Common Difference

Q.

If the $m^{th}$ term of a H.P. be n and $n^{th}$ be m, then the $r^{th}$ term will be

1. 

2. 

3. 

4. 

Q.

Suppose that $F(n+1)=\frac{2F\left(n\right)+1}{2}$for n=1, 2, 3, ....... and F(1)=2. Then, F(101) equals

1. none of these

2. 50

3. 52

4. 54

Q. 13. If arg (z+a) =pi/6 and arg (z-a)=2pi/3 :a beongs to positive real number then z= ??

Q.

Find the real numbers x and y such that: (x + iy)(3 + 2i) = 1 + i

1. 

2. 

3. 

4. 

Q.

How do you find absolute min and max in calculus.

Q.

If the first 2 terms of H.P are $\frac{4}{11}$ and $\frac{2}{3}$ respectively, then the largest term is _____

1. 4th term

2. Can't be determined

3. 2nd term

4. 3rd term

Q.

The negation of the statement “p: For every real number x, x³ > x².” is:

1. There exists real number c such that x³ = x².

2. There exists real number c such that x² > x.

3. There exists real number c such that x³ > x.

4. There exists a real number x such that x³ < x².

Q.

Let P and Q be two finite sets such that n(P – Q) = 20, n(P U Q) =95, n(P ∩ Q) =35, then n(Q – P) is

1. 150

2. 75

3. 55

4. 40

Q. 40. The nth term of an A.P is (an=2n+1), find its sum?

Q.

The quantifier used in the statement “For every real number x, x is less than x + 5” is:

1. Or

2. For every

3. There exist

4. And

Q.

Find the indicated terms in each of the sequences in where $n^{th}$ terms are :

$a_n=4n-3;a_{17},a_{24}$

Q. 28 belongs to Y such that the sum of all positive integers of Y is 2y.this statement is true or false

Q. The value of $\underset{x\to \infty }{\lim }\frac{n!}{\left(n+1\right)!-n!}$ is

(a) 0

(b) 1

(c) −1

(d) none of these

Q.

Let $a_1,a_2,a_3,\dots \dots .,a_{49}$ be in A.P. such that $\sum _{k=0}^{12}{a}_{4k+1}=416$ and $a_9+a_{43}=66$. If ${a_1}^2+{a_2}^2+.......+{a_{17}}^2=140m$, then $m$ is equal to

1. $68$

2. $34$

3. $33$

4. $66$

Q.

I + f + f' is an even integer, where I is an integer, f and f' are proper positive

fractions (0 < f, f' < 1). Then the integer I is an odd number.

1. False

2. True

Q. Identify the quantifier in the following statements and write the negation of the statements.
'For every real number x, x is less than x+1.'

Q. If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x < −1 or x > 3.

Q. In an AP first term is 2 and the sum of first 5 terms is 1/4 of next five terms show that 20th term is 112 In an AP first term is 2 and the sum of first 5 terms is 1/4 of next five terms show that 20th term is -112

Q. Write the component statements of the following compound statements and check whether the compound statement is true or false:
(i) To enter into a public library children need an identity card from the school or a letter from the school authorities.
(ii) All rational numbers are real and all real numbers are not complex.
(iii) Square of an integer is positive or negative.
(iv) x = 2 and x = 3 are the roots or the equation 3x² − x − 10 = 0.
(v) The sand heats up quickly in the sun and does not cool down fast at night.

Q. Expand $(1-2x{)}^5$

Q.

Find the indicated terms in each of the sequences in where $n^{th}$ terms are :

$a_n=\frac{n(n-2)}{n+3};a_{20}$

Q.

${}^n{C}_0$ + 2.${}^n{C}_1$ + 3.${}^n{C}_2$ +..............(n+1)${}^n{C}_n$ =

1. (n+1)

2. (n+2)

3. (n+2)

4. (n+1)

Q.

Find the real numbers x and y such that : (x + iy)(3 + 2i) = 1 + i

1. 

2. 

3. 

4. 

Q.

Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.

Prove that

Q. If the first term of a G.P. a₁, a₂, a₃, ... is unity such that 4 a₂ + 5 a₃ is least, then the common ratio of G.P. is
(a) −2/5
(b) −3/5
(c) 2/5
(d) none of these

Q. If z is any complex number satisfying \vert z-3-2i\vert≤2, then the minimum value of \vert2z-6+5i\vert is

Q.

Let the sequence $a_1,a_2,a_3.....a_n$ form an A.P. Then
$a_1^2-a_2^2+a_3^2-a_4^2+.....+a_{2n-1}^2-a_{2n}^2$ is equal to

1. None of these

2. $\frac{n}{2n-1}[a_1^2-a_{2n}^2]$

3. $\frac{n}{n-1}[a_0^2-a_{2n}^2]$

4. $\frac{n}{n+1}[a_1^2+a_{2n}^2]$

Q. The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.

Q. If $a_1,a_2,a_3$ .... a_{n} are in A. P., then the common difference is--

1. $a_2-a_1$
2. $a_4-a_3$
3. $a_7-a_6$
4. $a_n-a_{n-1}$

Q.

Which option is not same as “if p then q”?

1. ~p implies q.

2. p is a sufficient condition for q.

3. p only if q.

4. ~q implies ~p