NDA II 2015 Mathematics
This test contains 120 question. Each question comprises four responses (answers). You need to select only ONE response for each question.
All questions carry equal marks.
Each question for which a wrong answer has been marked, one-third of the marks assigned to that question will be deducted as penalty.
If a candidate gives more than one answer, it will be treated as a wrong answer even if one of the given answers happens to be correct and there will be same penalty as above to
that question.
If a question is left blank, i.e., no answer is given by the candidate, there will be no penalty for that question.
All questions carry equal marks.
Each question for which a wrong answer has been marked, one-third of the marks assigned to that question will be deducted as penalty.
If a candidate gives more than one answer, it will be treated as a wrong answer even if one of the given answers happens to be correct and there will be same penalty as above to
that question.
If a question is left blank, i.e., no answer is given by the candidate, there will be no penalty for that question.
- Question 1
Let X be the set of all the 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:Option A: an equivalence relationOption B: reflexive and transitive but not symmetricOption C: symmetric and transitive but not reflexiveOption D: reflexive and symmetric but not transitiveVIEW SOLUTION
- Question 2
The matrix will have inverse for every real number x except forOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 3
If the value of the determinant is positive, where a ≠ b ≠ c, then the value of abcOption A: cannot be less than 1Option B: is greater than –8Option C: is less than –8Option D: must be greater than 8VIEW SOLUTION
- Question 4
Consider the following statements with respect to the determinant
where α, β are complementary angles
1. The value of the determinant is .
2. The maximum value of the determinant is .
Which of the above statements is/are correct?Option A: 1 onlyOption B: 2 onlyOption C: Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- Question 5
What is (1000000001)2 – (0.0101)2 equal to?Option A: (512.6775)10Option B: (512.6875)10Option C: (512.6975)10Option D: (512.0909)10VIEW SOLUTION
- Question 6
If then the matrix X for which 2X + 3A = 0 holds true isOption A:Option B:Option C:Option D:VIEW SOLUTION
- 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
Select the correct answer out of the given options:Option A: 1 onlyOption B: 2 onlyOption C: 3 onlyOption D: NoneVIEW SOLUTION
- Question 8
If then which of the following is/are correct?
1. A and B commute
2. AB is a null matrix
Select the correct answer out of the following options:Option A: 1 onlyOption B: 2 onlyOption C:
Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- Question 9
The number of real roots of the equation x2 – 3 |x| + 2 = 0 is:Option A: 4Option B: 3Option C: 2Option D: 1VIEW SOLUTION
- Question 10
If the sum of the roots of the equation ax2 + bx + c = 0 is equal to the sum of their squares, thenOption A: a2 + b2 = c2Option B: a2 + b2 = a + bOption C: ab + b2 = 2acOption D: ab – b2 = 2acVIEW SOLUTION
- Question 11
If then which of the following is/are correct ?
1.
2.
Select the correct answer using the code give below:Option A: 1 onlyOption B: 2 onlyOption C: Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- Question 12
A, B, C and D are four sets such that Consider the following:
1. are always disjoint.
2. and are always disjoint.
Which of the above statements is/are correct?Option A: 1 onlyOption B: 2 onlyOption C: Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- 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 isOption A: k–nOption B: k–1Option C: knOption D: nkVIEW SOLUTION
- Question 14
- Question 15
If the roots of the equation x2 – nx + m = 0 differ by 1, thenOption A: n2 – 4m – 1 = 0Option B: n2 + 4m – 1 = 0Option C: m2 + 4n + 1 = 0Option D: m2 – 4n – 1 = 0VIEW SOLUTION
- 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:Option A: NAAGIOption B: NAAIGOption C: IAAGNOption D: IAANGVIEW SOLUTION
- Question 17
The number of ways in which a cricket team of 11 players can be chosen out of a batch of 15 players, so that the captain of the team is always included would be:Option A: 165Option B: 364Option C: 1001Option D: 1365VIEW SOLUTION
- Question 18
In the expansion of , the value of the constant term (independent of x) is :Option A: 5Option B: 8Option C: 45Option D: 90VIEW SOLUTION
- Question 19
The value of sin25º + sin210º + sin215º + sin220º + …… + sin290º is:Option A: 7Option B: 8Option C: 9Option D:VIEW SOLUTION
- Question 20
On simplifying , we get:Option A: sin3AOption B: cos3AOption C: sinA + cosAOption D: 3VIEW SOLUTION
- Question 21
- Question 22
Two poles are 10 m and 20 m high. The line joining their tops makes an angle of 15º with the horizontal. The distance between the poles is approximately equal to:Option A: 36.3 mOption B: 37.3 mOption C: 38.3 mOption D: 39.3 mVIEW SOLUTION
- Question 23
If , then which one of the following is correct?Option A: f(f(f(g(g(f(x)))))) = g(g(f(g(f(x)))))Option B: f(f(g(g(g(f(x)))))) = g (g (f(g(f(x)))))Option C: f(g(f(g(g(f(g(x)))))) = g(g (f (g(f(x)))))Option D: f(f(f(g(g(f(x)))))) = f(f(f(g(f(x)))))VIEW SOLUTION
- Question 24
Consider the following:
Which of the above statements is/are correct?Option A: 1 onlyOption B: 2 onlyOption C: Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- Question 25
If A is an orthogonal matrix of order 3 and then which of the following is/are correct?
1.
2. AB = BA
Select the correct answer using the code given below:Option A: 1 onlyOption B: 2 onlyOption C: Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- Question 26
If a, b, c are the sides of triangle ABC, then where p > 1, is:Option A: always negativeOption B: always positiveOption C: always zeroOption D: positive if 1 < p < 2 and negative if p > 2VIEW SOLUTION
- Question 27
If a, b, c are real numbers, then the value of the determinant isOption B: (a – b) (b – c) (c – a)Option C: (a + b + c)2Option D: (a + b + c)3VIEW SOLUTION
- Question 28
If the point z1 = 1 + i where is the reflection of a point z2 = x + iy in the line , then the point z2would be:Option A: 1 + 4iOption B: 4 + iOption C: 1 – iOption D: – 1 –iVIEW SOLUTION
- Question 29
If sinx + siny = a and cosx + cosy = b, then is equal to:Option A:Option B:Option C:Option D: None of theseVIEW SOLUTION
- 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 30º and 45º respectively. Which one of the following gives the best approximation to the height of the tower?Option A: 3.90 mOption B: 4.00 mOption C: 4.10 mOption D: 4.25 mVIEW SOLUTION
- Question 31
Consider the expansion of (1 + x)2n+1
If the coefficients of xr and xr+1 are equal in the expansion, then r is equal to:Option A: nOption B:Option C:Option D: n + 1VIEW SOLUTION
- Question 32
Consider the expansion of (1 + x)2n+1
The average of the coefficient of the two middle terms in the expansion is:Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 33
Consider the expansion of (1 + x)2n+1
The sum of the coefficients of all the terms in the expansion is:Option A: 22n–1Option B: 4n–1Option C: 2 × 4nOption D: None of the aboveVIEW SOLUTION
- Question 34
The nth term of an A.P is then the sum of first 105 terms is:Option A: 270Option B: 735Option C: 1409Option D: 1470VIEW SOLUTION
- Question 35
A polygon has 44 diagonals. The number of its sides is:Option A: 11Option B: 10Option C: 8Option D: 7VIEW SOLUTION
- Question 36
If p, q, r are in one geometric progression and a, b, c are in another geometric progression, then ap, bq, cr are in:Option A: Arithmetic progressionOption B: Geometric progressionOption C: Harmonic progressionOption D: None of the aboveVIEW SOLUTION
- Question 37
Consider triangle ABC satisfying
The sides of the triangle are in
Option A: G.P.Option B: A.P.Option C: H.P.Option D: Neither in G.P. nor in A.P. nor in H.P.VIEW SOLUTION
- Question 38
Consider triangle ABC satisfying
sin A, sin B, sin C are inOption A: G.P.Option B: A.P.Option C: H.P.Option D: Neither in G.P. nor in A.P. nor in H.P.VIEW SOLUTION
- Question 39
If then which of the following is/are correct?
1. The value of p × r is 2.
2. p, q and r are in G.P.
Select the correct answer using the code given below:Option A: 1 onlyOption B: 2 onlyOption C: Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- Question 40
The number of ways in which 3 holidays tickets can be given to 20 employees of an organisation if each employee is eligible for any one or more of the tickets, isOption A: 1140Option B: 3420Option C: 6840Option D: 8000VIEW SOLUTION
- Question 41
- Question 42
The coefficient of x99 in the expansion of (x – 1)(x – 2)(x – 3) ……. (x – 100) isOption A: 5050Option B: 5000Option C: – 5050Option D: –5000VIEW SOLUTION
- Question 43
represents a circle withOption A: Centre (–3, –1) and radius 3Option B: Centre (–3, 1) and radius 3Option C: Centre (–3, –1) and radius 4Option D: Centre (–3, 1) and radius 4VIEW SOLUTION
- Question 44
The number of 3-digit even number that can be formed from the digits 0, 1, 2, 3, 4 and 5, repetition of digits being not allowed, isOption A: 60Option B: 56Option C: 52Option D: 48VIEW SOLUTION
- Question 45
- Question 46
- Question 47
If a line is perpendicular to the line 5x – y = 0 and forms a triangle of area 5 square units with co-ordinate axes, then its equation isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 48
Consider any point P on the ellipse in the first quadrant. Let r and s represents its distance from (4, 0) and (–4, 0) respectively, then (r + s) is equal toOption A: 10 unitOption B: 9 unitOption C: 8 unitOption D: 6 unitVIEW SOLUTION
- Question 49
A straight line x = y + 2 touches the circle The value of r isOption A:Option B:Option C: 2Option D: 1VIEW SOLUTION
- Question 50
The three lines 4x + 4y = 1, 8x – 3y =2, y = 0 areOption A: the sides of an isosceles triangleOption B: concurrentOption C: mutually perpendicularOption D: the sides of an equilateral triangleVIEW SOLUTION
- Question 51
The line intersects the x-axis at A and y-axis at B. Then the circumcentre of triangle OAB where O is the origin isOption A: (2, 3)Option B: (3, 3)Option C: (4, 3)Option D: None of theseVIEW SOLUTION
- Question 52
The eccentricity of the hyperbola 16x2 – 9y2 = 1 isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 53
The product of the perpendiculars from two points (± 4, 0) to the line isOption A: 25Option B: 16Option C: 9Option D: 8VIEW SOLUTION
- Question 54
If the centre of the circle passing through the origin is (3, 4), then the intercepts cut off by the circle on x-axis and y-axis respectively areOption A: 3 units and 4 unitsOption B: 6 units and 4 unitsOption C: 3 units and 8 unitsOption D: 6 units and 8 unitsVIEW SOLUTION
- Question 55
The lines 2x = 3y = – z and 6x = – y = –4zOption A: are perpendicularOption B: are parallelOption C: intersect at an angle 45ºOption D: intersect at an angle 60ºVIEW SOLUTION
- Question 56
Two straight lines passing through the point A(3, 2) cut the line 2y = x + 3 and x-axis perpendicularly at P and Q, respectively. The equation of the line PQ isOption A: 7x + y – 21 = 0Option B: x + 7y + 21 = 0Option C: 2x + y – 8 = 0Option D: x + 2y + 8 = 0VIEW SOLUTION
- Question 57
The radius of the sphere 3x2 + 3y2 + 3z2 – 8x + 4y + 8z – 15 = 0 isOption A: 2Option B: 3Option C: 4Option D: 5VIEW SOLUTION
- Question 58
The direction ratios of the line perpendicular to the lines with direction ratios <1, –2, –2 > and < 0, 2, 1 > areOption A: < 2, –1, 2 >Option B: < –2, 1, 2 >Option C: < 2, 1, –2 >Option D: < –2, –1, –2 >VIEW SOLUTION
- Question 59
What are the co-ordinates of the foot of the perpendicular drawn from the point (3, 5, 4) on the plane z = 0?Option A: (0, 5, 4)Option B: (3, 5, 0)Option C: (3, 0, 4)Option D: (0, 0, 4)VIEW SOLUTION
- Question 60
The length of the intercepts on the coordinate axes made by the plane 5x + 2y + z – 13 = 0 areOption A: 5, 2, 1 unitOption B:Option C:Option D: 1, 2, 5 unitVIEW SOLUTION
- Question 61
The area of the square, one of whose diagonals is isOption A: 12 square unitOption B: 12.5 square unitOption C: 25 square unitOption D: 156.25 square unitVIEW SOLUTION
- Question 62
ABCD is a parallelogram and P is the point of intersection of the diagonals. If O is the origin, then is equal toOption A:Option B:Option C:Option D: Null vectorVIEW SOLUTION
- Question 63
If and are the position vectors of the point B are C, respectively, then the position vector of the point D such that isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 64
If the position vector of the point (5, n) is such that , then the value/values of n can beOption A: ± 8Option B: ± 12Option C: 8 onlyOption D: 12 onlyVIEW SOLUTION
- Question 65
- Question 66
Consider the following inequalities in respect of vectors and :
Which of the above is/are correct?Option A: 1 onlyOption B: 2 onlyOption C: Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- Question 67
If the magnitude of difference of two unit vectors is , then the magnitude of sum of the two vectors isOption A:Option B: 1 unitOption C: 2 unitOption D: 3 unitVIEW SOLUTION
- Question 68
If the vectors lie on a plane, where are distinct non-negative numbers, then isOption A: Arithmetic mean of andOption B: Geometric mean of andOption C: Harmonic mean of α andOption D: None of the aboveVIEW SOLUTION
- Question 69
The vectors are such that Which of the above is/are correct?
Select the correct answer using the code give below:Option A: 1 onlyOption B: 2 onlyOption C: Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- Question 70
The value of where a + b = 0 isOption A: 2b – asin (b – a)Option B: a + 3bcos (b – a)Option C: sin a – (b – a) cos bVIEW SOLUTION
- Question 71
- Question 72
Consider the function
What is the value of a for which f(x) is continuous at x = –1 and x = 1?Option A: –1Option B: 1Option D: 2VIEW SOLUTION
- Question 73
The function is not defined at x = π. The value of f(π) so that f(x) is continuous at x = π isOption A:Option B:Option C: –1Option D: 1VIEW SOLUTION
- Question 74
Consider the following functions:
Which of the above functions is/are derivable at x = 0?Option A: 1 onlyOption B: 2 onlyOption C: Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- Question 75
The domain of the function isOption A: [0, ∞)Option B: (–∞, 0)Option C: [1, ∞)Option D: (–∞, 0]VIEW SOLUTION
- Question 76
Consider the following statements:
1. The function f(x) = x2 + 2cosx is increasing in the interval (0, π)
2. The function is decreasing in the interval (–∞, ∞)
Which of the above statements is/are correct?Option A: Only 1Option B: Only2Option C: Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- Question 77
The derivative of ln(x + sinx) with respect to (x + cosx) isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 78
If where then is equal toOption A:Option B: 2Option C: sin x + cos xOption D: sin x – cos xVIEW SOLUTION
- Question 79
The function is monotonically increasing itOption A: x < 0 onlyOption B: x > 2 onlyOption C: 0 < x < 2Option D:VIEW SOLUTION
- Question 80
- Question 81
If be two functions given by
f(x) = 2x – 3 and g(x) = x3 + 5, then (fog)–1(x) is equal toOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 82
- Question 83
- Question 84
- Question 85
Consider the following statements:
1. f(x) = ln x is an increasing function on (0, ∞).
2. is an increasing function on (1, ∞).
Which of the above statements is/are correct?Option A: 1 onlyOption B: 2 onlyOption C: Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- Question 86
- Question 87
Consider the following statements:
Statement 1: The function such that f(x) = x3 for all is one-one.
Statement 2: for all if the function f is one-one.
Which one of the following is correct in respect of the above statements?Option A: Both the statements are true and Statement 2 is the correct explanation of Statement 1.Option B: Both the statements are true and Statement 2 is not the correct explanation of Statement 1.Option C: Statement 1 is true but Statement 2 is false.Option D: Statement 1 is false but Statement 2 is true.VIEW SOLUTION
- Question 88
is equal to where c is the constant of integrationOption A: 1 + ex + cOption B: ln (1 + e–x) + cOption C: ln (1 + ex) + cOption D: 2 ln (1 + e–x) + cVIEW SOLUTION
- Question 89
- Question 90
The area bounded by the coordinate axes and the curve isOption A: 1 square unitOption B: square unitOption C: square unitOption D: square unitVIEW SOLUTION
- Question 91
Consider the function where x > 0
At what value of x, does the function attain maximum value?Option A: eOption B:Option C:Option D:VIEW SOLUTION
- Question 92
Consider the function where x > 0
The maximum value of the function isOption A: eOption B:Option C:Option D:VIEW SOLUTION
- Question 93
Consider such that f(0) = 0 and f(3) = 15
The value of k isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 94
Consider such that f(0) = 0 and f(3) = 15
is equal toOption A: –1Option B:Option C:Option D: 1VIEW SOLUTION
- Question 95
Consider the function f(x) = 2x3 – 9x2 – 12x + 1
The function f(x) is an increasing function in the intervalOption A: (–2, –1)Option B: (–∞, –2)Option C: (–1, 2)Option D: (–1, ∞)VIEW SOLUTION
- Question 96
Consider the function f(x) = 2x3 – 9x2 – 12x + 1
The function f(x) is a decreasing function in the intervalOption A: (–2, –1)Option B: (–∞, –2) onlyOption C: (–1, ∞) onlyOption D:VIEW SOLUTION
- Question 97
Consider the integrals
Which one of the following is correct?Option A: A = 2BOption B: B = 2AOption C: A = BOption D: A = 3BVIEW SOLUTION
- Question 98
- Question 99
Consider the function
Which is continuous everywhere.
The value of A isOption A: 1Option C: –1Option D: –2VIEW SOLUTION
- Question 100
- Question 101
The degree of the differential equation isOption A: 2Option B: 3Option C: 4Option D: 5VIEW SOLUTION
- Question 102
The solution of is where c is an arbitrary constantOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 103
The differential equation of the family of circles passing through the origin and having centres on the x-axis isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 104
The order and degree of the differential equation of parabolas having vertex at the origin and focus at (a, 0) where a > 0, are respectivelyOption A: 1, 1Option B: 2, 1Option C: 1, 2Option D: 2, 2VIEW SOLUTION
- Question 105
is true for allOption A: Polynomial functions fOption B: Trigonometric functions fOption C: Exponential function fOption D: Logarithmic functions fVIEW SOLUTION
- Question 106
Three digits are chosen at random from 1, 2, 3, 4, 5, 6, 7, 8 and 9 without repeating any digit. What is the probability that the product is odd?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 107
Two events A and B are such that P(not B) = 0.8, P(A∪B) = 0.5 and P(A|B) = 0.4. Then P(A) is equal toOption A: 0.28Option B: 0.32Option C: 0.38Option D: None of theseVIEW SOLUTION
- Question 108
If the mean and variance of a Binomial variate X are 2 and 1, respectively, then the probability that X takes a value greater than 1 isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 109
Seven unbiased coins are tossed 128 times. In how many throws would you find at least three heads?Option A: 99Option B: 102Option C: 103Option D: 104VIEW SOLUTION
- Question 110
A coin is tossed five times. What is the probability that heads are observed more than three times?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 111
The geometric mean of the observations x1, x2, x3,…….,xn is G1. The geometric mean of the observations y1, y2, y3,……yn is G2. The geometric mean of the observations isOption A: G1G2Option B: ln (G1G2)Option C:Option D:VIEW SOLUTION
- Question 112
The arithmetic mean of 1, 8, 27, 64, … Up to n terms is given byOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 113
An unbiased coin is tossed until the first head appears or until four tosses are completed, whichever happens earlier. Which of the following statements is/are correct?
1. The probability of no head observed is
2. The probability that the experiment ends with three tosses is
Select the correct answer using the code given below:Option A: 1 onlyOption B: 2 onlyOption C: Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- Question 114
If , then what is the probability that x2 – 3x + 2 ≥ 0?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 115
A bag contains 4 white and 2 black balls and another bag contains 3 white and 5 black balls. If one ball is drawn from each bag, then the probability that one ball is white and one ball is black isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 116
The problem in statistics is given to three students A, B and C whose chances of solving it independently are respectively. The probability that the problem will be solved isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 117
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probabilities of an accident involving a scooter driver, car driver and a truck driver are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. The probability that the person is a scooter driver isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 118
A coin is tossed 5 times. The probability that tails appear an odd number of times, isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 119
The regression coefficients of a bivariate distribution are –0.64 and –0.36. Then the correlation coefficient of the distribution isOption A: 0.48Option B: – 0.48Option C: 0.50Option D: – 0.50VIEW SOLUTION
- Question 120
What is the probability that the sum of any two different single digit natural numbers is a prime number?Option A:Option B:Option C:Option D: None of the theseVIEW SOLUTION