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Permutations and Combinations Test 10

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Permutations and Combinations Test 10
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  • Question 1
    1 / -0
    A group of 1200 persons consisting of captains and soldiers is travelling in a train. For every 15 soldiers there is one captain. The number of captains in the group is: 
    Solution
    Out of 16 men, there is a captain.
    Number of captains in 1200 men = 1200÷16=75\displaystyle \\ 1200\div 16=75.
  • Question 2
    1 / -0
    Choose the correct statement(s):
    AA: Every sequence is a progression.
    BB: Every progression is a sequence.
    Solution
    The difference between a progression and a sequence is that a progression has a specific rule to calculate its next term from its previous term, whereas a sequence can be based on a logical rule like 'a group of prime numbers'.
    Thus, every progression is a sequence but every sequence is not a sequence.
  • Question 3
    1 / -0
    A, B, C and D are four 4-digit numbers each having the digit 9 only once and in the place shown None of the other digits are known
    What can be said about A, B, C and D?

    Solution
    A is the largest of the 4
    numbers because its place
    value is 9000
  • Question 4
    1 / -0
    Observe the following pattern for obtaining the sums then find the sum of numbers from 781 to 790?
    1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
    11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 155
    21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 = 255
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 = 555
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
    101 + 102 + 103 + 104 + 105 + 106 + 107 + 108 + 109 + 110 = 1055
    251 + 252 + 253 + 254 + 255 + 256 + 257
    258 + 259 + 260 = 2555
    Solution
    781 + 782 + 783 + 784 + 785
    786 + 787 + 788 + 789
    + 790 = 7855
  • Question 5
    1 / -0
    Given three vertices of a triangle whose coordinates are A (1, 1), B (3, -3) and (5, -3). Find the area of the triangle.
    Solution
    We know that the area of the triangle whose vertices are (x1,y1),(x2,y2),\displaystyle (x_{1},y_{1}),(x_{2},y_{2}), and (x3,y3)(x_{3},y_{3}) is 12x1(y2y3)+x2(y3y1)+x3(y1y2)\cfrac{1}{2}\left | x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1}) +x_{3}(y_{1}-y_{2})\right |
    The three vertices of the triangle are A(1,1),B(3,3),C(5,3)A(1,1), B(3,-3), C(5,-3)
    Area of triangle =1(3(3)+3(31)+5(1(3))2=\dfrac{|1(-3-(-3)+3(-3-1)+5(1-(-3))|}{2}
    =1(0)+3(4)+4(4)2=\dfrac{|1(0)+3(-4)+4(4)|}{2}
    =12+202=\dfrac{|-12+20|}{2}
    =82=\dfrac{8}{2}
    =4=4 square units.

  • Question 6
    1 / -0
    The area of triangle whose vertices are A(3,1),B(5,3)A (-3, -1), B(5, 3) and C(2,8)C(2, -8) is ____  sq. units\text{ sq. units}.
    Solution
    We know that the area of the triangle whose vertices are (x1,y1),(x2,y2),\displaystyle (x_{1},y_{1}),(x_{2},y_{2}), and (x3,y3)(x_{3},y_{3}) is 12x1(y2y3)+x2(y3y1)+x3(y1y2)\cfrac{1}{2}\left | x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1}) +x_{3}(y_{1}-y_{2})\right |

    The given vertices of the triangle are A(3,1),B(5,3)A(-3,-1), B(5,3) and C(2,8)C(2,-8).
    So, by using the above formula,
    Area of the triangle=12[3(3(8))+5(8(1))+2(13)]=12[33358]=[76]2=762[Area can never be negative so, we ignore negative sign]=38 sq. units\begin{aligned}{}\text{Area of the triangle} &= \frac{1}{2} {[ - 3(3 - ( - 8)) + 5( - 8 - ( - 1)) + 2( - 1 - 3)]}\\ &= \frac{1}{2}{[ - 33 - 35 - 8]}\\ &= \frac{{[ - 76]}}{2}\\& = \frac{{76}}{2}\quad\quad\quad\quad\dots[\text{Area can never be negative so, we ignore negative sign}]\\& = 38\text{ sq. units}\end{aligned}

    So, the area of the triangle is equal to 38 sq. units38\text{ sq. units}.
  • Question 7
    1 / -0
    If there are five consecutive integer in a series and the first integer is 11, what is the value of the last consecutive integer?
    Solution
    Let the five consecutive integers be x,x+1,x+2,x+3x, x + 1, x + 2, x + 3 and x+4x + 4.
    Therefore, the value of first integer is 11 i.e., x=1x = 1
    So, the last integer is x+4x + 4
    The value of last integer =5= 5.
  • Question 8
    1 / -0
    A sequence of numbers in which each term is related to its predecessor by same law is called
    Solution

    A sequence of numbers in which each term is related to its predecessor by same law is called progression
    Example: 1, 2, 3, 4.... is an example of sequence or progression.
    Since the given sequence follows a same rule or law through out the sequence and there is a relation between each term and it's previous one.
  • Question 9
    1 / -0
    10,10, __,15,15,20,20,25,25,..., 15, 15, 20, 20, 25, 25,.... What number should fill the blank?
    Solution
    Here addition with repetition series, each number in the series repeats itself, and then increases by 55 to get the next number.
    The next number is 1010.
  • Question 10
    1 / -0
    ________ can be defined as arrangement of terms in which sequence of terms follow some conditions.
    Solution
    Progression can be defined as arrangement of terms in which sequence of terms follow some conditions.
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