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# NDA II - Mathematics Question Paper 2016

## 120 Questions MCQ Test NDA (National Defence Academy) Past Year Papers | NDA II - Mathematics Question Paper 2016

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This mock test of NDA II - Mathematics Question Paper 2016 for Defence helps you for every Defence entrance exam. This contains 120 Multiple Choice Questions for Defence NDA II - Mathematics Question Paper 2016 (mcq) to study with solutions a complete question bank. The solved questions answers in this NDA II - Mathematics Question Paper 2016 quiz give you a good mix of easy questions and tough questions. Defence students definitely take this NDA II - Mathematics Question Paper 2016 exercise for a better result in the exam. You can find other NDA II - Mathematics Question Paper 2016 extra questions, long questions & short questions for Defence on EduRev as well by searching above.
QUESTION: 1

Solution:

QUESTION: 2

### If c > 0 and 4a + c < 2b, then ax2- bx + c = 0 has a root in which one of the following intervals?

Solution:

Let f(x) = ax2-bx + c

f(2) = 4a - 2b + c < 0 (given)

f(0) = c > 0 (given)

QUESTION: 3

### If A = {x ∈ R : x2 + 6x - 7 < 0} and B = {x ∈ R : x2 + 9x + 14 > 0} , then which o f the following is/are correct? Q. Select the correct answer using the code given below:

Solution:

QUESTION: 4

If A is semy square matrix of order 3 and det A= 5, then what is det[(2A)-1] equal to?

Solution:

QUESTION: 5

What is ω100 + ω200 + ω300 equal to, where ω is the cube root of unity?

Solution:

QUESTION: 6

where 2 = x + iy is a complex number, then which one of the following is correct?

Solution:

QUESTION: 7

What is [x y z]  equal to ?

Solution:

QUESTION: 8

Out of 15 points in a plane, n points are in the same straight line. 445 triangles can be formed by joining these points. What is the value of n?

Solution:

QUESTION: 9

then what is the imaginary part of z equal to?

Solution:

QUESTION: 10

If both the roots of the equation x2- 2kx + k2 - 4 = 0 lie between -3 and 5, then which one of the following is correct?

Solution:

QUESTION: 11

What is the number of distinct solutions of the equation z2 + |z| = 0(where z is a complex number)?

Solution:

QUESTION: 12

How many geometric progressions is/are possible containing 27,8 and 12 as three of its/their terms?

Solution:

Let ‘a’ be the first term & ‘x’ be the common ratio. Also, suppose 27, 8 & 12 be the pth qth, & rth term of the GP.

There are infinitely many solutions for the eq. (1).

QUESTION: 13

Let R be a relation from A = {1,2,3,4} to B = {1,3,5} such that R=[( a ,b ) :a<b, where a ∈A and b ∈B]. What is RoR-1 equal to?

Solution:

QUESTION: 14

A five-digit number divisible by 3 is to be formed using the digits 0,1,2,3 and 4 without repetition of digits. What is the number of ways this can be done?

Solution:

Sincesum of digits = 10 (which is not divisible by 3)

∴ No numbers can be formed.

QUESTION: 15

What is equal to?

Solution:

QUESTION: 16

Consider the following

Let a ,x ,y ,z,b be in AP , where x + y + z = 15. Let a,p, q ,r,b be in HP, where

Q. What is the value of ab?

Solution:

QUESTION: 17

Consider the following

Let a ,x ,y ,z,b be in AP , where x + y + z = 15. Let a,p, q ,r,b be in HP, where

Q. What is the value of xyz?

Solution:

QUESTION: 18

Consider the following

Let a ,x ,y ,z,b be in AP , where x + y + z = 15. Let a,p, q ,r,b be in HP, where

Q. What is the value of pqr?

Solution:

Since a, p, q, r, b or 1, p, q, r, 9 are in H.P.

QUESTION: 19

Consider the following

The sixth term of an AP is 2 and its common difference is greater than 1.

Q. What is the common difference of the AP so that the product of the first, fourth and fifth terms is greatest?

Solution:

Let first term = a & common difference = x

QUESTION: 20

Consider the following

The sixth term of an AP is 2 and its common difference is greater than 1.

Q. What is the first term of the AP so that the product of the first, fourth and fifth terms is greatest?

Solution:

Since, a = 2 - 5x

QUESTION: 21

Consider the following

Q. What is the value of c?

Solution:

QUESTION: 22

Consider the following

Q. What is the value of a + b + c + d ?

Solution:

a + b + c + d = 63

QUESTION: 23

Consider the following

The interior angles of a polygon of n sides are in AP. The smallest angle is 120° and the common difference is 5°.

Q. How many possible values can n have?

Solution:

Here, a = 120° and d=5.

Sum of angles of polygon =( n - 2)180°

QUESTION: 24

Consider the following

The interior angles of a polygon of n sides are in AP. The smallest angle is 120° and the common difference is 5°.

Q. What is the largest interior angle of the polygon?

Solution:

QUESTION: 25

then what is the value of the determinant of m cosθ — n sinθ ?

Solution:

QUESTION: 26

then which of the following are correct ?

1.

2. The value of the determinant of the matrix  is 1.

3. The determinant of f(x) is an even function.

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

Solution:

QUESTION: 27

Which of the following are correct in respect of the system of equations x + y + z =8, x -y +2z = 6 and 3x -y +5z = k?

1. They have no solution, if k = 15.

2. They have infinitely many solutions, if k = 20.

3. They have unique solution, if k = 25

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

Solution:

( ∵ system is inconsistent i.e., it has no solution)

( ∵ system has infinitely many solutions)

QUESTION: 28

then which of the following is/are correct?

1. AB(A-1 B-1) is a unit matrix.

2. (AB)-1 = A-1 B-1

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

Solution:

QUESTION: 29

then which one of the following is correct?

Solution:

QUESTION: 30

If the number 235 in decimal system is converted into binary system, then what is the resulting number ?

Solution:

QUESTION: 31

Consider the following

Let α and β be the roots of the equation

Under what condition does the above equation have real roots ?

Solution:

QUESTION: 32

Consider the following

Let α and β be the roots of the equation

Under what condition is

Solution:

QUESTION: 33

What is  equal to, where ω is the cube root of unity?

Solution:

QUESTION: 34

In an examination, 70% students passed in Physics, 80% students passed in Chemistry, 75% students passed in Mathematics and 85% students passed in Biology, and x% students failed in all the four subjects. What is the minimum value of x ?

Solution:
QUESTION: 35

Consider the following

For the system of linear equations 2x + 3y + 5z = 9 , 7x + 3y - 2z= 8and 2x + 3y+ λz = μ

Q. Under what condition does the above system of equations have infinitely many solutions ?

Solution:

For infinitely many solutions:

QUESTION: 36

Consider the following

For the system of linear equations 2x + 3y + 5z = 9 , 7x + 3y - 2z= 8and 2x + 3y+ λz = μ

Q. Under what condition does the above system of equations have unique solutions?

Solution:

QUESTION: 37

What is the number of odd integers between 1000 and 9999 with no digit repeated?

Solution:

Case I

When unit digit can be 1,3,5 or 7 & digit at thousand’s place can be 1,2,3,4,5,6,7 or 8.
No. of ways digits can be filled are:

Total no’s = 7 * 8 x 7 x 4 = 1568.

Case II

When unit digit can be 9 & digit at thousand’s place can be 1,2,3,4,5,6,7 or 8.
No. of ways digits can be filled are:

Total no’s = 8 x 8 x 7 x 1 = 448.

CaseIII

When unit digit can be 1,3,5 or 7 & digit at thousand’s place can be 9.
No. of ways digits can be filled are:

Total no’s = 1 x 8 x 7 x 4 = 224.

∴ Number of odd digits between 1000 & 9999 with no digit repeated = 1568 + 448 + 224 = 2240.

QUESTION: 38

What is the greatest value of the positive integer n satisfying the condition

Solution:

QUESTION: 39

Consider the following

2x2 + 3x - α - 0 has roots -2 and β while the equation x2 - 3mx + 2m2 = 0 has both roots positive, where α > 0 and β > 0. 39.

Q. What is the value of α ?

Solution:

QUESTION: 40

Consider the following

2x2 + 3x - α - 0 has roots -2 and β while the equation x2 - 3mx + 2m2 = 0 has both roots positive, where α > 0 and β > 0. 39.

If β , 2 , 2m are in GP, then what is the value of

Solution:

QUESTION: 41

sin A + 2 sin 2A + sin 3A is equal to which of the following?

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

Solution:

Let A = 30°

QUESTION: 42

If x = sin70°.sin50° and y = cos60°.cos80°, then what is xy equal to ?

Solution:

QUESTION: 43

then what is the value of

Solution:

QUESTION: 44

What is the value of

Solution:

QUESTION: 45

then what is the value of

Solution:

QUESTION: 46

What is the value of cos(2 cos-110.8) ?

Solution:

QUESTION: 47

The top of a hill when observed from the top and bottom of a building of height h is at angles of elevation p and q respectively. What is the height of the hill?

Solution:

Let height of hill = H & horizontal distance between building & hill = d

QUESTION: 48

then what is the value of sin 81 ° ?

Solution:

After squaring all the options available, we come to a conclusion that option (a) is correct.

QUESTION: 49

A moving boat is observed from the top of a cliff of 150 m height. The angle of depression of the boat changes from 60° to 45° in 2 minutes. What is the speed of the boat in metres per hour?

Solution:

QUESTION: 50

What is  equal to?

Solution:

QUESTION: 51

An equilateral triangle has one vertex at (0,0) and another at . What are the coordinates of the third vertex?

Solution:

QUESTION: 52

What is the equation of the right bisector of the line segment joining (1,1) and (2,3)?

Solution:

Equation of given line is

and slope of perpendicular =

The perpendicular is also bisector, therefore it will pass through its mid-point.

=> Coordinates o f m id-point of given line are :

So, equation of perpendicular bisector is :

QUESTION: 53

What is the radius of the circle passing through the point (2,4) and having centre at the intersection of the lines x - y = 4 and 2x + 3y + 7 = 0?

Solution:

(these are coordinates of centre of the circle)

QUESTION: 54

What is the equation of the hyperbola having latus rectum and eccentricity 8 and  respectively?

Solution:

Let th e equation o f hyperbola be

QUESTION: 55

If the point (a, a) lies between the lines |x + y| = 2, then which one of the following is correct?

Solution:

QUESTION: 56

What is the equation of the straight line which passes through the point of intersection of the straight lines x + 2y = 5 and 3x + 7y = 17 and is perpendicular to the straight line 3x+4y= 10?

Solution:

Intersecting lines are : x+2y = 5 & 3x + 7y = 17

On solving these we get : x = 1 & y = 2

Equation of perpendicular line is

QUESTION: 57

If (a, b) is at unit distance from the line 8x + 6y + 1 = 0, then which of the following conditions are correct?

1. 3 a - 4 b - 4 = 0

2. 8a+6b+11=0

3. 8a+6b - 9 = 0

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

Solution:

QUESTION: 58

If the ellipse 9x2 + 16y2 = 144 intercepts the line 3 x + 4y = 12, then what is the length of the chord so formed?

Solution:

QUESTION: 59

A straight line cuts off an intercept of 2 units on the positive direction of x-axis and passes through the point (-3, 5). What is the foot of the perpendicular drawn from the point (3,3) on this line?

Solution:

The given line passes through (-3, 5) and (2, 0). Its equation is

QUESTION: 60

What is the eccentricity of rectangular hyperbola?

Solution:

QUESTION: 61

Consider the following

Let Q be the image o f the point P ( - 2 , 1, - 5 ) in the plane 3x-2y+2z+ 1=0.

1. The coordinates of Q are (4, -3, -1).

2. PQ is of length more than 8 units.

3. The point (1, -1, -3) is the mid-point of the line segment PQ and lies on the given plane.

Q. Which of the above statements are correct?

Solution:

QUESTION: 62

Consider the following

Let Q be the image o f the point P ( - 2 , 1, - 5 ) in the plane 3x-2y+2z+ 1=0.

1. The direction ratios of the line segment PQ are <3, -2,2 >.

2. The sum of the squares of direction cosines of the line segment PQ is unity.

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

Solution:

From (i) above, 1 is correct.

We know that,

Sum of direction cosines of the line segment PQ = 1.

QUESTION: 63

Consider the following

A line L passes through the point P(5, - 6,7) and is parallel to the planes x + y + z = 1 and 2 x - y - 2 z = 3.

Q. What are the direction ratios of the line of intersection of the given planes?

Solution:

Let a, b, c be the direction ratios of the line.

Then its equation is

QUESTION: 64

Consider the following

A line L passes through the point P(5, - 6,7) and is parallel to the planes x + y + z = 1 and 2 x - y - 2 z = 3.

What is the equation of the line L ?

Solution:

From (ii) equation of the line is

QUESTION: 65

Consider the following

where is parallel to and  is perpendicular to .

Q. What is equal to?

Solution:

QUESTION: 66

Consider the following

where  is parallel to  and  is perpendicular to .

then which of the following equations is/are correct?

1. y - x = 4

2. 2z - 3 = 0

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

Solution:

So, neither 1 nor 2 is correct.

QUESTION: 67

Consider the following

be three vectors such that

Q. What is  equal to?

Solution:

QUESTION: 68

Consider the following

be three vectors such that

Q. What is the angle between  and  ?

Solution:

QUESTION: 69

In a right-angled triangle ABC, if the hypotenuse AB = p, then what is  equal to?

Solution:

QUESTION: 70

A force  is applied at the point (1, - 1 , 2 ) . What is the moment of the force about the point (2, -1, 3)?

Solution:

Let point .Pis (1,-1,2)

and point Q is (2, -1,3)

=> Position vector of P w.r.t. Q is

QUESTION: 71

What is the domain of the function

Solution:

We know that

For domain, | x |- x > 0

Case2:x Case 1 : x > 0 => x - x = 0 (not possible)

=>-x - x > 0 => -2x > 0

=>x<0 So, x ∈ (-∞, 0)

QUESTION: 72

Consider the following in respect of the function

1. lim f(x) does not exist.

x—>1

2. f(x) is differentiable at x = 0

3. f(x) is continuous at x = 0 Which of the above statements is/are correct?

Solution:

QUESTION: 73

Let f : A → R, where A = R \ {0} is such that  On which one of the following sets is f(x) continuous?

Solution:

QUESTION: 74

Which one of the following statements is correct in respect of the function f(x) = x3sinx?

Solution:

So, neither maximum nor min. at x=0.

QUESTION: 75

What is the area bounded by the curves

Solution:

For y> 0 =>y = 1 - x2

For y < 0 =>.y = x2- 1

For y = 0=>x = ± 1

So area under the curve = 4 x Area under the region OABO (symmetry)

QUESTION: 76

Consider the following function

Which of the following statements is/are correct?

1. f(x) is increasing in the interval [-1,2].

2. f(x) is decreasing in the interval (2,3].

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

Solution:

=> f ( x ) is decreasing in the interval (2,3]

QUESTION: 77

Consider the following function

Which of the following statements are correct?

1. f(x) is continuous at x = 2.

2. f(x) attains greatest value at x = 2.

3. f(x) is differentiable at x = 2.

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

Solution:

For continuity at x = 2.
RHL

So f ( x ) attains greatest value at x = 2.

QUESTION: 78

Consider the following

Q. What is f'(x) equal to when x> 1 ?

Solution:

When x > 1

f(x) = 1

f(x)=0

QUESTION: 79

Consider the following

Q. What is f'(x) equal to when 0<x<1?

Solution:

When 0 <x< 1

f(x) = ( 2 x -l)2

f' (x ) = 2 (2 x -1) .2 = 4 (2x-l)

f'(x) = 8x -4

QUESTION: 80

Consider the following

Which of the following equations is/are correct?

1. f[-2) = f(5)

2. f "(-2) + f "(0.5) + f "(3) = 4

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

Solution:

Only statement 1 is correct.

QUESTION: 81

Consider the following

Let f (x) = [x], where [.] is the greatest integer function and g(x) = sin x be two real valued functions over R.

Q. Which of the following statements is correct?

Solution:

=> f ( x ) is not continuous at x = 0 and also g(x) is continuous at x = 0. (every trignometric function is continuous).

QUESTION: 82

Consider the following

Let f (x) = [x], where [.] is the greatest integer function and g(x) = sin x be two real valued functions over R.

Q. Which one of the following statements is correct?

Solution:

QUESTION: 83

Consider the following

Let f (x) = [x], where [.] is the greatest integer function and g(x) = sin x be two real valued functions over R.

Which of the following statements are correct?

1. (fof)(x) = f(x).

2. (gog) (x) = g(x) only when x = 0.

3. (go (fog)) (x) can take only three values.

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

Solution:

QUESTION: 84

Consider the following

be a real valued function.

Q. Which one of the following statements is correct?

Solution:

which is a strictly decreasing function.

QUESTION: 85

Consider the following

be a real valued function.

Which of the following statements is/are correct?
1. f(x) is right continuous at x = 0.
2. f(x) is discontinuous at x = 1.

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

Solution:

For right hand continuity at x = 0

f (O) = O

= >f{x) is not right continuous at x = 0.

For discontinuity at x = 1

So f is d iscontinuous.

QUESTION: 86

Consider the following

Consider the parabola y = x2 + 7x + 2 and the straight line y = 3x - 3 .

Q. What are the coordinates of the point on the parabola which is closest to the straight line?

Solution:

Parabola Eq : y = x2 + 7x +2

Line eq. :y = 3x - 3

Since all the points given in the options lie on the parabola.

Thus we will calculate the distance from the given line to these points :

QUESTION: 87

Consider the following

Consider the parabola y = x2 + 7x + 2 and the straight line y = 3x - 3 .

Q. What is the shortest distance from the above point on the parabola to the line?

Solution:
QUESTION: 88

Consider the following

Which of the following statements is/are correct?

1. g(x) is differentiable at x = 0.

2. g(x) is differentiable at x = 2.

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

Solution:

Thus g(x) is not differentiable at x = 2.

QUESTION: 89

Consider the following

Q. What is the value of the differential coefficient of g(x) at x = - 2 ?

Solution:

QUESTION: 90

Consider the following

Which of the following statements are correct?
1. g(x) is continuous at x = 0.
2. g(x) is continuous at x = 2.
3. g(x) is continuous at x = -1.

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

Solution:

QUESTION: 91

Letf ( x ) be a function such that  What is

Solution:
QUESTION: 92

What is  dx equal to?

Solution:

Take option (a)

Take option (b):

QUESTION: 93

What are the degree and order respectively of the differential equation satisfying

Solution:

QUESTION: 94

What is the curve which passes through the point (1,1) and whose slope is

Solution:

QUESTION: 95

If xdy = ydx + y2 dy,y > 0 and y(1)= 1, then what is y (-3) equal to?

Solution:

QUESTION: 96

What is the order of the differential equation

Solution:

QUESTION: 97

Which one of the following differential equations represents the family of straight lines which are at unit distance from the origin?

Solution:

y = mx + c (Equation of straight line)

and mx - y + c = 0 is at unit distance from origin.

QUESTION: 98

What is  equal to?

Solution:

Let us differentiate all the options one by one to get the expression in the question whose integral is to be found.

Here xesinx is the common term in all the options. So, let us differentiate it first.

Differentiation of option (b) is

QUESTION: 99

then what is the value of K ?

Solution:

QUESTION: 100

What is equal to?

Solution:

QUESTION: 101

A special dice with numbers 1, - 1 , 2 , - 2 , 0 and 3 is thrown thrice. What is the probability that the sum of the numbers occurring on the upper face is zero?

Solution:

Total no. of elementary events = 63.
Favourable no. of elementary events

= coefficient of x0 in [x + x-l + x0 + x-2 + x2 + x3 ]3

Required probability

QUESTION: 102

There is 25% chance that it rains on any particular day. What is the probability that there is at least one rainy day within a period of 7 days?

Solution:

The probability of rain in one day

Probability of getting at least one rainy day within a period of 7 days

QUESTION: 103

A salesman has a 70% chance to sell a product to any customer. The behaviour of successive customers is independent. If two customers A and B enter, what is the probability that the salesman will sell the product to customer A or B?

Solution:

QUESTION: 104

A student appears for tests I, II and III. The student is considered successful if he passes in tests I, II or III or all the three. The probabilities of the student passing in tests I, II and III are m, n and 1/2 respectively. If the probability of the student to be successful is 1/2, then which one of the following is correct?

Solution:

QUESTION: 105

Three candidates solve a question. Odds in favour of the correct answer are 5:2,4:3 and 3:4 respectively for the three candidates. What is the probability that at least two of them solve the question correctly?

Solution:

QUESTION: 106

Consider the following statements:
1. The mean and median are equal in symmetric distribution.
2. The range is the difference between the maximum value and the minimum value in the data.
3. The sum of the areas of the rectangles in the histogra is equal to the total area bounded by the frequency polygon and the horizontal axis

Q. Which of the above statements are correct?

Solution:

Mean = Median (in symmetric distribution) Range = (Max. value - Min. value) And sum of areas of rectangles in the histogram is always equal to the total area bounded by frequency polygon and the horizontal axis.

QUESTION: 107

The scores of 15 students in an examination were recorded as 10,5,8,16,18,20,8,10,16,20,18,11,16,14 and 12. After calculating the mean, median and mode, an error is found. One of the values is wrongly written as 16 instead of 18. Which of the following measures of central tendency will change?

Solution:

Mean o f the scores

Mean of the correct scores

i.e., Mean changes.
Median is same for both cases i.e., 14.
Mode is proportional to mean.

QUESTION: 108

For 10 observations on price (x) and supply (y), the following data was obtained :

Q. What is line of regression of y on x?

Solution:

Line of regression of y on x is :

QUESTION: 109

In a study of two groups, the following results were obtained:

Q. Which of the following statements is correct?

Solution:

For Group A:

Coefficient of variation

=> Group A is less variable.

QUESTION: 110

Consider the following statements in respect of class intervals of grouped frequency distribution:
1. Class intervals need not be mutually exclusive.
2. Class intervals should be exhaustive.
3. Class intervals need not be of equal width.

Q.Which of the above statements are correct?

Solution:
QUESTION: 111

A medicine is known to be 75% effective to cure a patient. If the medicine is given to 5 patients, what is the probability that at least one patient is cured by this medicine?

Solution:

Probabilty of medicine to cure a patient

Probability of curing at least one patient

QUESTION: 112

For two events , A and B, it is given that P (A )    are the complementary events of A and B, then  equal to?

Solution:

QUESTION: 113

A machine has three parts, A, B and C, whose chances of being defective are 0.02, 0.10 and 0.05 respectively. The machine stops working if any one of the parts becomes defective. What is the probability that the machine will not stop working?

Solution:

Probability that machine stops working

(∵ A, B & C are independent events)

∴ Probability that the machine will not stop working

QUESTION: 114

Three independent events, A, A2 and A3 occur with probabilities  What is the probability that at least one of the three events occurs?

Solution:

Probability that at least one of these events occur is P( A1 ∪ A2 ∪ A3). Also Aly A2 & A3 are independent events.

QUESTION: 115

Two variates, x and y, are uncorrelated and have standard deviationsa σand σy respectively. What is the correlation coefficient between x+y and x - y ?

Solution:

Therefore x and y are uncorrelated.

QUESTION: 116

A random sample of 20 people is classified in the following table according to their ages:

Q. What is the meanage of this group of people?

Solution:

QUESTION: 117

If the covariance between x and y is 30, variance of x is 25 and variance of y is 144, then what is the correlation coefficient?

Solution:

cov(x,y) = 30

var(x) = 25; var(j) = 144

QUESTION: 118

A coin is tossed three times. Consider the following events:

C: At least two heads appear

Q. Which one of the following is correct?

Solution:

By checking the options

(d) A ∩ ( B' ∪ C' ) = B' ∩ C' is correct.

QUESTION: 119

In a series of 3 one-day cricket matches between teams A and B of a college, the probability of team A winning or drawing are 1/3 and 1/6 respectively. If a win, loss or draw gives 2, 0 and 1 point respectively, then what is the probability that team A will score 5 points in the series?

Solution:

Req. Prob. = P(5 Points) =P(two win and one draw)

=P(WWD) + P(WDW) P(DWW)

QUESTION: 120

Let the random variable X follow B (6, p). If 16 P(X=4)= P(X=2), then what is the value of p?

Solution:

X Following B (6,p) = 16P (x=4) = P(x=2)