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Mathematics Test - 2

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Mathematics Test - 2
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  • Question 1
    1 / -0

    What is the equation of the straight line passing through the point (2, 3) and making an intercept on the positive y-axis equal to twice its intercept on the positive x-axis?

    Solution

    Slope of line = m = 0-2xx-0=-2xx=-2

    Equation of line which passes through (2,3) and having slope = -2 is
    y-y1 =mx-x1

    i.e.y-3=-2x-2

    y-3=-2+4

    2x+y=7 

  • Question 2
    1 / -0

    How many numbers between 100 and 1000 can be formed with the digits 5, 6, 7, 8, 9, if the repetition of digits is not allowed?

    Solution

    5P3=5!5-3!=5×4×3×2!2!=60

  • Question 3
    1 / -0

    In the equation:

    \(\cos ^{-1} \frac{1-{a}^{2}}{1+{a}^{2}}-\cos ^{-1} \frac{1-{b}^{2}}{1+{b}^{2}}=2 \tan ^{-1} {x} \text { value of } {x} \text { is: }\)

    Solution

    We know that,

    \(\tan ^{-1} x-\tan ^{-1} y=\tan ^{-1}\left(\frac{x-y}{1+x y}\right)\)

    \(2 \tan ^{-1} x=\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)\)

    \(\Rightarrow \cos \left(2 \tan ^{-1} x\right)=\left(\frac{1-x^{2}}{1+x^{2}}\right)\)

    Given:

    \(\cos ^{-1} \frac{1-{a}^{2}}{1+{a}^{2}}-\cos ^{-1} \frac{1-{b}^{2}}{1+{b}^{2}}=2 \tan ^{-1} {x}\)

    Using above formulas,

    \(\cos ^{-1}\left(\cos \left(2 \tan ^{-1} a\right)\right)-\cos ^{-1}\left(\cos \left(2 \tan ^{-1} b\right)\right)=2 \tan ^{-1} x\)

    \(\Rightarrow2\left(\tan ^{-1} a-\tan ^{-1} b\right)=2 \tan ^{-1} x\)

    \(\Rightarrow\tan ^{-1} a-\tan ^{-1} b=\tan ^{-1} x\)

    \(\Rightarrow\tan ^{-1}\left(\frac{a-b}{1+a b}\right)=\tan ^{-1} x\)

    \(\Rightarrow{x}=\left(\frac{a-b}{1+a b}\right)\)

  • Question 4
    1 / -0

    If the ratio AM to GM of two positive numbers a and b is 5 : 3, then a : b is equal to?

    Solution

    AMGM=53

    a+b2ab=53

    a+b2ab=53

    ab+12ab=53

    Letab=y

    y+12y=53

    3y+3=10y

    Squaring both side 

    9y2+9+18y=100y

    9y282y+9=0

    9y281yy+9=0

    9y(y9)1(y9)=0    [divide numerator & denominator by b]

    9y(y9)1(y9)=0

    (9y1)=0

    y9=0

    y=19y=9

    i.e. y=91

    i.e. ab=19 or ab=91

  • Question 5
    1 / -0

    What is the number of triangle that can be formed by choosing the vertices from a set of 12 points in a plane, seven of which lie on the same straight line? 

    Solution

    No. of triangle = 12C3 - 7C3

    12!9! 3!-7!4! 3!

    = 12×11×103×2-7×6×53×2

    = 220 - 35

    = 185.

  • Question 6
    1 / -0

    If a flag- staff of 6 m height placed on the top of a tower throws a shadow of 2√3 m along the ground, then what is the angle that the sun makes with the ground?

    Solution

    tanθ=623=33=3

    tan θ=tan 60°

    θ=60°

  • Question 7
    1 / -0

    If the cardinality of a set A is 4 and that of a set B is 3, then what is the cardinality of the set A ∆ B?

    Solution

    Here we are given that the cardinality of set \(\mathrm{A}\) is 4 and that of a set \(\mathrm{B}\) is 3 .

    We must know the meaning of cardinality. It is actually the number of elements that are there in the set. As here we are

    given that the cardinality of set \(\mathrm{A}\) is 4 and that of a set \(\mathrm{B}\) is 3 this means that there are 4 elements in the set \(\mathrm{A}\) and there

    are 3 elements in the set \(B\).

    So we can write it as:

    \(n(A)=4\)

    \(n(B)=3\)

    So we need to find the cardinality of \(A \Delta B\) set.

    Let us see through Venn diagram that:

    We know that \(A \Delta B\) means all the elements that are there in the set A or the set \(\mathrm{B}\) but not in their intersection or we can

    say that all the elements that are there in either set \(\mathrm{A}\) or set \(\mathrm{B}\) nut we don't need to include the elements common to

    both the set \(\mathrm{A}\) and \(\mathrm{B}\).

    We can write that:

    \(n(A \Delta B)=n(A)+n(B)-n(A \cap B)\)

    Here we know the values of \(n(A)\) and \(n(B)\) but not the elements that are common to \(\mathrm{A}\) and \(\mathrm{B}\).

    Hence we do not know what \(n(A \cap B)\) is. Therefore the value of \(n(A \Delta B)\) cannot be determined.

    Hence we can say that \(\mathrm{D}\) ) is the correct option.

    Note:

    Here in these types of problems we must know all the symbols that are used in the sets related problems. If we are given

    the set \((B-A)\) then this would mean that all the elements that are there in set \(\mathrm{B}\) but not in the set \(\mathrm{A}\). These problems

    can be easily solved by using the Venn diagram.

  • Question 8
    1 / -0

    The binary number expression of the decimal number 31 is?

    Solution


    \(\because(31)_{10}=(11111)_{2}\)

  • Question 9
    1 / -0

    If 0 < a < 1, the value of log10a is negative. This is justified by

    Solution

    let log10 a = x

    ⇒ 10-x = a       [By the definition of log]

    So, 0 < a < 1

    ⇒ 0 < 10-x < 1

  • Question 10
    1 / -0

    In a zoo, there are rabbits and pigeons. If heads are counted there are 340 heads and if legs are counted there are 1060 legs. How many pigeons are there?

    Solution

    Suppose there are all the pigeons then the total number of heads is 340

    Total number of legs = 340 × 2 = 680

    Extra legs = 1060 – 680 = 380

    As we know rabbit has two extra legs.

    So, number of rabbit = 3802 = 190

    No. of pigeons = 340 – 190 = 150

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