A line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line has intercepts $p$ and $q$. Which of the following is true?

If $a,b$ are the roots of $x^2+px+1=0$ and $c,d$ are the roots of $x^2+qx+1=0$, the value of $E=(a-c)(b-c)(a+d)(b+d)$ is

If $f(x)+f(1-x)=2$, then the value of $f\left(\dfrac{1}{2001}\right)+f\left(\dfrac{2}{2001}\right)+\cdots+f\left(\dfrac{2000}{2001}\right)$ is

Suppose $a,b,c$ are in A.P. with common difference $d$. Then $e^{1/c},e^{1/b},e^{1/a}$ are

Let $\alpha$ and $\beta$ be the roots of $x^2+x+1=0$. The equation whose roots are $\alpha^{19}$ and $\beta^{19}$ is

In the expression $(x+1)(x+4)(x+9)(x+16)\cdots(x+400)$ the coefficient of $x^{19}$ is

The value of $y=0.36\log_{0.25}\left(\dfrac13+\dfrac1{3^2}+\cdots\right)$ is

If $H_1,H_2,\ldots,H_n$ are $n$ harmonic means between $a$ and $b$, $a\ne b$, then the value of $\dfrac{H_1+a}{H_1-a}+\dfrac{H_n+b}{H_n-b}$ is equal to

For $a>0,\ a\ne1$, the number of values of $x$ satisfying $2\log_x a+\log_{ax} a+3\log_{a^2x} a=0$ is

An eight digit number divisible by $9$ is to be formed by using $8$ digits out of the digits $0,1,\ldots,9$ without replacement. The number of ways in which this can be done is

The number of ordered pairs $(m,n)$, $m,n\in{1,2,\ldots,100}$ such that $7^m+7^n$ is divisible by $5$ is

If $a,b,c$ are the roots of the equation $x^3-3px^2+3qx-1=0$, then the centroid of the triangle with vertices $\left(a,\frac1a\right),\left(b,\frac1b\right),\left(c,\frac1c\right)$ is the point

Equation of the common tangent touching the circle $(x-3)^2+y^2=9$ and the parabola $y^2=4x$ above the $x$-axis is

The number of roots of the equation $|x^2-x-6|=x+2$ is

A pair of unbiased dice is rolled together till a sum of either $5$ or $7$ is obtained. The probability that $5$ comes before $7$ is

A letter is taken at random from the letters of the word STATISTICS and another letter is taken at random from the letters of the word ASSISTANT. The probability that they are the same letter is

A bag contains $6$ red and $4$ green balls. A fair die is rolled and a number of balls equal to that appearing on the die is chosen from the bag at random. The probability that all the balls selected are red is

The value of $\lambda$ for which the volume of the parallelepiped formed by the vectors $\vec i+\lambda\vec j+\vec k,\ \vec j+\lambda\vec k,\ \lambda\vec i+\vec k$ is minimum is

A six-faced die is a biased one. It is thrice more likely to show an odd number than to show an even number. It is thrown twice. The probability that the sum of the numbers in the two throws is even is

A letter is known to have come from either TATANAGAR or CALCUTTA. On the envelope, just two consecutive letters, TA, are visible. The probability that the letter has come from CALCUTTA is

If $\cos\alpha+\cos\beta=a$, $\sin\alpha+\sin\beta=b$ and $\theta$ is the arithmetic mean between $\alpha$ and $\beta$, then $\sin2\theta+\cos2\theta$ is equal to

If $(1+\tan1^\circ)(1+\tan2^\circ)\cdots(1+\tan45^\circ)=2^n$, then the value of $n$ is

The value of $\sin12^\circ\sin48^\circ\sin54^\circ$ is

The value of $\lambda$ such that the four points whose position vectors are $3\vec i-2\vec j+\lambda\vec k,\ 6\vec i+3\vec j+\vec k,\ 5\vec i+7\vec j+3\vec k$ and $2\vec i+2\vec j+6\vec k$ are coplanar is

Let $\vec A=2\vec i+\vec j-2\vec k$ and $\vec B=\vec i+\vec j$. If $\vec C$ is a vector such that $\vec A\cdot\vec C=|\vec C|$, $|\vec C-\vec A|=2\sqrt2$ and the angle between $\vec A\times\vec B$ and $\vec C$ is $30^\circ$, then $|(\vec A\times\vec B)\times\vec C|$ is equal to

A rigid body is rotating at the rate of $3$ radians per second about an axis $AB$, where $A(1,-2,1)$ and $B(3,-4,2)$. The velocity of the point $P(5,-1,-1)$ of the body is

If $\vec A+\vec B+\vec C=\vec0$, $|\vec A|=3$, $|\vec B|=5$, $|\vec C|=7$, then the angle between $\vec A$ and $\vec B$ is

There are six houses in a row. Mr. Lal has Mr. Babu and Mr. Anil as neighbours. Mr. Bhatia has Mr. Gupta and Mr. Sharma as neighbours. Mr. Gupta’s house is not next to Mr. Babu or Mr. Anil and Mr. Sharma does not live next to Mr. Anil. Who are Mr. Babu’s next-door neighbours?

A watch which gains 10 seconds in 5 minutes was set correct at 9 a.m. When the watch indicated 20 minutes past 7 o’clock the same evening, the true time is:

A boy observes the reflection of a clock in a mirror. The time observed by the boy in the mirror is 8 hours 45 minutes. What is the actual time shown in the clock?

Gold is 19 times as heavy as water and copper is 9 times as heavy as water. In what ratio should these be mixed to get an alloy 15 times as heavy as water?

In an objective type examination, 120 objective type questions are there; each with 4 options P, Q, R and S. A candidate can choose either one of these options or can leave the question unanswered. How many different ways exist for answering this question paper?

You are given two (unmarked) containers of capacity 9 liters and 4 liters, and a huge tank of water. Need is to get a measure of exactly 6 liters of water. A move is either filling a container completely or emptying a container (either fully or partly). The smallest number of moves needed to do this task is

What is the next letter in the series O T T F F S S E N _______

What is the diameter of the largest circle that can be drawn on a chessboard so that its entire circumference gets covered by the black squares and no part of the circumference falls on any white space, given that the chessboard has black and white squares of size one inch?

A car is filled with 4½ liters of fuel for a round trip. If the amount of fuel taken while going is 1/4th more than the amount taken for coming, what is the amount of fuel consumed while coming back?

$\text{Which of the following are greater than } x \text{ when } x=\frac{9}{11}?$ $(I)\ \frac{1}{x}$ $(II)\ \frac{x+1}{x}$ $(III)\ \frac{x+1}{x-1}$

Four friends – Arjan, Bhuvan, Guran and Lakha were comparing the number of sheep they owned. Guran has 10 more sheep than Lakha. Arjan gave one third to Bhuvan, Bhuvan gave one fourth of what he then held to Guran, Guran passed one fifth of his holding to Lakha. After this, all had equal number of sheep. How many sheep did each possess? (Minimum possible)

In a class, six students $P,Q,R,S,T,U$ are the top six rank holders. $R$ did not get the 4th rank. $P$’s rank is higher than $U$’s and $R$’s but lower than $Q$’s. Four students have ranks lower than $S$, and five students have ranks above $T$. Who is ranked 5th in the class?

Three players – Aalu, Kachaalu and Bhalu were playing poker. At least one cheated. Exactly one always spoke the truth, one always lied, and one alternated between truth and lie. Which of the following can never be true?

If $x$ and $y$ are the two digits of the number $565xy$ such that the number is divisible by $80$, then $x+y$ equals

If both $7^2$ and $3^3$ are factors of $a\times11^3\times6^2\times13^{11}$, then the smallest possible value of $a$ is

Let $x,y,z$ be distinct integers. $x$ and $y$ are odd positive integers and $z$ is an even positive integer. Which of the following cannot be true?

From a height of $16$ meters a ball fell down and each time it bounces half the distance back. What is the total distance traveled?

If a man walks at the rate of $4$ kmph, he misses a train by $6$ minutes. If he walks at $5$ kmph, he reaches the station $6$ minutes before the train. Find the distance to the station.

Floating point numbers in a computer are represented using a 10-bit mantissa (including a sign bit) a 7-bit exponent (including a sign bit). What is the approximate value of the maximum number, which can be represented? Assume that the mantissa is stored in the normalized form, that is, without leading zeroes.

The capacity of a memory unit is defined by the number of words multiplied by the number of bits per word. How many separate address and data line are needed for a memory of 4K × 16?

The main disadvantage of direct mapping of cache organization is that

Let A [1… 10] be an array. Let A [i] = 2i for 1 ≤ i ≤ 10. After the assignment j = A[A[5]] is executed, the value of A[j] is equal to

The first instruction of bootstrap loader program of an operating system is stored in

The function AB'C + A'BC + ABC' + A'B'C + AB'C' is equivalent to

The addition of 4 bit, 2’s complement binary numbers 1101 and 0100 results in

The addition of 4 bit, 2’s Given, √(224)ᵣ = (13)ᵣ, the value of radix r iscomplement binary numbers 1101 and 0100 results in

Let A = 11111010 and B = 00001010 be two 8 bit 2’s complement numbers. Their product in 2’s complement is

Identify the logic function performed by the circuit

Read the following sentence and choose one underlined word or phrase that would not be appropriate in standard English. One of "the chair’s legs" was broken and the upholstery "needed" mending.