NDA II - Mathematics Question Paper 2018


120 Questions MCQ Test NDA (National Defence Academy) Past Year Papers | NDA II - Mathematics Question Paper 2018


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This mock test of NDA II - Mathematics Question Paper 2018 for Defence helps you for every Defence entrance exam. This contains 120 Multiple Choice Questions for Defence NDA II - Mathematics Question Paper 2018 (mcq) to study with solutions a complete question bank. The solved questions answers in this NDA II - Mathematics Question Paper 2018 quiz give you a good mix of easy questions and tough questions. Defence students definitely take this NDA II - Mathematics Question Paper 2018 exercise for a better result in the exam. You can find other NDA II - Mathematics Question Paper 2018 extra questions, long questions & short questions for Defence on EduRev as well by searching above.
QUESTION: 1

What is the value of log7log equal to?

Solution:


= log 77 − log 7 23
= 1 − 3 log7 2

QUESTION: 2

If an infinite GP has the first term x and the sum 5, then which of the following is correct?

Solution:

Infinite GP sum formula = ar + ar² + ar³ + .........
Where a is the first term and r is the common ratio
If |r| < 1, the sum converges
If |r| > 1, the sum doesn't exist and series diverges.
Given:
First term = a = x
Hence from the equation of |r| < 1

where |r| < 1

Thus, 0 < x < 10.

QUESTION: 3

Consider the following expressions:


Which of the above are rational expressions? 

Solution:

A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. In 2nd and 3rd expressions, there is no denominator. So, correct answer is (a).

QUESTION: 4

A square matrix A is called orthogonal if
Where A’ is the transpose of A.

Solution:

An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. The determinant of any orthogonal matrix is either +1 or −1.
By property,
If A–1 = AT , then A is orthogonal matrix.

QUESTION: 5

If A, B and C are subsets of a Universal set, then which one of the following is not correct?

Where A’ is the complement of A.

Solution:

Since, option (a)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C ) and option (d) (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C) are Distributive Laws.
Option (b) A′ ∪ (A ∪ B) = (B ′ ∩ A)′ ∪ A is complement law thus option (c) is incorrect.

QUESTION: 6

Let x be the number of integers lying between 2999 and 8001 which have at least two digits equal. Then x is equal to

Solution:

Every 100 from 3001 to 4000 will have 44 integers where digits 0 to 9 repeat except from 3301 to 3400. (Also 4401 to 4500 and 5501 to 5600 and so on) wherein all 100 integers will have digits 0–9 repeating.
Between 3001 and 4000
44 × 9 + 100 = 496
(Numbers with repeating digits.)
Up to 8000 it will be
5 × 496 = 2480.
Considering 3000 as well, it will be 2481 numbers.
(or)
The thousands place can be filled using any of the digits 3 to 7.
So that is 5 ways. Now, since the numbers can’t repeat, the hundreds place, tens place, units place can be filled by 9 ways (0 to 9 is 10 ways minus the number used in the thousands place amounts to 9 ways), 8 ways and 7 ways respectively (since the number already used can’t be used, the number of ways of filling any place keeps on decreasing).
The no. with all distinct digits = 5 × 9 × 8 × 7
= 2520
x = 5001 – 2520 = 2481

QUESTION: 7

The sum of the series  is equal to

Solution:

Apply the geometric sum formula to find that

Given a geometric series with initial value a and common ratio r with |r| < 1 the sum is given by the formula

In the given series, the initial value is a=3 and the common ratio, found by dividing any term by its preceding term, is r = - 1/3.
 we can apply the formula to obtain

QUESTION: 8

A survey was conducted among 300 students. If was found that 125 students like to play cricket, 145 students like to play football and 90 students like to play tennis. 32 students like to play exactly two games out of the three games.

Q. How many students like to play all the three games?

Solution:

Total number of students = 300


⇒ 300 = 125 + 145 + 90


Again,


From (i) and (ii)

QUESTION: 9

How many students like to play exactly only one game?

Solution:

Exactly one  = n(A) + n(B) + n(c) −

= 125 + 145 + 90 − 2[32 + 3 × 14] + 3 × 14
= 360 − 106 = 254

QUESTION: 10

If α and β(≠ 0) are the roots of the quadratic Equation x2 + αx − β = 0, then the quadratic expression −x2 + αx + β = 0, where x ∈ R has

Solution:

For the quadratic equation, x2 + αx − β = 0
Sum of roots, α + β = −α,
Product of roots, αβ = −β

QUESTION: 11

What is the coefficient of the middle term in the binomial expansion of (2+3x)4?

Solution:

The middle term depends upon the value of n.
When n is even, then total number of terms in the expansion of (x + y)n is  n + 1
So, there is only one middle term i.e. term is the middle term.

On substituting the value of x, y in the above formulae.

QUESTION: 12

For a square matrix A, which of the following


Select the correct answer using the code given below:

Solution:

Statement 1 and 2 are correct.
Because, Properties of Inverses.
If A is non-singular, then so is A−1 and (A−1)−1 =A
Statement 3 is incorrect because

QUESTION: 13

Which one of the following factors does the expansions of the determinant 
 contain?

Solution:


Now,

So, x − 3 is the expansion of determinant.

QUESTION: 14

What is the adjoint of the matrix

Solution:

In the matrix 

Matrix of coefficient of A = 
[∵ cos(−θ) = cos θ and sin(−θ) = −sin θ]

QUESTION: 15

What is the value of

where i =√-1?

Solution:

Cube root of unity
x3 = 3
⇒ x = 1


QUESTION: 16

There are 17 cricket players, out of which 5 players can bowl. In how many ways can a team of 11 players be selected so as to include 3 bowlers?

Solution:

Three bowlers can be selected from the five players and eight players can be selected from 12 players i.e. (17 − 5) = 12 is the number of ways of selecting the cricket eleven, then
P = C(5, 3) × C(12, 8)

QUESTION: 17

What is the value of log9 27 + log8 32?

Solution:

Let log9 27 = x
Then, 9x = 27 ⇔ (32)x = 33
So, 2x = 3

Let log8 32 = y.
Then, 8y = 32

So,3y = 5

QUESTION: 18

If A and B are two invertible square matrices of same order, then what is (AB)−1 equal to?

Solution:

Let AB = C
A−1AB = A−1C
IB = A−1 C as the identity matrix I = A−1A

QUESTION: 19

If a + b + c = 0, then one of the solutions of

Solution:


On eliminating respective rows and columns and expanding.

If a + b + c = 0
x = 0 is one of the solutions.

QUESTION: 20

What should be the value of x so that the matrix   does not have an inverse?

Solution:

If the matrix does not have an inverse, then 

(2 × x) − (4 × (−8)) = 0
2x + 32 = 0
x = −16

QUESTION: 21

The system of equations
2x + y – 3z = 5
3x – 2y + 2z = 5 and
5x – 3y – z = 16

Solution:

The equations are 

Convert to matrix 

Solving the matrix along the rows and columns.

System is consistent with unique solution.

QUESTION: 22

Which one of the following is correct in respect of the cube roots of unity?

Solution:

Assume that the cube root of 1  z.
Then, cubing both sides we get, z3 = 1

Therefore, either z – 1 = 0

Therefore, 

Therefore, the three cube roots of unity are
 among them 1 is real number and the other two are conjugate complex numbers and they are also known as imaginary cube roots of unity.
∆ is equilateral.

QUESTION: 23

If u, v and w (all positive) are the terms of a GP, the determinant of the matrix

Solution:

Since u, v, and w are the terms of a GP whose first term is A and common ratio is R




QUESTION: 24

Let the coefficient of the middle term of the binomial expansion of (1 + x)2n be α and those of two middle terms of the binomial expansion of (1 + x)2n– 1 be β and γ . Which one of the following relations is correct?

Solution:

QUESTION: 25

Let  and S be the subset of A × B, defined by  Which one of the following is correct?

Solution:


S is not a function (By vertical line Test)

QUESTION: 26

Let Tr be the rtℎ term of an AP for r = 1, 2, 3,…… If for some distinct positive integers m and n we have  then what is Tmn equal to?

Solution:







Thus, Tmn = 1

QUESTION: 27

Suppose f(x) is such a quadratic expression that it is positive for all real x. If  then for any real x. Then for any real x.

Solution:


QUESTION: 28

Consider the following in respect of matrices A, B and C of same order:

Where A’ is the transpose of the matrix A. Which of the above are correct?

Solution:

According to the rules of transposition:

QUESTION: 29

The sum of the binary numbers (11011)2 , (10110110)2 and (10011x0y)2 is the binary number (101101101)2 . What are the values of x and y?

Solution:

QUESTION: 30

Let matrix B be the adjoint of a square matrix A, l be the identity matrix of same order as A. If k(≠ 0) is the determinant of the matrix A, then what is AB equal to?

Solution:

A square matrix where every element is unity is called an identity matrix.
A square matrix in which elements in the diagonal are all 1 and rest are all zeroes is called an identity matrix. In other words, the square matrix  is an identity matrix, if    when i ≠ j

QUESTION: 31

If (0.2)x = 2 and log10 2 = 0.3010, then what is the value of x to the nearest tenth?

Solution:

Given, (0.2)x = 2
Taking log on both sides

QUESTION: 32

The total number of 5-digit numbers that can be composed of distinct digits from 0 to 9 is

Solution:

There are 9 choices for the first digit, since 0 can't be used. For the second digit, you can use any of the remaining 9 digits.
For the third digit you can use any of the 8 digits not already used.
For the next digit, there are 7 choices. And for the final digit there are 6 choices left.
Multiplying the values together gives the stated answer: 9 × 9 × 8 × 7 × 6 = 27216
If the last 4 don't include 0, you only have 5 choices left for the first one.
Since the number of distinct-4-digits arrangements which don't include a 0 is 9C4 × 4!

= 9 × 8 × 7 × (30 + 60 − 36)
= 9 × 9 × 8 × 7 × 6 = 27216

QUESTION: 33

What is the determinant of the matrix

Solution:

Given



(Replacing z by x)

QUESTION: 34

If A, B and C are the angles of a triangle and​    then which one of the following is correct?

Solution:

Given



R2 → R2 + R1

⇒ sinA = sinB = sinC ⇒ Δ

QUESTION: 35

Consider the following in respect of matrices A and B of same order:

Where I is the identity matrix and O is the null matrix.
Which of the above is/are correct?

Solution:

Matrix product is commutative if both are diagonal matrices of same order.

QUESTION: 36

What is  equal to?

Solution:

QUESTION: 37

If are in AP, where cos α ≠ 1, then what is the value of sin2 θ + cos α ?

Solution:


QUESTION: 38

If A + B + C = 180°, then what is sin 2A – sin 2B – sin 2C equal to?

Solution:

sin2A − sin2B − sin2C
= 2cos(A + B) · sin(A − B) + sin(2A + 2B)
= 2cos(A + B)sin(A − B) + 2sin(A + B)cos(A + B)
= 2cos(A + B)[sin(A − B) + sin(A + B)]
= −2cosC[2sinA · cosB]
= −4sin A cos B cos C

QUESTION: 39

A balloon is directly above one end of a bridge. The angle of depression of the other end of the bridge from the balloon is 48°. If the height of the balloon above the bridge is 122 m, then what is the length of the bridge?

Solution:


QUESTION: 40

A is an angle in the fourth quadrant. If satisfies the trigonometric equation  which one of the following is a value of A?

Solution:

Checking through options
A = 300° = −60°

= 1

QUESTION: 41

The top of a hill observed from the top and bottom of a building of height h is at angles of elevation π/6 and π/3 respectively. What is the height of the hill?

Solution:



QUESTION: 42

What is/are the solution(s) of the trigonometric equation cosec x + cot x = √3, where 0 < x < 2π ?

Solution:




Possible values of x 
∴ Option (b) is correct.

QUESTION: 43

If, θ = π/8, then what is the value of (2 cos θ + 1)10 (2 cos 2θ − 1)10 (2cos θ −1)10 (2cos 4θ − 1)10?

Solution:

Given θ = π/8
(2 cos θ + 1)10 (2 cos 2θ − 1)10 (2cos θ
− 1)10 (2cos 4θ − 1)10

QUESTION: 44

If cos α and cos β(0 < α < β < π) are the roots of the quadratic equation 4x2 – 3 = 0, then what is the value of sec α × sec β?

Solution:

If cos α and cos β(0 < α < β < π) are the roots.
Quadratic equation 4x2 – 3 = 0, is

QUESTION: 45

Consider the following values of x:
1. 8
2. -4
3. 1/6
4. 
Q. Which of the above values of x is/are the solution(s) of the equation tan−1(2x) + tan−1(3x) = π/4?

Solution:

Given equation, tan−1(2x) + tan−1(3x) = π/4


 (Rejected negative value)
⇒ x =1/6

QUESTION: 46

If the second term of a GP is 2 and the sum of its infinite term is 8, then the GP is

Solution:

Given, ar 

QUESTION: 47

If a, b, c are in AP or GP HP, then  is equal to 

Solution:




QUESTION: 48

What is the sum of all three-digit numbers that can be formed using all the digits 3, 4 and 5, when repetition of digits is not allowed?

Solution:

543 + 534 + 453 + 435 + 354 + 345 = 2664
(or)
The total such numbers possible = 6. Each of 3, 4 and 5 will appear twice in the hundredth, tenth and unit place.
Sum = 2((300 + 400 + 500) + (30 + 40 + 50) + (3 + 4 + 5)) = 2664

QUESTION: 49

The ratio of roots of the equations ax2 + bx + c = 0 and px2 + qx + r = 0 are equal. If D1 and D2 are respective discriminants, then what is D1/Dequal to?

Solution:

Let α1, β1 are roots of ax2 + bx + c = 0 and α2, β2 are roots of px2 + qx + r = 0

D1 = b2 − 4ac, D2 = q2 − 4pr
Also, we know that

Using componendo and dividendo we get


QUESTION: 50

If A = sin2 θ + cos4 θ, then for all real θ, which one of the following is correct?

Solution:


QUESTION: 51

The equation of a circle whose end points of a diameter are (x1 , y1) and (x2 , y2) is

Solution:

Let A(x1, y1) and B(x2,y2) be the endpoints of a diameter of the circle.
Let P(x,y) be an arbitrary point on the circumference.
Then we have angle APB = 90° (angle in a semicircle is a right angle).
Hence AP is perpendicular to BP.
So, slope of AP x slope of BP = -1


Which is the required equation of the circle.

QUESTION: 52

The second degree equation x2 + 4y2– 2x– 4y + 2 = 0 represents

Solution:



It is a point.

QUESTION: 53

The angle between the two lines lx + my + n = 0 and l’x + m’y + n’ = 0 is given by tan–1 θ. What is θ equal to?

Solution:

Given two lines:

θ =?

QUESTION: 54

Consider the following statements:
1. The distance between the lines y = mx + c1 and y = mx + 
2. The distance between the lines ax + by + c1 and ax + by + c2 = 0 
3. The distance between the lines x = c1 and x = c2 is |c1– c2| .
Which of the above statements are correct?

Solution:

In order to find the distance between two parallel lines,
y = mx + c1 and y = mx + c2

Considers two parallel lines

Now the distance between two parallel lines can be found with the following formula:

General formula for distance between any line is 
Let any point on the line, y = mx + c1 be P(x1, y1)

Thus, the distance between the lines x = c1 and x = c2 is |c1– c2 |.

QUESTION: 55

What is equation of straight line passing through the point of intersection of the lines  and parallel to the line 4x + 5y – 6 = 0?

Solution:

Point of intersection 
Let equation of line be 4x + 5y + k = 0

∴ Equation of line is 20x + 25y − 54 = 0

QUESTION: 56

What is the distance of the point (2, 3, 4) from the plane 3x – 6y + 2z + 11 =0?

Solution:

The length p of the ⊥ drawn from a point (x1, y1, z1) ax + by + cz + d = 0 is given by 

Now, distance of the point (2, 3, 4) from the plane 3x − 6y + 2z + 11 = 0 is 

QUESTION: 57

Coordinates of the points O, P, Q and R are respectively (0, 0, 0), (4, 6, 2m), (2, 0, 2n) and (2, 4, 6). Let L, M, N and K be points on the sides OR, OP, PQ and QR respectively such that LMNK is a parallelogram whose two adjacent sides LK and LM are each of length √2. What are the values of m and n respectively?

Solution:


LM = (2 − 1)2 +(3 − 2)2 + (m − 3)2 = 2
⇒ 1 + 1 + (m − 3)2 = 2
⇒ (m − 3)2 = 0
⇒ m = 3
Similarly, by LK = 2, we can get n = 1.

QUESTION: 58

The line  is given by

Solution:



So, answer (d) is correct.

QUESTION: 59

Consider the following statements:
1. The angle between the planes

2. The distance between the planes
6x – 3y + 6z + 2 = 0 and 2x – y + 2z + 4 = 0 is 10/9

Which of the above statements is/are correct? 

Solution:

Angle between planes

Distance between planes

QUESTION: 60

Consider the following statements:
Statement I: If the line segment joining the points P(m, n) and Q(r, s) subtends an angle α at the origin, then cos  
Statements II: In any triangle ABC, it is true that a2 = b2 + c2– 2bc cos A.

Q. Which one of the following is correct in respect of the above two statements?

Solution:


Statement 1 is false, statement 2 is true.

QUESTION: 61

What is the area of the triangle with vertices  

Solution:



QUESTION: 62

If y-axis touches the circle x2 + y2 + gx + fy + c/4 = 0, then the normal at this point intersects the circle at the point

Solution:

The circle x2 + y2 + gx + fy + c/4
= 0 

QUESTION: 63

Let   holds if and only if

Solution:



= a2 + b2


Therefore,  are perpendicular.

QUESTION: 64

 then what is  is equal to?

Solution:

QUESTION: 65

A unit vector perpendicular to each of the vectors 

Solution:



QUESTION: 66

If  then what is the value of 

Solution:


= 2 × 25 = 50

QUESTION: 67

Let  be three mutually perpendicular vectors each of unit magnitude. If   then which one of the following is correct?

Solution:


For simplicity let us take 
Now magnitude of .

QUESTION: 68

What is  equal to?

Solution:


 


 

QUESTION: 69

A spacecraft at is subjected to a force by firing a rocket. The spacecraft is subjected to a moment of magnitude

Solution:

QUESTION: 70

In a triangle ABC, if taken in order, consider the following statements:

How many of the above statements are correct? 

Solution:

From ∆ law of vector addition:
Triangle law of vector addition states that when two vectors are represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector.

Only statement (1) is correct.

QUESTION: 71

Let the slope of the curve y = cos −1(sinx) be tanθ Then the value of θ in the interval (0, π) is

Solution:

QUESTION: 72

If f(x) =  defines a function of R, then what is its domain?

Solution:

If   defines a function of R f(x) is defined if x ≥ 1 and x ≠ 4
∴ Domain = [1,4) ∪ (4, ∞]

QUESTION: 73

Consider the function

Which one of the following is correct in respect of the function?

Solution:

The function


⇒ f(x) is not continuous at x = 0

QUESTION: 74

For the function f(x) = |x − 3|, which of the following is not correct?

Solution:

For the function f(x) = |x − 3|
f(x) = |x − 3| is continuous everywhere and not differentiable at x = 3
Thus, Option (a) is incorrect.

QUESTION: 75

If the function  is continuous at each point in its domain, then what is the value of f(0)? 

Solution:

The function  (continuous at each point in its domain)

QUESTION: 76

If  then what is 

Solution:



As x=1

QUESTION: 77

If  then what is dy/dx equal to?

Solution:


QUESTION: 78

Which one of the following is correct in respect of the function

Solution:

QUESTION: 79

What is  equal to?

Solution:


Applying the limits,

QUESTION: 80

A function f: A → R is defined by the equation where A = (1, 4). What is the range of the function?

Solution:



∴ Range f(x) = [1,5)

QUESTION: 81

What is  equal to, where [. ] is the greatest integer function?

Solution:

[. ]is the greatest integer function

QUESTION: 82

What is  equal to?

Solution:


QUESTION: 83

What is  equal to?
Where c is the constant of integration.

Solution:

Let t = sinx
⇒ dt = cosxdx

QUESTION: 84

What is  equal to?
Where c is the constant of integration

Solution:

QUESTION: 85

What is  equal to?

Solution:


QUESTION: 86

In which one of the following intervals is the function f(x) = x2 − 5x + 6 decreasing?

Solution:

QUESTION: 87

The differential equation of the family of curves y = pcos(ax) + qsin(ax), where p, q are arbitrary constants, is

Solution:

y = pcos ax + qsin ax

QUESTION: 88

The equation of the curve passing through the point (−1, −2) which satisfies   is

Solution:


Putting (-1, -2), we get

QUESTION: 89

What is the order of the differential equation whose solution is y = a cosx + b sinx + ce−x + d where a, b, c and d are arbitrary constants?

Solution:


QUESTION: 90

What is the solution of the differential equation ln (dy/dx) = ax + by?

Solution:


Integrating both sides,

QUESTION: 91

If u = eax sinbx and v = eax cosbx then what is equal to?

Solution:


QUESTION: 92

If   then which one of the following is correct?

Solution:


Again, Differentiating,

QUESTION: 93

A flower-bed in the form of a sector has been fenced by a wire of 40 m length. If the flower-bed has the greatest possible area, then what is the radius of the sector?

Solution:


Area is maximum, when l + 2r = 20
⇒ r = 10

QUESTION: 94

What is the minimum value of [x(x − 1) +1]1/3 where 0 < x < 1?

Solution:

[x(x − 1) + 1]1/3 is minimum when x2 − x + 1is minimum, 0 ≤ x ≤ 1
Minimum value of x2 − x + 1 = 3/4
⇒Required value = (3/4)1/3

QUESTION: 95

If   y = |sinx||x|, then what is the value of dy/dx at x = π/6?

Solution:

y = (−sinx)−x
 · (−1)]

QUESTION: 96

What is  equal to, where π/4<x <π/2?

Solution:

To find 
For 
 cosx + sinx

QUESTION: 97

What is  equal to?
Where c is the constant of integration.

Solution:

QUESTION: 98

Let  f(x + y) = f(x)f(y) and  f(x) = 1 + xg(x)ϕ(x), where  b. Then what is f(x) equal to?

Solution:



= abf(x)

QUESTION: 99

What is the solution of the differential equation 
where c is an arbitrary constant.

Solution:




⇒ t − logt = 2y + C1

 

QUESTION: 100

What is 

Solution:



On applying the limits

= −3

QUESTION: 101

If two dice are thrown and at least one of the dice shows 5, then the probability that the sum is 10 or more is

Solution:

n(S) = 36
A = {(1,5), (2,5), (3,5)(4,5)(5,5), (6,5), (5,1), (5,2),(5,3), (5,4), (5,6)}
B = {(5,5), (6,4), (4,6), (6,5), (5,6), (6,6)}

QUESTION: 102

The correlation coefficient computed from a set of 30 observations is 0.8. Then the percentage of variation not explained by linear regression is

Solution:

r = 0.8

That means answer is 20%

QUESTION: 103

The average age of a combined group of men and women is 25 years. If the average age of the group of men is 26 years and that of the group of women is 21 years, then the percentage of men and women in the group is respectively

Solution:

Let No. of Men = M
Let No. of Women = W
26 M + 21 W = 25 (M + W)
M = 4W
M ∶ W = 4 ∶ 1
Therefore, in terms of percentage, 80% men and 20% women are in the group.

QUESTION: 104

If  sin β is the harmonic mean of sin α and cos α, is the arithmetic mean of sin α and  cos α, then which of the following is/are correct?

Select the correct answer using the code given below: 

Solution:

Statement 1 : 

= 2sinαcosα
= sin2α
Statement2: 

QUESTION: 105

Let A, B and C be three mutually exclusive and exhaustive events associated with a random experiment. If P(B) = 1.5P(A) and P(C) = 0.5P(B) then P(A) is equal to

Solution:

P(A) + P(B) + P(C) = 1

QUESTION: 106

In a bolt factory, machines X, Y, Z manufacture bolts that are respectively 25%, 35% and 40% of the factory’s total output. The machines X, Y, Z respectively produce 2%, 4% and 5% defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by machine X?

Solution:

Let E be the probability of drawing defective bolt. Let E1, E2 and E3 be the event of drawing a bolt produced by the Machines A, B and C respectively. Then 

Therefore, P(E2/E) =

Required probability

= 5/39

QUESTION: 107

8 coins are tossed simultaneously. The probability of getting at least 6 heads is

Solution:

Required probability

= 37/256

QUESTION: 108

Three groups of children contain 3 girls and 1 boy; 2 girls and 2 boys; 1 girl and 3 boys. One child is selected at random from each group. The probability that the three selected consist of 1 girl and 2 boys is

Solution:

Required Probability


= 13/32

QUESTION: 109

Consider the following statements:
1. If 10 is added to each entry on a list, then the average increases by 10.
2. If 10 is added to each entry on a list, then the standard deviation increases by 10.
3. If each entry on a list is doubled, then the average doubles.
Which of the above statement are correct?

Solution:

1. When same number is added to each number in a group, then the average is also increased by that number.
2. Similar to point 1, when each number is doubled in a group, the average is also doubled.
3. But standard deviation is used to represent the variation of the set of data from its mean value. So, adding the same number in each value does not change the value of standard deviation.
Thus, option (c) is correct.

QUESTION: 110

The variance of 25 observations is 4. If 2 is added to each observation, then the new variance of the resulting observations is

Solution:

Consider,

Where,
i = 1, 2, ….. 25
Mean of Y:

From equation (1) and (2), we get,

On multiplying 1/n, we get,

Since, the variance remains same. The variance of the new observation is also 4.

QUESTION: 111

If  xi > 0, yi > 0 (i = 1, 2, 3, ...... n) are the values of two variables X and Y with geometric mean P and Q respectively, then the geometric mean of x/y is

Solution:



= P/Q

QUESTION: 112

If the probability of simultaneous occurrence of two events A and B is p and the probability that exactly one of A, B occurs is q, then which of the following is/are correct?

Select the correct answer using the code given below:

Solution:


QUESTION: 113

If the regression coefficient of Y on X is –6, and the correlation coefficient between X and Y   then the regression coefficient of X on Y would be

Solution:

QUESTION: 114

The set of bivariate observation (x1, y1), (x2, y2),..... (xn, yn) are such that all the values are distinct and all the observations fall on a straight line with non-zero slope. Then the possible values of the correlation coefficient between x and y are

Solution:

If set of bi-variate observation fall in a straight line with non-zero slope, then the values of correlation coefficient is -1 and 1 only.

QUESTION: 115

Two integers x and y are chosen with replacement from the set {0, 1, 2,.....,10}. The probability that |x – y| > 5 is

Solution:

Probability 
= 30/121

QUESTION: 116

An analysis of monthly wages paid to the workers in two firms A and B belonging to the same industry gives the following result:

The average of monthly wages and variance of distribution of wages of all the workers in the firms A and B taken together are 

Solution:

Combined Average

Combined Variance

QUESTION: 117

Three dice having digits 1, 2, 3, 4, 5 and 6 on their faces are marked I, II and III and rolled. Let x, y and z represent the number on die-I die-II and die-III respectively. What is the number of possible outcomes such that x > y > z?

Solution:

Three dice having digits 1, 2, 3, 4, 5 and 6 on their faces are rolled.
No. of possible outcomes
= 10 + 6 + 3 + 1 
= 20

QUESTION: 118

Which one of the following can be obtained from an give?

Solution:

Median can be obtained from given.
In order to find the median in an ogive,
i) Plot the points on the graph and join them with lines.
ii) Find the value of N/2.
iii) Mark this value in the Cumulative frequency scale (y axis).
iv) Join this value to the line formed by plotting the points with dotted line.
v) Now connect this point to the point on the x axis with a straight dotted line.
vi) The point where this line meets the x axis will be the median.

QUESTION: 119

In any discrete series (when all values are not same), if x represents mean deviation about mean and y represents standard deviation, then which one of the following is correct?

Solution:

If x represents mean deviation about mean and y represents standard deviation,
x = M · D · y = S. D

QUESTION: 120

In which one of the following cases would you except to get a negative correlation?

Solution:

A negative correlation means that there is an inverse relationship between two variables -when one variable decreases, the other increases. The vice versa is a negative correlation too, in which one variable increases and the other decreases. These correlations are studied in statistics as a means of determining the relationship between two variables.
On seeing the statements, the statement (c) is expected to be a negative correlation.
Not every change gives a positive result. These different examples of negative correlation show how many things in the real world react inversely.