Arithmetic Progression

Q. Let the numbers $2,b,c$ be in an A.P. and $A=\left[\begin{array}{ccc}1& 1& 1\\ 2& b& c\\ 4& b^2& c^2\end{array}\right]$. If $\text{det}\left(A\right)\in [2,16]$, then $c$ lies in the interval :

1. $[3,2+2^{3/4}]$
2. $[4,6]$
3. $[2,3]$
4. $[2+2^{3/4},4]$

Q. Let there be odd number of stones placed one by one at an interval of $10\text{m}$ along a straight road. All the stones has to be assembled at the middle stone. A person start from one end and can only carry one stone at a time. If the distance covered by the person is $3\text{km}$ in this job, then the number of stones is

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

1. $5:6:7$
2. $5:9:13$
3. $4:5:6$
4. $3:4:5$

Q.

If the angles of a triangle be in the ratio 1 : 2 : 7, then the ratio of its greatest side to the least side is

1. 1:2

2. 2:1

3. $(\sqrt{5}+1):(\sqrt{5}-1)$

4. $(\sqrt{5}-1):(\sqrt{5}+1)$

Q.

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the sides of the triangle are

1. 1, 2, 3

2. 2, 3, 4

3. 3, 4, 5

4. 4, 5, 6

Q. If the numbers $3^{2a-1},14,3^{4-2a}$ $(0<a<1)$ are in A.P., then the value of $(2a+1)(3a-1)$ is

1. $\frac{7}{2}$
2. $\frac{11}{2}$
3. $1$
4. $4$

Q. The number of terms in the sequence $3,7,11,\dots ,407$ is

1. $102$
2. $101$
3. $103$
4. $104$

Q. If $a^2,b^2,c^2$ are in A.P. consider two statements
(i) $\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$ are in A.P.
(ii) $\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}$ are in A.P.

1. (i) and (ii) both correct
2. (i) and (ii) both incorrect
3. (i) and (ii) both incorrect
4. (i) incorrect (ii) correct

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. If $a,b,c\in R^{+}$ are such that $2a,b,4c$ are in A.P. and $c,a,\text{and}b$ are in G.P., then

1. $a^2,ac\text{and}{c}^2\text{are in A.P.}$
2. $c,a\text{and}a+2c\text{are in A.P.}$
3. $c,a\text{and}a+2c\text{are in G.P.}$
4. $\frac{a}{2},c\text{and}c-a\text{are in G.P.}$

Q.

If the sides of a right-angled triangle form an A.P. Then the sines of the acute angle are

1. $\frac{3}{4},\frac{4}{5}$

2. $\sqrt{3},\frac{1}{\sqrt{3}}$

3. $\sqrt{\frac{\sqrt{5}-1}{2}},\sqrt{\frac{\sqrt{5}+1}{2}}$

4. $\sqrt{\frac{\sqrt{3}-1}{2}},\sqrt{\frac{\sqrt{3}+1}{2}}$

Q. For any three positive real numbers $a,b$ and $c,$ $9(25a^2+b^2)+25(c^2–3ac)=15b(3a+c).$ Then:

1. $b,c$ and $a$ are in G.P.
2. $a,b$ and $c$ are in A.P.
3. $a,b$ and $c$ are in G.P.
4. $b,c$ and $a$ are in A.P.

Q. If $\alpha ,\beta ,\gamma ,\delta$ are in $A.P.$ and $\underset{0}{\overset{2}{\int }}f\left(x\right)dx=-4$, where $f\left(x\right)=\left|\begin{array}{ccc}x+\alpha & x+\beta & x+\alpha -\gamma \\ x+\beta & x+\gamma & x-1\\ x+\gamma & x+\delta & x-\beta +\delta \end{array}\right|$ then common difference $d$ is:

1. $1$
2. $-1$
3. $2$
4. $-2$

Q. $\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{a.}& \sqrt{x+2}>\sqrt{8-x^2}& \text{p.}& (-\infty ,-\frac{3}{2})\cup (-\frac{1}{4},\frac{1}{4})\cup (\frac{3}{2},\infty )\\ \text{b.}& ||x|-1|<1-x& \text{q.}& (-\infty ,0)\cup [1,2]\\ \text{c.}& \left|\frac{2-3|x|}{1+|x|}\right|>1& \text{r.}& (2,2\sqrt{2}]\\ \text{d.}& \frac{\sqrt{2-x}+4x-3}{x}\ge 2& \text{s.}& (-\infty ,0)\end{array}$

Which of the following is the correct option ?

1. $\left(a\right)\to \left(r\right)\left(b\right)\to \left(s\right)\left(c\right)\to \left(p\right)\left(d\right)\to \left(q\right)$
2. $\left(a\right)\to \left(q\right)\left(b\right)\to \left(s\right)\left(c\right)\to \left(p\right)\left(d\right)\to \left(r\right)$
3. $\left(a\right)\to \left(r\right)\left(b\right)\to \left(p\right)\left(c\right)\to \left(s\right)\left(d\right)\to \left(q\right)$
4. $\left(a\right)\to \left(q\right)\left(b\right)\to \left(s\right)\left(c\right)\to \left(r\right)\left(d\right)\to \left(p\right)$

Q.

If $a_1,a_2,a_3.....a_n$ are in A.P. Where $a_i>0$ for all i, then the value of
$\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+.........+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}=$

1. $\frac{n-1}{\sqrt{a_1}+\sqrt{a_n}}$

2. $\frac{n-1}{\sqrt{a_1}+\sqrt{a_n}}$

3. $\frac{n-1}{\sqrt{a_1}-\sqrt{a_n}}$

4. $\frac{n-1}{\sqrt{a_1}-\sqrt{a_n}}$

Q. If ${\log }_2(5\times 2^x+1),{\log }_4(2^{1-x}+1)$ and $1$ are in A.P., then $x$ equals

1. ${\log }_{2}5$
2. $1-{\log }_{5}2$
3. ${\log }_{5}2$
4. $1-{\log }_{2}5$

Q. Let $a_1,a_2,a_3\cdots$ be terms of A.P.
$If\frac{a_1+a_2+\cdots a_p}{a_1+a_2+\cdots +a_q}=\frac{p^2}{q^2},p\ne q,$then $\frac{a_6}{a_{21}}$ equal

1. $\frac{41}{11}$
2. $\frac{7}{2}$
3. $\frac{2}{7}$
4. $\frac{11}{41}$

Q. If $1,lo{g}_9(3^{1-x}+2)lo{g}_3({4.3}^x-1)$ are in A.P., then x equals

1. $lo{g}_{3}4$
2. $1+lo{g}_{3}4$
3. $1-lo{g}_{3}4$
4. $lo{g}_{4}3$

Q. $\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}a,b,c,d\text{are in A.P. such that}a+b+c+d=32\text{and}& \text{(P)}& a+b=8\\ & 15ad=7bc\text{, then}(a<b<c<d)& & \\ \text{(B)}& \text{If the first three terms of the sequence}\frac{1}{16},a,b,\frac{1}{6}\text{are in G.P.}& \text{(Q)}& a-b=12\\ & \text{and the last three terms are in H.P., then}(a>0,b>0)& & \\ \text{(C)}& \text{If}{S}_n=(\sqrt{3}+1{)}^{2n}+(\sqrt{3}-1{)}^{2n},n\in N\text{and}& \text{(R)}& a+b=4\\ & S_{n+1}=a{S}_n+b{S}_{n-1}\text{, then}& & \\ \text{(D)}& \text{If}{}^a{C}_b=84,{}^a{C}_{b-1}=36\text{and}{}^a{C}_{b+1}=126\text{, then}& \text{(S)}& a+b=12\\ & & \text{(T)}& a+d=16\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(C)}\to \text{(Q)},\text{(R)}$
2. $\text{(C)}\to \text{(R)},\text{(S)}$
3. $\text{(D)}\to \text{(P)},\text{(S)}$
4. $\text{(D)}\to \text{(Q)},\text{(S)}$

Q.

If the sides of a right-angled triangle form an A.P. Then the sines of the acute angle are

1. $\frac{3}{4},\frac{4}{5}$

2. $\sqrt{3},\frac{1}{\sqrt{3}}$

3. $\sqrt{\frac{\sqrt{5}-1}{2}},\sqrt{\frac{\sqrt{5}+1}{2}}$

4. $\sqrt{\frac{\sqrt{3}-1}{2}},\sqrt{\frac{\sqrt{3}+1}{2}}$

Q. For any three positive real numbers a, b and c, if $9(25a^2+b^2)+25(c^2-3ac)=15b(3a+c)$, then

1. b, c and a are in A.P
2. b, c and a are in G.P
3. a, b, and c are in A.P
4. a, b and c are in G.P