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Let A be the first term and D the common difference of A.P. 

Tp = a = A + (p − 1)D = (A − D) + pD ... (1) 

Tq = b = A + (q − 1)D = (A − D) + qD ... (2) 

Tr = c = A + (r − 1)D = (A − D) + rD ...(3) 

Here we have got two unknowns A and D which are to be eliminated. 

We multiply (1), (2) and (3) by q−r, r−p and p−q respectively and add: 

a(q - r) = (A – D)(q - r) + Dp(q - r) 

b(r - p) = (A - D) (r - p) + Dq(r - p) 

c(p - q) = (A - D) (p - q) + Dr(p - q) 

a(q − r) + b(r − p) + c(p − q) 

= (A − D)[q − r + r − p + p − q] + D[p(q − r) + q(r − p) + r(p − q)] 

= (A – D) (0) + D [pq - pr + qr – pq + rp – rq] 

= 0 

a_n = a + (n - 1)d 

4 = a + 6 x (-4) 

a = -28

a_n = a + (n - 1)d 

0 = 27 + (n - 1)(-3) 

30 = 3n 

n = 10 

10^th

Let the cost of 1 chair = Rs. x 

And the cost of 1 table = Rs. y 

3x + y = 1500 

6x + y = 2400 

\(\frac{3}{6}=\frac{1}{k}\) ≠ \(\frac{3}{8}\)

\(\frac{3}{6}=\frac{1}{k}\)

k = 2

Solution:
No, every positive integer cannot be only of the form 4q + 2.
Justification:
Let a be any positive integer. Then by Euclid’s division lemma, we have
a = bq + r, where 0 ≤ r < b

Putting b = 4, we get
a = 4q + r, where 0 ≤ r < 4
Hence, a positive integer can be of the form,
4q, 4q + 1, 4q + 2 and 4q + 3.​

Solution:
True.
Justification:
At least one out of every three consecutive positive integers is divisible by 2.
Therefore, The product of three consecutive positive integers is divisible by 2.
At least one out of every three consecutive positive integers is divisible by 3.
Therefore, The product of three consecutive positive integers is divisible by 3.
Since the product of three consecutive positive integers is divisible by 2 and 3.
Hence, It is divisible by 6 also.

Solution:
True.
Justification:
Let a, a + 1 be two consecutive positive integers.
By Euclid’s division lemma, we have
a = bq + r, where 0 ≤ r < b
For b = 2 , we have
a = 2q + r, where 0 ≤ r < 2 ...(i)
Putting r = 0 in (i), we get
a = 2q, which is divisible by 2.
a + 1 = 2q + 1, which is not divisible by 2.
Putting r = 1 in (i), we get
a = 2q + 1, which is not divisible by 2.
a + 1 = 2q + 2, which is divisible by 2.
Thus for 0 ≤ r < 2, one out of every two consecutive integers is divisible by 2.
Hence, The product of two consecutive positive integers is divisible by 2.

Correct answer is (D) 2q + 1

Total defective bulbs in the lot = 4 

Total number of bulbs in the lot = 20. 

Now probability that drawn bulb is defective

= \(\frac{Total \,defective\,bulbs\,in\, the\, lot}{Total\, number\, of\, bulbs\,in\, the\, lot} = \frac{4}{20} = \frac{1}{5}\)

Hence, the probability that drawn bulb is defective is \(\frac{1}{5}\)

Solution:
Any odd integer n is of the form 4m + 1 or 4m + 3.

=> n² – 1 = (4m + 1)² – 1
= 16m² + 8m
= 8(2m² + m),
which is divisible by 8.
Also, n² – 1 = (4m + 3)² – 1
= 16m² + 24m + 8
= 8(2m² + 3m + 1),
which is divisible by 8.
Hence, n² – 1 is divisible by 8 for any odd integer n.

n² - 1 is divisible by 8, if n is an odd integer 

Comparison between Induced and Autonomous Investments:
(i) Induced investment is income-elastic (i.e., rise in level of national income implies rise in level of investment) whereas Autonomous investment is income-inelastic.
(ii) Induced investment is positively related to national income but the Autonomous investment is unrelated to national income.
(iii) Induced investment is determined by consideration of profit, whereas Autonomous investment is determined by consideration of social welfare.
(iv) Induced investment curve is positively sloped but Autonomous investment curve is horizontal straight line parallel to X-axis, According to Keynes during short period firms plan to invest the same amount every year. Ex-ante investment demand (I) may be written as I indicating autonomous.

1)  Guantanamo Bay is a prison located in an area near Cuba and was controlled by American Navy. Here, about 600 people were secretly picked up by US forces from all over the world and put in this Guantanamo Bay prison 
 
2) The American government said that they were enemies of US and were linked to the attack on New York on 11^th September 2001. In most cases, the governments of these countries were not asked or even informed about their imprisonment 
 
3)  Families of the prisoners, media or even UN representatives were not allowed to meet them. The US army just arrested them, interrogated them and decided whether to keep them or not. There was no trial before any magistrate in US nor could these prisoners approach courts in their own country 
 
4) These were against the rights of people which were denied by the US in Guantanamo Bay incident

A bag contains cards and each and bearing numbers 1, 3, 5, ……., 35. A cards is drawn at random from the bag. 

(i). 1, 3, 5, 7, 9, 11 and 13 are prime numbers in the sequence 1, 3, 5, …., 35 which are less than 15. 

Total prime number less than 15 in given sequence = 7. 

∵ Last term is 35 in the given sequence. 

∴ 35 = 1+ (n – 1)2 

⇒ 2(n–1) = 35 – 1 = 34 

⇒ n –1 = 17

⇒ n = 17 + 1 = 18. 

∴ Total number of cards = 18. 

The probability of getting a prime member less than 15

= \(\frac{Total\, prime\, number\, less\, than\, 15\, in \, given\,sequence}{Total\, cards\, in \, the\, bag} = \frac{7}{18}\)

(ii). Numbers divisible by 3 and 5 are multiples of 15. 

Therefore, only 15 are such number in the sequence 1, 3, 5, ……., 35 which are divisible by both 3 and 5. 

Hence, total number divisible by 3 and 5 in the sequence = 1. 

(∵15, 30, 45 are multiple of 15 but last term of sequence is 35 & 30 is not odd number) 

The probability of getting a number divisible by 3 and 5

= \(\frac{Total\, number\, divisible\, by\, 3 and\, 5\, in\, given\, sequence}{Total\, cards\, in\, the\,bag} = \frac{1}{18}\).

after 1 year,

height of tree = 250cm + (1/5)250cm

= 250 + 50 = 300cm

After 2 years,

height of tree = 300 + (1/5)300cm = 360cm

After three years,

height of tree = 360 + (1/5)360cm = 360 + 72

= 432cm (or) 4.32m

Area of triangle = 0.5 x Base x Height

6 = 0.5 x 3 x PQ

PQ = 6 ÷ 1.5

PQ = 4cm

Since, this is a right-angled triangle, applying Pythagoras theorem,

PR = \(\sqrt{3^2 + 4^2}\)

= \(\sqrt{9+16}\)

= 5 cm

Concept

Prime numbers are numbers which have only two factors namely, 1 and themselves. For example, 2, 3, 19 etc.

Calculation

The prime numbers less than 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

∴ The number of prime numbers less than 50 is 15

Calculation

Numbers lying between 100 and 1000 will be three-digit numbers

Let the number be xyz

⇒ xyz = 100x + 10y + z

The sum of the digits = x + y + z

Subtracting the sum of the digits from the number we get,

(100x + 10y + z) - (x + y + z) = 99x + 9y

⇒ 9(11x + y) which is always divisible by 9

Calculation:

Smallest six digits = 100000

Greatest four digits = 9999

Required answer: 100000 – 9999

⇒ 90001

∴ The required answer is 90001

First Term of the AP (a) = 5

Common difference (d) = 8 - 5 = 3

Last term = a₄₀ = a + (40 - 1) d

= 5 + 39 × 3 = 122

Also a₃₁ = a + 30d = 5 + 30 × 3 = 95

Sum of last 10 terms = \(\frac{n}{2}\)(a₃₁ + a₄₀)

= \(\frac{10}{2}\)(95 + 122)

= 5 × 217 = 1085

∵ Mode = 38.

∴ The modal class is 30-40.

Mode = \(l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h\)

= 30 + \(\frac{16-12}{32-12-x}\) x 10 = 38

\(\frac{4}{20-x}\) x 10 = 8

8 (20 - x) = 40

20 - x = 5

X = 15

The common difference is 9 - 4 = 5

If the first term is 6 and common difference is 5, then new AP is,

6, 6 + 5, 6 + 10…

= 6,11,16….

(1) f(x) = \(\frac{x}{1-x^3}\) 

f(-x) = \(\frac{-x}{1-(-x)^3}\) 

= \(\frac{-x}{1+x^3}\) ≠ -f(x) or f(x)

∴ f(x) is neither even nor odd function.

(2) f(x) = \(\frac{x^2}{1+x}\) 

∴ f(-x) = \(\frac{(-x^2)}{1+(-x)}\) 

= \(\frac{x^2}{1-x}\) ≠ -f(x) or f(x).

(3) f(x) = x-|x|

∴ f(-x) = -x-|-x|

= -x-|x|

= -(x+|x|) ≠ -f(x) or f(x)

∴ f(x) is neither even nor odd function.

Let l be the side of the cube and L, B, H be the dimensions of the cuboid

Since l³ = 64 cm³ ∴ l = 4 cm

Total surface area of cuboid is 2[LB + BH + HL],

Where L = 12, B = 4 and H = 4

= 2 (12 × 4 + 4 × 4 + 4 × 12) cm² = 224 cm²

3x² − 7x − 6 = 0

⇒ 3x² − 9x + 2x − 6 = 0

⇒ 3x(x − 3) + 2(x − 3) = 0

⇒ (x − 3)(3x + 2) = 0

∵ x = 3, \(-\frac{2}{3}\)

OR

Since the roots are real and equal,

∴ D = b² − 4ac = 0

⇒ k² – 4×3×3 = 0 (∵ a = 3, b = k, c = 3)

⇒ k² = 36

⇒ k = 6 or −6

Kokila gave a pen to each child in the classroom on her birthday,

India is the largest democracy in the world,

I’ve got to solve some problems before I go to sleep,

According to the definition of reciprocal fraction which is reciprocal of 3/7 is 7/3.

\(f(x) = log_4(x + \frac 1x)\)

Domain of f(x), \(x + \frac 1x > 0\)

⇒ \(\frac{x^2 + 1}x > 0\)

⇒ \(x \in (0, \infty)\)    \((\because x^2 + 1 > 0)\)

\(\because x + \frac 1x = (\sqrt x)^2 + (\frac 1{\sqrt x})^2 - 2\sqrt x \times \frac 1{\sqrt x} + 2\)

\(= (\sqrt x - \frac1{\sqrt x})^2 + 2\)

\(\ge 2\)

\(\therefore log_4 (x + \frac 1x) \ge log_42\)   (\(\because\) log  is an increasing function)

⇒ \(f(x) \ge \frac 12 \)    \(\left(\because log_42 = \frac{log\,2}{log\,4} = \frac{log\,2}{2log\,2} = \frac12\right)\)

⇒ \(f(x) \in [0.5, \infty)\).

Let an integer be 'x'  

Then, its consecutive integer is (x + 1).

According to the question, 

x(x + 1) = 90

x² + x = 90

x² + x - 90 = 0

x² + 10x - 9x - 90 = 0

x(x + 10) -9(x + 10) = 0

(x + 10)(x - 9) = 0

x + 10 = 0 i.e. x = -10 and (x +1) = (-10 +1) = -9 

or, x - 9 = 0 i.e. x = 9 and (x + 1) = (9 + 1) = 10 

So, the required integers are -9 and -10 or 10 and 9.

Sexual Reproduction produces variable offspring creating diversity and variation among populations. It is important for plants as it provides variation to the progeny that helps in better survival and helps it to gain its own uniqueness within the species and remove the unwanted genes. 

Answer: Acid pH of intestine

i don't know the answer

Nuclear fission :The splitting of a heavy nucleus (A > 230) into two medium-mass nuclei in a nuclear reaction with the release of.huge amount of energy due to mass defect is called nuclear fussion. 

Radioactivity :The process of spontaneous (i.e. without external means' by itself) distintegration of the nuclei of heavy elements with the emission of certain types of radiations is called radioactivity.

Atomic Nucleus: The central core of the atom which contains all the atom's positive charge and most of its mass is known as atomic nucleus. The size of the nucleus is much smaller than that of the atom. Intact, then size of the nucleus is 10,000 times smaller than the size of the atom. But more than 99.9% of the whole mass of the atom is concentrated in the nucleus. Therefore, a nucleus Occupies a very small space in the atom. The nucleus has its own structure and consists of photons and neutrons.

Consider a parallel plate cpacitor having plates of area a separated by a distance d. Then, capacitence of this air capacitor is given by

Cair = ε₀A/d

If the space between the (dates is filled completely with dielectric of relative permitivity K, then capacitance of the same capacitor is given by

cm =ε₀kA/d

Dividing (ii) by (i), we have

k = cm/air

= Capacitance of capacitor with dielectric between the plates/capacitance of the same capacitor with air between the plates.