Computer

Combinatorics Questions with Answers for Computer Science

Question 1.
(a) Solve the recurrence equations:
T(n) = T(n – 1) + n
T(1) = 1
(b) What is the generating function G(z) for the sequence of Fibonacci numbers?

Answer/Explanation

Explanation:

(a) T(n) = T(n – 1) + n

T₂ – T_{n – 1} n
For Homogeneous solution:
T_n -T_{n – 1} = 0
t- 1 = 0
t = 1
Therefore, homogenous solution is
T_n = C(1)ⁿ = C
For Particular solution:
Let particular solution be (d₀ + d₁n)n
⇒ (d₀ + d₁n)n – (d₀ + d₁n – 1))(n – 1) = n
⇒ d₀n + d₁n² – d₀n + d₀ – d₁(n – 1)² = n
⇒ d₀n + d₁n² – d₀n + d₀ – d₁(n²– 2n + 1) = n
⇒ d₀ + 2d₁n – d₁ = n
d₀ – d₁ = 0 and 2d₁ = 1
⇒ d₀ = d₁ and d₁ = \(\frac{1}{2}\)
∴ d₀ = d₁ = \(\frac{1}{2}\)
So particular solution is,
\(\left(\frac{1}{2}+\frac{1}{2} n\right)\) × n = \(\frac{n(n+1)}{2}\) = \(\frac{n^{2}+n}{2}\)
So complete solution is,
T(n) C +\(\frac{1}{2}\)n{n +1)
Given, T( 1) = 1
1 = C + \(\frac{1}{2}\) × 1 × (1 + 1)
1 = C+ 1
C= 0
Therefore complete solution of the recurrence relation is T(n) = \(\frac{n(n+1)}{2}\).

(b) The Fibonacci numbers are defined as a₀ = 1, a₁ = 1
a_r = a_{r – 1} + a_{r – 2} r ≥ 2

⇒ A(x) – a₀ – a₁ x = (A(x) – a₀) + x²A(x)
Since, a₀ = 1 and a₁ = 1
⇒ A(x) – 1 – x = x(A(x) – 1) + x²A(x)
⇒ A(x) = \(\frac{1}{1-x-x^{2}}\)

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Question 2.
Solve the recurrence equations:
T(n) = T\(\left(\frac{\mathrm{n}}{2}\right)\) + 1
T(1) = 1

Answer/Explanation

Explanation:
T(n) = T\(\left(\frac{\mathrm{n}}{2}\right)\) + 1
⇒ T(n) – T\(\left(\frac{\mathrm{n}}{2}\right)\) = 1
Let, n = 2^k.
⇒ T(2^k) – T\(\left(\frac{2^{\mathrm{k}}}{2}\right)\) = 1
⇒ T(2^k) – T(2^{k – 1} = 1
Let, T(2^k) – x_k
x_k – x_{k – 1} = 1
For Homogeneous solution:
x_k – x_{k – 1} = 0
t – 1 = 0
t = 1
So homogeneous solution is: x_k = C(1)^k = C
For Particular solution:
Let particular solution be d₁k
d₁k – d₁(k- 1) = 1
⇒ d₁ = 1
Particular solution is k
∴ Complete solution is
x_k = C + k
T(2^k) = C + k
T(n) = C + log₂n
Given, T(1) = 1
⇒ C= 1
∴ Complete solution is: T(n) = log₂n + 1

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Question 3.
How many substrings can be formed from a character string of length n?

Answer/Explanation

Explanation:
Let the string be of length 4 : abcd
Number of substrings of length 0=1 (only ε )
Number of substrings of length 1 = 4
a, b, c, d
Number of substrings of length 2 = 3
ab, bc, cd
Number of substrings of length 3 = 2
abc, bcd
Number of substrings of length 4=1
abcd
∴ Total number of substrings = 1 +(4+ 3 + 2 + 1)
= 1 + (Sum of 4 natural number)
= 1 + \(\frac{4 \times(4+1)}{2}\)
= 11
Therefore, total number of substrings (maximum) that can be formed from a character string of
length 1 + \(\frac{n(n+1)}{2}\).

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Question 4.
The number of binary strings of n zeros and k ones that no two ones are adjacent is
(a) ⁿ⁺¹C_K
(b) ⁿC_K
(c) ⁿC_K+1
(d) None of these

Answer/Explanation

Answer: (a) ⁿ⁺¹C_K
Explanation:
First arranging all n zeros in a row. There is only 1 way for arranging n zeros in a row. By arranging n zeros in a row, we get (n + 1) positions to place ones.
So number of ways arranging k ones in (n + 1) positions = ^{n + 1}C_k
∴ Required number of binary strings of n zeroes and k ones that no two ones are adjacent = 1 × ^{n + 1}C_k = ^{n + 1}C_k.

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Question 5.
The number of substrings (of all lengths inclusive) that can be formed from a character string of length n is
(a) n
(b) n²
(c) \(\frac{n(n-1)}{2}\)
(d) \(\frac{n(n+1)}{2}\)

Answer/Explanation

Answer: (d) \(\frac{n(n+1)}{2}\)
Explanation:
For a string of length n:
The number of substrings of length 1 = n
The number of substrings of length 2 = n – 1
The number of substrings of length 3 = n – 2 and so on…
The number of substrings of length n is 1 So total number of substrings ; = n + (n-1) + …+1
= Sum of n natural numbers.
= \(\frac{\mathrm{n}(\mathrm{n}+1)}{2}\)

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Question 6.
In a room containing 28 people, there are 18 people who speak English, 15 people who speak Hindi and 22 people who speak Kannada. 9 persons speak both English and Hindi, 11 people speak both Hindi and Kannada whereas 13 persons speak both Kannada and English. How many people speak all three languages?
(a) 9
(b) 8
(c) 7
(d) 6

Answer/Explanation

Answer: (d) 6
Explanation:
E : Persons speaks English
H : Persons speaks Hindi
K : Persons speaks Kannada
Assume that everyone in the room speaks at least one of the languages.
Given, n(E ∪ H ∪ K) = 28
n(E) = 18
n(H) =15
n(K) = 22
n(E ∩ H) = 9
n(H ∩ K) = 11
n(K ∩ E) = 13
n(K ∩ E ∩ H) = ?
By principle of inclusion and exclusion
n(E ∪ H ∪ K) = n(E) + n(H) + n(K) – n(E ∩ H) – n(H ∩ E) – n(K ∩ E) + n(K ∩ E ∩ H)
28 = 18 + 15 + 22 – 9 – 11 – 13 + n(K ∩ E ∩ H) n(k ∩ E ∩ H) = 28 – 55 + 33 = 61 – 55 = 6
Therefore, the number of people who speak all three languages are 6.

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Question 7.
Solve the following recurrence relation
x_n = 2x_n-1, n>1
x₁ = 2

Answer/Explanation

Explanation:
x_n = 2x_{n – 1} – 1
For Homogeneous solution:
x_n = 2x_{n – 1} = 0
t – 2 = 0
t= 2
So homogeneous solution is
x_n = C(2)ⁿ
For Particular solution:
Let particular solution be d_o.
d₀ = 2d₀ – 1
d₀ = 1
So particular solution is 1
∴ Complete solution = Homogeneous solution + Particular solution
Complete solution = C(2)ⁿ + 1
⇒ x_n = C(2)ⁿ + 1
Given initial condition is x₁ = 2
2 = C(2)¹ + 1
1 = 2C
C = \(\frac { 1 }{ 2 }\)
x_n = \(\frac { 1 }{ 2 }\)(2)ⁿ + 1
∴ x_n = 2^{n – 1} + 1

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Question 8.
Two girls have picked 10 roses, 15 sunflowers and 14 daffodils. What is the number of ways they can divide the flowers among themselves?
(a) 1638
(b) 2100
(c) 2640
(d) None of these

Answer/Explanation

Answer: (c) 2640
Explanation:
Number of ways for distributing r similar things among n different things = ^{(n – 1 + r)}C_r
The number of ways for distributing 10 roses among the two girls = ^{(2 – 1 + 10)}C₁₀ =11.
Similarly number of ways for distributing 15 sunflowers among two girls = ^{(2 – 1 + 15)}C₁₅
= ¹⁶C₁₅ = ¹⁶C₁ = 16
Number of ways for distributing 14 daffodils among the two girls = ^{(2 – 1 + 14)}C₁₄ = ¹⁵C₁₄ = ¹⁵C₁ =15
∴ Total number of ways = 11 × 16 × 15 = 2640

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Question 9.
The minimum number of cards to be dealt from an arbitrarily shuffled deck of 52 cards to guarantee that three cards are from some same suit is
(a) 3
(b) 8
(c) 9
(d) 12

Answer/Explanation

Answer: (c) 9
Explanation:
Let the number of cards to be dealt from an arbitrarily shuffled deck of 52 cards be n.
Number of suits = 4
Required number of cards from the same suit = 3.
So by the pigeonhole principle

So the minimum number of cards to be dealt from an arbitrarily shuffled deck of 52 cards to guarantee that three cards are from the same suit is 9.

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Question 10.
How many 4 digit even numbers have all 4 digits distinct?
(a) 2240
(b) 2296
(c) 2620
(d) 4536

Answer/Explanation

Answer: (b) 2296
Explanation:
The digits are given to be distinct i.e. no repetition. 4 digit even numbers cannot start with 0 and must end with 0, 2, 4, 6 or 8.
Since there is a condition for 0 in starting as well as ending we will count the even numbers ending with 0 separately.
So the total number of 4 digits even number = 4 digit even numbers ending with zero + 4 digit even numbers ending with 2, 4, 6, 8.
4 digit even numbers ending with 0

4 digit even numbers ending with 2, 4, 6, 8

So the total number of 4 digit even numbers = 504+1792 = 2296.

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Question 11.
The solution to the recurrence equation T(2^k) = 3T(2^k-1) + 1, T(1) = 1 is
(a) 2^k
(b) \(\frac{\left(3^{\mathrm{k}+1}-1\right)}{2}\)
(c) 3^log2k
(d) 2^log3k

Answer/Explanation

Answer: (b) \(\frac{\left(3^{\mathrm{k}+1}-1\right)}{2}\)
Explanation:
T(2^k) = 3T(2^{k – 1}) + 1
Let, T(2^k) = x_n
⇒ x_n = 3x_{n – 1} + 1
⇒ x_n – 3x_{n – 1} = 1
So for Homogenous solution
x_n – 3x_{n – 1} = 0
n – 3 = 0
n = 3
Homogenous solution is
x_n = C₁(3)ⁿ
T(2^k) = C₁(3)^k
For Particular solution
Let d be the particular solution
d – 3d = 1
2d = -1
d = \(-\frac{1}{2}\)
Therefore, the complete solution is
T(2^k) = C₁(3)^k \(-\frac{1}{2}\)
Given T(1) = 1
1 = C₁(3)⁰ \(-\frac{1}{2}\)
1 = C₁ \(-\frac{1}{2}\)
C₁ = \(\frac { 3 }{ 2 }\)
So the complete solution is
T(2^k) = C₁ (3)^k \(-\frac{1}{2}\)
Given, T(1) = 1
1 = C₁(3)⁰ \(-\frac{1}{2}\)
1 = C₁ \(-\frac{1}{2}\)
C₁ = \(\frac { 3 }{ 2 }\)
So the complete solution is
T(2^k) = \(\frac { 3 }{ 2 }\)(3)^k \(-\frac{1}{2}\)
T(2^k) = \(\frac{3^{\mathrm{k}+1}-1}{2}\)

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Question 12.
The minimum number of colors required to color the vertices of a cycle with n nodes in such a way that no two adjacent nodes have the same color is
(a) 2
(b) 3
(c) 4
(d) n – 2\(\left[\frac{n}{2}\right\rfloor\)+2

Answer/Explanation

Answer: (d) n – 2\(\left[\frac{n}{2}\right\rfloor\)+2
Explanation:
The minimum number of colors required to color the vertices of a cycle with n nodes = 2, when n is even = 3, when n is odd
Therefore n – 2 \(\left\lfloor\frac{\mathrm{n}}{2}\right\rfloor\) +2 gives 2 when n is even. and 3 when n is odd.

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Question 13.
Let A be a sequence of 8 distinct integers sorted in ascending order. How many distinct pairs of sequences, B and C are there such that (i) each are sorted in ascending order, (ii) B has 5 and C has 3 elements, and (iii) the result of merging B and C gives A?
(a) 2
(b) 30
(c) 56
(d) 256

Answer/Explanation

Answer: (c) 56
Explanation:
This corresponds to an ordered partition of 8 elements into two groups, the first with 5 elements and second with 3 elements. The number of ways of doing this is p(8; 5, 3) = \(\frac{8 !}{5 ! 3 !}\) = 56

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Question 14.
n couples are invited to a party with the condition that every husband should be accompanied by his wife. However, a wife need not be accompanied by her husband. The number of different gatherings possible at the party is
(a) \(\left(\begin{array}{c}
2 n \\
n
\end{array}\right)\) * 2ⁿ
(b) 3ⁿ
(c) \(\frac{(2 n) !}{2^{n}}\)
(d) \(\left(\begin{array}{c}
2 n \\
n
\end{array}\right)\)

Answer/Explanation

Answer: (b) 3ⁿ
Explanation:
For each of the n couples invited to the party one of three things is possible:
1. Both husband and wife attend the party.
2. Wife only attends the party.
3. Neither husband nor wife attends the party. Since there are n such couples, a total number of possibilities = 3ⁿ.

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Question 15.
Mala has a colouring book in which each English letter is drawn two times. She wants to paint each of these 52 prints with one of k colours, such that the colour-pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of k that satisfies this requirement?
(a) 9
(b) 8
(c) 7
(d) 6

Answer/Explanation

Answer: (d) 6
Explanation:
The problem reduces to finding how many distinct ordered color pairs (C₁, C₂) are possible with k colors.
Since the first color C₁ can be any one of the k colors and the second color C₂ also can be any one of the k colors (both prints of a letter can be colored with the same color), the total no. of such order color pairs is equal to k × k = k₂.
Since each pair of letters must be colored with different color pairs, at least 26 color pairs are required to do this.
Therefore the requirement is k² ≥ 26.
The minimum value of k that satisfies this equation is k = 6.

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Question 16.
In how many ways can we distribute 5 distinct balls, B₁, B₂……., B₅ in 5 distinct cells, C₁ C₂, …, C₅ such that Ball B_i is not in cell C_i,∀_i = 1,2,…,5 and each cell contains exactly one ball?
(a) 44
(b) 96
(c) 120
(d) 3125

Answer/Explanation

Answer: (a) 44
Explanation:
We want every one of the 5 balls to be in the wrong box. This is nothing but the number of derangements of a set of 5 elements = D₅. i.e. we need to compute D₅

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Question 17.
Let n = p²q, where p and q are distinct prime numbers. How many numbers m satisfy 1 ≤ m ≤ n and gcd (m, n) = 1? Note that gcd (m, n) is the greatest common divisor of m and n.
(a) p(q – 1)
(b) pq
(c) (p² – 1)(q – 1)
(d) p(p – 1)(q – 1)

Answer/Explanation

Answer: (d) p(p – 1)(q – 1)
Explanation:
The number of numbers from 1 to n, which are relatively prime to n i.e., gcd (m, n) = 1, is given by the Euler Totient function Φ(n). If n is broken down into its prime factors as n = p₁^n₁ . p₂^n₂ ……
where p₁,p₂ etc. are distinct prime numbers, then

Φ(n) = Φ( p₁^n₁) Φ(p₂^n₂)
Here, n = p² q
So, Φ(n) = Φ(p²) × Φ(q)
Now, using the property Φ(p^k) = p^k – p^{k – 1}
Φ(p²) = p² – p¹ = p² – p and
Φ(q) = q¹ – q⁰ = q – 1
Substituting these in eq. (i), we get
Φ(n) = (p² – p) (q – 1)
= p(p – 1) (q – 1)

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Question 18.
What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs (a, b) and (c, d) in the chosen set such that a=c mod 3 and b = d mod 5
(a) 4
(b) 6
(c) 16
(d) 24

Answer/Explanation

Answer: (c) 16
Explanation:
The number of combinations of pairs (a mod 3, b mod 5) is 3 × 5 = 15 (since a mod 3 can be 0, 1, or 2) and b mod 5 can be 0, 1, 2, 3 or 4)
∴ If 16 different ordered pairs are chosen at least 2 of them must have (a mod 3, b mod 5) as same (basic pigeon hole principle).
Let such two pairs be (a, b) and (c, d) then
a mod 3 ≡ c mod 3 ⇒ a ≡ c mod 3, and b mod 5 ≡ d mod 5 ⇒ b ≡ d mod 5

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Question 19.
Let G(x) = 1/(1 – x)² = \(\sum_{x=1}^{∞}\)g(i)xⁱ where |x| < 1. what is g(i)?
(a) i
(b) i + 1
(c) 2i
(d) 2ⁱ

Answer/Explanation

Answer: (b) i + 1
Explanation:

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Question 20.
For each element in a set of size 2n, an unbiased coin is tossed. The 2n coin tossed are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is
(a) \(\left(\begin{array}{c}
2 n \\
\mathrm{n}
\end{array}\right) / 4^{\mathrm{n}}\)
(b) \(\left(\begin{array}{c}
2 n \\
n
\end{array}\right) / 2^{n}\)
(c) \(1 /\left(\begin{array}{c}
2 \mathrm{n} \\
\mathrm{n}
\end{array}\right)\)
(d) \(\frac { 1 }{ 2 }\)

Common Data for Q.21 & Q.22
Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at (i, j) then it can move to either (i + 1, j) or (i, j + 1).

Answer/Explanation

Answer:
(a) \(\left(\begin{array}{c}
2 n \\
\mathrm{n}
\end{array}\right) / 4^{\mathrm{n}}\)
Explanation:
The probability that exactly n elements are chosen = The probability of getting n heads out of 2n tosses
= ²ⁿC_n (1/2)ⁿ (1/2)^{2n – n}(Binomial formula)
= ²ⁿC_n (1/2)ⁿ (1/2)ⁿ
= \(\frac{{ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}}}{2^{2 \mathrm{n}}}\) = \(\frac{{ }^{2 n} \mathrm{C}_{\mathrm{n}}}{\left(2^{2}\right)^{\mathrm{n}}}\) = \(\frac{{ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}}}{4^{\mathrm{n}}}\)

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Question 21.
How many distinct paths are there for the robot to reach the point (10,10) starting from the initial position (0, 0)?
(a) \(\left(\begin{array}{l}
20 \\
10
\end{array}\right)\)
(b)2²⁰
(c)2¹⁰
(d) None of these

Answer/Explanation

Answer:
(a) \(\left(\begin{array}{l}
20 \\
10
\end{array}\right)\)
Explanation:
Consider the following diagram:

The robot can move only right or up as defined in the problem. Let us denote the right move by ‘R’ and the up move by ‘U’. Now to reach (3, 3) from (0, 0), the robot has to make exactly 3 ‘R’ moves and 3 ‘U’ moves in any order. Similarly to reach (10, 10) from (0, 0), the robot has to make 10 ‘R’ moves and 10 ‘U’ moves in any order. The number of ways this can be done is the same as the number of permutations of a word consisting of 10 ‘R’s and 10 ‘U’s.
Applying formula of permutation with limited repetitions we get the answer as \(\frac{20 !}{10 ! 10 !}\) = ²⁰C₁₀.

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Question 22.
Suppose that the robot is not allowed to traverse the line segment from (4, 4) to (5, 4). With this constraint, how many distinct paths are there for the robot to reach (10,10) starting from (0,0)?
(a) 2⁹
(b) 2¹⁹
(c) \(\left(\begin{array}{l}
8 \\
4
\end{array}\right) \times\left(\begin{array}{c}
11 \\
5
\end{array}\right)\)
(d) \(\left(\begin{array}{l}
20 \\
10
\end{array}\right)-\left(\begin{array}{l}
8 \\
4
\end{array}\right) \times\left(\begin{array}{c}
11 \\
5
\end{array}\right)\)

Answer/Explanation

Answer:
Explanation:
The robot can reach (4, 4) from (0,0) in ⁸C₄ ways as argued in the previous problem. Now after reaching (4,4), the robot is not allowed to go to (5,4). Let us count how many paths are there from (0,0) to (10, 10) if the robot goes from (4, 4) to (5, 4) and then we can subtract this from a total number of ways to get the answer.
Now there are 8C₄ ways for robot to reach (4, 4) from (0, 0) and then robot takes the ‘U’ move from (4, 4) to (5, 4). Now from (5, 4) to (10, 10) the robot has to make 5 ‘U’ moves and 6 ‘R’ moves
in any order which can be done in \(\frac{11 !}{5 ! 6 !}\) ways = ¹¹C₅ ways.

∴ The number of ways robot can move from (0,0) (10, 10) via (4, 4) – (5, 4) move is
⁸C₄ × ¹¹C₅ = \(\left(\begin{array}{l}
8 \\
4
\end{array}\right)\) × \(\left(\begin{array}{c}
11 \\
5
\end{array}\right)\)

∴ Number of ways robot can move from (0,0) to (10, 10) without using (4, 4) to (5, 4) move is

\(\left(\begin{array}{l}
20 \\
10
\end{array}\right)\) – \(\left(\begin{array}{l}
8 \\
4
\end{array}\right)\) x \(\left(\begin{array}{l}
11 \\
5
\end{array}\right)\) ways.
Which is an option (d).

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Question 23.

where k is positive integer. Then
(a)P = Q – k
(b)P=Q + k
(c) P = Q
(d) P = Q + 2k

Answer/Explanation

Answer:
Common Data for Q.24 & Q.25
Let x_n denote the number of binary strings of length n that contain no consecutive 0s.
Explanation:
P = Σ i = 1 + 3 + 5 + 7 + … + (2k – 1)
1 ≤ i ≤ 2k, i is odd
Q= Σ i=2+4+ 6+…2k, 1 < i < 2k, 1 ≤ i ≤ 2k, i is even.
P is in A.P with a = 1, d = 2 and n = k
Q is in A.P with a = 2, d = 2 and n = k

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Question 24.
Which of the following recurrences does x_n satisfy?
(a) x_n = 2x_{n – 1}
(b) x_n = x_[n/2] + 1
(c) x_n = x_[n/2] + n
(d) x_n = x_{n – 1} + x_{n – 2}

Answer/Explanation

Answer: (d) x_n = x_{n – 1} + x_{n – 2}
Explanation:
The ✓ represents those strings with no consecutive 0’s.

Now, substituting n = 3 in all of the answers only choice (d) x_n = x_{n – 1} + x_{n – 2} satisfies the numbers obtained from the tree counting.

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Question 25.
The value of x₅ is
(a) 5
(b) 7
(c) 8
(d) 13

Answer/Explanation

Answer: (d) 13
Explanation:
x₁ = 2
x₂ = 3
⇒ x₃ = x₁ + x₂ = 2 + 3 = 5
⇒ x₄ = ₂ + x₃ = 3 + 5 = 8
⇒ x₅ = ₃ + x₄ = 5 + 8 = 13
So option (d) is correct.

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Question 26.
The exponent of 11 in the prime factorization of 300! is
(a) 27
(b) 28
(c) 29
(d) 30

Answer/Explanation

Answer: (c) 29
Explanation:
In 300!, distinct numbers divisible by 11 = \(\left[\frac{300}{11}\right\rfloor\) = 27.
Since these 27 numbers contain, the same number which can be further divisible by 11 i.e.
\(\left[\frac{27}{11}\right\rfloor\) = 2(121 and 242)
So, total number of 11’s = 27 + 2 = 29 Hence exponent of 11 is 29.

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Question 27.
In how many ways can b blue balls and r red balls be distributed in re distinct boxes?
(a) \(\frac{(n+b-1) !(n+r-1) !}{(n-1) ! b !(n-1) ! r !}\)
(b) \(\frac{(n+(b+r)-1) !}{(n-1) !(n-1) !(b+r) !}\)
(c) \(\frac{n !}{b ! r !}\)
(d) (n + (b + r) -1)! /n!(b + r – 1)

Answer/Explanation

Answer: (a) \(\frac{(n+b-1) !(n+r-1) !}{(n-1) ! b !(n-1) ! r !}\)
Explanation:
A number of ways to distribute ‘b’ blue balls in n distinct boxes:

Since each box can contain 0 to n blue balls, so number of ways
= ^{( n + b – 1)}C_{(n – 1)}
=\(\frac{(n+b-1) !}{(n-1) ! \times b !}\)

Number of ways to distribute ‘r’ red balls in n distinct boxes:

Since each box can contain 0 to n red balls. So number of ways = ^{( n + r – 1)}C_{(n – 1)}

=\(\frac{(n+r-1) !}{(n-1) ! \times r !}\)

Since both are independent of each other So total ways = Number of ways to distribute ‘b’ blue balls x Number of ways to distribute ‘r’ red balls

= \(\frac{(n+b-1) !(n+r-1) !}{(n-1) ! \times b !(n-1) ! \times r !}\)

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Question 28.
A pennant is a sequence of numbers, each number being 1 or 2. An n-pennant is a sequence of numbers with a sum equal to n. For example, (1,1,2) is a 4-pennants. The set of all possible 1-pennant is {(1)}, the set of all possible 2-pennants is {(2), (1,1)}and the set of all 3-pennants is {(2,1), (1,1,1), (1,2)}. Note that the pennant (1,2) is not the same as the pennant (2,1). The number of 10-pennants is ______.

Answer/Explanation

Explanation:
In a 10-pennant let there we x₁ ones and x₂ twos. So we need to find all the solutions of
x₁ + 2x₂ = 10

put, x₁ = 0 ⇒ x₂ = \(\frac { 10 }{ 2 }\) = 5

So, (0, 5) is a solution i.e. a 10-pennant could have 0 ones and 5 twos. The number of ordered permutations of 0 ones and 5 twos = 5!/ 5! = 1

Now x₁ cannot be 1 since in that case x₂ = \(\frac { 10 }{ 2 }\) = 4.5
(is not an integer).

put, x₁ = 2 ⇒ x₂ = \(\frac { 8 }{ 2 }\) = 4

So, (2, 4) is a solution i.e. a 10-pennant could have 2 ones and 4 twos. The number of ordered permutations of 2 ones and 4 twos

\(\frac{6 !}{2 ! 4 !}\) = 15

Similarly (4, 3), (6, 2), (8, 1) and (10,0) are the other four solutions and the number of pennants for each is respectively

\(\frac{7 !}{4 ! 3 !}\) = 35, \(\frac{8 !}{6 ! 2 !}\) =28, \(\frac{9 !}{8 ! 1 !}\) = 9, \(\frac{10 !}{10 ! 10 !}\) = 1 = 89

So the total number of 10-pennants = 1 + 15 + 35 + 28 + 9 + 1 = 89

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Question 29.
Each of the nine words in the sentence
“The quick brown fox jumps over the lazy dog”
is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is _____. (The answer should be rounded to one decimal place).

Answer/Explanation

Explanation:
“The quick brown fox jumps over the lazy dog” (3) (5) (5) (3) (5) (4) (3) (4) (3)
Now let x be the number of letters in the word that is randomly picked.
Now, we make a probability distribution table for x

From this table we can easily find the expected value of x.

E(x) = Σx p(x)
= 3 × \(\frac { 4 }{ 9 }\) + 4 × \(\frac { 2 }{ 9 }\) + 5 × \(\frac { 3 }{ 9 }\) = \(\frac { 35 }{ 9 }\) = 3.88 = 3.9
(after rounding to one decimal accuracy).

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Question 30.
The number of distinct positive integral factors of 2014 is _______.

Answer/Explanation

Explanation:
Factorizing 2014 in primes by successively dividing by primes, we get
2014 = 2¹ × 19¹ × 53¹
Now We use the formula:
If n = p₁^n₁ . p₂^n₂…..p_r^n_r is the prime factorization of n, then the number of distit factors of n is given by (n₁ + 1) × (n₂ + 1) …….. (n_r + 1).
Now since 2014 = 2¹ × 19¹ × 53¹, the number of distinct factors of 2014 = (1 + 1) (1 + 1) (1 + 1) = 2 × 2 × 2 = 8

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Question 31.
\(\sum_{x=1}^{99} \frac{1}{x(x+1)}\) = _______.

Answer/Explanation

Explanation:

= 1 – \(\frac { 1 }{ 100 }\) (all the terms in above series except the first and last terms cancels out)
= \(\frac { 99 }{ 100 }\) = 0.99

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Question 32.
Let a_n represent the number of bit strings of length n containing two consecutive 1s. What is the recurrence relation for a_n?
(a) a_{n -2} + a_{n – 1} + 2^{n – 2}
(b) a_{n -2} + 2a_{n – 1} + 2^{n – 2}
(c) 2a_{n -2} + a_{n – 1} + 2^{n – 2}
(d) 2a_{n -2} + 2a_{n – 1} + 2^{n – 2}

Answer/Explanation

Answer: (a) a_{n -2} + a_{n – 1} + 2^{n – 2}
Explanation:
a₁ = 0 [n0 strings of length 1 contain two consecutive 1’s]
a₂ = 1 [∴ strings = 11]
a₃ = 3 [∴ strings are : 011, 110, 111]
a₄ = 8 [∴ strings are : 0011, 0110, 0111, 1011, 1100, 1101, 1110, 1111]

Option (a):
a_n = a_{n – 2} + a_{n – 1} + 2^{n – 2}
⇒ a₄ = a_{4 – 2} + a_{4 – 1} + 2^{4 – 2}
= a₂ + a₃ + 2²
= 1 + 3 + 4 = 8 which is true.

Option (b):
a_n = a_{n – 2} + 2a_{n – 1} + 2^{n – 2}
⇒ a₄ = a₂ + 2a₃ + 2²
= 1 + 2 × 3 + 4 = 11 which is False.

Option (c):
a_n = 2a_{n – 2} + a_{n – 1} + 2^{n – 2}
⇒ a₄ = 2a₂ + a₃ + 2²
= 2 × 1 + 3 + 4 = 9 which is false.

Option (d):
a_n = 2a_{n – 2} + 2a_{n – 1} + 2^{n – 2}
⇒ a₄ = 2a₂ + 2a₃ + 2²
= 2 × 1 + 2 × 3 + 4 = 12 which is false.

∴ Option (a) : a_n = a_{n – 2} + a_{n – 1} + 2^{n – 2} is correct.

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Question 33.
The number of divisors of 2100 is _______.

Answer/Explanation

Explanation:
Dividing 2100 successively by prime numbers, we get that, the prime factors of 2100 are 2 × 2 × 3 × 5 × 5 × 7
⇒ 2² × 3¹ × 5² × 7¹
If N = p^k₁ × q^k₂ × r^k₃…… × y^k_n
then the number of factors of N are (k₁+1) (k₂+1) … (k_n+1)
∴ For 2100 we have (2 + 1) (1 + 1) (2 + 1) (1 + 1) = 36 factors.

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Question 34 .
The number of 4 digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set {1, 2, 3} is ____.

Answer/Explanation

Explanation:
To satisfy the non-decreasing order condition allow 1, 2, 3 after 1, allow only 2, 3 after 2 and allow only 3 after 3. The following tree diagram gives all the allowed numbers satisfying the given condition.

All the tick-marked numbers satisfy the non-decreasing order condition.
⇒ A number of 4-digit numbers = Number of tick marks = 15.

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Question 35.
Let a_n be the number of re-bit strings that do NOT contain two consecutive l’s. Which one of the following is the recurrence relation for a_n?
(a) a_n = a_{n – 1} + 2a_{n – 2}
(b) a_n = a_{n – 1} + a_{n – 2}
(c) a_n = 2a_{n – 1} + a_{n – 2}
(d) a_n = 2a_{n – 1} + 2a_{n – 2}

Answer/Explanation

Answer: (d) a_n = 2a_{n – 1} + 2a_{n – 2}
Explanation:
Let a_n be the number of n-bit strings that do not contain two consecutive 1’s. we wish to develop a recurrence relation for a_n.
Consider 1 bit strings 0, 1 So, a₁ = 2
Consider 2 bit strings 00, 01, 10, 11
Out of minimum only 00, 01, 10 do not contain two consecutive 1’s.
So, a₂ = 3

Out of minimum six strings only 000, 001, 010, 100 and 101 five strings satisfy do not contain two consecutive 1’s. So, a₃ = 5. Three numbers a₁, a₂, a₃ satisfy clearly only (b) a_n = a_{n – 1} + a_{n – 2} is correct.

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Question 36.
The coefficient of x¹² in (x³ + x⁴ + x⁵ + x⁶ +…)³ is _______.

Answer/Explanation

Explanation:
We wish to find coefficient of x¹² in (x³ + x⁴ + x⁵ +…)³

Now to make x¹² we need to put r = 3 So coefficient of x¹² is ^{3 + 2}C₃ = ⁵C₃ = ⁵C₂ = 10

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Question 37.
Consider the recurrence relation a₁ = 8, a_n = 6n² + 2n + a_{n – 1} Let a₉₉ = K × 10⁴. The value of K is ______.

Answer/Explanation

Explanation:
Given, a_n = 6n² + 2 n + a_{n – 1} and a₁ = 8
We wish to find a₉₉
Now,
a₂ = 6 × 2₂ + 2 × 2 + a₁
a₃ = 6 × 3₂ + 2 × 3 + a₂
= 6 × 3₂ + 2 × 3 + 6 × 2₂ + 2 × 2 + a₁………
a₉₉ = 6 × 99₂ + 2 × 99 + 6 × 98² + 2 + 98 ….
…. + 6 × 2² + 2 × 2 + a₁
Since,
a₁ = 8
a₉₉ = 6 × 99² + 2 × 99 + 6 × 98² + 2 × 98…….
……+ 6 × 22 + 2 × 2 + 8
= 6 × 99² + 2 × 99 + 6 × 98² + 2 × 98…
…6 × 2² + 2 × 2 + 6 × 1² + 2 × 1
= 6(1²+2²+3²… 99²)+ 2.(1 + 2 + 3…99)
= 6. \(\frac{(99(99+1)(2 \times 99+1))}{6}\) + 2 \(\left(\frac{99(99+1)}{2}\right)\)
= 99 × 100 × 199 + 99 × 100
= 100 × 99 (199 + 1) = 100 × 99 × 200
= 2 × 99 × 10⁴ = 198 × 10⁴
So if a₉₉ = K × 10⁴ then K = 198.

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Question 38.
If the ordinary generating function of a sequence \(\left\{a_{n}\right\}_{n=0}^{\infty} \text { is } \frac{1+z}{(1-z)^{3}}\) , then a₃ – a₀ is equal to _______.

Answer/Explanation

Explanation:
Given that generating function:

Now we read to find a₀ and ₃ which are nothing but the coefficient of x⁰ and x³ respectively.
a₀ = coefficient x⁰ = ^{(0 + 2)}C₂ = ²C₂ = 1
a₃ = coefficient x³ = ^{(3 + 2)}C₂ = ^{(2 + 2)}C₂
= ⁵C₂ + ⁴C₂ = 16
So, a₃ – a₀ = 16 – 1 = 15

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Question 39.
Which one of the following is a closed form expression for the generating function of the sequence {a_1n}, where a_n = 2n + 3 for all n = 0 1,2,…..?
(a) \(\frac{3}{(1-x)^{2}}\)
(b) \(\frac{3 x}{(1-x)^{2}}\)
(c) \(\frac{2-x}{(1-x)^{2}}\)
(d) \(\frac{3-x}{(1-x)^{2}}\)

Answer/Explanation

Answer: (d) \(\frac{3-x}{(1-x)^{2}}\)
Explanation:
Given, a_n = 2n + 3
Since generating function for 1 is \(\frac{1}{1-x}\) and n is \(\frac{x}{(1-x)^{2}}\), the generating function for a_n is
A(x) = \(\frac{2 x}{(1-x)^{2}}\) + \(\frac{3}{1-x}\)
= \(\frac{2 x+3(1-x)}{(1-x)^{2}}\) = \(\frac{3-x}{(1-x)^{2}}\)
which is option (d).

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Question 40.
Let U = {1,2, ….., n} and A = {(x , X), x ∈ X and X ⊆ U}. Consider the following two statements om |A|.

Which of the following is correct?
(a) Both I and II
(b) Neither I nor II
(c) Only II
(d) Only I

Answer/Explanation

Answer: (a) Both I and II
Explanation:
A – {(x, X), x ∈ X and X ⊆ U}
The number of k element subsets of a set U with n elements = \(\begin{aligned}
&n \\
&k
\end{aligned}\) = ⁿC_k
The number of possible ordered pairs (x, X) where x ∈ X is k. ⁿC_k for a given value of k from 1 to n.
So total number of ordered pairs in A

So II is correct, (Note that k = 0 is excluded since the empty set has no elements and cannot form an ordered pair such as (x, X)).
But since by the combinational identity

So I am also correct.
So both I and II are correct.

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Question 41.
The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is_________.

Answer/Explanation

Explanation:
Since both L’s are indistinguishable.
First L’s can be arranged in 3 positions 2, 3, or 5 in ³C₂ = 3 ways as follows:
_ L _ L _
or _ L _ _ L
or _ _ _ L L
Now the letters I, A, C can be deranged in 2 x 2! ways. Example in _ L _ L _. C cannot occupy 5^th position, so only 2 ways.
The remaining I and A can be arranged in the remaining 2 positions in 2! ways = 2 ways.
So answer is 3 × 2 × 2! = 12.

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Question 42.
There are 6 jobs with distinct difficulty levels, and 3 computers with distinct processing speeds. Each job is assigned to a computer such that:

• The fastest computer gets the toughest job and the slowest computer gets the easiest job.
• Every computer gets at least one job.

The number of ways in which this can be done is ______.

Answer/Explanation

Explanation:
Let computers be A, B, and C
The toughest job assigned to the fastest computer (Say, A) is 1 way.
The easiest job assigned to the shortest computer (Say, B) is 1 way.
The remaining 4 jobs are to be assigned to 3 computers so that computer C gets at least one job since A and B have already been assigned a job.
A number of ways 4 jobs assigned to 3 computers = 3⁴.
The number of ways 4 jobs assigned to 3 computers so that computer C does not get any job = 2⁴. Required number of ways = 3⁴ – 2⁴ = 81 – 16 = 65 ways

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1) Which of these is a standard interface for serial data transmission?

(a) ASCII
(b) RS232C
(c) 2
(d) Centronics

Answer

Answer: (b) RS232C

Explanation: The RS232C is a standard interface for serial data transmission that defines the protocol and physical interface for transmitting serial data fairly easily between associated appliances and computers.

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2) Which type of topology is best suited for large businesses which must carefully control and coordinate the operation of distributed branch outlets?

(a) Ring
(b) Local area
(c) Hierarchical
(d) Star

Answer

Answer: (d) Star

Explanation: The star topology is the best network topology for large businesses because it is simple to control and coordinate from the central computer.

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3) Which of the following transmission directions listed is not a legitimate channel?

(a) Simplex
(b) Half Duplex
(c) Full Duplex
(d) Double Duplex

Answer

Answer: (d) Double Duplex

Explanation: Double duplex is not a legitimate channel for transmission in computer network.

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4) “Parity bits” are used for which of the following purposes?

(a) Encryption of data
(b) To transmit faster
(c) To detect errors
(d) To identify the user

Answer

Answer: (c) To detect errors

Explanation: The parity bit is also known as the check bit, and has a value of 0 or 1. It is used for error detection for blocks of data.

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5) What kind of transmission medium is most appropriate to carry data in a computer network that is exposed to electrical interferences?

(a) Unshielded twisted pair
(b) Optical fiber
(c) Coaxial cable
(d) Microwave

Answer

Answer: (b) Optical fiber

Explanation: The optical fiber is made of glass or plastic. In this cable, the transmission of data occurs in the form of light rather than the electric current, so this cable provides higher data transfer speed than other cables.

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6) A collection of hyperlinked documents on the internet forms the ?.?

(a) World Wide Web (WWW)
(b) E-mail system
(c) Mailing list
(d) Hypertext markup language

Answer

Answer: (a) World Wide Web (WWW)

Explanation: World Wide Web (WWW) creates a collection of hyperlinked documents on the Internet.

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7) The location of a resource on the internet is given by its?

(a) Protocol
(b) URL
(c) E-mail address
(d) ICQ

Answer

Answer: (b) URL

Explanation: A URL (Uniform Resource Locator) is a database connection that describes the database’s location on a computer network and the retrieval process. A URL is a different form of URI (Uniform Resource Identifier) although the two words are used interchangeably by many people.

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8) The term HTTP stands for?

(a) Hyper terminal tracing program
(b) Hypertext tracing protocol
(c) Hypertext transfer protocol
(d) Hypertext transfer program

Answer

Answer: (c) Hypertext transfer protocol

Explanation: The term HTTP stands for Hypertext transfer protocol.

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9) A proxy server is used as the computer?

(a) with external access
(b) acting as a backup
(c) performing file handling
(d) accessing user permissions

Answer

Answer: (a) with external access

Explanation: A proxy server is a computer that acts as a gateway between a user’s computer and the Internet. The proxy server is also called application level gateway. By this the client computer can establish indirect network connection to another network.

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10) Which one of the following would breach the integrity of a system?

(a) Looking the room to prevent theft
(b) Full access rights for all users
(c) Fitting the system with an anti-theft device
(d) Protecting the device against willful or accidental damage

Answer

Answer: (b) Full access rights for all users

Explanation: None

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11) Which software prevents the external access to a system?

(a) Firewall
(b) Gateway
(c) Router
(d) Virus checker

Answer

Answer: (a) Firewall

Explanation: A firewall is a network securing software that prevents unauthorized users and dangerous elements from accessing the network. Software firewall acts as a filter for our network which prevents harmful information.

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12) Which one of the following is a valid email address?

(a) btechgeeks@geek.com
(b) gmail.com
(c) tpoint@.com
(d) btechgeeks@books

Answer

Answer: (a)btechgeeks@geeks.com

Explanation: A proper email address is a combination of the email prefix and email domain, both in appropriate formats. The domain appears to the right side of the @ symbol and the prefix appears to the left side of the @ symbol.

For example, in this email address btechgeeks@geeks.com, “btechgeeks” is the prefix, and “geeks.com” is the domain.

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13) Which of the following best describes uploading information?

(a) Sorting data on a disk drive
(b) Sending information to a host computer
(c) Receiving information from a host computer
(d) Sorting data on a hard drive

Answer

Answer: (b) Sending information to a host computer

Explanation: None

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14) Which one of the following is the most common internet protocol?

(a) HTML
(b) NetBEUI
(c) TCP/IP
(d) IPX/SPX

Answer

Answer: (c) TCP/IP

Explanation: TCP/IP is the most common internet protocol because it is the most widely used network protocol.

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15) Software programs that allow you to legally copy files and give them away at no cost are called which of the following?

(a) Probe ware
(b) Timeshare
(c) Shareware
(d) Public domain

Answer

Answer: (d) Public domain

Explanation: Public domain software can be modified, distributed or sold by anyone without any attention, but no one can ever own it.

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16) The term FTP stands for?

(a) File transfer program
(b) File transmission protocol
(c) File transfer protocol
(d) File transfer protection

Answer

Answer: (c) File transfer protocol

Explanation: The term FTP stands for File transfer protocol.

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17) At what speed does tele-computed refer?

(a) Interface speed
(b) Cycles per second
(c) Baud rate
(d) Megabyte load

Answer

Answer: (c) Baud rate

Explanation: In telecommunication, baud rate is a specific unit of the speed rate. It is one of the significant functions that determine the speed of the communication over the data channel.

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18) Which one of the following is not a network topology?

(a) Star
(b) Ring
(c) Bus
(d) Peer to Peer

Answer

Answer: (d) Peer to Peer

Explanation: Peer to Peer network is a network to which all computers are used the same resources and rights as other computers. Its network designed primarily for the small local area.

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19) The maximum length (in bytes) of an IPv4 datagram is?

(a) 32
(b) 1024
(c) 65535
(d) 512

Answer

Answer: (c) 65535

Explanation: None

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20) Which of the following statements could be valid with respect to the ICMP (Internet Control Message Protocol)?

(a) It reports all errors which occur during transmission.
(b) A redirect message is used when a router notices that a packet seems to have been routed wrongly.
(c) It informs routers when an incorrect path has been taken.
(d) The “destination unreachable” type message is used when a router cannot locate the destination.

Answer

Answer: (b) A redirect message is used when a router notices that a packet seems to have been routed wrongly.

Explanation: None

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21) The IP network 192.168.50.0 is to be divided into 10 equal sized subnets. Which of the following subnet masks can be used for the above requirement?

(a) 255.243.240
(b) 255.255.0.0
(c) 255.255.0
(d) 255.255.255

Answer

Answer: (c) 255.255.255.0

Explanation: This address belongs to class C, so 8 bits are reserved for the host ID. 24 bits are reserved for network ID.

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22) When the mail server sends mail to other mail servers it becomes ___ ?

(a) SMTP client
(b) SMTP server
(c) Peer
(d) Master

Answer

Answer: (a) SMTP client

Explanation: The SMTP client is an organization that allows sending emails using the SMTP server. SMTP servers can’t send emails to other SMTP servers separately. It is based on client-server architecture.

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23) The length of an IPv6 address is?

(a) 32 bits
(b) 64 bits
(c) 128 bits
(d) 256 bits

Answer

Answer: (c) 128 bits

Explanation: An IPv6 address is 128 bits long, which has 2^128 address space.

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NCERT 9TH CLASS COMPUTER SCIENCE BOOK OBJECTIVE QUESTION:
Chapter 1. Basics of internet technology.
1. ARPANET stands for-
(a) Advanced Real Projects Air Network
(b) Advanced Research Preparation Agency Network
(c) Advanced Recruitment Process Agency Network
(d) Advanced Research Projects Agency Network
2. In 1990s, the internetworking of which three networks resulted into Internet?
(a) WWW, GPS and other private networks
(b) ARPANET, NSFnet and other private networks
(c) ARPANET, NSFnet and other public networks
(d) ARPANET, GPS and NSFnet
3. Web search engines works with the help of two programs. Which are they?
(a) Web crawler and Cascading Style Sheet
(b) Spider and Indexer
(c) Web server and web crawler
(d) None of the above
4. Web Site is a collection of ______________.
(a) Audio and video files
(b) Pictures
(c) Web pages
(d) All of the above
5. AOL, iGoogle, Yahoo are examples of ______________.
(a) Web Site
(b) Web Page
(c) Web Portal
(d) None of the above
6. ______________is distributed computing over a network, and involve a large number of
computers connected via real-time communication network such as the Internet.
(a) Cloud Computing
(b) Thin Client Computing
(c) Fat Client Computing
(d) Dumb terminal Computing
7. A ______________ is a web site like any other, but it is intended to offer personal opinions
of people on their hobbies, interests, commentaries, photo, etc.
(a) Protocol (b) Blog
(c) Webpage (d) Journals

8. ______________ protocol defines how messages are formatted and transmitted, and
what actions Web servers and browsers should take in response to various commands.
(a) FTP (b) TCP/IP
(c) HTTP (d) SMTP
9. URLs are of two types:
(a) Absolute & Relative (b) Static & Dynamic
(c) Absolute & Dynamic (d) None of the above
10. DNS is an acronym for ______________.
(a) Domain Name Security (b) Domain Number System
(c) Document Name System (d) Domain Name System

Answer.

1.d 2.b 3.b 4.d 5.c 6.a 7.b 8.c 9.a 10.d

Chapter 2
Web Services
A. Multiple Choice Questions:
1. In web services, the communication takes place between-
(a) Two electronic devices. (b) Two human beings
(c) Two spiders (d) None of the above
2. Now a days, an email can be sent via,
(a) PC (b) Tablet
(c) Smart phone (d) All of the above
3. An e-mail address consists of four parts:
(a) The username (b) The symbol
(c) The hostname (d) The top level domain

4. The following is not the name of e-shopping portal:
(a) Amazon (b) e-bay
(c) twitter (d) naaptol
5. The benefit of E-leaning is:
(a) Class work can be scheduled around work and family
(b) Reduces travel time and travel costs for off-campus students
(c) Students can study anywhere they have access to a computer and Internet connection
(d) All of the above

Answer. 1.a 2.d 3. 4.c 5.d

Chapter 3
Introduction to GIMP
1. GIMP is image an _________ editing tool.
(a) Open source (b) Proprietary
(c) Shareware (d) Both (a) & (b)
2. The ________ tool allows you to select a drawing with the free-hand.
(a) Fuzzy Selection (b) Lasso
(c) Text d. Bucket Fill
3. The Fuzzy Select (Magic Wand) tool helps you to select areas of the _______.
(a) Canvas (b) All the layers
c. Current Layer (d) Image with different colours
4. Shift+ C is the shortcut to _______ the image.
(a) Duplicate (b) Delete
(c) Copy (d) Crop
5. To prepare the images for the web means
(a) Reducing the size of an image (b) Zipping the image
(c) Crop the image (d) All of the above
6. GIMP’s basic palette, consisting of ________ colours.
(a) Three (b) Two
(c) One (d) Many
7. _________ are transparent sheets stacked one on top of the other.
(a) Layers b. Canvas
(c) Windows (d) None of the these
8. Scale option is used to change the _________ of an image.
(a) Colour (b) Size
(c) Area (d) All of the above
9. It is the most important part of the image window where the image is displayed.
(a) Canvas (b) Window
(c) Scale (d) Size
10. The ______ tool places text into an image.
(a) Fuzzy Selection (b) Text
(c) Eraser (d) Bucket Fill

Answer. 1.a 2.b 3.c 4.d 5.a 6.b 7.a 8.b 9.a 10.b

Chapter 4
Introduction to HTML
1. Html uses
(a) Pre specified tags (b) User defined tags
(c) Fixed tags (d) Tags used for linking
2. What is the correct html tag for inserting a line break?
(a) (b)
(c) (d) 3. Choose the correct HTML tag to make the text bold
(a) (b)
(c) (d)
4. Choose the correct html tag for the largest heading
(a)

(b)
(c)(d)

5. How many blank line breaks will a browser create if you enter four
tag
(a) 5 (b) None
(c) 3 (d) 4

Answer. 1.a 2.b 3.b 4.a 5.d

NCERT 9TH CLASS COMPUTER SCIENCE BOOK OBJECTIVE QUESTION next chapter 5

Chapter 5
HTML II
….
2. Which one of the following is the list type that will create a bulleted list?
(a) Unordered (b) Option
(c) Decorated (d) Ordered
3. Which html tag is use to render text as italics?
(a) (b)
(c) (d) Text style=”italics”
4. We can view the source code of html in
(a) Notepad (b) Power point
(c) Excel (d) Paint
5. To wrap the text to the next line, the tag used for
(a)
(b)

(c) (d)

Answer. 1.a 2.a 3.b 4.a 5.aNCERT 9TH CLASS COMPUTER SCIENCE BOOK OBJECTIVE QUESTION next chapter 6

Chapter 6
Security Threats and Safety
Measures
1. Which of the following is an anti-virus program
(a) Norton (b) Quick heal
(c) K7 (d) All of these
2. All of the following are examples of real security and privacy threats except:
(a) Hackers (b) Spam
(c) Virus (d) Worm
3. Trojan horses are very similar to virus in the matter that they are computer programs
that replicate copies of themselves.
(a) True (b) False
4. _____________ monitors user activity on internet and transmit that information in the
background to someone else.
(a) Malware (b) Adware
(c) Spyware (d) None of these
5. Viruses are __________.
(a) Man made (b) Naturally occur
(c) Machine made (d) All of the above
6. Firewall is a type of ____________.
(a) Virus (b) Worm
(c) Security threat (d) None of the above
7. Unsolicited commercial email is known as ____________.
(a) Spam (b) Virus
(c) Malware (d) Spyware

Answer. 1.d 2.a 3.b 4.c 5.a 6.d 7.a

Computer MCQ Questions

Here is a list of Computer GK questions in English for class 5,

1) _________ converts of images of typed, handwritten or printed text into

machine-encoded text, whether from a scanned document, or a photo of a document

a) MICR

b) OCR

c) Barcode Reader

d) None of the above

2) _______ maintains power to the connected devices of a computer when power gets

disconnected.

a) Central Processing Unit (CPU)

b) Monitor

c) IC Chips

d) UPS

3) ____________ typically consists of code developed by cyberattackers, designed to

cause extensive damage to data and systems.

a) Software

b) Hardware

c) Freeware

d) Malware

4) ________ are irrelevant or unsolicited messages sent over the internet, typically to a large number of users, for the purposes of advertising, phishing, etc.

a) Email messages

b) Spam

c) Virus

d) None of the above

5) The right full form of URL is _______

a) Uniform Receiver Location

b) Unified Resource Locator

c) Uniform Resource Locator

d) None of the above

6) _______ makes use of additive manufacturing process that creates a physical object

from a digital design.

a) Laser Printer

b) Inkjet Printer

c) 3D Printer

d) None of the above

7) ______ is typically used for making presentations.

a) Microsoft Word

b) Microsoft Power Point

c) Microsoft Excel

d) Microsoft Outlook

8) _______ is usually used for making final documents or reports

a) Microsoft Word

b) Microsoft Power Point

c) Microsoft Excel

d) Microsoft Outlook

9) The World Wide Web (WWW) was invented by _______.

a) Bill Gates

b) Larry Page

c) Steve Rogers

d) Tim Berners-Lee

10) ________ is considered the official birthday of the internet.

a) January 1, 1983

b) January 1, 1980

c) April 20, 1987

d) None of the above

11) Choose the correct representation, while adding 2 cells in Excel ______

a) Sum (A1; B1) =

b) = Sum (A1 : B1)

c) = A1; B1

d) None of the above

12) _______ is a new generation of storage devices used in computers.

a) SSD

b) PCI Card

c) Joystick

d) None of the above

13) Which of the following is the function of an Operating System?

a) Device Management

b) File Management

c) Processor Management

d) All of the above

14) Kaspersky Lab is a multinational cybersecurity and anti-virus provider

headquartered in _______

a) Helsinki

b) Seattle

c) Moscow

d) Turin

15) Which of the following are the main types of User Interface?

a) Graphical User Interface

b) Menu Driven Interface (mdi)

c) natural language Interface (nli)

d) All of the above

Answers:

1. b)	2. d)	3. d)	4. b)	5. c)
6. c)	7. b)	8. a)	9. d)	10. a)
11. b)	12. a)	13. d)	14. c)	15. d)

Computer MCQ Questions

Computer General Knowledge Questions for Class 4

Here is a list of Computer GK questions in English for class 4,

1) MacOS is the operating system created by _____

a) Apple

b) Microsoft

c) Google

d) IBM

2) High Sierra was released in 2017. It is a/an ______

a) Operating System

b) Graphic User Interface

c) Modem

d) Router

3) ________ is known as the father of computers.

a) Clifford Berry

b) Alan Turing

c) Douglas Englebart

d) Charles Babbage

4) ______ invented Java Programming language in 1994.

a) Dennis Ritchie

b) Yukihiro Matsumoto

c) Bjarne Stroustrup

d) James Gosling

5) 1 TB (Terabyte) is equal to ______

a) 100 GB

b) 1000 GB

c) 10 GB

d) 10000 GB

6) Dennis Ritchie is known as the father of ______ language

a) Python

b) C

c) Java

d) None of the above

7) ______ is not a programming language

a) HTML

b) Java

c) Python

d) C++

8) bit is short for ______

a) binary tree

b) Bivariate Theory

c) Binary digit

d) None of the above

9) ASCII Code is a ________.

a) 10 bit code

b) 9 bit code

c) 7 bit code

d) 5 bit code

10) A motherboard is also known as mainboard, main circuit board, baseboard, logic board

a) True

b) False

11) A hard disk drive (HDD), is a data storage device that stores and retrieves digital data .

a) False

b) True

12) Pratyush is a supercomputer established at _______

a) Indian Institute of Tropical Meteorology (IITM)

b) Indian Institute of Science

c) Indian Institute of Astrophysics

d) None of the above

13) World’s fastest supercomputer is ______

a) Fugaku Supercomputer

b) Mihir

c) Cray XC 40

d) None of the above

14) The tasks performed by Excel is similar to _______

a) Word

b) Powerpoint

c) Microsoft Outlook

d) Spreadsheet

15) Central Processing Unit (CPU) has which of the following components?

a) Control Unit

b) ALU (Arithmetic Logic Unit)

c) Memory or storage unit

d) All of the above

Answers:

1. a)	2. a)	3. d)	4. d)	5. b)
6. b)	7. a)	8. c)	9. c)	10. a)
11. b)	12. a)	13. a)	14. d)	15. d)

Computer MCQ Questions

1. Which one of the Following is C.P.U

(a)

(b)

(c)

(d)

Answer: B

2. Given Below is a Picture Identify this Picture

(a) CPU
(b) Mouse
(c) Keyboard
(d) Monitor

Answer: (d) Monitor

3. Given Below is a Picture Identify this Picture

(a) CPU
(b) Mouse
(c) Keyboard
(d) Monitor

Answer: (b) Mouse

4. Given Below is a Picture Identify this Picture

(a) CPU
(b) Mouse
(c) Keyboard
(d) Monitor

Answer: (c) Keyboard

5. Which one of the following parts of a computer looks like an animal?
(a) Monitor
(b) CPU
(c) Mouse
(d) Keyboard

Answer: (c) Mouse

6. A ________ is a pointing part of a computer
a) Mouse
b) Monitor
c) Keyboard
Answer: Mouse

7. We cannot use a _________ for writing on a computer
a) both
b) pencil
c) pen
Answer: both

8. Always sit on the
a) table
b) bench
c) chair
Answer: table

9. Double click means pressing the left mouse button
a) twice
b) thrice
c) once
Answer: twice

10. _________ means pressing the right mouse button once.
a) Right-click
b) Left click
c) Single click
Answer: Right-click

11. A _________ keeps the mouse free from dust and dirt
a) mouse pad
b) rubber pad
c) paper pad
Answer: mouse pad

12. Keep the index finger on the ________ button
a) left
b) right
c) middle
Answer: left

13. Keep the middle finger on the ________ button
a) Right
b) Left
c) Scroll wheel
Answer: Right

14. Move the mouse over a ________
a) Mouse pad
b) Rubber pad
c) Rubber pad
Answer: Mouse pad

15. Always keep the _________ on the mouse pad
a) mouse
b) mouse
c) CPU
Answer: mouse

16. _________ has a tail.
a) Mouse
b) CPU
c) Monitor
Answer: Mouse

17. A mouse is a part of a _________
a) washing machine
b) computer
c) fan
Answer: Computer

18. A keyboard has _________ keys.
a) Both
b) Number
c) Alphabet
Answer: Both

19. Cursor is a _________ blinking line on the monitor
a) small
b) big
c) slanting
Answer: small

20. There are usually __________ keys on the keyboard
a) 104
b) 106
c) 110
Answer: 104

21. __________ keys are found at the top of the keyboard
a) Function
b) Number
c) Alphabet
Answer: Function

22. __________ key is use to write in capital letters
a) Caps Lock
b) Enter
c) Backspace
Answer: Caps Lock

23. _________ erases letters to the left of the cursor
a) Backspace
b) Delete
c) Both
Answer: Backspace

24. Question: Arrow keys are used to move the cursor in __________ directions
a) Both
b) LEFT/RIGHT
c) UP/DOWN
Answer: Both

25. Mouse is connected to _________
a) CPU
b) UPS
c) Keyboard
Answer: CPU

Computer MCQ Questions

MCQ Questions Answers For Class 6th Computer:

Mcq On Computer Languages With Answers For Class 6

1- ………….. Was The First World Electronic Digital Computer.
Tide Predictor
FACTA
Colossus
None Of These
Correct Ans: …..

Colossus

2- …………. Computer Is Also Known As Grand Father Computer.
Mainframe
Super
Workstation
Both 1 And 2
Correct Ans: …..
Super

3- Digital Diary Is A Type Of ……………..
Mini
Micro
Mainframe
Workstation
Correct Ans: …..
Micro

4- Tide Predictor Ia A Type Of …………….. Computer.
Analog
Digital
Hybrid
None Of These
Correct Ans: …..
Analog

5- FACTA Is A Type Of ………….. Computer.
Hybrid
Analog
Digital
Both 1 And 3
Correct Ans: …..
Hybrid

6- Mainframe Is Also Known As ………….. Of Computer.
Mother
Brother
Sister
Father
Correct Ans: …..
Father

7- FACTA Was Invented In ……………….
1955
1972
1970
1976
Correct Ans: …..
1976

Objective Questions Answers For Class 6th Computer: Categories Of Computer And Computer Languages Class 6 Mcq

8- The Set Of Instructions Given To The Computer.
Program
Commands
Hardware
Software
Correct Ans: …..
Program

9- The ………….. Language Is Directly Understood By The Computer.
Basic
Machine
Hard
Easy
Correct Ans: …..
Machine

10- The Instructions In Machine Language Are Written With The Help Of …………….. .
Digits 0 To 9
English Alphabets
0 And 1
None Of These
Correct Ans: …..
0 And 1

11- …………. Language Uses Mnemonic Codes.
Machine
High Level
Assembly
Both 1 And 2
Correct Ans: …..
Assembly

12- A ………… Converts The Complete Program Into Machine Language In One Go .
Translator
Compiler
Interpreter
None Of These
Correct Ans: …..
Compiler

13- The Programs Written In High Level Language Are Called ………….. .
Program
Source Code
Code
Only 1
Correct Ans: …..
Source Code

14- …………. Converts Program In Assembly Language In To Machine Language .
Assembler
Interpreter
Compiler
Hardware
Correct Ans: …..
Assembler

15- ………….. Is Not A High Level Language.
Basic
Java
MAC
Only 1
Correct Ans: …..
MAC

Multiple Choice Questions MCQ Answers For Class 6 Computer:
16- Which Of The Following Tab Has Options To Work With Mail Merge ?
Home
Mailings
Insert
Both 1 And 3
Correct Ans: …..
Mailings

17- The Recipients List Is Saved In ………… Folder.
Main Data Sources
My Data Sources
Merge Data Sources
None Of These
Correct Ans: …..
My Data Sources

18- There Are …………….. Vertical Alignments To Align The Text In A Table .
4
5
6
3
Correct Ans: …..
3

19- Under Which Tab Is The Table Option Found ?
Mailing
Home
Both 1 And 2
Insert
Correct Ans: …..
Insert

20- The Table Tools Tab Consists Of Design Tab And ……….. Tab.
Format
Layout
Draw
None Of These
Correct Ans: …..
Layout

21- ………. Contains The Main Body Text .
Data Source
Merged Documents
Main Document
Basic Documents
Correct Ans: …..
Main Document

22- Insert Merge Field Is Present Under ………… Tab.
Home
Page Layout
Mailing
Both 1 And 2
Correct Ans: …..
Mailing

23- There Are ………….. Ways To Create A New Table
2
3
6
4
Correct Ans: …..
4

General Knowledge GK Questions For Class 6 Computer:

24 — The Tide Predictor Was Developed In ………….
1873
1874
1875
1895
Correct Ans: …..
1873

25 – Laptop And Palmtop Are Examples Of ………. Computer.
Mainframe
Mini
Micro
Super
Correct Ans: …..
Micro

26 – FACTA Was Developed By ……….
Davie Hendrick
William Thompson
Ada Lovelace
Both 1 And 2
Correct Ans: …..
Davie Hendrick

27 – …………. Computer Is A Combination Of Analog And Digital Computer .
Mini
Micro
Hybrid
Mainframe
Correct Ans: …..
Hybrid

28 – …………. Computer Can Perform Calculations And Logical Operations .
Digital
Analog
Mini
Micro
Correct Ans: …..
Analog

29 – The Tide Predictor Was Developed By ………..
Lady Ada
William Thompson
Davie Hendrick
Only 1
Correct Ans: …..
William Thompson

30 – ………….. Computer Is Also Known As Father Computer.
Mainframe
Mini
Micro
Super
Correct Ans: …..
Mainframe

31 – ………… Computer Operate On 0 And 1 Only.
Digital
Analog
Hybrid
Both 1 And 2
Correct Ans: …..
Digital

NCERT Class 6 Computer MCQ Questions Answers:

32 – The Full Form Of HLL Is ………….
Hang Level Language
Hot Level Language
High Level Language
Only 1
Correct Ans: …..
High Level Language

33 – ……… Was The First Programmer.
Lady Ada Lovelace
William Thomson
Davie Hendrick
Only 2
Correct Ans: …..
Lady Ada Lovelace

34 – …………. Was The First Super Computer Of India .
Colossus
FACTA
Param 8000
Only 1
Correct Ans: …..
Param 8000

35 – There Are ……….. Types Of Computer Languages .
3
4
5
6
Correct Ans: …..
3

36 – Interpreter, Compiler, And Assembler Are The ……….. Types Of Translator.
Three
Four
Two
Five
Correct Ans: …..
Three

37 – A Table Consists Of ……….
Rows
Columns
Both 1 And 2
Only 1
Correct Ans: …..
Both 1 And 2

38 – Mail Merge Has ………. Components.
Three
Four
Five
Six
Correct Ans: …..
Three

39 – ………. Is Your Starting Point Of The Mail Merge Procedure .
Main Document
Data Source
Merged Document
Only 3
Correct Ans: …..
Main Document

NCERT MCQ Solutions Class 6 Computer Questions:

40 – There Are ……. Types Of Translator .
Four
Three
Two
Five
Correct Ans: …..
Three

41 – C++ Is A ……… Language .
High Level
Fourth Generation
Low Level
Both 1 And 2
Correct Ans: …..
High Level

42- Mail Merge Is A Feature Of ……………… .
MS Excel
MS Power Point
Both 1 And 2
MS Word
Correct Ans: …..
MS Word

43- ……….. Language Is A Machine Dependent Language.
High Level
Fourh Generation
Normal Level
Low Level
Correct Ans: …..
Low Level

44- ……… Is The Fastest Super Computer Of The World.
Tianhe 1A
Super 8000
Param 8000
Super 800
Correct Ans: …..
Tianhe 1A

45- There Are …………. Types Of Micro Computers.
4
5
6
3
Correct Ans: …..
3

46- Top , …………. And Bottom Are The Name Of Vertical Alignments In A Table.
Up
Center
Down
Both 1 And 3
Correct Ans: …..
Center

47- Computers Only Understand ……….. Language.
Assembly
High Level
Machine
Low Level
Correct Ans: …..
Machine

Class 6 Computer Chapter 1 MCQ Questions:

48- We Can Change Table Design By ………….. .
Design Tab
Lay Out Tab
Out Tab
None Of These
Correct Ans: …..
Design Tab

49- Table Button Option Is Available In …………… .
Insert Tab
Design Tab
Lay Out Tab
Both 2 And 3
Correct Ans: …..
Insert Tab

50- The Software That Translates The Instructions Of Different Language Into Machine Language Is Known As …….. .
HLL
Translator
Assembly
Hardware
Correct Ans: …..
Translator

51- Workstation Is A Powerful , ………. Computer.
Multi User
Two User
Single User
None Of These
Correct Ans: …..
Single User

53- Full Form Of CPU Is ……………. .
Common Processing Unit
Change Processing Unit
Central Processing Unit
Control Processing Unit
Correct Ans: …..
Central Processing Unit

54- Full Form Of ROM Is …………… .
Read Only Memory
Real Only Memory
Rom Only Memory
Right Only Memory
Correct Ans: …..
Read Only Memory

55- Basic And Java Are The Example Of …………. .
Fourth Generation
Fifth Generation
Low Level
HLL
Correct Ans: …..
HLL

Class 6 Chapter 2 MCQ Questions Social Computer:

56- Full Form Of LAN Is ……….. .
Local Area Network
Line Area Network
Long Area Network
Low Area Network
Correct Ans: …..
Local Area Network

57- Mainframe , Mini And Micro Computers Are Classified On The Basis Of ……….. .
Working Principle
Size
Area
Both 1 – 3
Correct Ans: …..
Size

58- This Translator Is Time Consuming .
Compiler
Assembler
Interpreter
None Of These
Correct Ans: …..
Interpreter

59- Full Form Of WAN Is …………
Won Area Network
Wise Area Nerwork
Both 1 And 2
Wide Area Network
Correct Ans: …..
Wide Area Network

60- The Execution Speed Of This Translator Is Fast.
Compiler
Interpreter
Hardware
Assembler
Correct Ans: …..
Compiler

Computer MCQ Questions

Digital Diary Is A Type Of questions also have asked.