Maths & Logical Reasoning Test - 4 - SSC CGL MCQ

Let the height of the tower be ‘h’ meters

In right angled △ABD,

⇒ tan75° = AB/BD

⇒ (√3 + 1) / (√3 – 1) = h/DB

⇒ DB = h × [(√3 – 1) / (√3 + 1)] ----(1)

In right angled △ABC,

⇒ tan15° = AB/BC

As we know,

BC = CD + DB and CD = 50√3

⇒ (√3 – 1) / (√3 + 1) = h/ (CD + DB)

⇒ 50√3 + DB = h × [(√3 + 1) / (√3 – 1)]

⇒ DB = h × [(√3 + 1) / (√3 – 1)] – 50√3 ----(2)

From equation (1) and equation (2),

⇒ h × [(√3 – 1) / (√3 + 1)] = h × [(√3 + 1) / (√3 – 1)] – 50√3

⇒ h × [(√3 + 1) / (√3 – 1) – (√3 – 1) / (√3 + 1)] = 50√3

⇒ h × [(4 + 2√3) – (4 – 2√3)] /2 = 50√3

⇒ h × 4√3/2 = 50√3

∴ h = 25 meters

Given:

The distance between two trains A & B is 530 km

The speed of train A is 90 km/h

Speed of train B is 90 km/h

Formula Used:

Distance = Time × Speed

Calculation:

Train A, 1 hours distance = 1 × 90 = 90 km

So, train A started time is 10:00 A.M.

Remaining distance = 530 - 90 = 440 km

Time = 440/(90 + 90)

Time = 440/180 = 22/9 hour

Train meet after 22/9 hours after 10:00 A.M.

⇒ Time they meet = 10:00 A.M. + 22/9 hours

22/9 hours = 2(4/9) hours = 2 hours, 26 minutes and 40 seconds

since,

4/9 hours × 60 minutes = 80/3 min = 26 (2/3) min

2/3 min = 2/3 × 60 = 120/3 = 40 sec

⇒ So, they meet at 12:26:40 P.M.

∴ Train meet after 12:26:40 P.M. hours after 10:00 A.M.

Given:-
cos² θ − sin² θ = −1/2
Formula used:-
(1) cos(2θ) = cos²θ - sin²θ
(2) cos (180 - θ) = -cos(θ)
In the second quadrant(90 to 180 degrees), the value of cos decreases from 0 to -1.
Calculation:-
⇒ cos² θ − sin² θ = −1/2
∵ cos(2θ) = cos²θ - sin²θ
⇒ cos(2θ) = -1/2
∵ cos(60) = 1/2
⇒ cos(2θ) = -cos60
∵ cos (180 - θ) = -cos(θ)
⇒ cos(2θ) = cos (180 -60)
⇒ 2θ = 120
⇒ θ = 60
As per the question,
⇒ cos²2θ + cot² (θ/2)
Put θ = 60 in the above function
⇒ cos²120 + cot² (30)
⇒ (-1/2)² + (√3)² = 3 + (1/4) = 13/4
∴ The required answer is 13/4.

Given :

A boat can go 5 km upstream and 4 km downstream in 36 minutes

Same boat goes 10 km downstream and 8 km upstream in 63 minutes

Concept used :

Speed = Distance/Time

Upstream speed = (speed of boat in still water) - (speed of flow of stream) ---- (1)

Downstream speed = (speed of boat in still water) + (speed of flow of stream) ----- (2)

Calculations :

Let the speed of boat in still water be x

Speed of flow of stream be y

Let the downstream and upstream speed be v and u

⇒ 5/u + 4/v = 36/60 ---- (3)

⇒ 8/u + 10/v = 63/60 ----- (4)

Multiply equation (2) with 8 and equation (3) with 5

⇒ 40/u + 32/v = 48/10 ----- (5)

⇒ 40/u + 50/v = 105/20 ----- (6)

Subtract equation (3) from equation (4), we get

⇒ 50/v - 32/v = 105/20 - 48/10

⇒ 18/v = 9/20

⇒ v = 40

Putting value of v in equation (3)

⇒ 5/u + 1/10 = 6/10

⇒ 5/u = 6/10 - 1/10

⇒ 5/u = 5/10

⇒ u = 10

Using equation (1) and (2), we get

⇒ x + y = 40

⇒ x - y = 10

⇒ x = 25 and y = 15

Speed of water in still water : speed of stream = 25 : 15

⇒ 5 : 3

∴ The ratio of the speed of the boat in still water to the speed of the current is 5 : 3

Given:

Mean = 40

Standard deviation = 10

N (Sample size) = 100

Formula used:

Standard deviation,

Calculation:

∑x_i = 40 × 100 = 4000

Correct sum of 100 observation = 4000 - 30 - 70 + 3 + 27 = 3930

Correct mean of the observation = 3930/100 = 39.30

According to the concept,

⇒

⇒ 100 = - 1600

⇒ = 1700

⇒ = 170000

According to the question, two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively.

Hence, the correct

⇒ 170000 - 30² - 70² + 3² + 27² = 164938

According to the formula,

Now, the correct standard deviation

⇒

⇒

⇒ 10.24

∴ The correct standard deviation is 10.24.

Given:

4/10 + 5/100 + 7/1000 = ?

Calculation:

4/10 + 5/100 + 7/1000 = ?

LCM of 10, 100 and 1000 is 1000.

⇒ ()

⇒ 457/1000

∴ The required value is 457/1000.

Given:

The ratio of the salary of Ram, Raj, and Rio is 5:7:9.

Rio receives Rs 280 more than Ram.

Calculation:

Let the common multiple for the salary of Ram, Raj, and Rio be x.

Salary of Ram = 5x

Salary of Raj = 7x

Salary of Rio = 9x

According to the question,

9x - 5x = 280

⇒ 4x = 280

⇒ x = 70

Salary of Raj = 7x

⇒ Salary of Raj = 7 × 70 = 490

∴ The salary of Raj is Rs. 490.

Given:

2x3 - 3x2 + 5x + 2 = 0

Concept Used:

The degree of a polynomial is given by the exponent of the highest power term.

Calculation:

2x3 - 3x2 + 5x + 2 = 0

In this polynomial, the highest power of x is 3 (in the term 2x³)

∴ The degree of the polynomial is 3.

Given:

The ratio of 3 years of C.I. and S.I. of one year on a certain sum of money is 4.75 ∶ 1.

Formula used:

(x + y)³ = x³ + y³ + 3xy(x + y)

S.I. = (P × R × T)/100

A = P(1 + R/100)ⁿ

Where A → amount

n → time

S.I. → Simple Interest

P → Principal

R → Rate

T → Time

Calculations:

Let the sum of money be Rs. P. and rate of interest be R% p.a.

Compound interest for 3 years = P[(1 + R/100)³ - 1]

⇒ P[1 + 3 × 1 × (R/100)(1 + R/100) + (R/100)³ -1]

⇒ P(R/100)(3 + 3 × R/100 + R²/10000)

Simple interest for one year = (P × R × 1)/100

According to the question,

⇒ {P(R/100)(3 + 3 × R/100 + R2/10000)}/(P × R × 1)/100 = 4.75/1.

⇒ 3 + 3 × (R/100) + R²/10000 = 4.75

⇒ R2/10000 + 3 × (R/100) - 1.75 = 0

⇒ R² + 300R - 17500 = 0

⇒ R² + 350R - 50R - 17500 = 0

⇒ R(R + 350) - 50(R + 350) = 0

⇒ (R + 350)(R - 50) = 0

Thus R = - 350% (not valid) , or R = 50%

∴ The rate is 50%.

x = 105.5
Hence, option B is correct.

Let the speeds be a and b

If Speed of B becomes 2b

From 1 and 2 we get, b = 35 km/h and a = 42 km/h
Hence, option D is correct.

Let,
In the mixture,
The quantity of P be 5a and the quantity of Q be 6a
The quantity of mixture = 5a + 6a = 11a
To get double the volume of the mixture of P and Q, after adding mixture Q,
The quantity of Q = 6a + (5a + 6a) = 17a
Let, the percentage of milk in mixture P be x%
Using allegation,


360 – 5x = 85
5x = 275
x = 55
So, the percentage of milk in P is 55%
Hence, Option D is correct.

Let CQ = x and AQ = 21 ‒ x
In ∆ABC,



In ∆ACD,




x = 12 cm

PQ = 13.7 cm
Hence, option D is correct.

Total number of ways = 2 × 2 × 2 × 2 × 2 = 32
Possibility of all tails = (T, T, T, T, T) = 1
So, the probability of at least one head

Hence, Option C is correct.

ar ΔAOB ~ ar ΔCOD


x² – 9x + 8 = 0
On solving, we get x = 8 cm.
Hence, Option D is correct.

Given Statements:

All plains are mountains.

No sea is a mountain.

Some valleys are seas.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some valleys are plains - False (Because, no definite relation is given between valleys and plains so it can only be a possibility).

II. No valley is a mountain - False (Because, no definite relation is given between valleys and mountains so it can only be a possibility).

III. No sea is a plain - True (Because, no sea is a mountain implies that no sea can not be a plan, as all plains are mountains).

So, Only conclusion III follows.

Hence, the correct answer is "Option 1".

The equation we get after interchanging the signs:

→ [{(14 - 6) + (4 × 3)} ÷ (6 + 4)] × 3 =?

→ [{8 + 12} ÷ 10] × 3 =?

→ [20 ÷ 10] × 3 =?

→ 2 × 3 =?

→ 6 =?

Hence, the correct answer is "6".

The pattern followed here is:

1) V and P are diagonally opposite and V is facing south.

2) T is facing W, but is not in the same row as that of P.

(It means T does not sit in same row where P is sitting)

3) W is between S and R.

4) R is facing U.

(It means R sits immediate right of T)

Clealry, P is facing Q.

Hence, the correct answer is "P".

The pattern in the given series is:

Logic: (First term ÷ 2) + (Second ÷ 3) = Third term

Let's apply the pattern in each row:

First row: 32, 51, 33
(First term ÷ 2) + (Second ÷ 3) = Third term
(32 ÷ 2) + (51 ÷ 3) = 33
16 + 17 = 33
33 = 33

Second row: 28, 33, 25
(First term ÷ 2) + (Second ÷ 3) = Third term
(28 ÷ 2) + (33 ÷ 3) = 25
14 + 11 = 25
25 = 25

Third row: 16, 27, ?
(First term ÷ 2) + (Second ÷ 3) = Third term
(16 ÷ 2) + (27 ÷ 3) = Third term
8 + 9 = Third term
17 = Third term

Hence the correct answer is 17.

The pattern followed here is:

Logic: First number × 10 = Second number & Second number × 20 = Third number.

Let's check each option one by one:

Option 1) (6 – 60 – 1200)

⇒ 6 × 10 = 60 & 60 × 20 = 1200 (Correct)

Option 2) (7 – 70 – 1600)

⇒ 7 × 10 = 70 & 70 × 20 = 1400 ≠ 1600 (Incorrect)

Option 3) (5 – 50 – 1000)

⇒ 5 × 10 = 50 & 50 × 20 = 1000 (Correct)

Option 4) (4 – 40 – 800)

⇒ 4 × 10 = 40 & 40 × 20 = 800 (Correct)

Hence, the correct answer is "(7 – 70 – 1600)".

Option 2.Only assumption II is implicit.

The statement shares that the traditional methods of teaching faced a real tough time during the Pandemic(offline mode of teaching is being referred to).

From what is supplied it cannot be assumed that the conventional methods of teaching are unfit and not suitable for the modern age.(Since the statement limits itself to the pandemic )Hence, assumption I is not implicit.

But, from what is given it can be assumed that the existing teaching system failed to perform during the Pandemic. Therefore, assumption II is implicit.

Hence, option 2 is correct.

Note: Robust means capable of performing without failure under a wide range of conditions

Given Statements:

All bottles are bags.

Some bags are pencils.

Some pencils are sharpeners.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some sharpeners are bottles - False (Because, there is no definite relation given between sharpeners and bottles, so it can be possible only).

II. Some bottles are pencils - False (Because, there is no definite relation given between Bottles and Pencil, so it can be possible only).

So, Neither conclusion I nor II follows.

Hence, the correct answer is "Option 2".

Given Statement: X < U = F > R ≥ P < T < W

According to the given information:

X is less than U, U equal to F, and F is greater than R. And R is greater equal to P, P is less than T and T is less than W.

Conclusions

I: X < T → False (X < U = F > R ≥ P < T; A definite relation cannot be established between X and T)

II:​ R > W → False (R ≥ P < T < W ; A definite relation cannot be established between R and W)

Hence the correct answer is "Neither conclusion I nor II is true".

The logic followed here is:

Square of the first number = second number.

i) 28 : 784 ⇒ 28² = 784;

ii) 19 : 361 ⇒ 192 = 361;

iii) 23 : 539 ⇒ 232 = 529 ≠ 539;

iv) 17 : 289 ⇒ 172 = 289;

Hence, '23 : 539' is different from the rest.

Hence, option D is correct.

Hence, Option C is correct.

The anger management is the key in emotional intelligence. Reducing conflict is the best strategy and therefore Option C is the best way forward since it suggests win-win solution.

Hence, Option C is the correct answer.

On applying above operations in option B we get,
18 + 36 ÷ 108 × 24 – 7 = 23
Now if we interchange ‘+’ and ‘×’
We get,
18 × 36 ÷ 108 + 24 – 7 = 23
6 + 24 – 7 = 23
23 = 23
Hence, L.H.S = R.H.S
Therefore (+ Means $) and (× Means %) must be interchanged.
Hence, Option B is correct.

Checking C1: Some White is Cat
We can see that in the Venn diagram, there is no definite relation between White and Cat, so conclusion 1 doesn’t follow.
Checking C2: .Some Cat is not Black
We can see that in the Venn diagram, there is no definite relation between Cat and Black, so conclusion 2 doesn’t follow.
Hence, Option C is correct.

18 × 2 + 4 = 40
40 × 2 + 4 = 84
84 × 2 + 4 = 172
172 × 2 + 4 = 348
348 × 2 + 4 = 700
Hence, Option C is correct.