Harmonic Progression

Q.

For the statements $p$ and $q$, consider the following compound statements:

(a) $(~q\wedge (p\to q\left)\right)\to ~p$

(b) $\left(\right(p\vee q\left)\right)\wedge ~p)\to q$

Then which of the following statements is correct?

1. (a) is a tautology but not (b)

2. (a) and (b) both are not tautologies

3. (a) and (b) both are tautologies

4. (b) is a tautology but not (a)

Q.

If $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3-6x^2+11x+6=0$, then $\sum {\alpha }^2\beta +\sum \alpha {\beta }^2$ is equal to

1. $80$

2. $168$

3. $90$

4. $-84$

Q. If x+y =16 and $x^2+y^2$ is minimum, then the values of x and y are

1. 3, 13
2. 4, 12
3. 6, 10
4. 8, 8

Q.

If a, 4, b are in A.P., and a, 2, b are in G.P., then a, 1, b are in:

1. A.P.

2. G.P

3. H.P

4. None

Q.

If ‘a. b, c’ are in arithmetic progression ‘b, c, d’ are in geometric progression & ‘c, d, e’ are in harmonic progression, then ‘a, c, e’ are in

1. Not particular order

2. Arithmetic Progression

3. Geometric Progression

4. Harmonic Progression

Q.

Subtract $\mathrm{a}\left(\mathrm{b}-5\right)$ from $\mathrm{b}\left(5-\mathrm{a}\right)$.

Q.

If $a_1,a_2,a_3,..$are in a harmonic progression with $a_1=5$ and $a_{20}=25$. The least positive integer $n$ for which $a_n<0$is

Q.

Which of the following Boolean expression is a tautology?

1. $\left(p\wedge q\right)\wedge \left(p\to q\right)$

2. $\left(p\wedge q\right)\vee \left(p\vee q\right)$

3. $\left(p\wedge q\right)\vee \left(p\to q\right)$

4. $\left(p\wedge q\right)\to \left(p\to q\right)$

Q.

If z = x + iy, $z^{\frac{1}{3}}$ = a - ib and $\frac{x}{a}$ - $\frac{y}{b}$ = $\lambda$$(a^2-b^2)$, then $\lambda$ is equal to

1. 2

2. 3

3. 4

4. 1

Q. If $x^2+9y^2+25z^2=xyz(\frac{15}{x}+\frac{5}{y}+\frac{3}{z})$ then,

1. $x,y$ and $z$ are in $H.P.$
2. $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in $A.P.$
3. $x,y$ and $z$ are in $G.P.$
4. $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in $G.P.$

Q. Let $A,B$ and $C$ are distinct positive integers, less than or equal to $10.$ If the arithmetic mean of $A$ and $B$ is $9$ and the geometric mean of $A$ and $C$ is $6\sqrt{2}$, then the harmonic mean of $B$ and $C$ is

1. $\frac{86}{19}$
2. $\frac{160}{17}$
3. $\frac{180}{19}$
4. $\frac{86}{17}$

Q.

If $1,{\log }_y\left(x\right),{\log }_z\left(y\right),-15{\log }_x\left(z\right)$ are in A. P. then the common difference of this A. P. is

1. $1$

2. $2$

3. $-2$

4. $3$

Q.

Add the following: $a-b+ab,b-c+bc,c-a+ac$

Q. If $x>1,y>1,z>1$ are in G.P., then $\frac{1}{1+Inx},\frac{1}{1+Iny},\frac{1}{1+Inz}$ are in

1. A.P.
2. H.P.
3. G.P.
4. None of these

Q. If $x,y,z$ are real and $9x^2+16y^2+25z^2-12xy-20yz-15zx=0,$ then $x,y,z$ are in

1. A.P.
2. G.P.
3. H.P.
4. None of the above

Q. If the roots of the equation $x^2-8kx+16(k^2-k+1)=0$ are real, distinct and have values at least $4$, then which of the following is (are) true?

1. $m,m^2-1,m^3-4$ are in A.P., where $m$ is the minimum value of $k.$
2. G.M. of $p$ and $p^2$ is $8$, where $p$ is the minimum value of $k^2.$
3. The reciprocals of the integral values of $k$ are in H.P.
4. The sum of the first ten integral values of $k$ is $55.$

Q. If the 8th term of an H.P. is 1/2 and the 14th term is 1/3, then the 20th term is

1. $\frac{1}{5}$
2. $\frac{1}{4}$
3. $-\frac{1}{2}$
4. $\frac{1}{6}$

Q. If $m$ arithmetic means $(A.Ms)$ and three geometric means $(G.Ms)$ are inserted between $3$ and $243$ such that $4^{th}$ $A.M.$ is equal to $2^{nd}$ $G.M.$, then $m$ is equal to

Q.

GM and HM of two numbers are $10$ and $8$, respectively. The numbers are

1. $5,20$

2. $4,25$

3. $2,50$

4. $1,100$

Q.

If $a,b,careinAP$and $\left|a\right|,\left|b\right|,\left|c\right|<1andx=1+a+a^2+...\infty ,y=1+b+b^2+...\infty ,z=1+c+c^2+...\infty$, then $x,y,z$ shall be in

1. $A.P$

2. $G.P$

3. $HP$

4. $Noneofthese$

Q. If $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P., prove that:

(ii) $a(b+c),b(c+a),c(a+b)$ are in A.P.

Q.

If $a,b,c$are in GP and $4a,5b,4c$are in AP such that $a+b+c=70$, then value of $b$ is

1. $5$

2. $10$

3. $15$

4. $20$

Q. If the roots of $10x^3-c{x}^2-54x-27=0$ are in H.P., then the value of $c$ is

Q. If $a,b,c$ are in $G.P$ and $x,y$, respectively be arithmetic means between $a,b$ and $b,c$ then the value of $\frac{a}{x}+\frac{c}{y}$ is

1. $2$
2. $1$
3. $1/2$
4. $Noneofthese$

Q.

If$\mathrm{a},\mathrm{b}\mathrm{and}\mathrm{c}$ are different positive real numbers , then $[a+b+c]\left[\left(\frac{1}{a}\right)+\left(\frac{1}{b}\right)+\left(\frac{1}{c}\right)\right]$ is

1. $<9$

2. $>9$

3. $>0$

4. $>24$

Q. If $x,y,z$ are $T_l,T_{2m},T_{3n}$ terms of an H.P. respectively then $\Delta =\left|\begin{array}{ccc}yz& xz& yz\\ l& 2m& 3n\\ 1& 1& 1\end{array}\right|$ is independent of

1. $m$
2. $n$
3. $y$
4. $x$

Q. Let the positive numbers a, b, c, d be in A.P. Then abc, abd, acd, acd, bcd are in

1. H.P
2. G.P
3. A.P
4. none of these

Q.

If $x=3,$find the value of

$2x+1$

Q. If $a^2,b^2,c^2$ are in $AP$, then which of the following are in $AP$?

1. $(b+c),(c+a),(a+b)$
2. $\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$
3. $\frac{1}{a^2},\frac{1}{a^2},\frac{1}{c^2}$
4. $\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$

Q.

Find the equation whose roots are reciprocals of the roots of $2x^2+5x+4=0$ .

1. 

2. 

3. 

4.