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Engineering
129 Views   |  

The number of non-zero terms is the expansion of

\left [ (1+3\sqrt{2}\: x)^{9} +(1-3\sqrt{2}\: x)^{9}\right ] is

  • Option 1)

    9

  • Option 2)

    10

  • Option 3)

    5

  • Option 4)

    None of these

 
As discussed in concept Properties of Binomial Theorem - - wherein Sum of odd terms or even Binomial coefficients    We get We have 5 non-zero terms Option 1) 9 This is incorrect option Option 2) 10 This is incorrect option Option 3) 5 This is correct option Option 4) None of these This is incorrect option
Engineering
116 Views   |  

Find the value of \frac{(1+i)^{n}}{(1-i)^{n-2}}

  • Option 1)

    i^{n+1}

  • Option 2)

    i^{n-1}

  • Option 3)

    2(i)^{n-1}

  • Option 4)

    None of the above

 
As learnt in concept Power of i in Complex Numbers - - wherein       Option 1) Incorrect option Option 2) Incorrect option Option 3) Correct option Option 4) None of the above Incorrect option
Engineering
157 Views   |  

find the sum of the 37th bracket of the following series

\left ( 1 \right )+\left ( 7+7^{2} \right +7^{3})+\left ( 7^{4} \right +7^{5}+7^{6}+7^{7}+7^{8})+\left ( 7^{9} \right +7^{10}+-------+7^{15})+------

  • Option 1)

    \frac{7^3^7}{6}\left ( 7^{73} -1\right )

  • Option 2)

    \frac{7^{73}-1}{6}

  • Option 3)

    7^{1260}\left ( \frac{7^{71}-1}{6}\right )

  • Option 4)

    None of these

 
As learnt in concept Sequence - Arragement of real numbers specified in a definite order, by some assigned law. - wherein Notations - or    In the 37th bracket,  The total number of terms =  So the final term of bracket is 1369. i.e.  None of the options is correct.    Option 1) Incorrect option  Option 2) Incorrect option  Option 3) Incorrect option  Option 4) None of these Correct option
Engineering
130 Views   |  

Find the square roots of 9-12i 

  • Option 1)

    \pm (2\sqrt3 - \sqrt 3 \:i)

  • Option 2)

    \pm (3 + 2i)

  • Option 3)

    \pm (\sqrt3 - 2\sqrt 3 \:i)

  • Option 4)

    \pm (3 - 2i)

 
As learnt in concept Definition of Complex Number -   & i2=-1 - wherein Real part of z = Re (z) = x & Imaginary part of z = Im (z) = y     here,  Thus square root =    Option 1) Correct option Option 2) Incorrect option  Option 3) Incorrect option  Option 4) Incorrect option 
Engineering
129 Views   |  

Find the set of values of 'a' for which the given condition is true (a-1) (a-3)(a+5) > 0

  • Option 1)

    \left ( -5,1 \right ) \cup \left ( 3,\infty \right )

  • Option 2)

    (-1, 5)

  • Option 3)

    \left [ -5,1 \right ] \cup \left [ 3,\infty \right)

  • Option 4)

    None of these. 

 
As learnt in concept Quadratic Expression Graph when a> 0 & D > 0 - Real and distinct roots of & - wherein     According to wavy curve method,    Option 1) Correct option Option 2) (-1, 5) Incorrect option Option 3) Incorrect option Option 4) None of these.  Incorrect option
Engineering
72 Views   |  

Find the Range 

p = \frac{2q+1}{q-5}

  • Option 1)

    p \neq 2

  • Option 2)

    q \neq 5

  • Option 3)

    p \neq 5

  • Option 4)

    None of these. 

 
As learnt in concept Range - The range of the relation R is the set of all second elements of the ordered pairs in a relation R. - wherein eg. R={(a,b),(c,d)}. Then Range is {b,d}     Here,  is the range.   Option 1) Correct option Option 2) Incorrect option Option 3) Incorrect option Option 4) None of these.  Incorrect option
Engineering
130 Views   |  

Find the number of ways in which a pack of 52 playing cards can be divided equally among four persons sitting around a circular table.

  • Option 1)

    \frac{52!\times 3!}{4!\times \left ( 13! \right )^{4}}

  • Option 2)

    \frac{52!}{4! \times\left ( 13! \right )^{4}}

  • Option 3)

    \frac{52!}{\left ( 13! \right )^{4}}

  • Option 4)

    \frac{52!\times3!}{\left ( 13! \right )^{4}}

 
As learnt in concept Groups of Unequal size - Number of ways in which (m + n + p) things can be divided into unequal groups containing m, n and p things is      - wherein    Number of ways =    Option 1) Incorrect option Option 2) Incorrect option Option 3) correct option Option 4) Incorrect option
Engineering
128 Views   |  

Find the middle term in the expansion of (1+x)^{2n}

  • Option 1)

    2n_{Cn+1}x^{n}

  • Option 2)

    2n_{Cn-1}x^{n}

  • Option 3)

    2n_{Cn}x^{n}

  • Option 4)

    2n+1_{Cn}x^{n}

 
As we learnt in Middle term in Binomial Expression if n is odd - If n is odd then and - wherein Two terms are middle terms and their binomial coefficients are equal.       Middle term in  is  Option 1) Incorrect option  Option 2) Incorrect option  Option 3) Correct option  Option 4) Incorrect option 
Engineering
135 Views   |  

Find the domain f \left ( b \right ) = \frac{1}{\sqrt{b-5}} + b^{2} +\frac{1}{\sqrt{b+7}}

  • Option 1)

    b\epsilon \left [ -7,5 \right ]

  • Option 2)

    b\epsilon \left ( 5, \infty \right )

  • Option 3)

    b\epsilon \left ( -\infty, 7 \right )

  • Option 4)

    None of these.

 
As learnt in concept Domain of function - All posible values of x for f(x) to be defined is known as domain. -     Here b-5>0 So, b>5. and b+7>0 So,b>-7 Thus,    Option 1) Incorrect option Option 2) Correct option Option 3) Incorrect option Option 4) None of these. Incorrect option
Engineering
107 Views   |  

Find the co-efficient of x^{19} in the expansion of (1-x)(1+x^{3})^{12}

  • Option 1)

    924

  • Option 2)

    462

  • Option 3)

    -462

  • Option 4)

    -924

 
As laernt in concept General Term in the expansion of (x+a)^n -   - wherein Where     We get  in  We will find  in  , we get r=6 Thus   Option 1) 924 Incorrect option Option 2) 462 Incorrect option Option 3) -462 Incorrect option Option 4) -924 Correct option
Engineering
99 Views   |  

Find the 4th term from the end in the expansion of (\frac{3}{x^{2}}-\frac{x^{3}}{6})^{7}

 

  • Option 1)

    -\frac{35}{48}x^{6}

  • Option 2)

    \frac{35}{24}x^{6}

  • Option 3)

    -\frac{35}{24} x^{6}

  • Option 4)

    \frac{35}{48} x^{6}

 
As learnt in concept General Term in the expansion of (x+a)^n -   - wherein Where        Option 1) Incorrect option Option 2) Incorrect option Option 3) Incorrect option Option 4) correct option
Engineering
109 Views   |  

Find co-efficient of x^{5}y^{4} is the expansion of \left ( \frac{1}{2}x-2y \right )^{90}

  • Option 1)

    126

  • Option 2)

    63

  • Option 3)

    84

  • Option 4)

    None of these

 
As.learnt in concept General Term in the expansion of (x+a)^n -   - wherein Where     General term =   Here r=4 Thus we get,    Option 1) 126 Incorrect option Option 2) 63 Correct option Option 3) 84 Incorrect option Option 4) None of these Incorrect option
Engineering
130 Views   |  

A sequence is numbers is written in the following position : 1,7,1,1,7,7,1,1,1,7,7,7, ------- till n terms

what is the value of \left (T _{5040} \right +T_{5042}+T_{5044}+T_{5046})

T_{n} is the nth term of the given sequence.

  • Option 1)

    4

  • Option 2)

    28

  • Option 3)

    10

  • Option 4)

    22

 
As learnt in concept Sequence - Arragement of real numbers specified in a definite order, by some assigned law. - wherein Notations - or     The number of terms that end are ; 2, 2+4, 2+4+6, 2+4+6+8..... So we get,  2+4+6+8+....+n terms = Now,  Then, and rest of the terms,  So, Sum= 22  Option 1) 4 Incorrect option Option 2) 28 Incorrect option Option 3) 10 Incorrect option Option...
Engineering
135 Views   |  

A convex polygon has 65 diagonals. The number of its sides is equal to - 

  • Option 1)

    13

  • Option 2)

    10

  • Option 3)

    22

  • Option 4)

    11

 
As learnt in concept Geometrical Permutations - The number of diagonals of n sided convex polygon is . - wherein Where n > 3    If number of sides =n Number of diagonals =  Thus n =13   Option 1) 13 Correct option Option 2) 10 Incorrect option Option 3) 22 Incorrect option Option 4) 11 Incorrect option
Engineering
367 Views   |  

A bookshelf can accommodate 6 books from left to right. If 10 identical books on each of the languages - French, German, Spanish and English are available, in how many ways can the bookshelf be filled such that books on the same language are not put adjacently?

  • Option 1)

    \frac{40P_{6}}{6!}

  • Option 2)

    10 x 95

  • Option 3)

    \frac{6P_{4}}{2!}

  • Option 4)

    4 x 35

 
As learnt in concept Number of Permutations without repetition - Arrange n objects taken r at a time equivalent to filling r places from n things.   - wherein Where      can be filled in 4 four ways.  can be filled in 3 ways each. Since the books cannot be similar, Thus, 4*3*3*3*3*3=     Option 1) Incorrect option Option 2) 10 x 95 Incorrect option Option 3) Incorrect option Option 4) 4 x...
Engineering
447 Views   |  

12 persons are to be arranged to a round table. If two particular persons among them are not to be side by side, the total number of arrangements is:

  • Option 1)

    9(10!)

  • Option 2)

    2(10!)

  • Option 3)

    45(8!)

  • Option 4)

    10!

 
As learnt in concept Conditions for Circular Permutation - The number of circular permutations of n distinct things = (n - 1)! -    Total permutations = when two particular persons are together, then number of arrangements= 10!*2 Thus required cases: 11!-2*10!= (11-2)* 10! = 9*10!   Option 1) 9(10!) Correct option Option 2) 2(10!) Incorrect option Option 3) 45(8!) Incorrect option Option...
Engineering
154 Views   |  

1,1,2,3,5,---------9th term of the sequence is

  • Option 1)

    13

  • Option 2)

    34

  • Option 3)

    21

  • Option 4)

    Non of these

 
As learnt in concept General term of the sequence - The nth term of sequence denoted by Tn. - wherein For any n, there shall be a unique value for the nth  term.    1,1,2,3,5,.... This is fibonacci sequence. where      Option 1) 13 Incorrect option Option 2) 34 Correct option Option 3) 21 Incorrect option Option 4) Non of these Incorrect option
Engineering
146 Views   |  

Let f\left ( \theta \right ) = \begin{vmatrix} cos^{2}\theta & cos\theta sin\theta & -sin\theta\\cos\theta sin\theta &sin^{2}\theta &cos\theta \\ sin\theta & -cos\theta &0\end{vmatrix} =0

Then f\left ( \frac{\pi}{6} \right )=

  • Option 1)

    0

  • Option 2)

    1

  • Option 3)

    2

  • Option 4)

    None

 
As learned in concept Value of determinants of order 3 - -         Option 1) 0 Incorrect option Option 2) 1 Correct option Option 3) 2 Incorrect option Option 4) None Incorrect option
Engineering
122 Views   |  

If w\left ( \ne 1 \right )\, \, \, is\, \, \, a\, \, \, cube\, \, \, root \, \, \,of \, \, \,unity\, \, \, then

\begin{vmatrix} 1 & 1+i+w^{2} & w^{2} \\ 1-i &-1 &w^{2}-1\\ -i& -i+w-1 & -1\end{vmatrix} =

  • Option 1)

    0

  • Option 2)

    1

  • Option 3)

    i

  • Option 4)

    w

 
As leant in concept Value of determinants of order 3 - -     Sum of the cube roots of unity is zero.  On simplifying, the result will be zero Option 1) 0 Correct option Option 2) 1 Incorrect option Option 3) i Incorrect option Option 4) w Incorrect option
Engineering
111 Views   |  

If A=\left [ a_{ij} \right ] is a square matrix of order n\times n  and k is a scalar ,then \left | KA \right |=

  • Option 1)

    K^{n}\left | A \right |

  • Option 2)

    K\left | A \right |

  • Option 3)

    Kn^{-1}\left | A \right |

  • Option 4)

    None

 
As we learned in concept Scalar multiplication of matrix - - wherein  is multiplied to every element of matrix     Matrix KA means every element is multiplied by K.  Thus the determinant is multiplied by    Option 1) Correct option Option 2) Incorrect option Option 3) Incorrect option Option 4) None Incorrect option
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