NDA Mathematics 4 September 2022
DEFENCE 2022 Previous Year
3 hDuration
300Total Marks
120Questions
1Sections
Instructions
General instructions for this test:
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- Total questions: 120 across 1 section(s); maximum marks: 300.
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No exam-specific instructions were provided for this paper.
Paper Structure
Mathematics
Mathematics
Q1.
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**Passage:**
The mean and standard deviation (SD) of marks obtained by 50 students of a class in 4 subjects are given below :
Subject
Mathematics
Physics
Chemistry
Biology
Mean Marks
40
28
38
36
SD
15
12
14
16
---
What is the coefficient of variation of marks in Mathematics ?
Q2.
mcq single
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**Passage:**
Consider the following grouped frequency distribution :
Class
0-10
10-20
20-30
30-40
40-50
50-60
Frequency
1
2
4
6
4
3
---
What is the median of the distribution ?
Q3.
mcq single
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**Passage:**
Consider the following grouped frequency distribution :
Class
0-10
10-20
20-30
30-40
40-50
50-60
Frequency
1
2
4
6
4
3
---
What is mean deviation about the median ?
Q4.
mcq single
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**Passage:**
Consider the following grouped frequency distribution :
Class
0-10
10-20
20-30
30-40
40-50
50-60
Frequency
1
2
4
6
4
3
---
What is the mean deviation about the mean ?
Q5.
mcq single
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**Passage:**
The mean and standard deviation (SD) of marks obtained by 50 students of a class in 4 subjects are given below :
Subject
Mathematics
Physics
Chemistry
Biology
Mean Marks
40
28
38
36
SD
15
12
14
16
---
Which one of the following subjects shows highest variability of marks ?
Q6.
mcq single
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The coefficient of correlation between ages of husband and wife at the time of marriage for a given set of 100 couples was noted to be 0.7. Assume that all these couples survive to celebrate the silver jubilee of their marriage. The coefficient of correlation at that point of time will be
Q7.
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Let R be a relation from N to N defined by R = {(x, y): x, y β N and x2 = y^(3)}. Which of the following are not correct?
1. (x, x) β R for all x β N
2. (x, y) β R β (y, x) β R
3. (x, y) β R and (y, z) β R β (x, z) β R
Select the correct answer using the code given below :
Q8.
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**Passage:**
A university awarded medals in basketball, football, and volleyball. Only x students (x < 6) got medal in all the three sports and the medals went to a total of 15x students. It awarded 5x medals in basketball, (4x + 15) medals in football and (x + 25) medals in volleyball.
---
How many received medals in exactly one of three sports ?
Q9.
mcq single
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**Passage:**
A university awarded medals in basketball, football, and volleyball. Only x students (x < 6) got medal in all the three sports and the medals went to a total of 15x students. It awarded 5x medals in basketball, (4x + 15) medals in football and (x + 25) medals in volleyball.
---
How many received medals in at least two of three sports ?
Q10.
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Consider the following statements :
1. If f is the subset of Z Γ Z defined by f = {(xy, x β y); x, y β Z}, then f is a function from Z to Z.
2. If f is the subset of N Γ N defined by f = {(xy, x + y); x, y β N}, then f is a function from N to N.
Which of the statements given above is/are correct?
Q11.
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Let z = [y] and y = [x] β x, where [.] is the greatest integer function. If x is not an integer but positive, then what is the value of z ?
Q12.
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Consider the following :
1. A β© B = A β© C β B = C
2. A βͺ B = A βͺ C β B = C
Which of the above is/are correct ?
Q13.
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Let A = {7, 8, 9, 10, 11, 12, 13; 14, 15, 16} and let f βΆ A β N be defined by f(x) = the highest prime factor of x.
How many elements are there in the range of f?
Q14.
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Consider the following statements :
1. The set of all irrational numbers between $\sqrt{2}$ and $\sqrt{5}$ is an infinite set.
2. The set of all odd integers less than 100 is a finite set.
Which of the statements given above is/are correct?
Q15.
mcq single
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**Passage:**
A university awarded medals in basketball, football, and volleyball. Only x students (x < 6) got medal in all the three sports and the medals went to a total of 15x students. It awarded 5x medals in basketball, (4x + 15) medals in football and (x + 25) medals in volleyball.
---
How many received medals in exactly two of the three sports ?
Q16.
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If f(x) = $\frac{x^2+x+|x|}{x}$, then what is $\displaystyle\lim_{x \rightarrow 0}$ f(x) equal to ?
Q17.
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What is $\displaystyle\lim_{x\rightarrow \frac{\pi}{2}} \frac{4xβ2\pi}{\cos x}$ equal to ?
Q18.
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What is $\displaystyle\lim_{h \rightarrow 0} \frac{\sin^2(x+h)β\sin^2x}{h}$ equal to ?
Q19.
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What is $\displaystyle\lim_{x \rightarrow 0} \frac{x}{\sqrt{1β\cos 4x}}$ equal to ?
Q20.
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**Passage:**
Let P be the sum of first n positive terms of an increasing arithmetic progression A. Let Q be the sum of first n positive terms of another increasing arithmetic progression B.
Let P βΆ Q = (5n + 4) βΆ (9n + 6)
---
What is the ratio of their 10th terms ?
Q21.
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**Passage:**
Let f(x) be a function satisfying f(x + y) = f(x) f(y) for all x, y β N such that f(1) = 2 :
---
If $\displaystyle\sum_{x=2}^n$βf(x) = 2044, then what is the value of n ?
Q22.
mcq single
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Consider the following statements :
1. 2 + 4 + 6 + ........ + 2n = n2 + n
2. The expression n2 + n + 41 always gives a prime number for every natural number n
Which of the above statements is/are correct ?
Q23.
mcq single
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**Passage:**
Let P be the sum of first n positive terms of an increasing arithmetic progression A. Let Q be the sum of first n positive terms of another increasing arithmetic progression B.
Let P βΆ Q = (5n + 4) βΆ (9n + 6)
---
If d is the common difference of A, and D is the common difference of B, then which one of the following is always correct ?
Q24.
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Let x be the mean of squares of first n natural numbers and y be the square of mean of first n natural numbers. If $\frac{\text{x}}{\text{y}} = \frac{55}{42}$, then what is the value of n ?
Q25.
mcq single
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**Passage:**
Let P be the sum of first n positive terms of an increasing arithmetic progression A. Let Q be the sum of first n positive terms of another increasing arithmetic progression B.
Let P βΆ Q = (5n + 4) βΆ (9n + 6)
---
What is the ratio of the first term of A to that of B ?
Q26.
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PQRS is a parallelogram. If $\overrightarrow{\text{PR}}=\vec{\text{a}}$ and $\overrightarrow{\text{QS}}=\vec{\text{b}}$, then what is $\overrightarrow{\text{PQ}}$ equal to ?
Q27.
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Let $\vec{\text{a}}$ and $\vec{\text{b}}$ are two unit vectors such that $\vec{\text{a}}+2 \vec{\text{b}}$ and $5\vec{\text{a}}β4\vec{\text{b}}$ are perpendicular. What is the angle between $\vec{\text{a}}$ and $\vec{\text{b}}$ ?
Q28.
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Let $\vec{\text{a}}$, $\vec{\text{b}}$ and $\vec{\text{c}}$ be unit vectors lying on the same plane. What is $\{(3\vec{\text{a}} + 2\vec{\text{b}}) Γ (5\vec{\text{a}} β 4\vec{\text{c}})\}β
(\vec{\text{b}} + 2\vec{\text{c}})$ equal to ?
Q29.
mcq single
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The position vectors of vertices A, B and C of triangle ABC are respectively $\hat{\text{j}}+\hat{\text{k}}, 3\hat{\text{i}}+\hat{\text{j}+5\hat{\text{k}}}$ and $3\hat{\text{j}}+3\hat{\text{k}}$. What is angle C equal to?
Q30.
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What are the values of x for which the angle between the vectors 2x^(2)$\hat{\text{i}}$ + 3x$\hat{\text{j}}$ + $\hat{\text{k}}$ and $\hat{\text{i}}$ β2$\hat{\text{j}}$ + x^(2)$\hat{\text{k}}$ is obtuse ?
Q31.
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What is tan^(β1) cot(cosec^(β1) 2) equal to ?
Q32.
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Three fair dice are thrown. What is the probability of getting a total greater than or equal to 15 ?
Q33.
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A number x is chosen at random from first n natural numbers. What is the probability that the number chosen satisfies x + $\frac{1}{\text{x}}$ > 2 ?
Q34.
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If n > 7, then what is the probability that C(n, 7) is a multiple of 7?
Q35.
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What is the probability of getting a composite number in the list of natural numbers from 1 to 50?
Q36.
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Two numbers x and y are chosen at random from a set of the first 10 natural numbers. What is the probability that (x + y) is divisible by 4 ?
Q37.
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Five coins are tossed once. What is the probability of getting at most four tails ?
Q38.
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The completion of a construction job may be delayed due to strike. The probability of strike is 0.6. The probability that the construction job gets completed on time if there is no strike is 0.85 and the probability that the construction job gets completed on time if there is a strike is 0.35. What is the probability that the construction job will not be completed on time ?
Q39.
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A card is drawn from a pack of 52 cards. A gambler bets that it is either a spade or an ace. The odds against his winning are
Q40.
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During war, one ship out of 5 was sunk on an average in making a certain voyage. What is the probability that exactly 3 out of 5 ships would arrive safely?
Q41.
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The probability that a person hits a target is 0.5. What is the probability of at least one hit in 4 shots ?
Q42.
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If P(A) = 0.5, P(B) = 0.7 and P(A β© B) = 0.3, then what is the value of P(A' β© B') + P(A' β© B) + P(A β© B') ?
Q43.
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Three fair dice are tossed once. What is the probability that they show different numbers that are in AP?
Q44.
mcq single
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**Passage:**
ABC is a triangular plot with AB = 16 m, BC = 10 m and CA = 10 m. A lamp post is situated at the middle point of the side AB. The lamp post subtends an angle 45Β° at the vertex B.
---
What is the height of the lamp post ?
Q45.
mcq single
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**Passage:**
There are two points P and Q due south of a leaning tower, which leans towards north. P is at a distance x and Q is at distance y from the foot of the tower (x > y). The angles of elevation of the top of the tower from P and Q are 15Β° and 75Β° respectively.
---
What is the length of the tower ?
Q46.
mcq single
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**Passage:**
There are two points P and Q due south of a leaning tower, which leans towards north. P is at a distance x and Q is at distance y from the foot of the tower (x > y). The angles of elevation of the top of the tower from P and Q are 15Β° and 75Β° respectively.
---
If ΞΈ is the inclination of the tower to the horizontal, then what is cot ΞΈ equal to ?
Q47.
mcq single
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**Passage:**
There are two points P and Q due south of a leaning tower, which leans towards north. P is at a distance x and Q is at distance y from the foot of the tower (x > y). The angles of elevation of the top of the tower from P and Q are 15Β° and 75Β° respectively.
---
At what height is the top of the tower above the ground level ?
Q48.
mcq single
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Consider the following statements in respect of the line passing through origin and inclining at an angle of 75Β° with the positive direction of x-axis :
1. The line passes through the point $\left(1, \frac{1}{2β\sqrt{3}}\right)$.
2. The line entirely lies in first and third quadrants.
Which of the statements given above is/are correct ?
Q49.
mcq single
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If P(3, 4) is the mid-point of a line segment between the axes, then what is the equation of the line?
Q50.
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The base AB of an equilateral triangle ABC with side 8 cm lies along the y-axis such that the mid-point of AB is at the origin and B lies above the origin. What is the equation of line passing through (8, 0) and parallel to the side AC ?
Q51.
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What is the value of $\cos \left(\frac{5 \pi}{17}\right)+\cos \left(\frac{7 \pi}{17}\right)+2 \cos \left(\frac{11 \pi}{17}\right) \cos \left(\frac{\pi}{17}\right)$ ?
Q52.
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What is the value of tan2 165Β° + cot2 165Β° ?
Q53.
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If f(Ξ±) = $\sqrt{\sec^2\alphaβ1}$, then what is $\frac{f(\alpha)+f(\beta)}{1βf(\alpha) f(\beta)}$ equal to ?
Q54.
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What is the value of $\sin\left(2\text{n}\pi+\frac{5\pi}{6}\right)\sin\left(2\text{n}\piβ\frac{5\pi}{6}\right)$ where n β Z ?
Q55.
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If 1 + 2(sin x + cos x)(sin x β cos x) = 0 where 0 < x < 360Β°, then how many values does x take ?
Q56.
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What is the maximum value of 3(sin x β cos x) + 4(cos^(3) x β sin^(3) x) ?
Q57.
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If sec x = $\frac{25}{24}$ and x lies in the fourth quadrant, then what is the value of tan x + sin x ?
Q58.
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What is the value of tan$\left(\frac{3\pi}{8}\right)$ ?
Q59.
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What is cos 36Β° β cos 72Β° equal to ?
Q60.
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What is the value of cosec$\left(β\frac{73 \pi}{3}\right)$ ?
Q61.
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Where does the function f(x) = $\displaystyle\sum_{j=1}^7$(x β j)2 attain its minimum value ?
Q62.
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Under which one of the following conditions does the function f(x) = (p sec x)2 + (q cosec x)2 attain minimum value ?
Q63.
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What is the area of the region enclosed in the first quadrant by x^(2) + y^(2) = Ο^(2), y = sin x and x = 0 ?
Q64.
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What is the area of the region bounded by x β |y| = 0 and x β 2 = 0 ?
Q65.
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What is the area of the region (in the first quadrant) bounded by y = $\sqrt{1β\text{x}^2}$, y = x and y = 0 ?
Q66.
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Consider the following statements :
1. The direction ratios of y-axis can be <0, 4, 0>
2. The direction ratios of a line perpendicular to z-axis can be <5, 6, 0>
Which of the statements given above is/are correct?
Q67.
mcq single
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Consider the points A(2, 4, 6), B(β2, β4, β2), C(4, 6, 4), and D(8, 14, 12). Which of the following statements is/are correct?
1. The points are the vertices of a rectangle ABCD.
2. The mid-point of A C is the same as that of BD.
Select the correct answer using the code given below :
Q68.
mcq single
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Consider the equation of a sphere x2 + y2 + z2 β 4x β 6y β 8z β 16 = 0.
Which of the following statements is/are correct ?
1. z-axis is tangent to the sphere.
2. The centre of the sphere lies on the plane x + y + z β 9 = 0.
Select the correct answer using the code given below :
Q69.
mcq single
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A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?
Q70.
mcq single
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An equilateral triangle is inscribed in a parabola x2 = $\sqrt{3}$y where one vertex of the triangle is at the vertex of the parabola. If p is the length of side of the triangle and q is the length of the latus rectum, then which one of the following is correct ?
Q71.
mcq single
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**Passage:**
Consider the binomial expansion of (p + qx)9 :
---
What is the ratio of the coefficients of middle terms in the expansion (when expanded in ascending powers of x)?
Q72.
mcq single
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**Passage:**
Consider the binomial expansion of (p + qx)9 :
---
What is the value of q if the coefficients of x3 and x6 are equal ?
Q73.
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What is $\displaystyle\sum_{r=0}^n$2^(r )C(n, r) equal to ?
Q74.
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**Passage:**
Consider the binomial expansion of (p + qx)9 :
---
Under what condition the coefficients of x2 and x4 are equal ?
Q75.
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If f(x) = ln (x + $\sqrt{1+\text{x}^2}$), then which one of the following is correct ?
Q76.
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What is the minimum value of the function f(x) = log~10~(x^(2) + 2x + 11) ?
Q77.
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**Passage:**
Let z = $\frac{1+\text{i}\sin \theta}{1β\text{i}\sin \theta}$ where i = $\sqrt{β1}$
---
What is angle ΞΈ such that z is purely real ?
where n is an integer
Q78.
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**Passage:**
Let z = $\frac{1+\text{i}\sin \theta}{1β\text{i}\sin \theta}$ where i = $\sqrt{β1}$
---
What is angle ΞΈ such that z is purely imaginary ?
where n is an integer
Q79.
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**Passage:**
Let z = $\frac{1+\text{i}\sin \theta}{1β\text{i}\sin \theta}$ where i = $\sqrt{β1}$
---
What is the modulus of z?
Q80.
mcq single
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**Passage:**
Let f(x) be a function satisfying f(x + y) = f(x) f(y) for all x, y β N such that f(1) = 2 :
---
What is $\displaystyle\sum_{x=1}^6$2^(x) f(x) equal to ?
Q81.
mcq single
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Consider the following statements in respect of the function f(x) = $\left\{\begin{array}{rc}|x|+1, & 0<|x| \leqslant 3 \\ 1, & x = 0\end{array}\right.$
1. The function attains maximum value only at x = 3
2. The function attains local minimum only at x = 0
Which of the statements given above is/are correct ?
Q82.
mcq single
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**Passage:**
Let f(x) be a function satisfying f(x + y) = f(x) f(y) for all x, y β N such that f(1) = 2 :
---
What is $\displaystyle\sum_{x=1}^5$f(2x β 1) equal to ?
Q83.
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βIf f(x) = 4x + 1 and g(x) = kx + 2 such that fog(x) = gof(x), then what is the value of k ?
Q84.
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What is the differential equation of the family of parabolas having a vertex at origin and axis along positive y-axis?
Q85.
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What is the solution of the differential equation (dy β dx) + cos x(dy + dx) = 0 ?
Q86.
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Consider the following statements:
1. The degree of the differential equation $\frac{\text{dy}}{\text{dx}} + \cos \left(\frac{\text{dy}}{\text{dx}}\right)$ = 0 is 1.
2. The order of the differential equation $\left(\frac{\text{d}^2\text{y}}{\text{dx}^2}\right)^3 + \cos \left(\frac{\text{dy}}{\text{dx}}\right) $ = 0 is 2.
Which of the statements above is/are correct?
Q87.
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What is the diameter of a circle inscribed in a regular polygon of 12 sides, each of length 1 cm ?
Q88.
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The centre of the circle passing through the origin and making positive intercepts 4 and 6 on the coordinate axes, lies on the line
Q89.
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Let p, q(p > q) be the roots of the quadratic equation x2 + bx + c = 0 where c > 0. If p2 + q2 β 11pq = 0, then what is p β q equal to ?
Q90.
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For how many quadratic equations, the sum of roots is equal to the product of roots?
Q91.
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The centre of an ellipse is at (0, 0), major axis is on the y-axis. If the ellipse passes through (3, 2) and (1, 6), then what is its eccentricity ?
Q92.
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If y = (xx)x, then which one of the following is correct ?
Q93.
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Let f(x) be a function such that f'(x) = g(x) and f''(x) = βf(x). Let h(x) = {f(x)}^(2) + {g(x)}^(2). Then consider the following statements :
1. h'(3) = 0
2. h(1) = h(2)
Which of the statements given above is/are correct ?
Q94.
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If y = $\frac{x \sqrt{x^2β16}}{2} β 8 \ln\left|x + \sqrt{x^2β16}\right|$, then what is $\frac{\text{dy}}{\text{dx}}$ equal to ?
Q95.
mcq single
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If y = ln2$\left(\frac{x^2βx+1}{x^2+x+1}\right)$, then what is $\frac{\text{dy}}{\text{dx}}$ at x = 0 equal to ?
Q96.
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If $\frac{d}{d x}\left(\frac{1+x^4+x^8}{1βx^2+x^4}\right)$ = ax + bx3, then which one of the following is correct?
Q97.
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**Passage:**
Consider the word 'QUESTION' :
---
How many 8-letter words with or without meaning, can be formed such that consonants and vowels occupy alternate positions?
Q98.
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**Passage:**
Consider the word 'QUESTION' :
---
How many 4-letter words each of two vowels and two consonants with or without meaning, can be formed ?
Q99.
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**Passage:**
Consider the word 'QUESTION' :
---
How many 8-letter words with or without meaning, can be formed so that all consonants are together?
Q100.
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If different permutations of the letters of the word 'MATHEMATICS' are listed as in a dictionary, how many words (with or without meaning) are there in the list before the first word that starts with C ?
Q101.
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How many four-digit natural numbers are there such that all of the digits are odd ?
Q102.
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A box contains 2 white balls, 3 black balls, and 4 red balls. What is the number of ways of drawing 3 balls from the box with at least one black ball?
Q103.
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What is $\displaystyle\int_{β\pi/2}^{\pi/2}$(ecos x sin x + esin x cos x)dx equal to ?
Q104.
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What is $\displaystyle\int_0^1 \ln \left(\frac{1}{x}β1\right)$dx equal to ?
Q105.
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If $\displaystyle\int_0^{\pi/2}$(sin^(4) x + cos^(4) x)dx = k, then what is the value of $\displaystyle\int_0^{20 \pi}$(sin^(4) x + cos^(4) x)dx ?
Q106.
mcq single
+2.5 / 0.83
**Passage:**
Let A = $\left(\begin{array}{ccc}0 & \sin^2 \theta & \cos^2 \theta \\ \cos^2 \theta & 0 & \sin^2 \theta \\ \sin^2 \theta & \cos^2 \theta & 0\end{array}\right)$ and A = P + Q where P is symmetric matrix and Q is skew-symmetric matrix.
---
What is Q equal to ?
Q107.
mcq single
+2.5 / 0.83
If A = $\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & \sin \theta & β\cos \theta\end{array}\right)$, then which of the following are correct?
1. A + adj A is a null matrix
2. Aβ1 + adj A is a null matrix
3. A β Aβ1 is a null matrix
Select the correct answer using the code given below :
Q108.
mcq single
+2.5 / 0.83
**Passage:**
Let A = $\left(\begin{array}{ccc}0 & \sin^2 \theta & \cos^2 \theta \\ \cos^2 \theta & 0 & \sin^2 \theta \\ \sin^2 \theta & \cos^2 \theta & 0\end{array}\right)$ and A = P + Q where P is symmetric matrix and Q is skew-symmetric matrix.
---
What is P equal to ?
Q109.
mcq single
+2.5 / 0.83
If X is a matrix of order 3 Γ 3, Y is a matrix of order 2 Γ 3 and Z is a matrix of order 3 Γ 2, then which of the following are correct?
1. (ZY)X is a square matrix having 9 entries.
2. Y(XZ) is a square matrix having 4 entries.
3. X(YZ) is not defined.
Select the correct answer using the code given below :
Q110.
mcq single
+2.5 / 0.83
**Passage:**
ABC is a triangular plot with AB = 16 m, BC = 10 m and CA = 10 m. A lamp post is situated at the middle point of the side AB. The lamp post subtends an angle 45Β° at the vertex B.
---
What is $\frac{\text{AB}}{\sin \text{C}}$ equal to ?
Q111.
mcq single
+2.5 / 0.83
**Passage:**
ABC is a triangular plot with AB = 16 m, BC = 10 m and CA = 10 m. A lamp post is situated at the middle point of the side AB. The lamp post subtends an angle 45Β° at the vertex B.
---
What is cos A + cos B + cos C equal to ?
Q112.
mcq single
+2.5 / 0.83
In a triangle ABC, a = 4, b = 3, c = 2. What is cos 3C equal to ?
Q113.
mcq single
+2.5 / 0.83
Consider the determinant
Ξ = $\left|\begin{array}{lll}\text{a}_{11} & \text{a}_{12} & \text{a}_{13} \\ \text{a}_{21} & \text{a}_{22} & \text{a}_{23} \\ \text{a}_{31} & \text{a}_{32} & \text{a}_{33}\end{array}\right|$
If a13 = yz, a23 = zx, a33 = xy and the minors of a13, a23, a33 are respectively (z β y), (z β x), (y β x) then what is the value of Ξ ?
Q114.
mcq single
+2.5 / 0.83
**Passage:**
Let Ξ be the determinant of a matrix A, where A = $\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right]$ and C11, C12, C13 be the cofactors of a11, a12, a13 respectively.
---
What is the value of $\left|\begin{array}{lll}\text{a}_{21} & \text{a}_{31} & \text{a}_{11} \\ \text{a}_{23} & \text{a}_{33} & \text{a}_{13} \\ \text{a}_{22} & \text{a}_{32} & \text{a}_{12}\end{array}\right|$ ?
Q115.
mcq single
+2.5 / 0.83
**Passage:**
Let Ξ be the determinant of a matrix A, where A = $\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right]$ and C11, C12, C13 be the cofactors of a11, a12, a13 respectively.
---
What is the value of $a_{21}C_{11}+a_{22}C_{12}+a_{23}C_{13}$?
Q116.
mcq single
+2.5 / 0.83
**Passage:**
Let Ξ be the determinant of a matrix A, where A = $\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right]$ and C11, C12, C13 be the cofactors of a11, a12, a13 respectively.
---
What is the value of a11C11 + a12C12 + a13C13 ?
Q117.
mcq single
+2.5 / 0.83
**Passage:**
Let A = $\left(\begin{array}{ccc}0 & \sin^2 \theta & \cos^2 \theta \\ \cos^2 \theta & 0 & \sin^2 \theta \\ \sin^2 \theta & \cos^2 \theta & 0\end{array}\right)$ and A = P + Q where P is symmetric matrix and Q is skew-symmetric matrix.
---
What is the minimum value of determinant of A ?
Q118.
mcq single
+2.5 / 0.83
What is $\int\frac{(\cos x)^{1.5}β(\sin x)^{1.5}}{\sqrt{\sin xβ
\cos x}}$dx equal to ?
Q119.
mcq single
+2.5 / 0.83
What is β«(xx)2 (1 + ln x)dx equal to ?
Q120.
mcq single
+2.5 / 0.83
What is β«ex{1 + ln x + x ln x}dx equal to ?