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2021 paper1

23 maths questions

Q1 3 marks Circles Circle Equation Derivation View
Consider a triangle $\Delta$ whose two sides lie on the x-axis and the line $x + y + 1 = 0$. If the orthocenter of $\Delta$ is $(1,1)$, then the equation of the circle passing through the vertices of the triangle $\Delta$ is
(A) $x^2 + y^2 - 3x + y = 0$
(B) $x^2 + y^2 + x + 3y = 0$
(C) $x^2 + y^2 + 2y - 1 = 0$
(D) $x^2 + y^2 + x + y = 0$
Q2 3 marks Areas Between Curves Area Between Curves with Parametric or Implicit Region Definition View
The area of the region $$\left\{ (x,y) : 0 \leq x \leq \frac{9}{4}, \quad 0 \leq y \leq 1, \quad x \geq 3y, \quad x + y \geq 2 \right\}$$ is
(A) $\frac{11}{32}$
(B) $\frac{35}{96}$
(C) $\frac{37}{96}$
(D) $\frac{13}{32}$
Q3 3 marks Conditional Probability Sequential/Multi-Stage Conditional Probability View
Consider three sets $E_1 = \{1,2,3\}$, $F_1 = \{1,3,4\}$ and $G_1 = \{2,3,4,5\}$. Two elements are chosen at random, without replacement, from the set $E_1$, and let $S_1$ denote the set of these chosen elements. Let $E_2 = E_1 \setminus S_1$ and $F_2 = F_1 \cup S_1$. Now two elements are chosen at random, without replacement, from the set $F_2$ and let $S_2$ denote the set of these chosen elements.
Let $G_2 = G_1 \cup S_2$. The value of $P(E_2 = F_2)$ is
(A) $\frac{1}{7}$
(B) $\frac{3}{7}$
(C) $\frac{1}{5}$
(D) $\frac{2}{7}$
Q4 3 marks Complex Numbers Argand & Loci Modulus Inequalities and Triangle Inequality Applications View
Let $\theta_1, \theta_2, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_1 + \theta_2 + \cdots + \theta_{10} = 2\pi$. Define the complex numbers $z_1 = e^{i\theta_1}$, $z_k = z_{k-1} e^{i\theta_k}$ for $k = 2, 3, \ldots, 10$, where $i = \sqrt{-1}$. Consider the statements $P$ and $Q$ given below:
$P: |z_2 - z_1| + |z_3 - z_2| + \cdots + |z_{10} - z_9| + |z_1 - z_{10}| \leq 2\pi$
$Q: |z_2^2 - z_1^2| + |z_3^2 - z_2^2| + \cdots + |z_{10}^2 - z_9^2| + |z_1^2 - z_{10}^2| \leq 4\pi$
Then,
(A) $P$ is TRUE and $Q$ is FALSE
(B) $Q$ is TRUE and $P$ is FALSE
(C) both $P$ and $Q$ are TRUE
(D) both $P$ and $Q$ are FALSE
Q5 2 marks Projectiles Projectile with Mid-Flight Event (Breakup or Bounce) View
A projectile is thrown from a point O on the ground at an angle $45^\circ$ from the vertical and with a speed $5\sqrt{2}$ m/s. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, 0.5 s after the splitting. The other part, $t$ seconds after the splitting, falls to the ground at a distance $x$ meters from the point O. The acceleration due to gravity $g = 10$ m/s$^2$.
The value of $t$ is ____.
Q6 2 marks Projectiles Projectile with Mid-Flight Event (Breakup or Bounce) View
A projectile is thrown from a point O on the ground at an angle $45^\circ$ from the vertical and with a speed $5\sqrt{2}$ m/s. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, 0.5 s after the splitting. The other part, $t$ seconds after the splitting, falls to the ground at a distance $x$ meters from the point O. The acceleration due to gravity $g = 10$ m/s$^2$.
The value of $x$ is ____.
Q7 2 marks Stationary points and optimisation Find critical points and classify extrema of a given function View
Let $f_1 : (0, \infty) \to \mathbb{R}$ and $f_2 : (0, \infty) \to \mathbb{R}$ be defined by $$f_1(x) = \int_0^x \prod_{j=1}^{21} (t - j)^j \, dt, \quad x > 0$$ and $$f_2(x) = 98(x-1)^{50} - 600(x-1)^{49} + 2450, \quad x > 0,$$ where, for any positive integer $n$ and real numbers $a_1, a_2, \ldots, a_n$, $\prod_{i=1}^n a_i$ denotes the product of $a_1, a_2, \ldots, a_n$. Let $m_i$ and $n_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_i$, $i = 1, 2$, in the interval $(0, \infty)$.
The value of $2m_1 + 3n_1 + m_1 n_1$ is ____.
Q8 2 marks Stationary points and optimisation Find critical points and classify extrema of a given function View
Let $f_1 : (0, \infty) \to \mathbb{R}$ and $f_2 : (0, \infty) \to \mathbb{R}$ be defined by $$f_1(x) = \int_0^x \prod_{j=1}^{21} (t - j)^j \, dt, \quad x > 0$$ and $$f_2(x) = 98(x-1)^{50} - 600(x-1)^{49} + 2450, \quad x > 0,$$ where, for any positive integer $n$ and real numbers $a_1, a_2, \ldots, a_n$, $\prod_{i=1}^n a_i$ denotes the product of $a_1, a_2, \ldots, a_n$. Let $m_i$ and $n_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_i$, $i = 1, 2$, in the interval $(0, \infty)$.
The value of $6m_2 + 4n_2 + 8m_2 n_2$ is ____.
Q9 2 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Let $g_i : [\pi/8, 3\pi/8] \to \mathbb{R}$, $i = 1, 2$, and $f : [\pi/8, 3\pi/8] \to \mathbb{R}$ be functions such that $$g_1(x) = 1, \quad g_2(x) = |4x - \pi|, \quad f(x) = \sin^2 x,$$ for all $x \in [\pi/8, 3\pi/8]$. Define $$S_i = \int_{\pi/8}^{3\pi/8} f(x) \cdot g_i(x) \, dx, \quad i = 1, 2.$$
The value of $\frac{16 S_1}{\pi}$ is ____.
Q10 2 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Let $g_i : [\pi/8, 3\pi/8] \to \mathbb{R}$, $i = 1, 2$, and $f : [\pi/8, 3\pi/8] \to \mathbb{R}$ be functions such that $$g_1(x) = 1, \quad g_2(x) = |4x - \pi|, \quad f(x) = \sin^2 x,$$ for all $x \in [\pi/8, 3\pi/8]$. Define $$S_i = \int_{\pi/8}^{3\pi/8} f(x) \cdot g_i(x) \, dx, \quad i = 1, 2.$$
The value of $\frac{48 S_2}{\pi^2}$ is ____.
Q11 4 marks Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
For any real number $x$, let $\lfloor x \rfloor$ denote the largest integer less than or equal to $x$. If $$I = \int_0^{10} \left\lfloor \frac{10x}{x+1} \right\rfloor dx,$$ then the value of $9I$ is ____.
Q12 4 marks Sine and Cosine Rules Multi-step composite figure problem View
In a triangle $ABC$, let $AB = \sqrt{23}$, $BC = 3$ and $CA = 4$. Then the value of $$\frac{\cot A + \cot C}{\cot B}$$ is ____.
Q13 4 marks Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\vec{u}$, $\vec{v}$ and $\vec{w}$ be vectors in three-dimensional space, where $\vec{u}$ and $\vec{v}$ are unit vectors which are not perpendicular to each other and $$\vec{u} \cdot \vec{w} = 1, \quad \vec{v} \cdot \vec{w} = 1, \quad \vec{w} \cdot \vec{w} = 4.$$ If the volume of the parallelepiped, whose adjacent sides are represented by the vectors $\vec{u}$, $\vec{v}$ and $\vec{w}$, is $\sqrt{2}$, then the value of $|3\vec{u} + 5\vec{v}|$ is ____.
Q14 4 marks Modulus function Differentiability of functions involving modulus View
The number of points at which the function $$f(x) = |2x+1| - 3|x+2| + |x^2 + x - 2|, \quad x \in \mathbb{R}$$ is NOT differentiable is ____.
Q15 4 marks Complex Numbers Argand & Loci Solving Complex Equations with Geometric Interpretation View
Let $z$ be a complex number satisfying $|z|^3 + 2z^2 + 4\bar{z} - 8 = 0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Which of the following statements is TRUE?
(A) $|z|^2 = 2$
(B) $|z|^2 = 4$
(C) $|z|^2 = 8$
(D) $|z|^2 = 16$
Q16 4 marks Differentiating Transcendental Functions Higher-order or nth derivative computation View
Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $$f(x) = \begin{cases} x^5 \sin\left(\frac{1}{x}\right) + 5x^2, & x < 0 \\ 0, & x = 0 \\ x^5 \cos\left(\frac{1}{x}\right) + \lambda x^2, & x > 0 \end{cases}$$ The value of $\lambda$ for which $f''(0)$ exists is ____.
(A) 0
(B) 1
(C) $-1$
(D) $\frac{1}{2}$
Q17 4 marks Probability Definitions Probability Using Set/Event Algebra View
Let $E$, $F$ and $G$ be three events having probabilities $$P(E) = \frac{1}{8}, \quad P(F) = \frac{1}{6}, \quad P(G) = \frac{1}{4},$$ and let $P(E \cap F \cap G) = \frac{1}{10}$.
For any event $H$, if $P(H^c)$ denotes its complement, then which of the following statements is(are) TRUE?
(A) $P(E \cap F \cap G^c) \leq \frac{1}{40}$
(B) $P(E^c \cap F \cap G) \leq \frac{1}{15}$
(C) $P(E \cup F \cup G) \leq \frac{13}{24}$
(D) $P(E^c \cap F^c \cap G^c) \leq \frac{5}{12}$
Q18 4 marks Differentiating Transcendental Functions Compute derivative of transcendental function View
Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be functions satisfying $$f(x+y) = f(x) + f(y) + f(x)f(y) \quad \text{and} \quad f(x) = xg(x)$$ for all $x, y \in \mathbb{R}$. If $\lim_{x \to 0} g(x) = 1$, then which of the following statements is(are) TRUE?
(A) $f$ is differentiable at every $x \in \mathbb{R}$
(B) If $g(0) = 1$, then $g$ is differentiable at every $x \in \mathbb{R}$
(C) The derivative $f'(1)$ is equal to 1
(D) The derivative $f'(0)$ is equal to 1
Q19 4 marks 3x3 Matrices Determinant of Parametric or Structured Matrix View
Let $M = \begin{pmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{pmatrix}$ and $\text{adj}(M) = \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix}$
where $a$ and $b$ are real numbers. Which of the following statements is(are) TRUE?
(A) $(a+b)^2 = 9$
(B) $\det(\text{adj}(M^2)) = 81$
(C) $\text{adj}(\text{adj}(M)) = \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix}$
(D) $\det(\text{adj}(2M)) = 2^8$
Q20 4 marks Complex Numbers Argand & Loci Distance and Region Optimization on Loci View
Let $S$ be the set of all complex numbers $z$ satisfying $|z^2 + z + 1| = 1$. Which of the following statements is(are) TRUE?
(A) $\left| z + \frac{1}{2} \right| \leq \frac{1}{2}$ for all $z \in S$
(B) $|z| \leq 2$ for all $z \in S$
(C) $\left| z + \frac{1}{2} \right| \geq \frac{1}{2}$ for all $z \in S$
(D) The set $S$ has exactly four elements
Q21 4 marks Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
For any real number $t$, let $\lfloor t \rfloor$ be the largest integer less than or equal to $t$. Then the number of points of discontinuity of the function $x \mapsto \lfloor x^2 - 3 \rfloor$ for $x \in (-\infty, 0)$ is ____.
Let $f: [-1, 3] \to \mathbb{R}$ be defined as $$f(x) = \begin{cases} |x| + [x], & -1 \leq x < 1 \\ x + |x|, & 1 \leq x < 2 \\ x + [x], & 2 \leq x \leq 3 \end{cases}$$ where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points of discontinuity of $f$ in the interval $(-1, 3)$ is ____.
Q22 4 marks 3x3 Matrices Determinant of Parametric or Structured Matrix View
Let $\alpha$, $\beta$ and $\gamma$ be real numbers such that the system of linear equations $$x + 2y + 3z = \alpha$$ $$4x + 5y + 6z = \beta$$ $$7x + 8y + 9z = \gamma - 1$$ is consistent. Let $|M|$ represent the determinant of the matrix $$M = \begin{pmatrix} \alpha & 1 & 0 \\ \beta & 2 & 1 \\ \gamma & 1 & 0 \end{pmatrix}.$$ Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0, 1, 0)$ from the plane $P$.
The value of $|M|$ is ____.
Q23 4 marks 3x3 Matrices Geometric Interpretation of 3×3 Systems View
Let $\alpha$, $\beta$ and $\gamma$ be real numbers such that the system of linear equations $$x + 2y + 3z = \alpha$$ $$4x + 5y + 6z = \beta$$ $$7x + 8y + 9z = \gamma - 1$$ is consistent. Let $|M|$ represent the determinant of the matrix $$M = \begin{pmatrix} \alpha & 1 & 0 \\ \beta & 2 & 1 \\ \gamma & 1 & 0 \end{pmatrix}.$$ Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0, 1, 0)$ from the plane $P$.
The value of $D$ is ____.