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18 maths questions

Q37 Standard trigonometric equations Evaluate trigonometric expression given a constraint View
Let $-\frac{\pi}{6} < \theta < -\frac{\pi}{12}$. Suppose $\alpha_1$ and $\beta_1$ are the roots of the equation $x^2 - 2x\sec\theta + 1 = 0$ and $\alpha_2$ and $\beta_2$ are the roots of the equation $x^2 + 2x\tan\theta - 1 = 0$. If $\alpha_1 > \beta_1$ and $\alpha_2 > \beta_2$, then $\alpha_1 + \beta_2$ equals
(A) $2(\sec\theta - \tan\theta)$
(B) $2\sec\theta$
(C) $-2\tan\theta$
(D) $0$
Q38 Combinations & Selection Selection with Group/Category Constraints View
A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is
(A) 380
(B) 320
(C) 260
(D) 95
Q39 Probability Definitions Finite Equally-Likely Probability Computation View
Let $S = \left\{x \in (-\pi, \pi) : x \neq 0, \pm\frac{\pi}{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3}\sec x + \operatorname{cosec} x + 2(\tan x - \cot x) = 0$ in the set $S$ is equal to
(A) $-\frac{7\pi}{9}$
(B) $-\frac{2\pi}{9}$
(C) $0$
(D) $\frac{5\pi}{9}$
Q40 Probability Definitions Conditional Probability and Bayes' Theorem View
A computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces $20\%$ and plant $T_2$ produces $80\%$ of the total computers produced. $7\%$ of computers produced in the factory turn out to be defective. It is known that $P$(computer turns out to be defective given that it is produced in plant $T_1$) $= 10P$(computer turns out to be defective given that it is produced in plant $T_2$), where $P(E)$ denotes the probability of an event $E$. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $T_2$ is
(A) $\frac{36}{73}$
(B) $\frac{47}{79}$
(C) $\frac{78}{93}$
(D) $\frac{75}{83}$
Q41 Stationary points and optimisation Prove an inequality using calculus-based optimisation View
The least value of $\alpha \in \mathbb{R}$ for which $4\alpha x^2 + \frac{1}{x} \geq 1$, for all $x > 0$, is
(A) $\frac{1}{64}$
(B) $\frac{1}{32}$
(C) $\frac{1}{27}$
(D) $\frac{1}{25}$
Q42 Vectors 3D & Lines Multi-Part 3D Geometry Problem View
Consider a pyramid $OPQRS$ located in the first octant $(x \geq 0, y \geq 0, z \geq 0)$ with $O$ as origin, and $OP$ and $OR$ along the $x$-axis and the $y$-axis, respectively. The base $OPQR$ of the pyramid is a square with $OP = 3$. The point $S$ is directly above the mid-point $T$ of diagonal $OQ$ such that $TS = 3$. Then
(A) the acute angle between $OQ$ and $OS$ is $\frac{\pi}{3}$
(B) the equation of the plane containing the triangle $OQS$ is $x - y = 0$
(C) the length of the perpendicular from $P$ to the plane containing the triangle $OQS$ is $\frac{3}{\sqrt{2}}$
(D) the perpendicular distance from $O$ to the straight line containing $RS$ is $\sqrt{\frac{15}{2}}$
Q43 First order differential equations (integrating factor) View
Let $f:(0,\infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f'(x) = 2 - \frac{f(x)}{x}$ for all $x \in (0,\infty)$ and $f(1) \neq 1$. Then
(A) $\lim_{x \rightarrow 0+} f'\left(\frac{1}{x}\right) = 1$
(B) $\lim_{x \rightarrow 0+} xf\left(\frac{1}{x}\right) = 2$
(C) $\lim_{x \rightarrow 0+} x^2 f'(x) = 0$
(D) $|f(x)| \leq 2$ for all $x \in (0,2)$
Q44 Matrices Linear System and Inverse Existence View
Let $P = \left[\begin{array}{ccc} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{array}\right]$, where $\alpha \in \mathbb{R}$. Suppose $Q = [q_{ij}]$ is a matrix such that $PQ = kI$, where $k \in \mathbb{R}, k \neq 0$ and $I$ is the identity matrix of order 3. If $q_{23} = -\frac{k}{8}$ and $\det(Q) = \frac{k^2}{2}$, then
(A) $\alpha = 0, k = 8$
(B) $4\alpha - k + 8 = 0$
(C) $\det(P\operatorname{adj}(Q)) = 2^9$
(D) $\det(Q\operatorname{adj}(P)) = 2^{13}$
Q45 Sine and Cosine Rules Multi-step composite figure problem View
In a triangle $XYZ$, let $x, y, z$ be the lengths of sides opposite to the angles $X, Y, Z$, respectively, and $2s = x + y + z$. If $\frac{s-x}{4} = \frac{s-y}{3} = \frac{s-z}{2}$ and area of incircle of the triangle $XYZ$ is $\frac{8\pi}{3}$, then
(A) area of the triangle $XYZ$ is $6\sqrt{6}$
(B) the radius of circumcircle of the triangle $XYZ$ is $\frac{35}{6}\sqrt{6}$
(C) $\sin\frac{X}{2}\sin\frac{Y}{2}\sin\frac{Z}{2} = \frac{4}{35}$
(D) $\sin^2\left(\frac{X+Y}{2}\right) = \frac{3}{5}$
Q46 Differential equations Solving Separable DEs with Initial Conditions View
A solution curve of the differential equation $\left(x^2 + xy + 4x + 2y + 4\right)\frac{dy}{dx} - y^2 = 0, x > 0$, passes through the point $(1,3)$. Then the solution curve
(A) intersects $y = x + 2$ exactly at one point
(B) intersects $y = x + 2$ exactly at two points
(C) intersects $y = (x+2)^2$
(D) does NOT intersect $y = (x+3)^2$
Q47 Composite & Inverse Functions Derivative of an Inverse Function View
Let $f:\mathbb{R} \rightarrow \mathbb{R}, g:\mathbb{R} \rightarrow \mathbb{R}$ and $h:\mathbb{R} \rightarrow \mathbb{R}$ be differentiable functions such that $f(x) = x^3 + 3x + 2, g(f(x)) = x$ and $h(g(g(x))) = x$ for all $x \in \mathbb{R}$. Then
(A) $g'(2) = \frac{1}{15}$
(B) $h'(1) = 666$
(C) $h(0) = 16$
(D) $h(g(3)) = 36$
Q48 Circles Circles Tangent to Each Other or to Axes View
The circle $C_1: x^2 + y^2 = 3$, with centre at $O$, intersects the parabola $x^2 = 2y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_1$ at $P$ touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$, respectively. Suppose $C_2$ and $C_3$ have equal radii $2\sqrt{3}$ and centres $Q_2$ and $Q_3$, respectively. If $Q_2$ and $Q_3$ lie on the $y$-axis, then
(A) $Q_2 Q_3 = 12$
(B) $R_2 R_3 = 4\sqrt{6}$
(C) area of the triangle $OR_2R_3$ is $6\sqrt{2}$
(D) area of the triangle $PQ_2Q_3$ is $4\sqrt{2}$
Q49 Circles Circle-Related Locus Problems View
Let $RS$ be the diameter of the circle $x^2 + y^2 = 1$, where $S$ is the point $(1,0)$. Let $P$ be a variable point (other than $R$ and $S$) on the circle and tangents to the circle at $S$ and $P$ meet at the point $Q$. The normal to the circle at $P$ intersects a line drawn through $Q$ parallel to $RS$ at point $E$. Then the locus of $E$ passes through the point(s)
(A) $\left(\frac{1}{3}, \frac{1}{\sqrt{3}}\right)$
(B) $\left(\frac{1}{4}, \frac{1}{2}\right)$
(C) $\left(\frac{1}{3}, -\frac{1}{\sqrt{3}}\right)$
(D) $\left(\frac{1}{4}, -\frac{1}{2}\right)$
Q50 3x3 Matrices Determinant of Parametric or Structured Matrix View
The total number of distinct $x \in \mathbb{R}$ for which $\left|\begin{array}{ccc} x & x^2 & 1+x^3 \\ 2x & 4x^2 & 1+8x^3 \\ 3x & 9x^2 & 1+27x^3 \end{array}\right| = 10$ is
Q51 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2 + (1+x)^3 + \cdots + (1+x)^{49} + (1+mx)^{50}$ is $(3n+1)\,{}^{51}C_3$ for some positive integer $n$. Then the value of $n$ is
Q52 Indefinite & Definite Integrals Maximizing or Optimizing a Definite Integral View
The total number of distinct $x \in [0,1]$ for which $\int_0^x \frac{t^2}{1+t^4}\,dt = 2x - 1$ is
Q53 Chain Rule Limit Evaluation Involving Composition or Substitution View
Let $\alpha, \beta \in \mathbb{R}$ be such that $\lim_{x \rightarrow 0} \frac{x^2 \sin(\beta x)}{\alpha x - \sin x} = 1$. Then $6(\alpha + \beta)$ equals
Q54 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View
Let $z = \frac{-1 + \sqrt{3}\,i}{2}$, where $i = \sqrt{-1}$, and $r, s \in \{1,2,3\}$. Let $P = \left[\begin{array}{cc} (-z)^r & z^{2s} \\ z^{2s} & z^r \end{array}\right]$ and $I$ be the identity matrix of order 2. Then the total number of ordered pairs $(r,s)$ for which $P^2 = -I$ is