NDA I 2017 Mathematic
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 S be the set of all persons living in Delhi. We say that x, y in S are related if the y were born in Delhi on the same day. Which one of the following is correct?Option A: The relation is an equivalent relationOption B: The relation is not reflective but it is symmetric and transitiveOption C: The relation is not symmetric bout it is reflexive and transitiveOption D: The relation is not transitive but it is reflexive ands ymmetricVIEW SOLUTION
- Question 2
Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Then the number of subsets of A containing two or three elements isOption A: 45Option B: 120Option C: 165Option D: 330VIEW SOLUTION
- Question 3
- Question 4
If the difference between the roots of the equation is strictly less than , where |k| ≥ 2, then k can be any element of the intervalOption A:Option B: (–3, 3)Option C:Option D: None of theseVIEW SOLUTION
- Question 5
If the roots of the equation are in the same ratio as those of the equation , then which one of the following is correct?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 6
The value of where n is not a multiple of 3 and isOption A: 1Option B: –1Option C: iOption D: –iVIEW SOLUTION
- Question 7
Three-digit numbers are formed from the digits 1, 2 and 3 in such a way that the digits are not repeated. What is the sum of such 3 digit numbers?Option A: 1233Option B: 1322Option C: 1323Option D: 1332VIEW SOLUTION
- Question 8
What is the sum of the series 0.3 + 0.33 + 0.333 +….n terms?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 9
If 1, ω, ω2 are the cube roots of unity, then is equal toOption A: –2Option B: –1Option D: 2VIEW SOLUTION
- Question 10
If the sum of m terms of an AP is n and the sum of n terms is m, then the sum of (m + n) term isOption A: mnOption B: m + nOption C: 2(m + n)Option D: – (m + n)VIEW SOLUTION
- Question 11
The modulus and the principal argument of the complex numbers are respectivelyOption A: 1, 0Option B: 1, 1Option C: 2, 0Option D: 2, 1VIEW SOLUTION
- Question 12
If the graph of a quadratic polynomial lies entirely above the x-axis, then which one of the following is correct?Option A: Both the roots are realOption B: One root is real and the other is complexOption C: Both the roots are complexOption D: Cannot sayVIEW SOLUTION
- Question 13
- Question 14
- Question 15
If cot α and cot β are the roots of the equation with b ≠ 0 then the value of cot (α + β) isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 16
The sum of the roots of the equation (where b and c are non-zero) is equal to the sum of the reciprocals of their square. Then are inOption A: APOption B: GPOption C: HPOption D: None of the aboveVIEW SOLUTION
- Question 17
The sum of the roots of the equation (where a and c are non-zero) is equal to the sum of the reciprocals of their squares. Then a, ca2, c2 are inOption A: APOption B: GPOption C: HPOption D: None of the aboveVIEW SOLUTION
- Question 18
The value of [C(7, 0) + C(7, 1)] + [C(7, 1) + C(7, 2) ] + ……+ [C(7, 6) + C(7, 7)] isOption A: 254Option B: 255Option C: 256Option D: 257VIEW SOLUTION
- Question 19
The number of different words (eight-letter words) ending and beginning with a consonant which can be made out of the letters of the word ‘EQUATION’ isOption A: 5200Option B: 4320Option C: 3000Option D: 2160VIEW SOLUTION
- Question 20
The fifth term of an AP of n terms, whose sum is n2 – 2n, isOption A: 5Option B: 7Option C: 8Option D: 15VIEW SOLUTION
- Question 21
The sum of all the two-digit odd numbers isOption A: 2475Option B: 2530Option C: 4905Option D: 5049VIEW SOLUTION
- Question 22
The sum of the first n terms of the series is equal toOption A:Option B: 1 – 2–nOption C:Option D:VIEW SOLUTION
- Question 23
Consider the following in respect of sets A and B:
1.
2.
3.
4.
Which of the above are correct?Option A: 1, 2 and 3Option B: 2, 3 and 4Option C: 1, 3 and 4Option D: 1, 2 and 4VIEW SOLUTION
- Question 24
In the binary equation where p, q and r are binary digits, what are the possible value of p, q and r, respectively?Option A: 0, 1, 0Option B: 1, 1, 0Option C: 0, 0, 1Option D: 1, 0, 1VIEW SOLUTION
- Question 25
- Question 26
The expansion of is done in the descending powers of x. If the sum of the fifth and sixth terms is zero, then is equal toOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 27
- Question 28
If B is a non-singular matrix and A is a square matrix, then the value of is equal toOption A: det(B)Option B: det(A)Option C: det(B–1)Option D: det(A–1)VIEW SOLUTION
- Question 29
If a ≠ b ≠ c, then one value of x which satisfies the equation is given by ;Option A: aOption B: bOption C: cVIEW SOLUTION
- Question 30
then what is AAT equal to (where AT is the transpose of A)?Option A: Null matrixOption B: Identity matrixOption C: AOption D: –AVIEW SOLUTION
- Question 31
The equations
x + 2y + 3z = 1
2x + y + 3z = 2
5x + 5y + 9z = 4Option A: have the unique solutionOption B: have Infinitely many solutionsOption C: are in inconsistentOption D: None of theseVIEW SOLUTION
- Question 32
- Question 33
What is the value of the determinant ?Option A: 1 + x + y + zOption B: 2xyzOption C: x2 y2 z2Option D: 2x2 y2 z2VIEW SOLUTION
- Question 34
If then = 0, then which one of the following is correct?Option A: is one of the cube root of unityOption B: x is one of the cube roots of UnityOption C: y is one of the cube roots of unityOption D: is one of the cube root of –1VIEW SOLUTION
- Question 35
Consider the set A of all matrices of order 3 × 3 with entire 0 or 1 only. Let B be the subset of A consisting of all matrices whose determinant is 1. Let C be the subset of A consisting of all matrices whose determinant is –1. Then which one of the following is correct?Option A: C is emptyOption B: B has as many elements as COption C: A = B ∪ COption D: B has thrice as many elements as CVIEW SOLUTION
- Question 36
- Question 37
- Question 38
- Question 39
- Question 40
- Question 41
From the top of a lighthouse, 100 m high, the angle of depression of a boat is . What is the distance between the boat and the lighthouse?Option A: 120 mOption B: 180 mOption C: 240 mOption D: 360 mVIEW SOLUTION
- Question 42
The maximum value of in the interval is attained atOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 43
- Question 44
- Question 45
If sin θ = 3sin (θ + 2α), then the value of tan(θ + α) + 2 tanα is equal toOption A: –1Option C: 1Option D: 2VIEW SOLUTION
- Question 46
- Question 47
Let x, y, z be positive real numbers such that x, y, z are in GP and are in AP. Then which one of the following is correct?Option A: x = y = zOption B: xz = 1Option C: x ≠ y and y = zOption D: x = y and y ≠ zVIEW SOLUTION
- Question 48
If tan(α + β) = 2 and tan (α – β) = 1, then tan (2α) is equal toOption A: –3Option B: –2Option C:Option D: 1VIEW SOLUTION
- Question 49
Consider the following of triangle ABC:
(a)
(b)
(c) sin (B + C) = cos A
(d) tan (B + C) = –cot A
Which of the above are correct?Option A: a and cOption B: a and bOption C: a and dOption D: b and cVIEW SOLUTION
- Question 50
If sec θ – cosec θ = , then what is (sin θ – cos θ) equal to?Option A: –2 onlyOption B: onlyOption C: Both –2 andOption D: Neither nor – 2VIEW SOLUTION
- Question 51
If a vertex of a triangle is (1, 1) and the midpoints of two sides of the triangle through this vertex are (–1, 2) and (–3, 2), then the centroid of the triangle isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 52
The incentre of a triangle with vertices A(1, ), B(0, 0), and C(2, 0) isOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 53
If the three consecutive vertices of a parallelogram are (–2, –1), (1, 0) and (4, 3), then what are the coordinates of the fourth vertex?Option A: (1, 2)Option B: (1, 0)Option C: (0, 0)Option D: (1, –1)VIEW SOLUTION
- Question 54
The two circles and intersect at two distinct points. Then which one of the following is correct?Option A: 2 < r < 8Option B: r = 2 or r = 8Option C: r < 2Option D: r > 2VIEW SOLUTION
- Question 55
What is the equation of the circle which passes through the points (3, –2) and (–2, 0) and having its centre on the line 2x – y – 3 = 0?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 56
What is the ratio in which the point divides the line joining the points A(–2, –2) and B(2, –4)?Option A: 1 : 3Option B: 3 : 4Option C: 1 : 2Option D: 2 : 3VIEW SOLUTION
- Question 57
What is the equation of the ellipse having foci (±2, 0) and the eccentricity ?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 58
What is the equation of the straight line parallel to 2x + 3y + 1 = 0 and passes through the point (–1, 2)?Option A: 2x + 3y – 4 = 0Option B: 2x + 3y – 5 = 0Option C: x + y – 1 = 0Option D: 3x – 2y + 7 = 0VIEW SOLUTION
- Question 59
What is the acute angle between the pair of straight lines ?Option A:Option B:Option C:Option D: 4VIEW SOLUTION
- Question 60
If the centroid of a triangle formed by (7, x), (y, –6) and (9, 10) is (6, 3), then the values of x and y are respectivelyOption A: 5, 2Option B: 2, 5Option C: 1, 0Option D: 0, 0VIEW SOLUTION
- Question 61
A straight line with direction cosines isOption A: parallel to x-axisOption B: parallel to y-axisOption C: parallel to z-axisOption D: equally inclined to all the axesVIEW SOLUTION
- Question 62
(0, 0 , 0), (a, 0, 0), (0, b, 0) and (0, 0, c) are four distinct points. What are the coordinates of the point, which is equidistant from the four points?Option A:Option B: (a, b, c )Option C:Option D:VIEW SOLUTION
- Question 63
The point P(3, 2, 4), Q(4, 5, 2), R(5, 8, 0) and S(2, –1, 6) areOption A: vertices of a rhombus which is not a squareOption B: non-coplanarOption C: collinearOption D: coplanar but not collinearVIEW SOLUTION
- Question 64
The line passing through the points (1, 2, –1) and (3, –1, 2) meets the yz-plane at which one of the following points?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 65
Under which one of the following conditions, are the lines x = ay + b; z = cy + d and x = ey + f; z = gy + h perpendicular?Option A: ae + cg – 1 = 0Option B: ae + bf – 1 = 0Option C: ae + cg + 1 = 0Option D: ag + ce – 1 = 0VIEW SOLUTION
- Question 66
If , are three coplanar vectors and , then which one of the following is correct?Option A: m = 2 and n = ±1Option B: m = ±2 and n = –1Option C: m = 2 and n = –1Option D: m = ±2 and n = 1VIEW SOLUTION
- Question 67
Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin. What is equal to?
Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 68
ABCD is a quadrilateral whose diagonals are AC and BD. Which one of the following is correct?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 69
If , then which one of the following is correct?Option A: are orthogonal in pairs andOption B: are non-orthogonal to each otherOption C: are orthogonal in pairs butOption D: are orthogonal in pairs butVIEW SOLUTION
- Question 70
If are perpendicular, then what is the value of λ?Option A: 2Option B: 3Option C: 4Option D: 5VIEW SOLUTION
- Question 71
- Question 72
- Question 73
- Question 74
The function f : X → Y defined by f(x) = cos x, where x ∈ X, is one-one and onto if X and Y are respectively equal toOption A: [0, π] and [–1, 1]Option B: and [–1, 1]Option C: [0, π] and [–1, 1]Option D: [0, π] and [0, 1]VIEW SOLUTION
- Question 75
- Question 76
- Question 77
Let be defined by . Consider the following:
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 78
Let f(x) = px + q and g(x) = mx + n. Then f(g(x)) = g(f(x)) is equivalent toOption A: f(p) = g(m)Option B: f(q) = g(n)Option C: f(n) = g(q)Option D: f(m) = g(p)VIEW SOLUTION
- Question 79
- Question 80
- Question 81
Let
If is
Option A: one-one and intoOption B: neither one-one nor ontoOption C: many one and ontoOption D: one-one and ontoVIEW SOLUTION
- Question 82
What is the length of the longest interval in which the function is increasing?Option A:Option B:Option C:Option D: πVIEW SOLUTION
- Question 83
If x dy = y(dx + y dy); y(1) = 1 and y(x) > 0, then what is y(–3) equal to?Option A: 3Option B: 2Option C: 1VIEW SOLUTION
- Question 84
What is the maximum value of the function f(x) = 4 sin2 x + 1?Option A: 5Option B: 3Option C: 2Option D: 1VIEW SOLUTION
- Question 85
Let f(x) be an indefinite integral of sin2 x.
Consider the following statements:
Statement 1: The function f(x) satisfies f(x + π) = f(x) for all real x.
Statement 2: for all real x.
Which one of the following is correct with respect to the above statements?Option A: Both the statements are true and Statement 2 is the correct explanation of Statement 1Option B: Both the statements are true but Statement 2 is not the correct explanation of Statement 1Option C: Statement 1 is true but Statement 2 is falseOption D: Statement 1 is false but Statement 2 is trueVIEW SOLUTION
- Question 86
What are the degree and order respectively of the differential equationOption A: 1, 2Option B: 2, 1Option C: 1, 4Option D: 4, 1VIEW SOLUTION
- Question 87
What is the differential equation corresponding to by eliminating a?Option A:Option B:Option C:Option D:VIEW SOLUTION
Where
- Question 88
What is the general solution of the differential equation ?Option A:Option B:Option C:Option D: None of theseVIEW SOLUTION
- Question 89
Let f(x + y) = f(x) f(y) for all x and y. Then what is f ’(5) equal to [where f ’(x) is the derivative of f(x)]?Option A: f(5) f ’(0)Option B: f(5) – f ’(0)Option C: f(5) f(0)Option D: f(5) + f ’(0)VIEW SOLUTION
- Question 90
If f(x) and g(x) are continuous functions satisfying f(x) = f(a – x) and g(x) + g(a – x) = 2, then what is equal to?Option A:Option B:Option C:VIEW SOLUTION
- Question 91
What is the solution of the differential equation ?Option A:Option B:Option C: y = ln x + cOption D: x = ln y + cVIEW SOLUTION
- Question 92
Let f(x) be defined as follows:
Which one of the following statements is correct with respect to the above function?Option A: It is discontinuous at x = –2 but continuous at every other point.Option B: It is continuous only in the interval (–3, –2).Option C: It is discontinuous at x = 0 but continuous at every other point.Option D: It is discontinuous at every pointVIEW SOLUTION
- Question 93
Consider the following statements
1. If both exist, then exists.
2. If exists, then both and must exist.
Which of the above statement is/are correct?Option A: 1 onlyOption B: 2 onlyOption C: Both 1 and 2Option D: Neither 1 nor 2VIEW SOLUTION
- Question 94
Which one of the following functions is neither even nor odd?Option A:Option B:Option C: |x|Option D:VIEW SOLUTION
- Question 95
- Question 96
Let .
Consider the following:
1. f(2a) = f(a) + 1
2.
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 97
What is the maximum area of a triangle that can be inscribed in a circle of radius a?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 98
Let . Which one of the following is correct?Option A: f(x) fluctuates in the intervalOption B: f(x) increases in the intervalOption C: f(x) decreases in the intervalOption D: None of theseVIEW SOLUTION
- Question 99
Suppose the function is differentiable for all x. Then n can be any element of the intervalOption A: [1, ∞)Option B: (0, ∞)Option C:Option D: None of theseVIEW SOLUTION
- Question 100
- Question 101
The variance of 20 observations is 5. If each observation is multiplied by 3, then what is the new variance of the resulting observations?Option A: 5Option B: 10Option C: 15Option D: 45VIEW SOLUTION
- Question 102
The mean of a group of 100 observations was found to be 20. Later it was found that four observations were incorrect, which were recorded as 21, 21, 18 and 20. What is the mean if the incorrect observations are omitted?Option A: 18Option B: 20Option C: 21Option D: 22VIEW SOLUTION
- Question 103
A committee of two persons is constituted from two men and two women. What is the probability that the committee will have only women?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 104
A question is given to three students A, B and C whose chance of solving it are respectively. What is the probability that the question will be solved?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 105
The mean weight of 150 students in a certain class is 60 kg. The mean weight of boys in the class is 70 kg and that of girls is 55 kg. What is the number of boys in the class?Option A: 50Option B: 55Option C: 60Option D: 100VIEW SOLUTION
- Question 106
For two dependent events, A and E, it is given that P(A) = 0.2 and P(B) = 0.5. If , then the value of conditional probabilities are respectivelyOption A:Option B:Option C:Option D: Information is insufficientVIEW SOLUTION
- Question 107
A point is chosen at random inside a circle. What is the probability that the point is closer to the centre of the circle then to its boundary?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 108
If two regression lines between height (x) and weight (y) are 4y – 15x + 410 = 0 and 30 x – 2y – 825 = 0, then what will be the correlation coefficient between height and weight?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 109
In an examination, 40% of candidates get second class. When the data are represented by a pie chart, what is the angle corresponding to the second class?Option A: 40°Option B: 90°Option C: 144°Option D: 320°VIEW SOLUTION
- Question 110
Consider the following statements:
Statement 1: Range is not a good measure of dispersion
Statement 2: Range is highly affected by the existence of extreme values
Which one of the following is correct in respect of the above statements?Option A: Both Statement 1 and Statement 2 are correct and Statement 2 is the correct explanation of Statement 1Option B: Both Statement 1 and Statement 2 are correct but Statement 2 is not the correct explanation of Statement 1Option C: Statement 1 is correct but Statement 2 is not correctOption D: Statement 2 is correct but Statement 1 is not correctVIEW SOLUTION
- Question 111
A card is drawn from a well shuffled ordinary deck of 52 cards. What is the probability that it is an ace?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 112
If the data is moderately non-symmetrical then which one of the following empirical relationships is correct?Option A: 2 × Standard deviation = 5 × Mean deviationOption B: 5 × Standard deviation = 2 × Mean deviationOption C: 4 × Standard deviation = 5 × Mean deviationOption D: 5 × Standard deviation = 4 × Mean deviationVIEW SOLUTION
- Question 113
Data can be represented in which of the following forms
I. Textual form
II. Tabular form
III Graphical form
Select the correct answer using the code given below.Option A: 1 and 2 onlyOption B: 2 and 3 onlyOption C: 1 and 3 onlyOption D: 1, 2 and 3VIEW SOLUTION
- Question 114
For given statistical data, the graphs for less than ogive and more than ogive are drawn. If the point at which the two curves intersect is P, then abscissa of point P gives the value of which one of the following measures of central tendency?Option A: MedianOption B: MeanOption C: ModeOption D: Geometric meanVIEW SOLUTION
- Question 115
Consider the following statements:
1. Two events are mutually exclusive if the occurrence of one event prevents the occurrence of the other.
2. The probability of the union of two mutually exclusive events is the sum of their individual probabilities
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 116
If the regression coefficient of x on y and y on x are respectively, then what is the correlation coefficient between x and y?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 117
A sample of 5 observations has mean 32 and median 33. Later it is found that an observation was recorded incorrectly as 40 instead of 35. If we correct the data, then which one of the following is correct?Option A: The mean and median remain the sameOption B: The median remains the same but the mean will decreaseOption C: The mean and median both will decreaseOption D: The mean remains the same but median will decreaseVIEW SOLUTION
- Question 118
If two fair dice are thrown, then what is the probability that the sum is neither 8 nor 9?Option A:Option B:Option C:Option D:VIEW SOLUTION
- Question 119
Let A and B are two mutually exclusive events with P(A) = and P(B) = . What is the value ofOption A:Option B:Option C:Option D:VIEW SOLUTION
- Question 120
The mean and standard deviation of a binomial distribution are 12 and 2 respectively. What is the number of trials?Option A: 2Option B: 12Option C: 18Option D: 24VIEW SOLUTION