MCQ: Harmonic Progressions - SSC CGL MCQ

# MCQ: Harmonic Progressions - SSC CGL MCQ

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## 15 Questions MCQ Test Quantitative Aptitude for SSC CGL - MCQ: Harmonic Progressions

MCQ: Harmonic Progressions for SSC CGL 2024 is part of Quantitative Aptitude for SSC CGL preparation. The MCQ: Harmonic Progressions questions and answers have been prepared according to the SSC CGL exam syllabus.The MCQ: Harmonic Progressions MCQs are made for SSC CGL 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for MCQ: Harmonic Progressions below.
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MCQ: Harmonic Progressions - Question 1

### are in HP, then which of the following is/are correct? 1. a, b, c are in AP 2. (b + c)2, (c + a)2, (a + b)2 are in GP. Select the correct answer using the code given below.

Detailed Solution for MCQ: Harmonic Progressions - Question 1

Concept:

• If a, b and c are three terms in GP then b2 = ab and vice-versa
• If three terms a, b, c are in HP, then and vice-versa
• When three quantities are in AP, the middle one is called as the arithmetic mean of the other two.
• If a, b and c are three terms in AP then
• and vice-versa
• If a, b and c are in A.P then 1/a, 1/b, 1/c are in H.P and vice-versa

Calculation:

(b + c), (c + a) and (a + b) are in A.P

⇒ 2(c + a) = b + c + a + b

⇒ 2c + 2a = 2b + a + c     -----(i)

⇒ 2c + 2a - 2b - a - c = 0

⇒ 2b = c + a

So, a, b, c are in AP

Now, Let (b + c)2, (c + a)2, (a + b)2 are in GP

(c + a)2 = √[(b + c).(a + b)]2

⇒ c2 + a2 + 2ac = (b + c).(a + b)

⇒ c2 + a2 + 2ac = ab + b2 + ac + bc

⇒ c2 + a2 - b2 + ac - ab - bc = 0

From here we are unable to check the relation between a, b and c

So, Our assumption was wrong

So, (b + c)2, (c + a)2, (a + b)2 are not in GP.

MCQ: Harmonic Progressions - Question 2

### If 1/4, 1/x, 1/10 are in HP, then what is the value of x?

Detailed Solution for MCQ: Harmonic Progressions - Question 2

Concept:

If a, b, c are in HP then

If a, b, c are in AP then 2b = a + c

A sequence of numbers is called a Harmonic progression if the reciprocal of the terms are in AP.

Calculation:

Given: 1 / 4, 1 / x, 1 / 10 are in HP

The sequence  1 / 4, 1 / x, 1 / 10 are in HP, so its reciprocals 4, x, 10 are in AP.

As we know, If a, b, c are in AP then 2b = a + c

⇒ 2x = 4 + 10

⇒ 2x = 14

∴ x = 7

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MCQ: Harmonic Progressions - Question 3

### If a, b, c are in geometric progression, then logax x, logbx x and logcx x are in

Detailed Solution for MCQ: Harmonic Progressions - Question 3

Concept:

If a, b, c are in geometric progression then b2 = ac

If b - a = c - b, than a, b, c are in AP.

If 1/a, 1/b, 1/c are in AP than a, b, c are in HP.

Calculation:

If a, b, c are in geometric progression then b2 = ac

So, by multiplying both side by x2 and taking log on both side to the base x

MCQ: Harmonic Progressions - Question 4

Three numbers 5, p and 10 are in Harmonic progression if p = ?

Detailed Solution for MCQ: Harmonic Progressions - Question 4

Concept:

Three numbers x, y, and z are in H.P if and only if y

Calculation:

Given: Three numbers 5, p and 10 are in Harmonic progression

Now, According to the concept used

MCQ: Harmonic Progressions - Question 5

How many two digit numbers are divisible by 7?

Detailed Solution for MCQ: Harmonic Progressions - Question 5

CONCEPT:

Let us suppose a be the first term and d be the common difference of an AP. Then the nth term of an AP is given by:an = a + (n - 1) × d.

Note: If l is the last term of a sequence, then l = an = a + (n - 1) × d.

CALCULATION:

Here we have to find two digit numbers which are divisible by 7.

i.e 14, 21,............,98 is an AP sequence with first term a = 14, common difference d = 7 and the last term l = 98.

As we know that, if l is the last term of a sequence, then l = an = a + (n - 1) × d

⇒ 98 = 14 + (n - 1) × 7

⇒ 84 = 7(n - 1)

⇒ 12 = n - 1

⇒ n = 13

MCQ: Harmonic Progressions - Question 6

The nth terms of the two series 3 + 10 + 17 + ... and 63 + 65 + 67 + .... are equal, then the value of n is:

Detailed Solution for MCQ: Harmonic Progressions - Question 6

Given:

nth term for the first series = nth term for the second series.

Concept:

Arithmetic Progression:

• An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.
• This fixed number is called the common difference of the AP and it can be positivenegative or zero.

Consider the AP, whose first term is 'a' and common difference is 'd' and there are n number of terms.

a, a + d, a + 2d, a + 3d........a + (n - 1)d

Hence, nth term of AP is given by

Tn = a + (n - 1)d

Calculation:

Consider the first series

3 + 10 + 17 + ...

a = 3 and d = 10 - 3 = 7

nth term of AP is given by,

(Tn)1 = 3 + (n - 1) × 7

⇒ (Tn)1 = 7n - 4       .......(1)

Consider the second series

63 + 65 + 67 + ....

a = 63 and d = 2

nth term of AP is given by,

(Tn)2 = 63 + (n - 1) × 2

⇒ (Tn)2 = 2n + 61      .......(2)

According to the question

(Tn)1 = (Tn)2

⇒ 7n - 4 = 2n + 61

⇒ 5n = 65

⇒ n = 13

Hence, for the given series, 13th term will be equal.

MCQ: Harmonic Progressions - Question 7

If xa = yb = zc and x, y and z are in GP, then a, b and c are in

Detailed Solution for MCQ: Harmonic Progressions - Question 7

Concept:

Calculation:

From (1),

∴ a, b and c are in H.P.

MCQ: Harmonic Progressions - Question 8

If 1/2,1/x,1/8 are in HP, then what is the value of x?

Detailed Solution for MCQ: Harmonic Progressions - Question 8

Concept:

If a, b, c are in HP then

If a, b, c are in AP then 2b = a + c

A sequence of numbers is called a Harmonic progression if the reciprocal of the terms are in AP.

Calculation:

Given: 1/2,1/x,1/8 are in HP

The sequence  1/2, 1/x, 1/8 are in HP, so its reciprocals 2, x, 8 are in AP.

As we know, If a, b, c are in AP then 2b = a + c

⇒ 2x = 2 + 8

⇒ 2x = 10

∴ x = 5

MCQ: Harmonic Progressions - Question 9

Which term of the GP 2, 6, 18, 54, ..... is 4374?

Detailed Solution for MCQ: Harmonic Progressions - Question 9

Concept:

Let us consider sequence a1, a2, a3 …. an is a G.P.

Calculation:

Given: The sequence 2, 6, 18, 54, ..... is a GP.

Here, we have to find which term of the given sequence is 4374.

As we know that, the the general term of a GP is given by: an = arn−1

Here, a = 2, r = 3 and let an = 4374

⇒ 4374 = (2) ⋅ (3)n - 1

⇒ 3n-1 = 2187

⇒ 3n-1 = 37

∴ n - 1= 7

So, n = 8

Hence, 4374 is the 8th term of the given sequence.

MCQ: Harmonic Progressions - Question 10

If the roots of the equation a (b - c) x2 + b (c - a) x + c (a - b) = 0 are equal, then which one of the following is correct?

Detailed Solution for MCQ: Harmonic Progressions - Question 10

Concept:

For any quadratic equation, ax2 + bx + c = 0. We have discriminant, D = b2 - 4ac, then the given quadratic equation has:

I. Distinct and real roots if D > 0.

II. Real and repeated roots, if D = 0.

III. Complex roots and conjugate of each other, D < 0.

If a, b and c are in HP, then the harmonic mean,

Calculation:

Given: a (b - c) x2 + b (c - a) x + c (a - b) = 0 has equal roots ⇒ Discriminant, D = 0.

By comparing the given equation with the quadratic equation, ax2 + bx + c = 0. We get, a’ = a (b - c) , b’ = b (c - a) and c’ = c (a - b)

As, Discriminant, D = 0.

⇒ D = b2 - 4ac = b2 × (c - a) 2 - 4 × a (b - c) × c (a - b) = 0

⇒ D = (bc + ab - 2ac) 2 = 0

⇒ bc + ab - 2ac = 0

⇒ b × (a + c) = 2ac

Hence a, b and c are in HP.

MCQ: Harmonic Progressions - Question 11

The arithmetic mean and geometric mean of two numbers are 14 and 12 respectively. What is the harmonic mean of the numbers?

Detailed Solution for MCQ: Harmonic Progressions - Question 11

Given:

Arithmetic mean = 14

Geometric mean = 12

Formula used:

Calculation:

By using the above formula,

HM = 122/14

⇒ 144/14 = 72/7

∴ The correct answer is 72/7.

MCQ: Harmonic Progressions - Question 12

If a1, a2, a3, ........ are in H.P., then the expression a1a2 + a2a3 +...... + an - 1 an is equal to:

Detailed Solution for MCQ: Harmonic Progressions - Question 12

Given that a1, a2, a3,....an are in H.

Let d be the common difference of the A.P., then

adding all the above we get.

MCQ: Harmonic Progressions - Question 13

If the first two terms of a H.P are 2/5 and 12/13 respectively, then the largest term is

Detailed Solution for MCQ: Harmonic Progressions - Question 13

Given:

If the first two terms of a H.P. are 2/5 and 12/13 respectively.

Concept:

The harmonic progression for any existing arithmetic progression given as

a, a+d, a+2d, a+3d, ...... a+(n-1)d is as follows,

Solution:

According to the question,

The an for given first term "a" and "d" in general is as follows,

Clearly, 5th term is the largest.

MCQ: Harmonic Progressions - Question 14

If H is the Harmonic Mean between a and b, evaluate

Detailed Solution for MCQ: Harmonic Progressions - Question 14

Concept:

The Harmonic Mean of a & b is

H = 2ab/a+b

Calculation:

let a = 1, b = 3,

H = 2ab/(a+b)

H = (2 × 1 × 3)/(1 + 3)

H = 6/4 = 3/2

Therefore,

MCQ: Harmonic Progressions - Question 15

If H is the harmonic mean of numbers 1, 2, 22, 23, ......2n-1 what is n/H equal to?

Detailed Solution for MCQ: Harmonic Progressions - Question 15

Calculation:

The harmonic mean of the the x1 , x2 , x3 ......xn is given by

The above is geometric mean of common ratio 1/2 and first term 1. The sum of above G.P would be

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## Quantitative Aptitude for SSC CGL

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