Harmonic Progression

Q.

If $A=\left[\begin{array}{cc}x& 1\\ 1& 0\end{array}\right]$ & $A^2$ is the identity matrix, Then $x=$

1. $-1$

2. $0$

3. $1$

4. $2$

Q. If $x={\sum }_{n=0}^{\infty }{a}^n,y={\sum }_{n=0}^{\infty }{b}^n,z={\sum }_{n=0}^{\infty }{c}^n$ where a, b, c, are in AP
and |a| <1, |b|<1, |c|<1 then x, y, z are in

1. H.P
2. AP
3. GP
4. Arithmetic- Geometric progression

Q. In an A.P., if S_n–3 = 5n² – 28n + 39, then its 9^th term is

एक स.श्रे. में, यदि S_n–3 = 5n² – 28n + 39, तब इसका 9^th पद है

1. 77
2. 78
3. 87
4. 97

Q. If a, b, c, d are in H.P. , then ab +bc+cd =

1. ad
2. 2ad
3. 3ad
4. 4ad

Q. Sum of the roots of the equation, $x^2+2x-4=0$ is

1. $\sqrt{2}$
2. $-\sqrt{2}$
3. $2$
4. $-2$

Q. 16. If (m+1)th term of AP is twice the (n+1)th term prove that (3m+1)th term is twice the (m+n+1)th term.

Q. If the roots of the equation $(a^2+b^2)x^2-2(ac+bd)x+(c^2+b^2)=0$ are equal then prove that $ad=bc$

Q.

The sum of the first n terms of an AP is 2n. The sum of the first 2n terms is 3n.find the sum of the first 3n terms.

Q. the fourth term of an ap is equal to three times the first term and the sixth term exceeds twice the third term by 1. find the first term and common difference.

Q. If $b^2,a^2,c^2$ are in $AP$ then $a+b,b+c,c+a$ will be in

1. $AP$
2. $GP$
3. $HP$
4. $Noneofthese$

Q.

If 17^th term of an A.P. exceeds its 10^th term by 7. Find the common difference.

Q.

If ln(a+c) , ln(a-c) , ln(a-2b+c) are in A.P, then

1. a, b, c are in A.P.

2. $a^2,b^2,c^2$ are in A.P.

3. a, b, c are in G.P.

4. a, b, c are in H.P.

Q.

The sum of first 7 terms of an AP is 10, and that of the next 7 tems is 17, find the 8th term of the AP

Q. 9. If (m+1) term of an AP is twice the (n+1) term , prove that(3m+1) term is twice the (m+n+1) term

Q. If the roots of the equation $a{x}^2+x+b=0$ be real and different, then the roots of the equation $x^2-4\sqrt{ab}x+1=0$ will be-

1. Rational
2. Real
3. Irrational
4. Imaginary

Q. Determine k so that 2+3, k, 5+8k are three consecutive terms in AP.

Q. What is the sum of the first n positive integers? [1 MARK]

Q. If the $A.M.$ between $p^{th}$ and $q^{th}$ terms of an $A.P.$ is equal to the $A.M.$ between $r^{th}$ and $s^{th}$ terms of the $A.P.$, then which of the following is true

1. $pr=qs$
2. $pq=rs$
3. $p+r=q+s$
4. $p+q=r+s$

Q. Assertion :The equation $x^2+(2m+1)x+(2n+1)=0$ where $m$ and $n$ are integers cannot have any rational roots. Reason: The quantity $(2m+1{)}^2-4(2n+1)$, where $m$ $n\in I$ can never be a perfect square.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. the second term of an A.P is (x-y) and the fifth term is (x+y) find the first term of the AP

Q.

if in an A.P T4+T8= 24 a T6 + T10 =34 then find the four terms of A.p