NDA Mock Test: Mathematics - 6 - NDA MCQ

A = {(a, b) ∈ R × R : |a − 5| < 1 and |b − 5| < 1}

⇒ a ∈ (4, 6) and b ∈ (4, 6)

Therefore, A = {(a, b): a ∈ (4, 6) and b ∈ (4, 6)}

Now, B = {a, b) ∈ R × R : 4 (a − 6) ² + 9(b−5) ² ≤ 36}

From the conditions above for set A, the maximum value of B is:

4 (a − 6) ² + 9(b−5) ² = 4 (4 − 6)² + 9(6−5)² = 25 ≤ 36

We can check for other values as well.

Elements of set A satisfy the conditions in Set B.

Hence, A is a subset of B.

Given:

Statement 1: (a + b) ∝ (a - b)

Statement 2: a ∝ b

Calculation:

Statement: 1 (a + b) ∝ (a - b)

⇒ (a + b) = k (a - b), (where k is the proportionality constant)

On squaring both sides, we get

(a + b)² = k² (a - b)²

⇒ = k²

⇒ = k2

By using componendo & dividendo method, we get

⇒ =

⇒ =

⇒ (a² + b²) = (2ab)

Thus, (a2 + b2) ∝ ab, Where & 2 is constant.

Hence, the statement 1 is correct.

Statement: 2 a ∝ b

⇒ a = kb, (where k is the proportionality constant)

⇒ a/b = k

By using componendo & dividendo method, we get

⇒ (a + b)/(a - b) = (k + 1)/(k - 1)

On squaring both sides,

⇒ (a + b)²/(a - b)² = (k + 1)²/(k - 1)²

⇒ (a² + b² + 2ab)/(a² + b² - 2ab) = (k + 1)2/(k - 1)2

Again using componendo & dividendo method, we get

⇒ 2 (a² + b²)/4ab = [(k + 1)2 + (k - 1)2]/[(k + 1)2 - (k - 1)2]

⇒ (a2 + b2) = ab [(k + 1)2 + (k - 1)2]/[(k + 1)2 - (k - 1)2]

Here, [(k + 1)2 + (k - 1)2]/[(k + 1)2 - (k - 1)2] is constant.

⇒ (a2 + b2) = ab

Therefore, (a2 - b2) ≠ ab

Hence, the statement 2 is incorrect.

∴ Only 1st statement is correct.

Given:

(a + b + c)x² - (2a + 2b)x + (a + b - c) = 0

Concept:

For quadratic equation Ax² + Bx + C = 0

Sum of roots (α + β) = -B/A ,

Product of root (αβ) = C/A

Calculation:

Statement: 1

If we take a = b = c = 1

⇒ (1 + 1 + 1)x² - (2 × 1 + 2 × 1)x + (1 + 1 - 1) = 0

3x² - 4x + 1 = 0

3x² - x - 3x + 1 = 0

(3x - 1)(x - 1) = 0

x = 1 & 1/3

Hence, statement 1 is correct.

Statement:2

If we take a = b = c = -1

⇒ (-1 -1 -1)x² - [2 × (-1) + 2 × (-1)]x + [-1 - 1 - (-1)] = 0

⇒ -3x² + 4x - 1 = 0

⇒ -3x² + 3x + x - 1 = 0

⇒ -3x(x - 1) + 1(x - 1) = 0

⇒ (-3x + 1)(x - 1) = 0

⇒ x = 1/3 & x = 1

Hence, statement 2 is incorrect.

∴ Only statement 1 is correct

Explanation:

Statement 1:

We know that, any angle in a major sector is always acute.

Therefore statement 1 is correct.

Statement 2:

We know that, if two sides of a pair of opposite sides of a cyclic quadrilateral are equal, then its diagonals are also equal.

∴ The correct option is C.

Given:

Age relations:

D = 3 × A

B - C = E - G = D - E

(D + G)/2 = 16, (A + E)/2 = 11 and (B + C)/2 = 11

Weight relations:

Weight of B = Weight of C

A = B - 10, D = E + 4, E = F + 4 and F = G + 4

Age : Weight of D = 9 : 20

Age : weight of A = 2 : 5

Formula used:

Average = Sum of observations/Number of observations

Calculations:

Age : weight of D = 9 : 20

So, Weight will be a multiple of 20 and weight can not exceed 40

Let the weight of D = 40 kg

So, age of D = 9 × 2 = 18 years

Also, age of A = D/3 = 18/3 = 6 years

So, the weight of A = 5 × 3 = 15 kg

Weight of B = A + 10 = 25 kg

Weight of B = Weight of C = 25 kg

Age of E - Age of G = 16 - 14 = 2 years

Also from the given conditions,

B - C = 2 -----(1)

B + C = 22 -----(2)

From equation(1) and equation(2)

⇒ 2C = 20

⇒ C = 20/2 = 10

So, Age of B = 22 - 10 = 12

Hence, the Age of F can not be determined.

Now, the Average weight of D and F = (40 + 32)/2

⇒ 72/2 = 36 = Age of E

Now, A has a minimum age, and D has a maximum age

So, the weight difference is the maximum for D and A.

Hence, 1, 2 and 3 are correct.

Given:

Statement 1

m = (1 - p) (p2 + p + 1)

n = (p + 1) (p2 - p + 1)

Statement 2

m = pn

Formula:

a³ - b³ = (a - b) (a² + b² + ab)

a³ + b³ = (a + b)(a² + b² - ab)

Calculation:

Statement: 1

m = (1 - p) (p2 + p + 1)

By using the above identities

m = (1 - p3)

n = (p + 1) (p2 - p + 1)

By using the above identities

n = (p3 + 1)

Since, we don't know wether p is +ve, -ve, integer or fraction

We can not conclude the relation on m & n

Statement: 2 m = pn

Here also, relation b/w m & n depend on only P. therefore

∴ The correct answer is option D.

Given:

One of the roots is

Statement-1 The product of the roots is

Statement-2 The sum of roots of the quadratic equation is -1.

Calculation:

According to the first statement

Let, the other root = a

⇒ a × =

⇒ a = ×

This means the first statement is sufficient to answer.

According to the second statement

Let, the other root = a

⇒ + a = -1

⇒ a = -1 -()

This means the second statement is sufficient to answer.

∴ The correct answer is option B

Given:

A 3 digit number which is divisible by 10.

Calculation:

Let the three digit number be "xy0"

x = digit of hundred's place and y = digit of ten's place

According to Statement-1

When the digits of hundred's and ten's place interchanged, then

⇒ xy0 - yx0 = 180

⇒ (100x + 10y + 0) - (100y + 10x + 0) = 180

⇒ 90x - 90y = 180

⇒ x - y = 2 -----(1)

Statement-1 alone is not sufficient to get the values.

Now from Statement-2

Digit in hundred's place is halved and ten's and unit place interchanged,

⇒ xy0 - (x/2)0y = 336

⇒ (100x + 10y + 0) - (100x/2 + 10 × 0 + y) = 336

⇒ 100x + 10y - 50x - y = 336

⇒ 50x + 9y = 336 -----(2)

We got two linear equations using both the statement,

∴ Both statements are required to find the value of x and y.

∴ Question can be answered by using both the statements together.

Given:

x = y = z and x, y, and z are real numbers

Formula used:

x2 + y2 + z2 - xy - yz - zx = 1/2[(x - y)² + (y - z)² + (z - x)²]

x3 + y3 + z3 - 3xyz = (x + y + z) (x² + y² + z² - xy - yz - zx)

Calculation:

From statement (1)

We can write the identity as,

x2 + y2 + z2 - xy - yz - zx = 1/2[(x - y)2 + (y - z)2 + (z - x)2]

Now put x = y = z, we get

⇒x2 + y2 + z2 - xy - yz - zx = 1/2 (0 + 0 + 0)

⇒ x2 + y2 + z2 - xy - yz - zx = 0

∴ The given question can be answered by this statement.

From statement (2)

x3 + y3 + z3 - 3xyz = (x + y + z) (x2 + y2 + z2 - xy - yz - zx)

i.e.

(x + y + z) (x2 + y2 + z2 - xy - yz - zx) = 0

thus, We observe two cases:

Case (I): (x + y + z = 0)

This doesn't necessarily imply x = y = z.

e.g. x = 1, y = 1 and z = -2 satisfied x + y + z = 0 but not equal

case (II): (x2 + y2 + z2 - xy - yz - zx) = 0

This is the same as Statement-I, which we've shown leads to (x = y = z).

Thus, from the statement (II) we can not say x = y = z

Hence, if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.

∴ The correct answer is option (A).

Given:

Statements x + z/y = z/x and z - y/x = x/z

Calculation:

From statement-1 i.e. x + z/y = z/x

⇒ x(x + z) = yz -----(1)

⇒ x² + xz = yz

⇒ x² = z(y - x) -----(2)

Statement-1 alone is not sufficient to answer the question.

From statement-2 i.e. z - y/x = x/z

⇒ x² = z(z - y) -----(3)

Equating eq(2) and eq(3)

⇒ z(y - x) = z(z - y)

⇒ y - x = z - y

⇒ z + x = 2y -----(4)

Divide eq(1) by eq(4)

⇒  x(x + z)/z + x = yz/2y

⇒ x = z/2, which means x : z = 1 : 2

Put x = z/2 in eq(4)

⇒ 2y = z + x = 1 + 2 = 3

⇒ y = 3/2

Hence, x : y : z = 1 : 3/2 : 2 = 2 : 3 : 4

∴ The question can be answered by using both statements together, but cannot be answered using either statement alone.

Concept:

For two number a & b

LCM × HCF = a × b

Calculation:

Given that

Statement 1: LCM of the two numbers is 144

Statement 2: One of the numbers is 72

We can see, to find the other natural number, we need HCF also so that we can apply in the relation

LCM × HCF = a × b

∴ The correct answer is option D

Formula used:

a3 + b3 + c3 - 3abc = (a + b +c)(a2 + b2 + c2 - ab - bc - ac)

Calculation:

a2 - bc = α ] × a

a3 - abc = aα ------(1)

b2 - ac = β ] × b

b3 - abc = bβ ------(2)

c2 - ab = γ ] × c

c3 - abc = cγ ------(3)

Adding equation (1),(2) and (3)

a3 + b3 + c3 - 3abc = aα + bβ + cγ

a2 - bc + b2 - ac + c2 - ab = α + β + γ

⇒ =

∴ = 1

Calculation:

(x - 1)³ = x³ - 3x² + 3x - 1

∴ Other factor will be x + 1
Confusion Points
Focus only on the constant terms of the expression (α - 4)x³ + βx² + (γ - 2)x. Try to find a factor to divide it by, such that subtracting it from -1 makes the expression equal to zero

Shortcut Trick
⇒ 11(x + y) = 55

⇒ x + y = 5 -----(1)

⇒ 9(x - y) = 45

⇒ x - y = 5 -----(2)

From equation (1) and (2)

⇒ x × y = 5 × 0 = 0

∴ The correct answer is 0.

Solution:-
Calculation:

Let 1st digit = x & 2nd digit = y

2-digit number = 10x + y

Reversing the order of the digits of the number = 10y + x

According to question

The sum of the number and the number obtained by reversing the order of the digits of the number = 55

⇒ 10x + y + 10y + x = 55

⇒ 11x + 11y = 55

⇒ x + y = 5 ---------(1)

The difference of the given number and the number obtained by reversing the order of the digits of the number = 45

⇒ 10x + y - 10y - x = 45

⇒ 9x - 9y = 45

⇒ x - y = 5 -----------(2)

Adding Equation (1) & (2)

⇒ x + y + x - y = 10

⇒ 2x = 10 ⇒ x = 5

Put the value of x in equation (1)

⇒ 5 + y = 5 ⇒ y = 0

The product of the digits = 5 × 0 = 0

∴ The correct answer is 0.

Given:

A and B can finish a work = 10 days

B and C can finish a work = 12 days

C and A can finish a work = 15 days

Concept Used:

Time taken = total work/efficiency to complete work

LCM of (10, 12,15) = 60

Calculation:

According to question

1/A + 1/B = 1/10 --------(1)

1/B + 1/C = 1/12 ---------(2)

1/C + 1/A = 1/15 ---------(3)

Adding equations (1),(2) & (3)

2(1/A + 1/B + 1/C) = 1/10 + 1/12 + 1/15

2(1/A + 1/B + 1/C) = (6 + 5 + 4)/60 = 15/60 = 1/4

(1/A + 1/B + 1/C) = 1/8

Efficiency to complete work of A,B & C = 8 days

A, B, and C together finish half of the work = 8/2 = 4 days

∴ The correct answer is 4 days.

Shortcut TrickTotal work = LCM of (10, 12,15) = 60

Efficiency of A + B = 60/10 = 6

Efficiency of B + C = 60/12 = 5

Efficiency of A + C = 60/15 = 4

Efficiency of 2 (A + B + C) = 6 + 5 + 4 = 15

Efficiency of A + B + C = 15/2 = 7.5

Time taken by A + B + C to complete half work = Half work/Efficiency

⇒ 30/7.5 = 4 days

∴ The correct answer is 4 days.

Shortcut Trick
R = 12/4 = 3% and T = 9/3 = 3 year

Interest = (10000 / 1000000) × 92727 = 927.27

∴ The correct answer is Rs. 927.27.

Alternate Method
Given:

Amount = Rs. 10,000

Rate,R = 12%

Time ,n = 9 months

Concept used:

For quarterly, Rate = R/4 & Time = n × 4

Calculation:

R = 12%/4 = 3% & n = 9 × 4 = 36 months ⇒ 3 years

The interest of 1^st year = 10000 × 3% = 300

The interest of 2^nd year = 300 + 300 × 3% = 309

The interest of 3^rd year = 309 + 309 × 3% = 318.27

Total interest after 3 years = 300 + 309 + 318.27 = Rs. 927.27

∴ The correct answer is Rs. 927.27

Given:

Sinθ = (a + b)/2√ab

Concept:

-1 ≤ sinθ ≤ +1

Calculation:

-1 ≤ (a + b)/2√ab ≤ +1

We solve one of them

(a + b)/2√ab ≤ +1

⇒ a + b ≤ 2√ab

⇒ a + b - 2√ab ≤ 0

⇒ (√a - √b)² ≤ 0

⇒ √a = √b = a = b

∴ The correct answer is a = b

Shortcut Trick We can solve this question by option to satisfy equation

put a = b

then Sinθ = (a + a)/ 2√(a × a)

Sinθ = 2a/2a = 1 (sinθ ≤ 1)

A = {2, 3, 4, 8, 10}
B = {3, 4, 5, 10, 12}
C = {4, 5, 6, 12, 14}
► (A ∩ B) = {3,4,10}
► (A ∩ C) = {4}
► (A ∩ B) U (A ∩ C) = {3,4,10}

y is defined  If -x ≥ 0, i.e.

By definition, The Signum function =  

The given function is defined as Signum function. i.e.

domain is R and Range is {-1 , 0 , 1}.

f(a+1)−f(a−1)
=(a+1)−(a+1)2−{(a−1)−(a−1)2} = 2−4a(a−1)2} = 2−4a

A minor, Mij, of the element aij is the determinant of the matrix obtained by deleting the ith row and jth column.

Δ = Sum of products of element of row(or column)  with their corresponding co-factors.
Δ = a11 A11 + a21 A21 + a31 A31

Concept:
If a number ω = a + ib is purely imaginary, then

• a = 0 
• and ω =

Calculation:
Given, z is a complex number such that is purely imaginary,
Let = a + ib, then a = 0
And

As per question,

⇒ n(n - 1) (n + 1 -n + 2)= 126
⇒ n (n – 1) = 42 ⇒ n (n – 1) = 7 × 6  ⇒ n = 7.

If we see the blocks in terms of lines then there are 2m vertical lines and 2n horizontal lines. To form the required rectangle we must select two horizontal lines, one even numbered (out of 2, 4, .....2n) and one odd numbered (out of 1, 3....2n–1) and similarly two vertical lines. The number of rectangles is
^mC₁ .  ^mC₁ . ⁿC₁ . ⁿC₁ = ^m2n₂

Lets first place the men (M). '*' here indicates the linker of round table
 
* M -M - M - M - M *
which is in (5-1)! ways
So we have to place the women in between the men which is on the 5 empty seats ( 4 -'s and 1 linker i.e * )
So 5 women can sit on 5 seats in (5)! ways or
1st seat in 5 ways
2nd seat 4
3rd seat 3
4th seat 2
5th seat 1
i.e 5*4*3*2*1 ways
So the answer is 5! * 4! = 2880

Total no of balls = 9
red balls = 4
yellow balls = 3
green balls = 2
Total no. of arrangements = 9!/(4!*3!*2!)
= 1260