NDA Mock Test: Mathematics - 9 - NDA MCQ

Calculation:

We need to count the number of trailing zeros in that product.

It is obtained by multiplying a factor of 10, which is equivalent to multiplying

by 2 and 5.

Since every even number contributes a factor of 2 and every multiple of 5

contributes a factor of 5, we need to determine the number of multiples of 5

among the numbers from 1 to 29.

The numbers 5, 10, 15, 20, and 25 have one factor of 5 each, while 25 contributes

an additional factor of 5. Hence, there are 6 factors of 5 in total.

Therefore, the largest power of 10 that divides the product of the numbers

from 29 to 1 is 10⁶.

∴ The largest power of 10 that divides the given product is 6.

Calculation:
We have 65⁹⁹
Remainder of 65⁹⁹/11
Remainder of (-1)⁹⁹/11
Remainder of -1/11 = 11 - 1 = 10
∴ Required remainder = 10.

Calculation:
Let, x number of persons taking milk only.

According to the question
60 + 15 + (x - 5) + (50 - x) + x = 150
120 + x = 150
x = 150 - 120 = 30
∴ The required value is 30.

Concept:

Total investment = Investment × Time of Investment

Calculation:

A's total investment = 2 × 6 = 12 unit

B's total investment = 3 × 5 = 15 unit

C's total investment = 4 × 4 = 16 unit

D's total investment = 5 × 3 = 15 unit

E's total investment = 6 × 2 = 12 unit

C has the highest total investment, which is 16 unit.

Since, the profit is directly proportional to the total investment.

Therefore, C would receive the highest amount of profit.

Concept:
Divisibility rule of 5: A number is divisible by 5 only if the unit digit is 0 or 5.
Calculation:
Let n - 1, n & n + 1 be 6, 7 & 8.
⇒ n² + 1 = 7² + 1 = 50
⇒ n² - 1 = 7² - 1 = 48
⇒ n² + n = 7² + 7 = 56
⇒ n² - 7 = 7² - 7 = 42
∴ 5 divides n² + 1

Concept:

Divisibility rule of 11: The given number can only be completely divided by 11 if the difference between the sum of digits at the odd position and the sum of digits at the even position in a number is 0 or 11.

Calculation:

Let the required number be n.

According to the question,

n the remainder 7 when divided by 11. Therefore, n - 7 will be completely divisible by 11.

Now, let's check the options

Option:1 99981 - 7 = 99974 which is not divisible by 11.

Option:2 99988 - 7 = 99981 which is not divisible by 11.

Option:3 99997 - 7 = 99990 which is divisible by 11.

Option:4 99999 - 7 = 99992 which is not divisible by 11.

∴ The required value is 99997.

Formula used:

Present Value = Future Value /

Calculation:

Given that

Future Value = Rs. 20,000

Interest Rate = 5%, Time = 5 years

Present Value of A's debt = 20,000 /[1 + (0.05 × 5)]

⇒ 20,000 / (1 + 0.25) = Rs. 16,000

Again,

Future Value = Rs. 12,000,

Interest Rate = 5% , Time = 4 years

Present Value of B's debt = 12,000 /[1 + (0.05 × 4)]

⇒ 12,000 / (1 + 0.20) = Rs. 10,000

Therefore, In the settlement,

A needs to give B (16,000 - 10,000) = Rs. 6,000.

Concept:
1. For the quadratic equation ax² + bx + c = 0
Sum of root (α + β) = -b/a
Product of root = c/a
2. a² + b² = (a + b)² - 2ab
3. a⁴ + b⁴ = (a² + b²)² - 2(ab)²
Calculation:
x² - 7x + 1 = 0
As α & β be roots of the quadratic equation
α + β = -(-7)/1
α + β = 7
αβ = 1
By using the above identity
α² + β² = (α + β)² - 2αβ = 7² - 2
α² + β² = 47
Now we can use the identity:
α⁴ + β⁴ = (α² + β²)² - 2α²β²
Substituting in the value of α² + β² and αβ = 1, we get:
α⁴ + β⁴ = (47)² - 2 = 2207
∴ The value of α⁴ + β⁴ is 2207.

Concept:
For the number "a", which can be written as,

The total number of positive factors is given as,
T(a) = (a₁ + 1)(a₂ + 1)
Here, p₁ and p₂ are the prime numbers.
Sum of factors =
Calculation:
360 = 2³ × 3² × 5¹
Here, 2, 3 & 5 are the only prime number.
Total number of factor = (3 + 1) × (2 + 1) × (1 + 1) = 24

Sum of factors =

Sum of factors = (2⁴ - 1) × [(3³ - 1)/2] × [(5² - 1)/4]
Sum of factors = (15) × (13) × 6
Sum of factors = 1170
∴ The number of factors of 360 is 24 and the sum of all factors is 1170.

Calculation:

XYXYXY = 100000X + 10000Y + 1000X + 100Y + 10X + Y

XYXYXY = (100000X + 1000X + 10X) + (10000Y + 100Y + Y)

XYXYXY = 101010X + 10101Y

XYXYXY = 10101 × (10X) + 10101(Y)

XYXYXY = 10101.[10X + Y]

Prime factors of 10101 are 3 x 7 x 13 x 37

XYXYXY = 3 x 7 x 13 x 37 (10X + Y)

∴ XYXYXY is divisible by 3, 7, 13 & 37.

Concept:
HCF: It is the product of the smallest power of the common factors involved in
the prime factorization.
HCF of (a^m - 1) & (aⁿ - 1) is [a^{HCF (m, n)} - 1]
Calculation:
3²⁹ - 9 = 3² (3²⁷ - 1)
3³⁸ - 9 = 3² (3³⁶ - 1)
By using the above concept
HCF = 3² [3^{HCF (27, 36)} - 1]
Let's find the HCF of (27, 36) first
27 = 3³
36 = 2² × 3²
HCF of (27, 36) = 3²
Now, the required value
HCF = 3² (3⁹ - 1) = 3¹¹ - 9
Mistake Points
Do not forget to multiply with 3² as this will also be the common prime factor.

Calculation:

Given that

Q = P + 3

Q = 2S

We can equate these and solve for P and S.

P + 3 = 2S

⇒ P = 2S - 3 ----(1)

We also know that

R = 2P and R - S = 30,

We can equate these and solve for P and S.

2P = S + 30

⇒ P = (S + 30)/2 ------(2)

2S - 3 = (S + 30) / 2

⇒ 4S - 6 = S + 30

⇒ 3S = 36

⇒ S = 12

⇒ P = 2 × 12 - 3 [From equation (1)

⇒ P = 24 - 3 = 21

⇒ Q = P + 3 = 21 + 3 = 24

Now, the sum of the ages of P and Q

P + Q = 21 + 24 = 45 years.

Concept:
LCM: LCM is the product of the greatest power of each prime factor, involved in the numbers.
Calculation:
LCM of 15, 25, 35, and 45.
15 = 3 × 5
25 = 5 × 5
35 = 5 × 7
45 = 3 × 3 × 5
LCM = 3² × 5² × 7 = 1575 min = 26.25 hr
It menas, bell ring at every 26.25 hr
Since, 72/26.25 = 2.74
Therefore, In 72 hr, bell rings 2 times.

Concept:

For the sum of four numbers to be even, you either need to have

• All four numbers be even
• All four numbers be odd
• Two of them be even and two of them be odd

Explanation:

a + b + c + d = 200 (even)

As discussed above

Case:1 All four numbers be even

S = 1 + 1 + 1 + 1 = 4

Case:2 All four numbers be odd

S = - 1 - 1 - 1 - 1 = - 4

Case:3 Two of them be even and two of them be odd

S = 1 + 1 - 1 - 1 = 0

∴ There will be three possible values of S.

Concept:
aⁿ - bⁿ, (where n is even number) is always divisible by (a - b) & (a + b)
Calculation:
We have 97³⁰ - 14³⁰
By using the above concept,
97³⁰ - 14³⁰ is divisible by
(97 - 14) = 83
And
(97 + 14) = 111 = 37 × 3
∴ 97³⁰ - 14³⁰ is divisible by both 37 and 83.

Calculation:
Statement:1 n³ - n is divisible by 6.
n³ - n = n(n² - 1) = n(n - 1)(n + 1)
This product is the product of three consecutive integers which is always divisible by 6
Statement:2 n⁵ - n is divisible by 5
n⁵ - n = n(n⁴ - 1)
We know that (a² - b²) = (a - b)(a + b)
n⁵ - n = n (n² - 1)(n² + 1)
n⁵ - n = n (n² - 1)[(n² - 4) + 5)]
n⁵ - n = n (n² - 1)(n² - 4) + 5 n (n² - 1)
n⁵ - n = n(n - 1)(n + 1)(n - 2)(n + 2) + 5 n (n - 1)(n + 1)
n⁵ - n = [(n - 2)(n - 1) n (n + 1)(n + 2)] + 5 n (n - 1)(n + 1)
Since the first part is the product of five consecutive numbers which will always be divisible by 5. Therefore,
(n² - n) is divisible by 5.
Statement:3 n5 - 5n3 + 4n is divisible by 120.
n⁵ - 5n³ + 4n
n(n⁴ - 5n² + 4)
n(n² - 4)(n² - 1)
n(n - 2)(n + 2)(n - 1)(n + 1)
(n - 2)(n - 1) n (n + 1)(n + 2)
Which is always divisible by 120.
∴ All three statements are correct.

Formula used:

1. (a³ - b³) = (a - b)³ + 3ab(a -b)
2. (a - b)² = a² - 2ab + b²
3. (a + b)² = a² + 2ab +b²
4. (a - b)(a + b) = a² - b²

Calculation:
Given that

By using the formula (2), (3) & (4)

⇒ x - y =
⇒ x - y = 2√3 -----(3)
We know that
(x³ - y³) = (x - y)3 + 3xy(x -y)
⇒ (x³ - y³) = (2√3)³ + 3 × 1 (2√3)
⇒ (x³ - y³) = (2√3)³ + 3 × 1 (2√3)
⇒ (x³ - y³) = 8 × 3√3 + 6√3
∴ (x³ - y³) = 30√3

Concept:
The factor of (aⁿ - bⁿ) is always divisible by (a - b) & (a + b).
Calculation:
1 - x - xⁿ + xⁿ⁺¹
(1 - x) - xⁿ(1 - x)
(1 - x)(1 - xⁿ)
Since (1 - xⁿ) is the factor. Therefore
Statement 2 is correct.
By using the above concept
(1 - xⁿ) = (1 - x)(1 + x). k
Where k is the remaining factor.
1 - x - xⁿ + xⁿ⁺¹ = (1 - x)(1 - x)(1 + x).k
1 - x - xⁿ + xⁿ⁺¹ = (1 - 2x + x²)(1 + x).k
∴ Statement 1 is also correct.

Concept used:
In question there are 4 variables (a, b, x and y) and only 2 equation. Hence we put any value to a and b to get value of x and y
Calculation:
Let a = 0 and b = 1

Again,

Now,

∴ The required value is b².

Formula used:

(x + )² = (x² + ) + 2

(x + ) =

Calculation:

Given that

tan⁸ θ + cot⁸ θ = m

Since, cot θ = 1/tan θ. Therefore,

tan⁸θ + (1/tan²θ) = m

Let tan θ = n then

n⁸ + 1/n⁸ = m ---(1)

We know that , (x + ) =

⇒ (n⁴ + 1/n⁴) =

⇒ (n⁴ + 1/n⁴) =

Again applying the same identity

⇒ (n² + 1/n²) =

Again applying the same identity

⇒ (n + 1/n) =

But, n = tan θ

⇒ tan θ + 1/tan θ =

∴ tan θ + cot θ =

Formula used:

cos(30°) = √3/2

sin(30°) = 1/2

cot(30°) = √3

tan (90° - θ) = cot θ

tan θ × cot θ = 1

Calculation:

4 cos² 30° + 2x sin 30° - cot² 30° - 6 tan 15° tan 75° = 0

Since, tan (90° - θ) = cot θ

⇒ 4 cos² 30° + 2x sin 30° - cot² 30° - 6 tan (90° - 75°) tan 75° = 0

⇒ 4 cos² 30° + 2x sin 30° - cot² 30° - 6 cot 75° tan 75° = 0

But, tan θ × cot θ = 1

⇒ 4 cos² 30° + 2x sin 30° - cot² 30° - 6 = 0

Using the trigonometric values

⇒ 4(√3/2)² + 2x(1/2) - (√3)² - 6= 0

⇒ 3 + x - 3 - 6 = 0

∴ x = 6

Formula used:

Area of triangle = (1/2) × base × height

Caculation:

Area of ΔABC = Area of ΔABC

⇒ (1/2) × AC × AB = (1/2) × p × BC

⇒ AC × AB = p × BC

⇒ 5 × 12 = p × 13

∴ 13p = 60

Formula used:
The area of each surface of the cube = (side)² [each surface of a cube is a square]
Calculation:
Let the initial surface area of the cube = 600
then the initial area of each surface = 100
Initial length of side = √100 = 10 unit
Final Surface area of the cube = 600 × 125/100 = 750
Then the final area of each surface = 125
(Final length)² = 125
(Final length) = √125
Therefore, according to the question
P = [(√125 - 10)/10] × 100
P = 10(5√5 - 10)
P = 10(5 × 2.236 - 10) = 11.8
∴ 10 < P < 12

Formula used:

The diagonal of a cuboid is given by: √(L² + B² + H²)

The surface area of a cuboid is given by: 2(LB + BH + HL)

(a + b + c)² = a² + b² + c² + 2(ab + bc + ac)

Calculation:

By using the above formula

√(L² + B² + H²) = 11 cm

2(LB + BH + HL) = 240cm²

⇒ LB + BH + HL = 120cm²

We know that

(a + b + c)² = a² + b² + c² + 2(ab + bc + ac)

(L + B + H)² = 11² + 2 × 120 = 121 + 240 = 361

Taking the square root of both sides

∴ L + B + H = √361 = 19cm

Formula used:

The total surface area (A_Total) = 2πr(h + r)

The curved (or lateral) surface area (A_curved) = 2πrh

Calculation:

Given that

2 × Total Surface Area = 3 × Curved Surface Area

By using the above formula

⇒ 2 × 2πr(h + r) = 3 × 2πrh

⇒ 4πr(h + r) = 6πrh

⇒ 2(h + r) = 3h

⇒ 2h + 2r = 3h

∴ h = 2r.

Formula used:
HCF = The product of these common factors is the HCF of the given numbers.
Calculation:
Length of the floor = 30 m 60 cm = 3060 cm
Width of the floor = 23 m 40 cm = 2340 cm
Factor of 3060 = 2² × 3² × 5 × 17
Factor of 2430 = 2² × 3² × 5 × 13
HCF of 3060 & 2340 = 2² × 3² × 5 = 180
Total number of tiles required = (3060 × 2340) / (180 × 180)
∴ Total number of tiles required = 221

Given:

AB = x

∠APO = θ

Concept used:

Diagonal of square = √2a

cotθ = base/perpendicular

Calculation:

According to the above concept,

Diagonal of square = √2a

⇒ AC = x√2

⇒ OA = =

Now, according to the above figure,

In ΔOAP,

⇒ cotθ = =

⇒ cotθ = 2√2

∴ The required value of cotθ is 2√2

Concept:

Divisibility rule of 25: Last two digits of the number should be divisible by 25

Divisibility rule of 9: A number is divisible by 9 if and only if the sum of its digits is divisible by 9.

Calculation:

Given that,

A number 277XY5 (where X, Y are digits) is divisible by 25.

By using the above concept, the possible values of Y are 2 & 7.

Statement: I The given number is divisible by 9.

The sum of digits of 277XY5 = 21 + X + Y

If Y = 2 then sum = 21 + X + 2 = 23 + X

If it is divisible by 9, then X = 4

If Y = 7 then sum = 21 + X + 7 = 28 + X

If it is divisible by 9, then X = 8

So, we have two possible values of X

Statement: II X > 5

If we consider only statement II only, we will have information

In 277XY5, Y = 2 or 7

277X25 & 277X75 are divisible by 25.

Clearly, both numbers are divisible by 25 independent of X.

Let's combine both statements

From statement I, we have X = 4 & 8

From statement II, we have X > 5

Therefore, the value of X = 8.

Choose this option if the Question can be answered by using both Statements together, but cannot be answered by using either statement alone.

Concept:

For the quadratic equation ax² + bx + c = 0

Sum of root (α + β) = -b/a

Produt of root (αβ) = c/a

The quadratic equation can also be written as

k[x² - x (α + β) + α β] = 0

Where k is any integer.

Calculation:

We have

ax² + bx + c = 0 ----(1)

One of the roots is 2 (let α)

Statement: 1 Ratio of c to a is 1.

Product of root

αβ = c/a = 1

2 × β = 1

⇒ β = 1/2

The quadratic equation will be

k[x² - x (2 + 1/2) + 2 × (1/2)] = 0

k[x² - 5x/2 + 1] = 0

k[2x² - 5x + 2] = 0

On comparing it to equation (1)

a = 2k, b = -5k, c = 2k

So there is no unique value of a, b & c.

Statement:2 Ratio of b to a is (-5/2)

2 + β = 5/2

β = 1/2

So again there is no unique value of a, b & c.

∴ Option 4 will be correct.

Calculation:

We have p2 + q2 + q

Statement I: 2p + q is odd

For any value of p, 2p will be even

Since 2p + q is odd,

q must be odd.

Statement II: q - 2p is odd

Here also, q must be odd to make q - 2p even.

∴ Option 4 is correct.