Mathematics (Paper-1) Mock Test - 2

a6a41 is the number.
(a + 6 + a + 4 + 1) should be divisible by 9.
Or, (2a + 11) should be divisible by 9.We know that a number is divisible by 9 if the sum of its digits is divisible by 9.
2a + 11 = 9 or 2a + 11 = 18 or 2a + 11 = 27
But 2a +11 = 9 or 2a + 11 = 18 is not possible, so
2a + 11 = 27 a = 8

Given: 17/197

In order to express it in standard form, we divide its numerator and denominator by the greatest common divisor.

For both 17 and 197, HCF is 1.

So, 17/197 is in standard form.

16^2.5 : 10³
So, (2)^4(2.5) : (2 × 5)³
= (2)¹⁰ : (2)³ × (5)³
= 2⁷ : 5³
= 128 : 125

Let the units digit be x.
Then, tens digit is x + 5.
Number = 10(x + 5) + x = 10x + 50 + x = 11x + 50
When the digits are reversed, the number is:
10x + x + 5 = 11x + 5
According to the question,
11x + 50 − 5(2x + 5) = 10x + x + 5
x + 25 = 11x + 5
10x = 20
x = 2
So, the number is 72.

Let the digit at unit's place be y and the digit at tens place be x.
A.T.Q.
x + y = 10....(1)
10x + y + 18 = 10y + x
This implies that 9y - 9x = -18
Therefore, y - x = 2......(2)
Adding equation (1) and (2), we get
2y = 12
y = 6
Put the value of y in equation (1).
Hence, x = 4
Thus, the original number = 10x + y = 10(4) + 6 = 46.

n² + n + 1 = n(n + 1) + 1
Now, if n is even, n + 1 is odd.
Thus, n(n + 1) is even = 2a (let)
n(n + 1) + 1 = 2a + 1, which is odd.
If n is odd, n + 1 is even.
Thus, n(n + 1) is even = 2b (let)
Thus, n(n + 1) + 1 = 2b + 1, which is odd.
Thus, the given expression is always odd.

Volume of tank = 59400 m³
Diameter = 60 m
Radius = Diameter / 2
= 60/2
= 30 m
Volume of the tank = πr²h
59400 = 22/7 × 30 × 30 × h
h = 59400 × 7/22 × 1/900
h = 21 m

Area of triangle:
A = 1/2 (B x H) where B is base, and H is height of triangle

Given: B = 8H
According to the triangle:

144 = 1/2 × (8H × H)
144 = 1/2 × (8H²)

144 = 4H²
36 = H²
So, H = 6 m and B = 8H = 8 × 6 = 48 m

Let the terms be x₁, x₂, x₃, ..., x₇.
Now from the information given in the question:
x₁ + x₂ + x₃ + x₄ + x₅ + x₆ + x₇ = 7 × 7 = 49
When 3 is added to each term, then
Total sum = 49 + 3 × 7 = 49 + 21 = 70
When each term is divided by 2, thenTotal sum = 70/2 = 35
So, the new mean, A = (35/7) = 5
So, the difference between them is:
7 - 5 = 2

Since the median is the middle value, so we will first have to rearrange the list in the increasing order:
13, 13, 13, 13, 14, 14, 16, 18, 21
There are nine numbers in the list, so the middle one will be the 5th number, as:
(9 + 1) ÷ 2 = 10 ÷ 2 = 5
13, 13, 13, 13, 14, 14, 16, 18, 21
So, the median is 14.

The multiples of 15 are:
15, 30, 45, 60
Favourable outcomes = 4
Total outcomes = 60
Now,
Probability = (Favourable outcomes)/(Total number of outcomes)
= 4/60 = 1/15

Since three metal cubes of volumes 125 cm³, 64 cm³ and 27 cm³ are melted to form a new cube,
Therefore, volume of the new cube = 125 + 64 + 27 = 216 cm³
Total volume = 216 cm³
a³ = 216 cm³
a³ = 6³
On comparing, we get
a = 6
Edge = 6 cm

Measure of each exterior angle of a regular polygon = 360º/n (where n = number of sides)
In a regular octagon, n = 8.
So, measure of each exterior angle of a regular octagon = 360º/8 = 45°

Let the CP be Rs. 100.
For 12% profit, SP should be Rs. 112.
After 55% discount, 45% of the MP = Rs. 112
Therefore,
MP = 100 / 45 × 112 = Rs. 248.9
Required % = 148.9 / 100 = 148.9%
 

(73)² - (27)²
Using (a² - b²) = (a - b)(a + b);
(73)² - (27)²
= (73 - 27) (73 + 27)
= (46) × (100)
= 4600

There are 2n non-perfect square numbers that lie between the squares of n and n + 1.
So, there are 196 non-perfect square numbers that lie between the squares of 98 and 99.

Every number can be expressed as the product of power of its prime factors.
If the power of all the prime factors is in the multiple of 3, then the number is said to be a perfect cube.
Using this concept, we have:
The prime factors of 13824 = 2⁹ × 3³ , which makes it a perfect cube.

Square numbers always end with digits 0, 1, 4, 5, 6, 9.
So, from looking at the units digits in the given options, we can say that only 48542 can definitely not be a square number.

Let x be the initial SP of the product.
According to the given condition:
1.2x - x = Rs. 800
0.2x = Rs. 800
x = Rs. 4000
The person purchased the product at (4000 - 800) = Rs. 3200.

Time taken by Ravi and Param to complete the work = 30 days
Time taken by Param to complete the work = 40 days
Ravi and Param's one-day work = 1/30
Param's one-day work = 1/40
Ravi's one-day work = {(Ravi and Param's one-day work} - {Param's one-day work}
= 1/30 - 1/4
= 4 – 3 / 120
= 1/120
So, Ravi alone can finish the work in 120 days.

Total time taken by the friends to reach = 7 a.m. to 1 p.m = 6 hr
Time for which the stopped = 4 × 30 min = 120 min = 2 hr
So, actual time taken to cover the distance = 6 - 2 = 4 hours
Average speed = (360/4) = 90 km/hr

y⁴ + 10y² + 25
Using (a + b)² = a² + b² + 2ab
= {(y²)² + 2 × 5y² + 25)}
= (y² + 5)²
Then, (y⁴ + 10y² + 25) ÷ (y² + 5)
= (y² + 5)² ÷ (y² + 5)
= y² + 5

We know,
(x + a)(x + b) = x² + (a + b)x + ab
(2x - 6)(2x - 5) = (2x)² + (-6 - 5)2x + (-6)(-5)
= 4x² - 22x + 30 ... (i)
(x + 6)(x - 5) = x² + (6 -5)x + (6)(-5)
= x² + x - 30 ... (ii)
Now, on adding (i) and (ii), we get
x² + x - 30 + 4x² - 22x + 30
= 5x² - 21x

a⁴ + 2a²b³ + b⁶
= (a²)² + 2a²b³ + (b³)²
= (a² + b³)²
= (a² + b³)(a² + b³)

It is given that 1200 participants in a college have enough food for 28 days.
Let the number of participants eliminated be 'x'.
After 4 days, number of participants = (1200 – x)
Now, the food will last for 32 more days.
Hence,
1200 × 24 = (1200 – x) × 32
⇒ (1200 – x) = 900
⇒ x = 1200 – 900 = 300