Section A

If \(\begin{bmatrix} 2 & 1\\ 3 & 2 \end{bmatrix} \mathbf{A} \begin{bmatrix} -3 & 2\\ 5 & -3 \end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\), then matrix \(\mathbf{A}\) equals

If \(A\) is a square matrix such that \(AA^T = I = A^TA\), then \(A\) is

Consider a rare disease \(X\) that affect one in a million people. A medical test is used to test the presence of the disease. The test is \(99\%\) accurate in the sense that if a person has no disease, the chance that the test shows positive is \(1\%\) and if the person has the disease then the chance that the test shows negative is also \(1\%\). Suppose a person is tested for the disease and the result is positive. What is the probability that the person has the disease \(X\) ?

The mean value of a random variable \(x\) with probability density

\(\displaystyle \mathit{p(x)} = \displaystyle \frac{1}{\sigma \sqrt{2\pi}}\exp \left[-\frac{x^2+\mu x}{2\sigma^2} \right]\)

is

A man starts his journey at \(01:00\) hours local time to reach another country at \(09:00\) hours local time on the same date. He starts a return journey on the same night at \(21:00\) hours local time to his original place, taking the same time to travel back. If the time zone of his country of visit lags by \(10\) hours, then what is the duration for which the man was away from his place in hours?

A random number generator outputs \(+1\) or \(-1\) with equal probability every time it is run. After it is run \(6\) times, what is the probability that the the sum of the answers generated is zero ? \\
(Assume that the individuals runs are independent.)

Suppose \(1,2,3\) are roots of the equation \(x^3 + ax^2 + bx = c\). Then the value of \(c\) is

How many zeroes are there at the end of \(999!\) ?

Garfield is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can Garfield select?

If matrix \(AB\) is a zero matrix, then which of the following is/are correct?

Study the pseudocode given below:

What value does the pseudocode return for any integer inputs \(n,m\) ?

Let \(f : \mathbb{R} \to \mathbb{R} \) be defined by \(f(x) = x(x-1)(x-2)\). Then

Description for the next 3 questions

The web plot below shows the sales of TV and laptop from the year 2004 to 2009 by a particular company. The figures given here are in millions.

Based on the data given, answer the following questions.

Find the ratio of the total number of televisions sold to the total number of laptops sold in all these years.

Which item and for which year shows the highest percentage increase in the sales in the previous year?

Which year shows the highest percentage decrease in the total sales of the two items with respect to its previous year sales?

How many whole numbers between \(100\) and \(400\) contain the digit \(2\)?\(137\)

We toss a fair coin \(n\) times. Let \(\boldsymbol{A}\) denote the event that the first toss is a tail, \(\boldsymbol{B}\) denote the event that the second toss is a tail and \(\boldsymbol{C}\) denote the event that the first two tosses are both heads or both tails. Then which of the following statement/s is/are true ?

There are two circles, one inscribed in a square and the other circumscribing the square. What is the ratio of the area of the circumcircle and the incircle?

Generate the output of the given pseudocode for \(num = 4\).

A shopkeeper was selling rice at a profit of \(20\%\). With a bit of tampering he made his weighing scale show \(1 \; kg\) for every \(950 \; gms\). His profit percentage is

Section B

A spherical irom ball of radius \(10\; cm\) coated with a layer of ice of uniform thickness melts at a rate of \(100\pi\;cm^3/min\). Find the rate at which the thickness of the ice decreases when the thickness is \(5\; cm\).

A large freight elevator can transport a maximum of \(980\) pounds. Suppose a load of cargo containing \(49\) boxes must be transported via the elevator. Experience has shown that the weight of boxes of this type of cargo follows a distribution with mean $\mu$ = \(25\) and standard deviation $\sigma$ = \(15\) pounds. Based on this information what is probability that all $49$ boxes can be safely loaded onto the freight elevator and transported ?

Prove that the equation \(\displaystyle \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 1\) has no solutions for positive integers \(x,y\) and \(z\).

The following table represents the final scorecard of a soccer tournament between 3 teams. Unfortunately some entries are missing. Study the table carefully.

Note: The matches were played round robin, i.e., each team played each other team once)

With just the given information how much of the table can you complete? Fill in the missing entries, as many as you can.

Timothy and Sasha play a game with two dice. But they do not use the numbers. Some of the faces are painted red and the others blue. Each player throws the dice in turn. Timothy wins when the two top faces are the same color. Sasha wins when the colors are different. Their chances are even. The first die has \(5\) red faces and \(1\) blue face. How many red and how many blue faces are there on the second die?

In a hockey match against Pakistan, India tied the scores at \(4 —4\) at the end of the game. If Pakistan never gained lead over India throughout the match and at most tied the scores, then how many possible scoreboards were possible for the match.

Here are two example scoreboards:

A coffee can contains some black beans and white beans. The following process is to be repeated as long as possible. Randomly select two beans from the can. If they are the same colour, throw them out, but put another black bean in (Assume that enough black beans are available to do this). If they are of different colours, place the white one back into the can and throw the black one away. Execution of this process reduces the number of beans in the can by one. Repetition of this process must terminate with exactly one bean in the can. What can you say about the colour of this last bean depending on the number of black beans and the number of white beans that were there in the can to start with?

Study the code given below:

What will the code return for fun $([12,43,62,7,19], 5)$ ? What does the code in general return for fun $(arr, n)$ , for any given array \(arr\) and \(n = \) number of elements in \(arr\) ?

Suppose the velocity \(V\) in \(km/hr\) of a motor ship is expressed as a function of the cost of the fuel consumed per hour, say ‘\(p\)’ rupees per hour as \(V = c.\frac{p}{p+1}\) where \(c\) is constant. Also, suppose the fixed operating cost of the ship, other than the fuel cost, is \(Rs\:q\) per hour of running. Find the velocity of cruising from Port \(A\) to Port \(B\), located at a distance of ‘\(s\)’ kms from \(A\), so that the cost of the cruise is minimum.

Find the dimensions of the rectangle with the largest area that can fit inside the graph of the parabola \(y = x^2\) and the straight line \(y = a\), for all \(a>0\).

If the following songs are played in shuffle mode, what is the probability that no two songs from the same band shall be played one after the other?

(1) Sugar – Maroon 5
(2) Animals – Maroon 5
(3) Maps – Maroons 5
(4) Demons – Imagine Dragons
(5) Radioactive – Imagine Dragons
(6) Whatever It Takes – Imagine Dragons
(7) Thunder – Imagine dragons
(8) Believer – Imagine Dragons
(9) It’s Time – Imagine Dragons
(10) High Hopes – Pink Floyd
(11) Another Brick in the Wall – Pink Floyd

A pizza shop is to supply pizzas to 3 different parties. The total number of pizzas that needed be delivered is 800 out of which \(70\%\) are to be delivered to party 3 and the rest equally divided between party 1 and party 2. Now, pizza could be of thin crust T or Deep Dish D and come in Normal Cheese NC or Extra Cheese EC. Hence there are \(4\) types of pizza:

Partial information about proportions of T and NC pizzas ordered by the three parties are given below.

Based on the information provided, answer the following questions.

(A) How many Thin Crust pizzas were to be delivered to Party 3?

(B) How many Normal Cheese pizzas were required to be delivered to Party 1?

(C) For Party 2, if \(50\%\) of the Normal Cheese pizzas were of Thin Crust variety, what was the difference between the numbers of $T —EC$ and D —EC$ pizzas to be delivered to Party 2?

(D) Suppose that a $T— NC$ pizza cost as much as a $D— NC$ pizza, but of the price of a $D —EC$ pizza. A $D —EC$ pizza costs INR \(50\) more than a $T—EC$ pizza, and the latter costs INR \(500\). If \(25\%\) of the Normal Cheese pizzas delivered to Party 1 were of Deep Dish variety, what was the total bill for Party 1?