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This mock test of NDA II - Mathematics Question Paper 2017 for Defence helps you for every Defence entrance exam.
This contains 120 Multiple Choice Questions for Defence NDA II - Mathematics Question Paper 2017 (mcq) to study with solutions a complete question bank.
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QUESTION: 1

If x + log_{10}(1 + 2^{x} ) = xlog_{10} + log_{10} 6 then x

is equal to

Solution:

QUESTION: 2

The remainder and the quotient of the binary division

(101110)_{2} ÷ (110)_{2} are respectively

Solution:

(101110)_{2} = (46)_{10} and (110)_{2} = (6)_{10}

Quotient = (7)_{10} = (111)_{2}

Remainder = (4)_{10} = (100)_{2}

QUESTION: 3

The matrix A has x rows and x + 5 columns. The matrix B has y rows and 11 − y columns. Both AB and BA exist. What are the values x and x respectively?

Solution:

For AB and BA to be exist

x + 5 = y and 11 − y = x

Solving these, x = 8 and y = 3

QUESTION: 4

If Sn = nP , where S_{n} denotes the sum of the first n terms of an AP, then the common difference is

Solution:

∴ common difference (d)

QUESTION: 5

The roots of the equation (q − r)x^{2} + (r − p)x + (p − q) = 0 are

Solution:

Sum of coefficients

= q − r +r − p+ p −q = 0 ⟹ 1is a root.

Another root

QUESTION: 6

If E is the universal set and A = B ∪ C, then the set

E − (E − (E − (E − (E − A)))) is same as the set

Solution:

E − (E − (E − (E − (E − A))))

= E − (E − (E − (E − A′)))

= E − (E − (E − A))

= E−(E−A′) = E−A = A

= (B ∪ C)′ = B′ ∩ C′

QUESTION: 7

If A={x:x is a multiple of 2},B={x:x is a multiple of 5} and C={x:x is a multiple of 10}, then A∩(B∩C) is equal to

Solution:

A={x:x is a multiple of 2}={2,4,6,8,10,12,14,....}

B={x:x is a multiple of 5}={5,10,15,20,25,....} and

C={x:x is a multiple of 10}={10,20,30,40,....}

Here, C⊂A and C⊂B

C=A∩B=A∩(B∩C)

=A∩C=C

QUESTION: 8

If α and β are the roots of the equation 1 + x + x^{2} = 0 , then the matrix product is equal to?

Solution:

α and β are roots of x^{2} + x + 1 = 0

⟹ α = ω, β = ω^{2}

QUESTION: 9

If |a| denotes the absolute value of an integer, then which of the following are correct?

I.|ab| = |a||b|

II. |a + b| ≤ |a| + |b|

III. |a − b| ≥ |a| − |b|

Q. Select the correct answer using the code given below.

Solution:

All are true.

QUESTION: 10

How many different permutations can be made out of the letters of the word “PERMUTATION”?

Solution:

“T” is repeated twice. So, Number of permutations

QUESTION: 11

If A = and k =1/2i, where i =

√−1 , then kA is equal to

Solution:

QUESTION: 12

The sum of all real roots of the equation

|x − 3|^{2} + |x − 3| − 2 = 0 is

Solution:

|x − 3|^{2} + |x − 3| − 2 = 0 Let |x − 3| = y

⟹ y = 1 or −2 (−2 Rejected as y is +ve)

⟹ y = 1

⟹ |x−3| = 1 ⟹ x−3 = 1or x−3 = 1

⟹ x = 4 or 2

∴ Sum of roots = 6

QUESTION: 13

If is given that the roots of the equation x^{2} − 4x − log_{3} P = 0 are real. For this, the minimum value of P is

Solution:

QUESTION: 14

If A is a square matrix, then the value of adj A^{T} − (adj A)^{T} is equal to

Solution:

!dj A^{T} = (adjA)^{T} ⟹ Adj A^{T} − (adjA)^{T} = 0

QUESTION: 15

The value of the product

… up to infinite terms is

Solution:

QUESTION: 16

The value of the determinant

forall values of θ, is

Solution:

QUESTION: 17

The number of terms in the expansion of (x + a)^{100} + (x − a)^{100} after simplification is

Solution:

(x + a)^{100} + (x − a)^{100}

Number of terms = 101-50 = 51

QUESTION: 18

In the expansion of (1 +)^{50} , the sum of the coefficients of odd powers of x is

Solution:

In (1 + x)^{50}

C_{1} + C_{3} + C_{5} + ⋯

[∵ C_{0} + C_{1} + C_{2} + ⋯ C_{n} = 2^{n}

QUESTION: 19

If a, b, c are non-zero real numbers, then the inverse of the matrix

Solution:

QUESTION: 20

A person is to count 4500 notes. Let a_{n} denote the number of notes he counts in the nth minute.

If a_{1} = a_{2} = a_{3} = . . = a_{10} = 150 , and

a_{10} , a_{11} , a_{12} . .. are in AP with the common difference −2 , then the time taken by him to count all the notes is

Solution:

Let us assume that total minutes = x

For first nine minutes = 150 × 9

Let the remaining minutes = y = x − 9 Now,

(150 × 9) [2 × 150 + (y − 1)(−2)] = 4500

(302 − 2y) = 3150 ⟹ y^{2} − 151y + 3150 = 0

⟹ (y − 126)(y − 25) = 0 ⟹ y = 25 or 126

(Rejected)

So, x = y + 9 = 25 + 9 = 34 minutes

QUESTION: 21

The smallest positive integer n for which

n = 1 , is

Solution:

QUESTION: 22

If we define a relation R on the set N × N as (a, b) R (c, d) ⟺ a + d = b + c for all (a, b), (c, d) ∈ N × N, then the

Solution:

(a, b) R (c, d) ⟺ a + b = b + c a+a = a+a

⟹ (a, a) R (a, a) ⟹ R is reflexive.

Next, Let (a, b) R (c, d) ⟹ a + b = b + c ⟹

c + b = d + a ⟹ (c, d) R(a, b)

⟹ R is symmetric.

Next, (a, b) R (c, d) and (c, d) R (e, f)

⟹ a+b = b+cand c+f = d+e

⟹ a+d+c+f = b+c+d+e

⟹ a + f = b + e ⟹ (a, b)R (e, f)

⟹ R is transitive ⟹ R is an equivalence relation.

QUESTION: 23

If y = x + x2 + x3 + ⋯ up to infinite terms, where x < 1 , then which of the following is correct?

Solution:

QUESTION: 24

If α and β are the roots of the equation 3x^{2} + 2x + 1 = 0 , then equation whose roots are α + β^{−1} and β + α^{−1} is

Solution:

QUESTION: 25

The value of

Up to infinite terms is

Solution:

QUESTION: 26

A tea party is arranged for 16 people along two sides of a long table with eight chairs on each side. Four particular men wish to sit on one particular side and two particular men on the other side. The number of ways they can be seated is

Solution:

QUESTION: 27

The system of equation kx + y + z = 1,

x + ky + z = k and x = y + kz = k2 has nosolution if k equals

Solution:

⟹ k(k^{2} − 1) − (k − 1) + (1 − k) = 0

⟹ k(k+1)−1−1] = 0 ⟹ k2 +k−2 = 0

⟹ 1, −2

For k =1, first two equations will become same. ⟹ k = −2

QUESTION: 28

If 1.3 + 2.3^{2} + 3.3^{3} + ⋯ + n. 3^{n}

Then a and b are respectively

Solution:

a = n + 1, b = 3

QUESTION: 29

In △ PQR, ∠R are the roots of the equation ax^{2} + bx + c = 0, then which one of the following is correct?

Solution:

QUESTION: 30

If , then the maximum value of |z| is equal to

Solution:

QUESTION: 31

The angle of elevation of a stationary cloud from a point 25 m above a lake is 15° and the angle of depression of its image in the lake 45°. The height of the cloud above the lake level is

Solution:

tan 15° = 2 − √3

QUESTION: 32

The value of tan 9° − tan 27° − tan 63° + tan 81° is equal to

Solution:

tan 9° − tan 27° − tan 63° + tan 81°

= (tan 9° + cot 9°) − (tan 27° + cot 27°)

QUESTION: 33

The value of √3 cosec 20° − sec 20° is equal to

Solution:

QUESTION: 34

Angle α is divided into two parts A and B such that A − B = x and

tan A ∶ tan B = p ∶ q. The value sin x is equal to

Solution:

α = A + B and x = A − B

QUESTION: 35

The value of is equal to

Solution:

QUESTION: 36

The angles of elevation of the top of a tower from the top and foot of a pole are respectively 30° and 45°. If hT is the height of the tower and hP is the height of the pole, then which of the following are correct?

Select the correct answer using the code given below.

Solution:

Let the distance between pole & tower is ‘b’.

⟹ Statement ′3′ is incorrect.

∴ Option ‘c’ right choice.

QUESTION: 37

In a triangle ABC, a − 2b + c = 0.

The value of cot is

Solution:

a + b = 2b

QUESTION: 38

is true if

Solution:

QUESTION: 39

In triangle ABC, if

Then the triangle is

Solution:

sin^{2}A + sin^{2}B + sin^{2}C

= 2 cos^{2} A + 2 cos^{2} B + 2 cos^{2} C

⟹ cos^{2} A + cos^{2} B + cos^{2} C = 1

(cos 2A + cos 2B + cos 2C) = 1

⟹ cos 2A + cos 2B + cos 2C = −1

⟹ 2 cos(A + B) cos(A − B) = −(1 + cos 2C)

⟹ −2 cos C cos(A − B) = −2cos^{2} C

⟹ cos(A − B) = cos C

⟹ A − B = C

Again, A + B + C = π

⟹ A +B+ A− B = π

⟹ A = π/2

⟹ △ is right angle.

QUESTION: 40

The principal value of sin^{−1} x lies in the interval

Solution:

QUESTION: 41

The points (a, b), (0,0), (-a,-b) and (ab, b^{2} ) are

Solution:

All given points lie on the line of equation ay = bx

All points are collinear

QUESTION: 42

The length of the normal from origin to the plane x = 2y − 2z = 9 is equal to

Solution:

x + 2y − 2z = 9

⟹ Length of normal =

QUESTION: 43

If α, β and γ are the angles which the vector (O being the origin) makes with positive direction of the coordinate axes, then which of the following are correct?

1. cos^{2} α + cos^{2} β = sin^{2}γ

2. sin^{2}α + sin^{2}β = cos. γ

3. sin^{2}α + sin^{2}β + sin^{2}γ = 2

Q. Select the correct answer using the code given below:

Solution:

cos^{2} α + cos^{2} β + cos^{2} γ = 1

cos^{2} α + cos^{2} β = 1 − cos^{2} γ = sin^{2}

⟹ Statement 1 is correct

cos^{2} α + cos^{2} β + cos^{2} = 1

⟹ 1 − sin^{2}α + 1 − sin^{2}β + 1 − sin^{2}γ = 1

⟹ sin^{2}α + sin^{2}β + sin^{2}γ = 2

⟹ Statement 3 is correct

QUESTION: 44

The angle between the lines x + y − 3 = 0 and x − y + 3 = 0 is α and the acute angle between the lines x − √3y + 2√3 = 0 and √3x - y + 1 = 0 is β. Which one of the following is correct?

Solution:

∠ between x + y − 3 = 0 & x − y + 3 = 0 is

90° ⟹ α = 90°

As β is acute,

therefore α > β

QUESTION: 45

Let * * be three vectors. If are both perpendicular to the vector , then what is the magnitude of

Solution:

from (i) and (ii)

QUESTION: 46

If are two unit vectors, then the vector is parallel to

Solution:

QUESTION: 47

acts on a particle to displace it from the point

. The work done by the force will be

Solution:

QUESTION: 48

For any vector is equal to

Solution:

QUESTION: 49

A man running round a racecourse notes that the sum of the distances of two flag-posts from him is always 10 m and the distance between the flag-posts is 8 m. The area of the path he encloses is

Solution:

QUESTION: 50

The distance of the point (1, 3) from the line 2x + 3y = 6, measured parallel to the line 4x + y = 4, is

Solution:

Equation of line L is

y − 3 = −4(x − 1) ⟹ y − 3 = −4x + 4

⟹ 4x + y = 7

Solving equations,

QUESTION: 51

If the vector

are coplanar, then the value of

is equal to

Solution:

⟹ a(b − 1)(c − 1) − (1 − a)(c − 1) − (1 − a)(b − 1) = 0

Dividing by (1 − a)(1 − b)(1 − c) , we get

QUESTION: 52

The point of intersection of the line joining the points (−3, 4, −8) and (5, −6, 4) with the XY plane is

Solution:

Equation of line is

⟹ x = 8λ −3,y = −10λ +4,z = 12λ − 8,

since line intersects XY plane, so, z = 0

QUESTION: 53

If the angle between the lines whose direction ratios are (2, −1, 2) and (x, 3, 5) is π/4 , then the smaller value of x is

Solution:

8x^{2} + 98 + 56x = 306 + 9x^{2}

x^{2} − 56x + 208 = 0

QUESTION: 54

The position of the point (1, 2) relative to the ellipse 2x^{2} + 7y^{2} = 20 is

Solution:

2(1)^{2} + 7(2)^{2} − 20 = 2 + 28 − 20 > 0

∴ point lies outside the ellipse.

QUESTION: 55

The equation of a straight line which cuts off an intercept of 5 units on negative direction of yaxis and makes an angle 120° with positive direction of x-axis is

Solution:

m = tan 120° = −√3

y+5 = −√3x⟹ y+√3x+5 = 0

QUESTION: 56

The equation of the line passing through the point (2, 3) and the point of intersection of lines 2x −3y +7 = 0 and 7x+ 4y+ 2 = 0 is

Solution:

Required Line

(2x − 3y + 7) + λ(−42y − 4y)(98 − 2) = 0 …(1)

Putting (2, 3)

⟹ (4 − 9 + 7) + λ(14 + 12 + 2) = 0

(28x − 7x) + (−42y − 4y)(98 − 2) = 0

21x − 46y + 96 = 0

QUESTION: 57

The equation of the ellipse whose centre is at origin, major axis is along x-axis with eccentricity 3/4 and latus rectum 4 units is

Solution:

b2 = 2a, c2

We know, a2 = b2 + c2 So,

Equation of ellipse

Putting values of a and b

QUESTION: 58

The equation of the circle which passes through the points (1, 0), (0, 6), and (3, 4) is

Solution:

Let equation of circle is

x^{2} − x + y^{2} + 6y + λ(−y − 6 + 6x) = 0

Putting (3, 4), we get

9 − 3 + 16 + 24 + λ(−4 − 6 + 18) = 0

⟹ 4x2 + 4y2 − 4x + 24y + 23y + 138 − 138x = 0

⟹ 4x2 + 4y2 − 142x + 47y + 138 = 0

QUESTION: 59

A variable plane passes through a fixed point (a, b, c) and cuts the axes in A, B, and C respectively. The locus of the centre of the sphere OABC, O being the origin is

Solution:

Equation of plane is

It passes through

Equation of sphere is given by

x^{2} + y^{2} + z^{2} − pz − qy − rz = 0 with its centre at (x_{c} , y_{c} , z_{c} ) such that

⟹ p = 2x_{c} , q = 2y_{c} , r = 2 z_{c}

∴ locus of centre

QUESTION: 60

The equation of the plane passing through the line of intersection of the planes x + y + z = 1,

2x + 3y + 4z = 7 and perpendicular to the plane

x − 5y + 3z = 5 is given by

Solution:

Let P_{1} = x + y + z − 1 = 0

P_{2} = 2x + 3y + 4z − 7 = 0

Equation of plane passing through the line of intersection of

P1 and P2 is given by

x + y + z − 1 + λ(2x + 3y + 4z − 7) = 0

⟹ x(1 + 2λ) + y(1 + 3λ) + z(1 + 4λ) − 1 − 7λ = 0

This is perpendicular to x − 5y − 3z − 5 = 0

⟹ 1(1 + 2λ) − 5(1 + 3λ) + 3(1 + 4λ) = 0

⟹ 1 + 2λ − 5 − 15λ + 3 + 12λ = 0

⟹ −λ − 1 = 0 ⟹ λ = −1

∴ Equation of plane is −x − 2y − 3z − 1 + 7 = 0 ⟹ x + 2y + 3z = 0

QUESTION: 61

The inverse of the function y = 5^{lnx} is

Solution:

y = 5^{log x}

⟹ log y = (log x)(log 5)

QUESTION: 62

A function is defined as follows:

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

Solution:

f(x) is discontinuous at x = 0

QUESTION: 63

Solution:

y = (cos x)^{y}

⟹ log y = y log cos x

Differentiating both sides,

QUESTION: 64

Consider the following:

1. x + x^{2} is continuous at x = 0

2. x + cos 1/x is discontinuous at x = 0

3. x^{2} + cos 1/x is continuous at x = 0

Q. Which of the following are correct?

Solution:

Both statements are correct.

QUESTION: 65

Consider the following statements:

1. dy/dx at a point on the curve gives slope of the tangent at that point.

2. If a(t) denotes acceleration of a particle, then

gives velocity of the particle.

3. If s(t) gives displacement of a particle at time t, then ds/dt gives its acceleration at that instant.

Q. Which of the above statements is/are correct?

Solution:

Statements I and II are correct.

QUESTION: 66

equal to

Solution:

QUESTION: 67

What is equal to?

Solution:

QUESTION: 68

A function defined by (0, ∞) by

Which one of the following is correct in respect of the derivative of the function, i.e., f’(x)?

Solution:

f ′(x) = −2x, 0 < x ≤ 1

QUESTION: 69

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

f(x) = x(x − 1)(x + 1)?

Solution:

QUESTION: 70

Consider the following statements:

1. Derivative of f(x) may not exist at some point.

2. Derivative of f(x) may exist finitely at some point.

3. Derivative of f(x) may be infinite (geometrically) at some point.

Q. Which of the above statements are correct?

Solution:

All statements are correct.

QUESTION: 71

The maximum value of is

Solution:

QUESTION: 72

The function f(x) = |x| = |x| − x^{3} is

Solution:

Neither even nor odd

QUESTION: 73

If

Q. Then which one of the following is correct?

Solution:

QUESTION: 74

The general solution of

represents a circle only when

Solution:

QUESTION: 75

If

Q. Then which one of the following is correct?

Solution:

QUESTION: 76

What is equal to?

Solution:

QUESTION: 77

The area bounded by the curve |x| + |y| = 1 is

Solution:

QUESTION: 78

If x is any real number, then belongs to

Q. which one of the following intervals?

Solution:

QUESTION: 79

The left-hand derivative of

f(x) = [x] sin(πx) at x = k where k is an integer and [x] is the greatest integer function, is

Solution:

L. H. D.

QUESTION: 80

If f(x) , then on the interval [0, π]

Q. which one of the following is correct?

Solution:

Also, tan[f(x)] is discontinuous for x = 2 in [0, π]

QUESTION: 81

The order and degree of the differential equation

are respectively

Solution:

QUESTION: 82

If then is equal to

Solution:

QUESTION: 83

The set of all points, where the function

f(x) = is differentiable, is

Solution:

Which is defined ∀ x ∈ R, except x = 0

⟹ f(x) is differentiable on (−∞, 0) ∪ (0, ∞)

QUESTION: 84

Match List-I with List-II and select the correct answer using the code given below the lists.

Code:

Solution:

f(x) = sin x + cos x

⟹ maximum value = √2

(A) → (2)

f(x)3 sin x + 4 cos x

⟹ maximum value = = 5

(B) → (3)

f(x) = 2 sin + cos x

⟹ maximum value =

C) → (4)

f(x) = sin +3 cos x

maximum value =

(D) → (1)

QUESTION: 85

If f(x) = x(√x − √x + 1), then f(x) is

Solution:

L. H. Lt = R. H. Lt = f(0) = 0

⟹ f(x) is continuous at x = 0

L. H. D = R. H. D = −1

⟹ f(x) is differentiable at x = 0

QUESTION: 86

Which one of the following graphs represents the function f(x) =

Solution:

QUESTION: 87

Let f(n) = , where [x] denotes the integral part of x. then the value of

is

Solution:

QUESTION: 88

∫(ln x)^{−1} dx − ∫(ln x)^{−2} dx is equal to

Solution:

QUESTION: 89

A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet. If the capacity of the jar is to be maximum then the diameter of the jar must be k times the height of the jar. The value of k is

Solution:

The height and the radius of the base of an open cylinder of given surface area and maximum volume are equal i.e.,

radius=height.

⟹ Diameter = 2 × height.

⟹ k = 2

QUESTION: 90

The value of is equal to

Solution:

Putting sin x − cos x = t

⟹ dt = (sin x + cos x) dx

when x = 0, t = −1

QUESTION: 91

Let g be the greatest integer function. Then the function f(x) = (g(x))^{2}− g(x^{2}) is discontinuous at

Solution:

g(x) = [x]

f(x) = [x]2 − [x]

f(x) is discontinuous at every integers except x = 1.

QUESTION: 92

The differential equation of minimum order by eliminating the arbitrary constants A and C in the equation y = A[sin(x + C) + cos(x + C)] is

Solution:

y = A[sin(x + c) + cos(x + c)]

= A[cos(x + c) − sin(x + c)]

= −A[sin(x + c) + cos(x + c)] = −y

QUESTION: 93

Consider the following statements:

Statement I: x > sin x for all x > 0

Statement II: f(x) = x − sin x is an increasing function for x > 0

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

Solution:

Both statements are correct but statement 2 is not the correct explanation of statement 1.

QUESTION: 94

The solution of the differential equation is

Solution:

we get

Solution of differential equation is

QUESTION: 95

If

then what is the value of f °g equal to?

Solution:

QUESTION: 96

The value of the determinant

is equal to

Solution:

= (α − β)(β − γ)(γ − α)

QUESTION: 97

The adjoint of the matrix A = is

Solution:

C_{11} = 1 C_{12} = −2 C_{13} = 6

C_{21} = 6 C_{22} = 1 C_{23} = −3

C_{31} = −2 C_{32} = 4 C33 = 1

QUESTION: 98

If A =, then which one of the following is correct?

Solution:

QUESTION: 99

Geometrically Re(z^{2} − i) = 2 , where i = √−1 and Re is the real part, represents

Solution:

This equation represents rectangular hyperbola.

QUESTION: 100

If p + q + r = a + b + c = 0 ,

then the determinant equals

Solution:

p+q+ r = a+b +c = 0

= pqr(a^{3} + b^{3} + c^{3}) − abc(p^{3} + q^{3} + r^{3})

= pqr (3abc) − abc (3pqr) =

0(∴ a^{3} + b^{3} + c^{3} = 3abc p^{3} + q^{3} + r^{3} = 3pqr)

QUESTION: 101

A committee of two persons is selected from two men and two women. The probability that the committee will have exactly one woman is

Solution:

QUESTION: 102

Let a dice be loaded in such a way that even faces are twice likely to occur as the odd faces. What is the probability that a prime number will show up when the dice is tossed.

Solution:

Possible primes are 2, 3, 5.

QUESTION: 103

Let the sample space consist of nonnegative integers up to 50, denote the numbers which are multipliers of 3 and Y denote the odd numbers. Which of the following is/are correct?

Q. Select the correct answer using the code given below.

Solution:

n(X) = 16, n(Y) = 25 and S = 51

QUESTION: 104

For two events A and B, let P(A) = 1/2,

P(A ∪ B) = 2/3 and P(A ∩ B) = 1/6. What

P(A ∩ B) equal to?

Solution:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

QUESTION: 105

Consider the following statements:

1. Coefficient of variation depends on the unit of measurement of the variable.

2. Range is a measure of dispersion

3. Mean deviation is least when measured about median.

Q. Which of the above statements are correct?

Solution:

Statement 1 and statement 2 are correct. Mean derivation is least when measured about mean, therefore statement 3 is wrong.

QUESTION: 106

Given that the arithmetic mean and standard deviation of a sample of 15 observations are 24 and 0 respectively. Then which one of the following is the arithmetic mean of the smallest five observations in the data?

Solution:

As standard deviation is ‘0’, therefore all observations will be equal to 24.

⟹ Average of any five observations = 24.

QUESTION: 107

Which one of the following can be considered as appropriate pair of values of regression coefficient of y on x and regression of x on y?

Solution:

Regression coefficient of y on x

= regression coefficient of x on y

⟹ (x, y) lies on (y = x) line.

QUESTION: 108

Let A and B be two events with

What is equal to?

Solution:

QUESTION: 109

In a binomial distribution, the mean is 2/3 and the variance is 5/9. What is the probability that x = 2?

Solution:

QUESTION: 110

The probability that a ship safely reaches a port is 1/3 . The probability that out of 5 ships, at least 4 ships would arrive safely is

Solution:

p(all reach safely) =

p(4 reach safely) =

p(at least 4 reach safely)

QUESTION: 111

What is the probability that at least two persons out of a group of three persons were born in the same month (disregard year)?

Solution:

p(none born in same month) =

p(at least two born in same month)

QUESTION: 112

It is given that σ_{Y} = 12 and rxy = 0.8. The regression equation of X on Y is

Solution:

σ x = 3, σ y = 10

r_{xy} = 0.8 Regression equation x on y is

⟹ x − 10 = 0.2(y − 90)

⟹ x = −8 + 0.2y

QUESTION: 113

, then what is P(B ∩ C) equal to?

Solution:

QUESTION: 114

The following table gives the monthly expenditure of two families:

In constructing a pie diagram to the above data, the radii of the circles are to be chosen by which one of the following ratios?

Solution:

Total expenditure of A = 10,000

Total expenditure of B = 8,100

So, area of A: area of B= 10,000: 8,100 = 100:81

⟹ radii of A: radii of B =√100 ∶ √81 = 10: 9

QUESTION: 115

If a variable takes values 0, 1, 2, 3, …, n

with frequencies 1, C(n, 1), C(n, 2), C(n, 3), … ,

C(n,n) respectively, then the arithmetic mean is

Solution:

The arithmetic mean will always be between minimum and maximum value so out of the given option ‘n/2’ is possible value.

QUESTION: 116

In a multiple-choice test, an examinee either knows the correct answer with probability p, or guesses with probability 1 − p.

The probability of answering a question correctly is 1/m , if he or she merely guesses.

If the examinee answers a question correctly, the probability that he or she really knows the answer is

Solution:

P (to know correct answer) = p

P (to guess correct answer) = (1 − p) × (1/m)

P (to answer correctly) =

So, required probability

QUESTION: 117

If x_{1} and x_{2} are positive quantities, then the condition for the difference between the arithmetic mean and the geometric mean to be greater than 1 is

Solution:

QUESTION: 118

Consider the following statements:

1. Variance is unaffected by change of origin and change of scale.

2. Coefficient of variance is independent of the unit of observations.

Q. Which of the statements given above is/are correct?

Solution:

Variance is independent of change of origin but not scale. So, Statement 1 is incorrect, Statement 2 is correct.

QUESTION: 119

Five sticks of length 1, 3, 5, 7 and 9 feet are given. Three of these sticks are selected at random. What is the probability that the selected sticks can form a triangle?

Solution:

n(S) = ^{5}C_{3} = 10

n(E) = ^{4}C_{3} − 1 = 3

QUESTION: 120

The coefficient of correlation when coefficients of regression are 0.2 and 1.8 is

Solution:

Coefficient of correlation

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