Mathematics Exam Instructions

• The document contains instructions and questions for a mathematics examination.

General Information

• The exam duration is 2 hours.
• The maximum possible score (Full Marks) is 100.

Instructions

• All questions are objective type, having four answer options each.

Category-Specific Marking Schemes

Category-1

• Carries 1 mark per question.
• Only one option is correct.
• Incorrect answers or combinations of multiple answers will result in a deduction of ¼ mark.

Category-2

• Carries 2 marks per question.
• Only one option is correct.
• Incorrect answers or combinations of multiple answers will result in a deduction of ½ mark.

Category-3

• Carries 2 marks per question.
• One or more options may be correct.
• Score = 2 * (number of correct answers marked) - (actual number of correct answers) if all correct answers are not marked and no incorrect answers are marked.
• If any wrong option is marked, or any combination including a wrong option is marked, the answer will be considered wrong, however no negative marking will be awarded and the question will receive zero marks.

Answering Guidelines

• Answers must be marked on the OMR sheet by darkening the appropriate bubble (A, B, C, or D).
• Use only Black/Blue ink ballpoint pen to fill the bubbles completely.
• Write the Question Booklet number and Roll Number carefully in the specified locations on the OMR sheet and fill bubbles appropriately.
• Write name (in block letters), examination center name, and signature (as on Admit Card) in the designated boxes on the OMR sheet.
• The OMR sheet may be invalidated if there are mistakes in filling the Question Booklet number/Roll number or discrepancies in name/signature/examination center. Folding or stray marks can also invalidate the OMR sheet.
• Candidates are responsible for invalidation due to incorrect marking or careless handling.

Prohibited Items

• Candidates are not allowed to carry written/printed material, calculators, log tables, wristwatches, or communication devices (mobile phones, Bluetooth devices, etc.) into the examination hall.
• Possession of prohibited items will result in reporting and cancellation of candidature.

Rough Work

• Rough work must be done in the Question Booklet itself.
• Additional blank pages are provided for rough work.

OMR Sheet and Question Booklet Handling

• The OMR Sheet must be handed over to the invigilator before leaving the Examination Hall.
• Candidates are allowed to take the Question Booklet after the examination is over.

Language Discrepancy

• The booklet contains questions in both English and Bengali.
• In case of discrepancies between the two versions, the information provided in the English version will be treated as final.

Candidate and Invigilator Signatures

• There are designated spaces for the candidate's signature (as in Admit Card) and the Invigilator's signature.

Mathematics Questions

• The document contains multiple mathematics questions spanning various topics.

Question 1

• If A = [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]] and θ = 2π/7, then A^100 = A × A × ... (100 times) equals one of the given matrix options.

Question 2

• If (1 + x + x² + x³)^5 = Σ(k=0 to 15) a_k * x^k, then Σ(k=0 to 7) (-1)^k * a_{2k} equals one of the given numerical options.

Question 3

• The coefficient of a¹⁰b⁷c³ in the expansion of (bc + ca + ab)¹⁰ is one of the given numerical options.

Question 4

• The numbers 1, 2, 3, ..., m are arranged in random order.
• The number of ways this can be done so that the numbers 1, 2, ..., r (r < m) appear as neighbors is one of the given factorial or multiplied factorial options.

Question 5

• If a matrix equation involving x, y, and z variables equals (x-y)(y-z)(z-x)(1/x+1/y+1/z), then k equals one of the given numerical options.

Question 6

• If a matrix equation A * B * A equals another matrix, then A is equivalent to one of the given matrix options.

Question 7

• Let f(x) = (cos(x)-1)/x³ / (2sin(x)-x)/tan(x)/x, then lim(x->0) f(x)/x² equals one of the given numerical options.

Question 8

• In R, a relation p is defined as follows: "a p b holds iff a² - 4ab + 3b² = 0."
• Determine if p is an equivalence relation, only symmetric, only reflexive, or only transitive.

Question 9

• Let f: R->R be a function defined by f(x) = (e^|x| - e^-x) / (e^x + e^-x). Determine if f is one-one and onto, one-one but not onto, onto but not one-one, or neither one-one nor onto.

Question 10

• Let A be the set of even natural numbers less than 8, and B be the set of prime integers less than 7.
• Determine the number of relations from A to B.

Question 11

• Two integers, r and s, are drawn one at a time without replacement from the set {1, 2, ..., n}.
• Then, find the probability that r <= k/s <= k where k is an integer less than n.

Question 12

• A biased coin with probability p (0 < p < 1) of getting heads is tossed until a head appears for the first time.
• If the probability that the number of tosses required is even is 2/5, then find p.

Question 13

• Simplify the expression cos² θ + cos² (θ + φ) - 2 cos θ cos φ cos(θ + φ).
• Determine if the resulting expression is independent of θ, φ, both, or dependent on both.

Question 14

• Two smallest squares are chosen one by one on a chessboard.
• Determine the probability that they have a side in common.

Question 15

• The equation r cos θ = 2a sin² θ represents a curve; identify its type.

Question 16

• If (1, 5) is the midpoint of a line segment between the lines 5x - y - 4 = 0 and 3x + 4y - 4 = 0, find the equation of the line.

Question 17

• In triangle ABC, the coordinates of A are (1, 2), and the equations of the medians through B and C are x + y = 5 and x = 4, respectively. Find the midpoint of BC.

Question 18

• If 0 < θ < π/2 and tan 3θ ≠ 0, then tan θ + tan 2θ + tan 3θ = 0 if tan θ tan 2θ = k; find k.

Question 19

• A line of fixed length a + b, where a ≠ b, moves so that its ends are always on two fixed perpendicular lines. Determine the locus of a point that divides the line into two parts of length a and b.

Question 20

• With the origin as a focus and x = 4 as the corresponding directrix, a family of ellipses is drawn. Identify the locus of an end of the minor axis.

Question 21

• Chords AB and CD of a circle intersect at right angles at the point P. If AP, PB, CP, and PD are 2, 6, 3, and 4 units, respectively, find the radius of the circle.

Question 22

• The plane 2x - y + 3z + 5 = 0 is rotated through 90° about its line of intersection with the plane x + y + z = 1. Find the equation of the plane in the new position.

Question 23

• If the relation between the direction ratios of two lines in R³ is given by l + m + n = 0 and 2lm + 2mn - ln = 0, then find the angle between the lines.

Question 24

• If AOAB is an equilateral triangle inscribed in the parabola y² = 4ax, a > 0, with O as the vertex, then determine the length of the side of AOAB.

Question 25

• If $U(n = 1, 2)$ denotes the $n^{th}$ derivative $(n = 1, 2)$ of $U(x) = \frac{Lx + M}{x^2 - 2Bx + C}$ (L, M, B, C are constants), then for which condition $PU_2 + QU_1 + RU = 0$.

Question 26

• For every real number $x \neq -1$, let $f(x) = \frac{x}{x+1}$. Write $f_1(x) = f(x)$ & for $n \geq 2$, $f_n(x) = f(f_{n-1}(x))$. Then $f_1(-2) f_2(-2) ..... f_n(-2)$ must be.

Question 27

• The equation $2^x + 5^x = 3^x + 4^x$ has various solution types; determine what kind.

Question 28

• Consider the function $f(x) = (x - 2) \log x$. Then, what can be said regarding the equation $x \log x = 2 - x"?

Question 29

• If $\alpha, \beta$ are the roots of the equation $ax^2 + bx + c = 0$, find the value of $\lim_{x\to\beta} \frac{1-\cos(ax^2+bx+c)}{(x-\beta)^2}$.

Question 30

• If $f(x) = \frac{e^x}{1 + e^x}$, $I_1 = \int_{f(-a)}^{f(a)} x g(x(1-x)) dx$ and $I_2 = \int_{f(-a)}^{f(a)} g(x(1-x)) dx$, then what is the value of $\frac{I_2}{I_1}$?

Question 31

• Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function and $f(1) = 4$. Then, what is the value of $\lim_{x \to 1} \frac{\int_4^{f(x)} 2t dt}{x - 1}$, given that $f'(1) = 2$?

Question 32

• What expression could define $\int \frac{\log_e(x + \sqrt{1 + x^2})}{\sqrt{1 + x^2}} dx = f(g(x)) + c$, then determine $f(x)$ and $g(x)$.

Question 33

• For any integer n, what is the value of $\int_0^\pi e^{\cos^2x} \cos^3 (2n + 1)x dx$?

Question 34

• Determine $\lim_{x\to\infty} y(x)$ under what condition/relation, given f be a differential function with $\lim_{x \to \infty} f(x) = 0$. If $y' + yf'(x) - f(x)f'(x) = 0$.

Question 35

• If $xy' + y - e^x = 0$, $y(a) = b$, then what is $\lim_{x \to a} y(x)$?

Question 36

• Find all the values of $'a'$ for which $\frac{1}{\sqrt{a}} \int_{a}^{a+1} \frac{1}{3-2(\sqrt{x+1}-\sqrt{x})}dx < 4$.

Question 37

• Find the area bounded by the curves $x = 4 - y^2$ and the Y-axis.

Question 38

• Given $f(x) = \cos x - 1 + \frac{x^2}{2!}$ where $x \in \mathbb{R}$, determine whether $f(x)$ is an increasing function, decreasing function, neither, or a constant.

Question 39

• Let $y = f(x)$ be any curve on the X-Y plane & $P$ be a point on the curve. Let $C$ be a fixed point not on the curve. If the length $PC$ is either a maximum or a minimum, then what is the relationship between $PC$ and the tangent at $P$?

Question 40

• If a particle moves in a straight line according to the law $x = a \sin(\sqrt{a}t + b)$, then the particle will come to rest at two points whose distance is what?

Question 41

• What is a unit vector in the XY-plane making an angle of 45 degrees with $\hat{i} + \hat{j}$ and an angle of 60 degrees with $3\hat{i} - 4\hat{j}$?

Question 42

• Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = |x^2 - 1|$, then where does $f(x)$ have local minima and maxima?

Question 43

If, for the series $a_1, a_2, a_3, \dots$, $\frac{a_i - a_{i + 1}}{a_i a_{i + 1}}$ bears a constant ratio with -1, then $a_1, a_2, a_3, \dots$ are in which kind of progression?

Question 44

Given an A.P. and a G.P. with positive terms, with the first and second terms of the progressions being equal. If $a_n$ and $b_n$ be the $n^{th}$ term of A.P. and G.P. respectively then when does $a_n>b_n$ for all $n>2$ or $a_n<b_n$ for all $n>2$ or $a_n=b_n$ for some $n>2$ or $a_n=-b_n$ for some odd $n$.

Question 45

• If $z_1$ and $z_2$ be two roots of the equation $ z^2+az+b=0,a^2<4b $ then the origin , $z_1$ and $z_2$ form an equilateral triangle if what is the relationship between a and b.

Question 46

• If $ cos \theta+i sin \theta,\theta \in R, $ is a root of the equation $a_0x^n+a_1x^{n-1}+.......+a_{n-1}x+a_n=0,$ the value of what : $a_1 sin \theta+a_2 sin 2\theta+........+a_n sin n\theta is $ .

Question 47

• If $(x^2)^{log_x (27)} = x+4$ then value of x is what ?

Question 48

• If $ P(x)=ax^2+bx+c $ and $ Q(x)=-ax^2+dx+c $ where $ ac\neq0, $ then $ P(x)Q(x)=0 $ has what ? ( given $a,b,c,d$ are real ).

Question 49

Let $N$ be the number of quadratic equations with coefficients from {0,1,2,...,9} such that 0 is a solution of each equation. what is N?

Question 50

If $a, b, c$ are distinct odd natural numbers, what can be specified for rational roots of equation $ ax^2+bx+c=0 $ or must be any value for $0,1,2$?

Question 51

• For the real numbers $x$ & $y$, we write $x$ $p$ $y$ iff $x-y+\sqrt{2}$ is an irrational number. Then find the relation $p$.

Question 52

• Let $A = \begin{bmatrix} 0 & 0 & -1 \ 0 & -1 & 0 \ -1 & 0 & 0 \end{bmatrix}$, then what type of Matrix is it? A Null Matrix or a skew symmetric matrix or $ A^{-1} $ does not exist, what relation for $A^2 = I$ , it exists?

Question 53

• If $1000! = 3^n \times m $ where $'m'$ is an integer not divisible by 3, then $ n = $.

Question 54

• Two circles pass through points (0, a) and (0, -a), touching the line y = mx + c, and intersecting orthogonally. Find the relation between a, c, and m.

Question 55

• Find the locus of the midpoint of the system of parallel chords parallel to the line y = 2x for the hyperbola 9x² - 4y² = 36.

Question 56

• Calculate the angle between two diagonals of a cube.

Question 57

• If A and B are acute angles such that sin A = sin² B and 2cos² A = 3cos² B, find the value of (A, B).

Question 58

• Find the limit as n approaches infinity of the sum (1/(n^(k+1))) * [2^k + 4^k + 6^k + ... + (2n)^k].

Question 59

• If $ y = tan^{-1} \left[ \frac{\log_e(\frac{x}{e^2})}{\log_e(ex^2)} \right] + tan^{-1} \left[ \frac{3 + 2\log_e x}{1 - 6 \log_e x} \right] $, what expression determines $\frac{d^2y}{dx^2}$?

Question 60

• Given the function f(x) = x(x - 1)(x - 2) ... (x - 100), determine which statement is correct.

Question 61

• Let $I(R) = \int_{0}^{R} e^{-R}\sin{x} dx $, $R>0$, then determine if $I(R) > \frac{\pi}{2R}(1 - e^{-R})$ or $I(R) < \frac{\pi}{2R}(1 - e^{-R})$ or $I(R) = \frac{\pi}{2R}(1 - e^{-R})$

Question 62

• In a plane a and b are the position vectors of two points A and B respectively. a point P with position vector r moves on that plane in such a way that | r - a | - | *r - b *|= *c *(real constant) find eccentricity.

Question 63

• Five balls of different colors are to be placed in three boxes of different sizes. The number of ways in which we can place balls in the boxes so that no box remains empty is:

Question 64

• Let $ A = [ 1 , -1, 0],[0,1,-1],[1,1,1 ] A = [ 2 , 1, 4 ],B= [ 1 ,1 ,7 ],B=[1,1;7]$, then for the validity the result AX =B what is X Matrix is:

Question 65

• If $ a_1,a_2,......a_n$ are A.P. common difference 0, then , the SUM of the series: sec $ alpha_1$ se $ alpha_2$+sec $ alpha_3$ se $ alpha_4$........sec $ alpha_{n-1} $ se $ alpha_n= K$ (tan $ alpha_n $- tan $ alpha_1)$: what is the relation to find it ?

Question 66

The function f: R®R defined by f(x) = ex + e-x is: (choose all that apply from) one-one, onto, bijective, not bijective.

Question 67

Determine x from what form i, a1 + b1x a1x + b1 c1 , ,c1= R (i = 1

Question 68

• If ABC is an isosceles triangle and the coordinates of the base points are B(1, 3) and C(-2, 7). The coordinates of A can be

Question 69

• A square with its each sides equal to a above the x axis and has vertex at the origin . One of the sides passing through the origin makes an angel alpha(0

Question 70

Choose which one the correct statement: (A) x+sin 2x is a periodic statement ; (B) x+sin 2x is not a periodic statement :(C)Cos root x plus oneis a periodic statement: (D)Cos root x plus one os not a periodic statement .

Question 71

• Extremes of point $ \in $$ (t^2- 5+4/24+e )dt $. Are what. .

Question 72

Identify how be related. To the line that what condition related.

Question 73

The acceleration $ f/t/\sec^2 $ with starting from red what equation it attend. Is relate $t+ \surd 1+2t$ . So what max velocity and time attend.

Question 74

if The quadric equation $ atbxc= 0$ A(2 ) has to root $B7*

Question 75

• is $x to positive integer The value of a form + $c^{1x}/ c^{2 896}+\02799= what value express