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Answer is 1, 4, 9, 16, 25, 36,

\(1,1\frac{1}{2},2,2\frac{1}{2},3,3\frac{1}{2},4,........\)

Answer is 2,3,5,7,11,13,17,

Answer is 3,6,9,12,15,

Solution: 

If the number 12ⁿ, for any natural number n, ends with the digit 0 or 5, then it is divisible by 5. That is, the prime factorization of 12ⁿ contains the prime 5. This is not possible because prime factorisation of 12ⁿ = (2² x 3)ⁿ = 2²ⁿ x 3ⁿ; so the only primes in the factorisation of 12ⁿ are 2 and 3 and the uniqueness of the fundamental theorem of arithmetic guarantees that there are no other primes in the factorization of 12ⁿ. So, there is no natural number n for which 12ⁿ ends with the digit zero.

(b) The thousandth place can be filled up in 9 ways with any one of the digits 1, 2, 3, ...., 9. After that the other three places can be filled up in ⁹P₃ ways, with any one of the remaining 9 digits including zero. Hence, the number of four digit numbers with distinct digits = 9x⁹p_{3 .}

(d) The required number of ways
= (10 +1)(9 +1)(7 +1) -1 = 879.

(c) Since each question can be selected in 3 ways, by selecting it or by selecting its alternative or by rejecting it. Thus, the total number of ways of dealing with 10 given questions is 3¹⁰ including a way in which we reject all the questions.
Hence, the number of all possible selections is 3^{10 – 1}.

750x . (π/180) = (25π/6)C

Conjugate i(2 + i) = 2i + i² = -1 + 2i 

∴ conjugate Z̄ = -1 -2i.

Let the six digit number be abcdef
'a' can take all values except 0 since it will not be a 6 digit number then. So this means it can take any of 5 values.
'b' can take one the remaining 5 digits.
'c' can take any of the remaining 4
Similarly 'd' can take 3, 'e' can take 2 and 5'f' can take 1 value

So this means the total numbers that can be formed are :
5*5*4*3*2*1 = 600

For the numbers divisible by 10:
The last digit should be always 0
In this case the total possibilities are: 5*4*3*2*1*1 = 120

Amit walks 8 km in 1 hour 20 min

Or 1.1/3 = 4/3 hours

∴ Speed = Distance/Time

= 8/(4/3) = (8 × 3)/4 = 6 km/h

(a) (i) Distance covered in 10 minutes

= (6 × 1000 × 10)/ 60 = 1000 m = 1 km

(ii) Distance covered in 30 seconds

= (6 × 1000 × 30)/(60 × 60)  = 50 m

(b) (i) Time taken in 2500 m = 2500/(1000 × 6)

= 5/12 hours = 5/12 × 60 = 25 minutes

(ii) Time taken in 6.5 km

= 6.5/6 = 65/60 hours

= 1 hours 5 minutes

Correct option is: C. \(\frac{tan2x}{2}+k\)

Correct option (a) 18

Answer is (b) 17

Answer is (a) 0

Option: (d) √422

Correct option:(D)

Correct option(d) 0

Answer is (b) cosec^-1 x

lim_(x→0) = sinax/ax .a/b = a/b

lim_(x→1) f(x) = f(1) ⇒ lim_(x→1) 4x + 3 = k + 1

⇒ 4 + 3 = k + 1 ⇒ k = 6

\(\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} (1 + 3x)^\frac{1}{3} = e^3 = f(0)\)

∴ f(x) is continuous at x = 0

Answer is (a) 1

Gamete mother cells/cells which undergo meiosis to form gametes.

Answer is (c) e^1/e

Development of the egg into a individual without fertilization.

Answer is (a) (dy/dx) = 1

Binary Fission :
(i) Nucleus divides into two parts. 

(ii) It occurs during normal conditions. 

(iii) It gives rise to two individuals. 

(iv) Cytoplasm divides after each nuclear division.

(v) Includes definite pattern of division.
e.g., Amoeba.
Multiple Fission :
(i) Nucleus divides into many parts. 

(ii) It takes place during unfavourable conditions (Encysted stage), 

(iii) It forms many individuals. 

(iv) Cytoplasm does not divide after every nuciear division.

(v) Has no definite pattern of division.
e.g., Plasmodium.

Answer is (d) π/4

Correct option:

(B) 2 

Correct option:

(C) 1

Answer is (c) log (ex)

Answer is (a) y" + py = 0

Answer is (a) ∫e^Pdx

Answer is (d) 4

Let (x, 0) be a point on the x axis which is at equal distance from A and B.

(2 – x)² + (3 – 0)² = (5 – x)² + (4 – 0)²

(2 – x)² + 9 = (5 – x)² + 16

(2 – x)² + (5 – x)² = 16 – 9

4 – 4x + x² – 25 + 10x – x² = 7

6x = 28

x = \(\frac{28}{6}=\frac{14}{3}\)

∴ Point on x axis is \((\frac{14}{3},0)\)

\(P=(\frac{3+7}{2},7)=(5,7)\)

\(Q=(7,\frac{9+7}{2})=(7,8)\)

\(R=(\frac{3+7}{2},9)=(5,9)\)

\(S=(3,\frac{9+7}{2})=(3,8)\)

PQ = \(\sqrt{(7-5)^2+(8-7)^2}\) = \(\sqrt{2^2+1^2}\) = \(\sqrt{5}\)

QR = \(\sqrt{(5-5)^2+(9-8)^2}\) = \(\sqrt{(-2)^2+1^2}\) = \(\sqrt{5}\)

RS = \(\sqrt{(3-5)^2+(8-9)^2}\) = \(\sqrt{(-2)^2+(1)^2}\) = \(\sqrt{5}\)

SP = \(\sqrt{(5-3)^2+(7-8)^2}\) = \(\sqrt{2^2+1^2}\) = \(\sqrt{5}\)

AP = \(\sqrt{(4-2)^2+(5-3)^2}=2\sqrt{2}\)

PC = \(\sqrt{(6-4)^2+(7-5)^2}=2\sqrt{2}\)

AP + PC = 4√2 = AC

∴ P (4, 5) is a point on AC .

BP = \(\sqrt{(4-5)^2+(5-4)^2}=\sqrt{2}\)

PD = \(\sqrt{(3-4)^2+(6-5)^2}=\sqrt{2}\)

BP + PD = √2 + √2 = 2√2 = BD

∴ P (4, 5) is also a point on BD.

Midpoints of BC = \((6,\frac{5+1}{2})\) = (6,3)

Area of the rectangle ABCD = AB × BC 

= 4 × 2 

= 8 m²

Midpoints of AC = \((\frac{2+6}{2},\frac{1+5}{2})\) = (4,3)

Rhombus (SQ = 4, RP = 2, diagonals are not equal, sides are equal hence it is a rhombus)

AC = \(\sqrt{(6-2)^2+(7-3)^2}\) = \(\sqrt{35}\) = \(4\sqrt2{}\)

BD = \(\sqrt{(3-5)^2+(6-4)^2}\) = \(\sqrt{8}\) = \(2\sqrt2{}\)

AC ≠ BD

AB = |7 – 3| = 4 BC = |9 – 7| = 2

Radius of the circle = 6

The point on x-axis which is at a distance of 5 units from (4,-5) is (4, 0).

(x – 4)² + 52 = 25,

∴ x = 4

Point on x axis is (4, 0)

Point on y axis is (0, y).

(0 – 4)² + (y + 5)² = 25,

∴ y = –8, –2

The point on y -axis which is at a distance of 5 units from (4, –5) are (0, –2), (0, –8)