Computer Science – Mathematical Foundation of Computer

A class of 30 students occupy a classroom containing 5 rows of seats, with 8 seats in each row. IF the students seat themselves at random, the probability that the sixth seat in the fifth row will be empty is

The probability that a number selected at random between 100 and 999 (both inclusive) will not contain the digit 7 is

0.152525252….. us sane as

A class is composed of 2 brothers and 6 other boys. In how many ways can all the boys be seated at a round table so that the two brothers are not seated together?

The

n^th

order difference of a polynomial of degree n is

Each coefficient in the equation

ax²

+ bx + c = 0 is determined by throwing an ordinary die.
The probability that the equation will have real roots is

The sum of all numbers greater than 10,000 formed by using the digits 0, 2, 4, 6, 8, no digit being repeated in any number is

For a game in which 2 partners oppose 2 other partners, six men are available. If every possible pair must play against every other pair, the number of games to be played is

Let the elements g, h belong to a group G. If O(h) is 2, then O

(ghg-1^-1)

is

At any time, the total number of persons on earth who have shaken hands an odd number of times has to be

Which of the following are irrational number?

The function f(x) = 1 x/(x+1) |

The domain of the function log (log sin (x)) is

The system of equation

x + 2y + 3z = 4

x +  λy + 2z = 3

x + 4y + µz = 3

has infinite number of solutions if

Let R be a symmetric and transitive relation on a set A. Then

The number of elements in the power set of the set
{{{}}, 1, {2, {2,3}} is

If 4(log₉3) + (9(log₂4) = 10(log_x81), then x is 

The length of the longest pole that can be made inside a hall of length 18m, breadth 6m, and height 4.5m is

Six x’s have to be placed in the squares in the adjacent figure, such that each row contains at least one x. This can be done in

Out of 100 students, 10 students used to drink milk(M), coffee(C) and tea(T): 20 M and C: 30 C and T:25 M and T: 12 M only: 5 C only and 8 T only. The number of students who did not drink any of these is

Given the relation R = {(1,2). (2.3)}. The minimum number of ordered pairs that must be added to this set so that the enlarged relation is reflexive, symmetric and transitive is

A box contains 2 black, 4 white and 3 red balls. One ball is drawn at random from the box and kept aside. From the remaining balls in the box, another ball is drawn at random and kept beside the first. This process is repeated till all the balls are drawn from the box. The probability that the balls drawn are in the sequence 2 black, 4 white, and 3 red is

The range of the function f(X) =

x²

/ (1+

x²

) is

In calculating the mean and variance of 10 readings, a student wrongly used 52 instead of the correct figure 25. If the mean be obtained was 45, then the correct mean is

Refer Qn. 24. If the variance he obtained was 16, then the correct variance is

If ⁿC_r-1 = 36: ⁿC_r = 84 and ⁿC_{r + 1} =126, then the value of ‘r’ is

In the interval [0,∏], the equation x=cos(x) has

Ten different letters are given.Five letter words are formed from these given letters. The number of words having at least repeated is

The value of the expression

⁴⁷C₄ + ⁵∑_j=1 ^(52-j)C₃ is equal to

The rank of the following (n+1)x(n+1) matrix, where a is a real number is

[1 a a² . . . .  a^n]

[1 a a² . . . . a^n]

[. .  .]

[. .  .]

[1 a a² . . . . a^n]

Let ‘S’ be the standard deviation of ‘n’ numbers. If each of the ‘n’ numbers is multiplied by a constant C, then the new standard deviation will be

Let A be a finite set of Size ‘n’. The number of elements in the power set of A x A is

Probability of an event A happening is 0.4. Probability that in 3 independent trials. event A happens at least once is

If x, y are two real numbers such that x > 0 and xy = 1, then x + y can’t be less than

Let f(x+y) = f(x) + f(y), for all x,y. If f(x) is continuous at x = 0, then

Let f(x+y)=f(x) + f(y), for all x, y. If f(5) = 2 and f'(0) =3, then f'(5) is equal to

In numerical methods, accuracy refers to the

Suppose A₁ A₂, . . . A₃₀ are 30 sets, each with 5 elements, and B_{1 ,}B₂ . . . B_n are ‘n’ sets. each with 3 elements.

Let ³⁰U_i=2 A_i = ⁿU_j=1 B_j = S.

Each element of S, belongs to exactly 10 of the A_i ‘s and to exactly 9 of the B’S. then ‘n’ is

Which of the following remarks about an ill-conditioned system of equations are true?

If the cube roots of unity are 1, Co, Co², then the roots of the equation (x -1)³ + 8 = 0, are

f(x) and g(x) are two functions differentiable in [0,1] such that f(0) = 2; g (0) = 0;f(1) = 6; and g(1) = 2. Then there must exist a constant C in

Let f be a one-to-one function with domain {x,y,z} and range {1, 2, 3}. It is given that exactly one of the following statements is true and the remaining 2 are false:
f(x) = 1
f(y) ‡ 1

f(z) ‡ 2

Then f^-1 (1) equals

Let f be a twice differentiable function such that

f”(x) = -f(x) and f'(x) = g(x). Let h(x) = (f(x))² + (g(x))². If h(5) = 11, the h(10) is

iⁱ, where i is √-1, is

If p, q, r are three real numbers, then

The number of 1’s in the binary representation of (3 x 4096 + 15 x 256 + 5 x 16 + 3) is

A determinant is chosen at random from the set of all determinants of order 2 with each element either 0 or 1 only. the probability that the value of the chosen determinant is positive is

The number of permutations of ‘n’ different things taken not more than ‘r’ at a time, with repetitions being allowed, is

A relation R is defined in N x N, such that (a,b) R (c,d)iff a + d = b + c. The relation R is

If log₅ 10 = log₇ x(log_nm), then the values of x, m, n are

If √5 + √7 + i, is one of the roots of the equation f(x) = 0 with national coefficients, then the degree of the given equation can’t be less than

Consider the equation x⁷ -2x⁵ + 7x⁴ + x³ – 9 = 0. The numbe of imaginary roots will be at least

If f(a) and f(b) are of the same sign, then the equation f(x) = 0

The equation x⁵ + x³ – 8x – 5 = 0 has

Any polynomial of even degree in which the last term is negative and the coefficient of the highest power is positive, has at least

When the polynomial f(x) is divided by (x-a) (x-β), a≠β then the remainder is given by

Log 0 is

If a₁, . . . a_n are the roots of the equation xⁿ + nax – b = 0 then (a₁ – a₂)(a₁ – a₃) . . . (a₁ – a) equals

The set of all natural numbers is not closed with respect to

If |a – b\ < n and \b - c| < m, then |a - c| is

The domain of the function 1/ √(1-x)(x-2) is

A and B play a coin tossing game. They toss a coin alternately. The first one to get a head wins. If A starts, the probability of A winning is

The number of trailing zeroes in 200!(i.e., factorial of 200) is

The determinant of a matrix has 720 terms (in the unsimplified form), The order of the matrix is

The error in using Simpson’s rule is of the order

The domain of the function 1/√|x|-x is

A bag contains 10 white balls and 15 black balls. Two balls are drawn in succession. The probability that one of them is black and the other white is

The iteration formula to find the square root of a positive real number b, using the Newton Rephson method is

If |x – 1| + |x – 2| + |x – 3| ≥ 6, then

The number of real roots of the equation |x|² – 3|x| + 2 = 0 is

-20√-√20-√. . . equals

Two events A and B have probabilities 0.25 and 0.5 respectively. The probability that both A and B occur simultaneously is 0.14. Then the probability that neither A nor B occurs is

A function f(x) differentiable in the interval 0 ≤ x ≤ 5, is such that f(0) = 4 and f(5) = 1
If g(x) = f(x) / (x + 1), then there exists some constant C, 0 < C < 5 such that g'(C) equals

Let A and B be sets with cardinalities ‘m’ and ‘n’ respectively. The number of possible one to one mappings (injections) from A to B, when m < n, is

Choose the correct option,
Let A = {1. 2, 3} is equal to

Let A = {1. {2}, 3}
Choose the correct option.

In the set of integers, a relation R is defined as aRb, if and only if b = |a|. This realion is

Let S = {1, 2, 3, 4} A relation R defined in S as, R = {(1, 2), (4, 3), (2, 2), (2, 1), (3, 1)} is

Let A = {(1, 2, 3}. Which of the following relations are functions (mappings)?

Consider the mapping f:x → Y. f is a bijection if and only if

For a function to be invertible, it has to be

The advantages of partial pivoting in the solution of a system of equations are

Choose the correct statements.

Choose the correct statements.

In any undirected graph, the sum of degrees of all the nodes

(PVQ) Λ (P→R) Λ (Q→S) is equivalent to

Which of the following are tautologies?

Identify the valid conclusion from the premises P V Q, Q → R, P → M, ˜M

T is a graph with ‘n’ vertices. If T is connected and has exactly n-1 edges, then

If one has to obtain the roots of x² – 2x  + log 2 = 0 to four decimal places, log 2 should be given to the accuracy of approximately

Choose the incorrect statement(s).

M is a square matrix of order ‘n’ and its determinant value is 5. If all the elements of M are multiplied by 2, its determinant value becomes 40. The value of ‘n’ is

in a computer an n-digit integer a_n a_n – 1 . . . a₁ is represented as a_n a_n – 1 . . . a_{r + 1} 00 . . .0. The error e is

1 – x²/2! + x⁴/4! – . . . . + (-1)ⁿ x²ⁿ/2n! + . . . is the expansion of

In the previous question, for 5-digit accuracy, if |x|<∏/2, the number of terms in the series that should be considererd is

Which of the following methods gives the least error when e^x is integrated from 0 to 0.4?

Which of the following laws doesn’t hold good in finite precision floating point arithmetic?

Surplus variables are usually introduced in an LPP model

In an LPP model in its standard form, three of the constraints are

x₁ + x₂ ≤ 2

2x₁ + 2x₂ ≤ 3

3x₁ + 3x₂ ≤ 8

Removal of which of the constraints will not affect the optimality?

An LPP having 2 optimal solutions must have

The number of iterations taken by simplex method for solving an LPP in its standard form with ‘m’ equations and ‘n’ unknowns (m < n) can't exceed

In the solution of an LPP using simplex method, the curreat cost of the objective function must

If the cost of the objective function (of an LPP in its standard form) which corresponds to one of the corners of the convex region bound by the constraints, is greater than the cost corresponding to all its adjacent corners, then

Revised simplex method

The dual simplex method starts with a

Which of the following simplex based techniques are ideal for sensitivity analysis?

Choose the correct statements.

Choose the correct statement(s)

Changing the right hand side of the constraints and the coefficient of the cost function

Let A be the set of all non singular matrices over real numbers and let * be the matrix multiplication operator. then

Newton-Raphson method

In the bisection method for finding the roots of an equation, the approximate relative error is always

Trapezoidal rule gives the exact solution when the curve is

If a function y’ = f(x) has an inverse function, then f(x) can’t be

For what value of c, will the vector i + cj be orthogonal to 2i – j?

The solution of the differential equation y” + 3y’ + 2y = 0, is of the form

If the proposition P → Q is true, then the truth value of the proposition P V (P → Q), is

The number of the divisors of 600 (including 1 and 600) is

The determinant value of the matrix (1 2 3)
(4 5 6) is
(5 7 9)

Which of the following elementary operations may affect the rank of a matrix?

Which of the following will not form an abelian group?

A group has 11 elements. The number of proper Sub-groups it can have is

Let A and B be two n X n real symmetric matrices. Then

Backward Euler method for solving the differential equation dy/dx =f(x,y), is specified by

The rank of the matrix
[0 0 -3]
[9 3 5] is
[3 1  1]

(G, *) is an abelian group, Then

In a compact single dimensional array representation for lover triangular matrices (i.e. all the elements above the diagonal are zero). of size n X n, non-zero elements (i.e. elements of the lower triangle) of each row are stored one after the other, starting from the first row, The index of the (i,j)th element of the lower triangular matrix in this new representation is

The number of sub-strings (of all lengths) that can be formed from a character string of length n is

In the set of natural numbers, the binary operators that are not associative and not commutative are

A relation R is defined as xRy, if x ≠ y, This relation R is

The number of subsets of {1,2, . . .,n} of odd cardinality is

The probability of an event B occurring is P. The probability that events A and B occur together is Q. The probability that A occurs, without B occurring, is R.Then the probability of A occurring is

Let A, B, C be independent events with probabilities o.8, 0.5, 0.3. The probability of occurrence of at least one of these three is

The subset of a countable set

Every element of some ring (R, +,*) is such that a*a=a. This ring

For the M/G/1 queuing system, the arrival pattern and service time follows

Consider the set {1, 2, 3, 4, 6, 8, 12, 24}, together with the two binary operations LCM (Least Common Multiple) and GCD (Greatest Common Divisor).Which of the following does this algebraic structure represent?

The set {1, 2, 3, 4, 6, 8, 12, 24}, together with LCM as the binary operation is not a group because

The set {1, 2, 3, 4, 6, 8, 12, 24}, together with GCD as the binary operation is not a group because

The following set
(a) Q(x) → P(x) V˜R(a) (b) R(a) V˜Q(a)
(c) Q(a) (d) ˜P(y)
Where x and y are universally quantified variables, a is a constant and P, Q, R are monodic Predicates, is

Let x and y be sets with cardinalities m and n receptively. If the number of possible functions that can be defined with domain X and co-domain Y is exactly 10, then

Let F: R² → R² be the mapping defined by F(x,y) = (x/3,y/4). What will be the image of X²/9 + y²/16 = 1 under F?

A function g is defined as g(x) = f(x)[f(x)+f(x)]. Which of the following remarks about the function g is right?

What is the total number of equivalent relations that can be defined on the set {1, 2, 3}?

Cube roots of unity form a cyclic group under multiplication. For this group,

The value of lim_x→0 x log x is

If x ε [0, 1], and f(x) and g(x) are defined as f(x) = sin (cos(x∏/4)) and g(x) =cos (sin (x∏/4)), then

The function f(x) is continuous in [0, 1] such that f(0) = -1, f(1/2) = 1 and f(1) = -1, We can conclude that

The sum of the infinite series ∑kx^k, Where -1 < x < 1, is

Which of the following is not a linear transformation?

If the determinant of an n X n matrix A is zero, then

A is 2 X 2 matrix with eigen values 2 and -3. The eigen values of the matrix A²

Among any n + 1 distinct positive integers less than or equal to 2n, we can always find

Let X₁ and X₂ be any two unit vectors in R³ The angle between the two planes X₁ . X = c and X₂ . X = 2c, where c is a constant is given by

If (x₁, x₂, x₃) x (1, 3, 1) = (2, 1, 6), where x denotes the vector product, then (x₁, x₂, x₃) is given by

Which of the following is a cube root of the complex number -27i?

Suppose a system has been evolved by extraterrestrial creatures having only 3 fingers. They use the figures 0, 1, 2 with 2 > 1 > 0. What will be the binary equivalent of 222 in this system?

If you want to retain the first 4 bits of given string of 8 bits and complement the last 4 bits then the correct mask and the operation should be

Which of the following logical operation almost resembles an arithmetic multiplication operation?

To change lower case to upper case letters in ASCII. the correct mask and operation should be (ASCII value of character A is 65 and character a is 97)

Consider the nested for loop
for I1 = 1 to N
for I2 = 1 to I1
for I3 = 1 to I2
.
.
for Ik = 1 to I (k-1)
PRINT I1, I2, I3, . . . Ik
How many times is the PRINT statements executed ?

Which of the following remark(s) will be true for any x?

Which of the following, lists f(x),g(x),x,x-1 and x+1 in a non-decreasing sequence?