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QB3 15 marks Number Theory Congruence Reasoning and Parity Arguments View

(a) Show that there are exactly 2 numbers $a$ in $\{2, 3, \ldots, 9999\}$ for which $a^2 - a$ is divisible by 10000. Find these values of $a$.
(b) Let $n$ be a positive integer. For how many numbers $a$ in $\{2, 3, \ldots, n^2 - 1\}$ is $a^2 - a$ divisible by $n^2$? State your answer suitably in terms of $n$ and justify.

QB4 12 marks Applied differentiation Properties of differentiable functions (abstract/theoretical) View

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function, where $\mathbb{R}$ denotes the set of real numbers. Suppose that for all real numbers $x$ and $y$, the function $f$ satisfies
$$f'(x) - f'(y) \leq 3|x - y|$$
Answer the following questions. No credit will be given without full justification.
(a) Show that for all $x$ and $y$, we must have $\left|f(x) - f(y) - f'(y)(x - y)\right| \leq 1.5(x - y)^2$.
(b) Find the largest and smallest possible values for $f''(x)$ under the given conditions.

QB5 12 marks Number Theory GCD, LCM, and Coprimality View

For an arbitrary integer $n$, let $g(n)$ be the GCD of $2n + 9$ and $6n^2 + 11n - 2$. What is the largest positive integer that can be obtained as the value of $g(n)$? If $g(n)$ can be arbitrarily large, state so explicitly and prove it.

QB6 12 marks Circles Chord Length and Chord Properties View

You are given the following: a circle, one of its diameters $AB$ and a point $X$.
(a) Using only a straight-edge, show in the given figure how to draw a line perpendicular to $AB$ passing through $X$. No credit will be given without full justification. (Recall that a straight-edge is a ruler without any markings. Given two points, a straight-edge can be used to draw the line passing through the given points.)
(b) Do NOT draw any of your work for this part in the given figure. Reconsider your procedure to see if it can be made to work if the point $X$ is in some other position, e.g., when it is inside the circle or to the ``left/right'' of the circle. Clearly specify all positions of the point $X$ for which your procedure in part (a), or a small extension/variation of it, can be used to obtain the perpendicular to $AB$ through $X$. Justify your answer.

Q1 4 marks Discrete Probability Distributions Multiple Choice: Direct Probability or Distribution Calculation View

Ten people sitting around a circular table decide to donate some money for charity. You are told that the amount donated by each person was the average of the money donated by the two persons sitting adjacent to him/her. One person donated Rs. 500. Choose the correct option for each of the following two questions. Write your answers as a sequence of two letters (a/b/c/d).
What is the total amount donated by the 10 people?
(a) exactly Rs. 5000
(b) less than Rs. 5000
(c) more than Rs. 5000
(d) not possible to decide among the above three options.
What is the maximum amount donated by an individual?
(a) exactly Rs. 500
(b) less than Rs. 500
(c) more than Rs. 500
(d) not possible to decide among the above three options.

Q3 4 marks Number Theory Prime Number Properties and Identification View

A positive integer $n$ is called a magic number if it has the following property: if $a$ and $b$ are two positive numbers that are not coprime to $n$ then $a + b$ is also not coprime to $n$. For example, 2 is a magic number, because sum of any two even numbers is also even. Which of the following are magic numbers? Write your answers as a sequence of four letters (Y for Yes and N for No) in correct order.
(i) 129
(ii) 128
(iii) 127
(iv) 100.

Q4 4 marks Implicit equations and differentiation Piecewise differentiability and continuity conditions View

Let $A$, $B$ and $C$ be unknown constants. Consider the function $f(x)$ defined by
$$\begin{aligned} f(x) &= Ax^2 + Bx + C \text{ when } x \leq 0 \\ &= \ln(5x + 1) \text{ when } x > 0 \end{aligned}$$
Write the values of the constants $A$, $B$ and $C$ such that $f''(x)$, i.e., the double derivative of $f$, exists for all real $x$. If this is not possible, write ``not possible''. If some of the constants cannot be uniquely determined, write ``not unique'' for each such constant.

Q6 4 marks Circles Tangent Lines and Tangent Lengths View

Fill in the blanks. Let $C_1$ be the circle with center $(-8, 0)$ and radius 6. Let $C_2$ be the circle with center $(8, 0)$ and radius 2. Given a point $P$ outside both circles, let $\ell_i(P)$ be the length of a tangent segment from $P$ to circle $C_i$. The locus of all points $P$ such that $\ell_1(P) = 3\ell_2(P)$ is a circle with radius \_\_\_ and center at (\_\_\_, \_\_\_).

Q7 4 marks Geometric Sequences and Series Determine the Limit of a Sequence via Geometric Series View

(i) By the binomial theorem $(\sqrt{2} + 1)^{10} = \sum_{i=0}^{10} C_i (\sqrt{2})^i$, where $C_i$ are appropriate constants. Write the value of $i$ for which $C_i (\sqrt{2})^i$ is the largest among the 11 terms in this sum.
(ii) For every natural number $n$, let $(\sqrt{2} + 1)^n = p_n + \sqrt{2} q_n$, where $p_n$ and $q_n$ are integers. Calculate $\lim_{n \rightarrow \infty} \left(\frac{p_n}{q_n}\right)^{10}$.

Q8 4 marks Permutations & Arrangements Forming Numbers with Digit Constraints View

The format for car license plates in a small country is two digits followed by three vowels, e.g. 04 IOU. A license plate is called ``confusing'' if the digit 0 (zero) and the vowel O are both present on it. For example $04\,IOU$ is confusing but $20\,AEI$ is not. (i) How many distinct number plates are possible in all? (ii) How many of these are not confusing?

Q9 4 marks Standard trigonometric equations Inverse trigonometric equation View

Recall that $\sin^{-1}$ is the inverse function of $\sin$, as defined in the standard fashion. (Sometimes $\sin^{-1}$ is called $\arcsin$.) Let $f(x) = \sin^{-1}(\sin(\pi x))$. Write the values of the following. (Some answers may involve the irrational number $\pi$. Write such answers in terms of $\pi$.)
(i) $f(2.7)$
(ii) $f'(2.7)$
(iii) $\int_0^{2.5} f(x)\, dx$
(iv) the smallest positive $x$ at which $f'(x)$ does not exist.

Q10 4 marks Complex Numbers Argand & Loci Modulus Inequalities and Triangle Inequality Applications View

Answer the three questions below. To answer (i) and (ii), replace ? with exactly one of the following four options: $<$, $=$, $>$, not enough information to compare.
(i) Suppose $z_1, z_2$ are complex numbers. One of them is in the second quadrant and the other is in the third quadrant. Then $\left|\left|z_1\right| - \left|z_2\right|\right| \quad ? \quad \left|z_1 + z_2\right|$.
(ii) Complex numbers $z_1$, $z_2$ and $0$ form an equilateral triangle. Then $\left|z_1^2 + z_2^2\right| \quad ? \quad \left|z_1 z_2\right|$.
(iii) Let $1, z_1, z_2, z_3, z_4, z_5, z_6, z_7$ be the complex 8th roots of unity. Find the value of $\prod_{i=1,\ldots,7}(1 - z_i)$, where the symbol $\Pi$ denotes product.

Q11 4 marks Conditional Probability Combinatorial Conditional Probability (Counting-Based) View

There are four distinct balls labelled $1, 2, 3, 4$ and four distinct bins labelled A, B, C, D. The balls are picked up in order and placed into one of the four bins at random. Let $E_i$ denote the event that the first $i$ balls go into distinct bins. Calculate the following probabilities.
(i) $\Pr[E_4]$
(ii) $\Pr[E_4 \mid E_3]$
(iii) $\Pr[E_4 \mid E_2]$
(iv) $\Pr[E_3 \mid E_4]$.
Notation: $\Pr[X] =$ the probability of event $X$ taking place. $\Pr[X \mid Y] =$ the probability of event $X$ taking place, given that event $Y$ has taken place.

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