Arithmetic Progression

Q.

The sum of n terms of two arithmetic progressions are in the ratio (3n + 8) : (7n + 15). Find the ratio of their 12th terms.

Q.

The product of three numbers in AP is 224 and the largest number is 7 times the smallest. Find the numbers.

Q.

Divide 32 into four parts which are in AP such that the product of extremes is to the product of means as 7 :15.

Q.

If the sum of first m terms of an AP is the same as the sum of its first n terms, show that the sum of its (m +n) terms is zero.

Or

The sum of n terms of two arithmetic progressions are in the ratio $(3n+8)$ : $(7n+15)$. Find the ration of their 12th terms.

Q.

If $Li{m}_{x\to 0}\frac{4+sin2x+Asinx+Bcosx}{x^2}$ exists, then the values A and B are

1. -4 and -2

2. -3 and -2

3. -1 and -4

4. -2 and -4

Q. Find the sum to n terms of the series 3+15+35+63+.... .

Q.

Sum to n terms of the series 2 + 5 + 14 + 41 +... is:

1. 

2. 

3. 

4. 

Q.

If both the roots of $x^2$ - ax + a = 0 are greater than 2 then :

1. a (0, 2)

2. a

3. None of these

4. a (4, )

Q.

The sums of n terms of two arithmetic progressions are on the ratio 5n + 4 : 9n + 6. Find the ratio of their ${18}^{th}$ terms.

Q.

$1.3+{2.3}^2+{3.3}^3+\cdots +n.3^n=\frac{(2n-1)3^{n+1}+3}{4}$

Q.

If the ratio of the roots of the equation $x^2$+px+q=0 be equal to the ratio of the roots of $x^2$+lx+m=0. Then

1. p² m = q² l

2. pm² = q² l

3. p² l = q² m

4. p² m = l² q

Q.

The sequence whose terms follow the certain pattern is called a

1. progression

2. series

3. real sequence

4. finite sequence

Q.

$(3\sqrt{3}+5{)}^n$ = p + f, where p is an integer and f is a proper fraction(fractional part of $(3\sqrt{3}+5{)}^n$). Find the value of $(3\sqrt{3}-5{)}^n$ in terms of p and f, if n is an even integer.

1. p - f

2. f

3. p

4. 1 - f

Q.

If $\sin \left(\mathrm{A}\right)=\frac{3}{4}$, calculate $\cos \left(\mathrm{A}\right)$ and $\tan \left(\mathrm{A}\right)$.

Q.

Find the sum of first n terms of the series $3+7+13+21+31+....$

Or

If a b and c are in GP and x, y are the arithmetic means of a, b and b, c respectively. prove that $\frac{a}{x}+\frac{c}{y}=2and\frac{1}{x}+\frac{1}{y}=\frac{2}{b}$.

Q.

Find $\underset{t\to 4}{lim}\frac{t-\sqrt{3t+4}}{4-t}$

1. 

2. 

3. 

4. 

Q.

If a and b are two real numbers, then the condition for √a.√b = √ab to be true is

1. a and b are negative

2. a and b are 0

3. a and b are positive

4. at least one of a and b is either 0 or positive

Q.

Between 1 and 31 m arithmetic means are inserted so that the ratio of the $7^{th}$ and ${(m-1)}^{th}$ means is 5 : 9. Then the value of m is

1. 13

2. 14

3. 15

4. 12

Q.

The ratio of the sum of m and n terms of an AP is $m^2:n^2.$ Show that the ratio of the mth and nth terms is (2m - 1) : (2n - 1).

Q. If $\frac{3+5+7+...+\mathrm{upto}n\mathrm{terms}}{5+8+11+....\mathrm{upto}10\mathrm{terms}}$ 7, then find the value of n.

Q.

If A = {3, 5}, then A x A x A is

1. {(3, 3), (3, 5), (5, 3), (5, 5)}

2. {(3, 3, 3), (3, 3, 5), (3, 5, 3), (5, 5, 5)}

3. {(3, 3, 3), (3, 3, 5), (3, 5, 3), (5, 3, 3), (3, 5, 5), (5, 3, 5), (5, 5, 3), (5, 5, 5)}

4. {(3, 3, 3), (5, 5, 5)}

Q. The 12th term of an arithmetic progression has the value 0. Which term of the AP is 4 times the 19th element? a.37 b.41 c.40 d.4

Q.

${\left(\frac{1+cos\varphi +isin\varphi }{1+cos\varphi -isin\varphi }\right)}^n$ =

1. 

2. 

3. 

4. 

Q.

If the sides of a triangle are in A.P. and the greatest angle of the triangle is double the smallest angle, then the ratio of the sides of the triangle is

1. 3:4:5
   

2. 5:6:7
   

3. 4:5:6
   

4. 7:8:9

Q. A person is to count 4500 currency notes. Let $a_n$ denote the number of notes he counts in the $n^{th}$ minute. If $a_1=a_2=...=a_{10}=150and{a}_{10},a_{11},...$ are in an AP with common difference - 2, then the time taken by him to count all notes is

1. 34 minutes
2. 125 minutes
3. 135 minutes
4. 24 minutes

Q. All the roots of the equation $x^3-x^2+ax+b=0$ are real and distinct. If they are in A.P., then

1. $a\in (-\infty ,\frac{1}{3})$
2. $a\in (-\infty ,-\frac{1}{3})$
3. $b\in (-\frac{1}{9},\infty )$
4. $b\in (-\frac{1}{27},\infty )$

Q.

$a^n-b^n$ is always divisible by

1. a+b

2. 2a-b

3. a-b

4. a-2b

Q. If $a_1,a_2,a_3,\cdots$ are in A.P. and
$a_1^2-a_2^2+a_3^2-a_4^2+\cdots +a_{2k-1}^2-a_{2k}^2$
$=M(a_1^2-a_{2k}^2)$. Then M =

1. $\frac{k-1}{k+1}$
2. $\frac{k}{2k-1}$
3. $\frac{k+1}{2k+1}$
4. None

Q. Which term of the following sequences

Q. 11.Cot 15/2 is equal to 1)1+2+3+4 (2)2+3+45 3)2+3+4+6 (4)3+4+5+7