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NDA II - Mathematics Question Paper 2015 - Defence MCQ


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30 Questions MCQ Test - NDA II - Mathematics Question Paper 2015

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NDA II - Mathematics Question Paper 2015 - Question 1

Let X be the set of all persons living in Delhi. The persons a and b in X are said to be related if the difference in their ages is at most 5 years. The relation is

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 1

a ~ b≤ 5
Let b ~ c ≤ 5
so a ~ c ≤ 10
as (a ~ b) ∈ Rand (b ~c) ∈R
but (a ~ c) ∉ R
So it is not a transitive relation.
By considering all the options, we come to the conclusion that only option (d) is correct.

NDA II - Mathematics Question Paper 2015 - Question 2

The matrix A=  will have inverse for every real number x except for

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 2

|A| = 1 [(x -1 )(x-3) - 7] - 3 [(x - 3) - 2] + 2[7-2(x- 1)]
= x2 -11x + 29 = 0

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NDA II - Mathematics Question Paper 2015 - Question 3

If the value of the determinant  is positive, where a ≠ b ≠ c, then the value of abc

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 3

=> a(bc-1) - l(c-1)+ 1(1 - b)>0
=> abc-a-c+l + l- b>0
=> abc + 2-(a + b + c)>0
=> abc >(a + b + c)-2
Let; a = -l;b = 0& c = l
Then; 0 > -2 [which is correct]
Hence, abc = 0
After considering all the option; (b) is correct option.

NDA II - Mathematics Question Paper 2015 - Question 4

Consider the following statements in respect of the determinant

where α, β are complementary angles
1. The value of the determinant is 
2. The maximum value of the determinant is 1/√2

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

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 4

NDA II - Mathematics Question Paper 2015 - Question 5

What is (1000000001)2 - (0.0101)2 equal to?

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 5

(1000000001)2

= 1 x2° + 0x21+0x22+          + 1 x29
= 1 + 0 + 0+....... +512
=(513)10
(0.0101)2 =0 x2-1 + 1 x2-2 + 0 x 2-3 + 1 X2-4

= 1/4 + 1/16 = 5/16 = (0.3125)10

(1000000001)2 - (0.0101)2 = 513 - 0.312

=(512.6875)10 

NDA II - Mathematics Question Paper 2015 - Question 6

If A = , then the matrix X for which 2X+3A= 0 holds true is 

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 6

NDA II - Mathematics Question Paper 2015 - Question 7

If Z1 and z2 are complex numbers with |z1|= |z2|, then which of the following is/are correct?
1. z1 =z2
2. Real part of z1 = Real part of z2.
3. Imaginary part of z1 = Imaginary part of z2

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

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 7

It is true for many values of a1, a2 & b1, b2. So a1 must not equal to a2, and b1 must not equal to b2

NDA II - Mathematics Question Paper 2015 - Question 8

If A =  and B =  then which of the following is/are correct ?
1. A and B commute.
2. AB is a null matrix.

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

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 8

asAB ≠ BA
So A and B are not commute.

NDA II - Mathematics Question Paper 2015 - Question 9

The number of real roots of the equation x2 - 3 |x| + 2 = 0 is

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 9

NDA II - Mathematics Question Paper 2015 - Question 10

If the sum of the roots of the equation ax2 + bx + c = 0 is equal to the sum of their squares, then

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 10

NDA II - Mathematics Question Paper 2015 - Question 11

 then which of the following is/ are correct?
1. (A ∩ B) = (-2,1)
2. (A\B) = (-7,-2)

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

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 11

x2 + 6x - 7 < 0
=> ( x + 7 ) ( x - 1 ) < 0
=> x = ( - 7 , l )
=> A= {- 7 ,- 6 ,- 5 ,- 4 ,- 3 ,- 2 ,- 1,0,1}
=> x2+ 9 x + 1 4 > 0
=> (x + 7 )(x + 2 )> 0
=> x = (-∞, - 7 ) ∪ ( - 2 , ∞)
=> B = R - { - 7 , - 6, - 5, - 4 , - 3, - 2 }

So A ∩ B = ( - 2 , l )
A/B = (- 7, - 2).

NDA II - Mathematics Question Paper 2015 - Question 12

A, B, C and D are four sets such th at A ∩ B = C ∩ D = ø Consider the following:
1. A ∪ C and B ∪ D are always disjoint.
2. A ∩ C a n d B ∩ D are always disjoint

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

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 12

LetA= {1,2}
B = {3 ,4 ,0 }
C = {5 ,6 ,0 }
D = { 7 , 8 }
Such that (A ∩ B) = (C ∩ D) = ø
=> (A ∪ C) = { 1 , 2 , 5 , 6 , 0 }
=> ( B ∪ D ) = {3 ,4 ,7 ,8 ,0 }
=> (A ∪ C )∩(B ∪ D )={0}
So (A ∪ C) and (B ∪ D) are not always dispoint
=> ( A ∩ C ) = ø and (B ∩ D ) = ø
So (A ∩ C) and (B ∩ D) are always disjoint.

NDA II - Mathematics Question Paper 2015 - Question 13

If A is an invertible matrix of order n and k is any positive real number, then the value of [det(kA)]-1 det A is

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 13

If A is matrix that is invertible then det- (kA) will be kn. det(A), where n is the order.

NDA II - Mathematics Question Paper 2015 - Question 14

The value o f the infinite product is

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 14

NDA II - Mathematics Question Paper 2015 - Question 15

If the roots of the equation
x2 - nx + m = 0
differ by 1, then

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 15

NDA II - Mathematics Question Paper 2015 - Question 16

If different words are formed with all the letters of the word 'AGAIN' and are arranged alphabetically among themselves as in a dictionary, the word at the 50th place will be

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 16

NDA II - Mathematics Question Paper 2015 - Question 17

The number of ways in which a cricket team of 11 players be chosen out of a batch of 15 players so that the captain of the team is always included, is

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 17

If captain is always included then we can choose 10 more players out of the remaining 14 players. So

NDA II - Mathematics Question Paper 2015 - Question 18

In the expansion of the value of constant term (independent of x) is

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 18

Equating the coefficient of x to zero. 

NDA II - Mathematics Question Paper 2015 - Question 19

The value of sin2 5° + sin2 10° + sin2 15° + sin220°+.... + sin2 90° is

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 19

NDA II - Mathematics Question Paper 2015 - Question 20

On simplifying  , we get, sin A cos A

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 20

= 3 - 3 sin2A - 3cos2A + 3
= 6 - 3 (cos2A + sin2A)
= 6 - 3 (1)
=3

NDA II - Mathematics Question Paper 2015 - Question 21

The value of tan  is

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 21

NDA II - Mathematics Question Paper 2015 - Question 22

Two poles are 10 m and 20 m high. The line joining their tops makes an angle of 15° with the horizontal. The distance between the poles is approximately equal to

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 22

NDA II - Mathematics Question Paper 2015 - Question 23

If  then which one of the following is correct

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 23

f(x)=x
g(x)= 1/x
Putting these values in the options, only (b) is correct.

NDA II - Mathematics Question Paper 2015 - Question 24

Consider the following:

Q. which of the above is/are correct ?

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 24

Statement (1) is correct Again
Statement (2) is incorrect.

NDA II - Mathematics Question Paper 2015 - Question 25

If A is an orthogonal matrix of order 3 and B = , then which of the following is/are correct?
1. |AB| = ±47
2. AB = BA

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

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 25

The determinent of a orthogonal matrix is always ±1 |A |= ± 1

Taking A as identity matrix we can prove AB = BA

NDA II - Mathematics Question Paper 2015 - Question 26

If a,b,c are the sides o f a triangle ABC, then  where p> 1, is

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 26

Consider any equilateral triangle:
c = b = a = 1 unit

By considering all the options carefully; we came to a conclusion that opton (b) is correct.

NDA II - Mathematics Question Paper 2015 - Question 27

If a,b,c are real numbers, then the value of the determinant  is

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 27

NDA II - Mathematics Question Paper 2015 - Question 28

If the point z1 = 1 + i w here i = V-1 is th e reflection of a point z2 = x + iy in the line iz  - iz = 5, then the point z2 is

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 28

Let Z = a+ bi

NDA II - Mathematics Question Paper 2015 - Question 29

If sin x + sin y = a and cos x + cos y = b, then  is equal to 

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 29

Squaring of eq (1) & (2) and adding -

NDA II - Mathematics Question Paper 2015 - Question 30

A vertical tower standing on a levelled field is mounted with a vertical flag staff of length 3 m. From a point on the field, the angles of elevation of the bottom and tip of the flag staff are 30° and 45° respectively. Which one of the following gives the best approximation to the height of the tower ?

Detailed Solution for NDA II - Mathematics Question Paper 2015 - Question 30

as ∠CPA=45°

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