2}$

• Option 3)

$\frac{r}{2}$

• Option 4)

$\frac{r}{4}$

As learnt in

General term of a GP -

$T_{n}= ar^{n-1}$

- wherein

$a\rightarrow$ first term

$r\rightarrow$ common ratio

And,

$ax+by+c=0$

$b=ar, c=ar^{2}$

$x+ry+r^{2}=0$

Also,

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

$2y^{2}-ry-r^{2}=0$

$Sum= \frac{r}{2}$

Option 1)

$-\frac{r}{4}$

This option is incorrect.

Option 2)

$-\frac{r}{2}$

This option is incorrect.

Option 3)

$\frac{r}{2}$

This option is correct.

Option 4)

$\frac{r}{4}$

This option is incorrect.

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Engineering
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$If \begin{vmatrix} y+z & x & x \\ y & z+x &y \\ z & z & x+y\end{vmatrix} =K(xyz)$

Then K is equal to

• Option 1)

4

• Option 2)

-4

• Option 3)

0

• Option 4)

None

As learnt in concept Value of determinants of order 3 - -       Option 1) 4 This option is correct. Option 2) -4 This option is incorrect. Option 3) 0 This option is incorrect. Option 4) None This option is incorrect.
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