Geometric Progression

Q. If $x,2y,3z$ are in A.P. where the distinct numbers $x,y,z$ are in G.P, then the common ratio of G.P. is

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

Q.

Find the value of 'a' if one root of the quadratic equation ($a^2$ - 5a + 3)$x^2$ + (3a - 1) x + 2 = 0 is twice as large as the other.

1. 2/3

2. 3/2

3. -2/3

4. -3/2

Q.

If $n$ geometric means be inserted between $a$ and $b$ then the $n^{th}$ geometric mean will be

1. $a{\left(\frac{b}{a}\right)}^{\frac{n}{n-1}}$

2. $a{\left(\frac{b}{a}\right)}^{\frac{n-1}{n}}$

3. $a{\left(\frac{b}{a}\right)}^{\frac{n}{n+1}}$

4. $a{\left(\frac{b}{a}\right)}^{\frac{1}{n}}$

Q.

If the vertices P, Q, R of a triangle PQR are rational points, which of the following points of the triangle PQR is

(are) always rational point(s)

1. Centroid

2. Incentre

3. Circumcentre

4. Orthocentre

Q.

If $x,y,z$ are in G.P and $a^x=b^y=c^z$, then

1. ${\log }_{a}c={\log }_{b}a$

2. ${\log }_{b}a={\log }_{c}b$

3. ${\log }_{c}b={\log }_{a}c$

4. None of these

Q.

The (m + n)th and the (m - n)th terms of a GP are p and q respectively. Show that the mth and the nth terms of the GP are $\sqrt{pq}$ and ${\left(\frac{q}{p}\right)}^{\left(\frac{m}{2n}\right)}$

Q. In the quadratic equation $a{x}^2+bx+c=0,\Delta =b^2-4acand\alpha +\beta ,{\alpha }^2+{\beta }^2,{\alpha }^3+{\beta }^3,$ are in G.P. where $\alpha ,\beta$ are the root of $a{x}^2+bx+c=0,$ then

1. $\Delta \ne 0$
2. $b\Delta =0$
3. $c\Delta =0$
4. $\Delta =0$

Q. For what values of x, the numbers are in G.P?

Q.

If every pair of the equations $x^2+px+qr=0,x^2+qx+rp=0,x^2+rx+pq=0$ have a common root, then the sum of three common roots is

1. 

2. 

3. 

4. 

Q.

The following figure is a graph of

1. Signum function

2. Constant function

3. Identity function

4. Modulus function

Q.

If the roots of the cubic $x^3+a{x}^2+bx+c=0$ are three consecutive positive integers. Then the value of $\frac{a^2}{b+1}$ is equal to __.

Q.

If a root of the equation $a{x}^2+bx+c=0$ be reciprocal of the equation then $a^{\prime }{x}^2+b^{\prime }x+c^{\prime }=0$, then

1. $(c{c}^{\prime }-a{a}^{\prime }{)}^2=(b{a}^{\prime }-c{b}^{\prime }\left)\right(a{b}^{\prime }-b{c}^{\prime }).$

2. $(c{c}^{\prime }+a{a}^{\prime }{)}^2=(b{a}^{\prime }-c{b}^{\prime }\left)\right(a{b}^{\prime }-b{c}^{\prime }).$

3. $(b{b}^{\prime }-a{a}^{\prime }{)}^2=(c{a}^{\prime }-b{c}^{\prime }\left)\right(a{b}^{\prime }-b{c}^{\prime }).$

4. $(c{c}^{\prime }-a{a}^{\prime }{)}^2=(b{a}^{\prime }-c{b}^{\prime }\left)\right(a{b}^{\prime }-b{c}^{\prime }).$

Q.

If sum of an infinite geometric series is $\frac{4}{5}$ and its first term is $\frac{3}{4}$, then its common ratio is

1. $\frac{7}{16}$

2. $\frac{9}{16}$

3. $\frac{1}{9}$

4. none of these

Q.

If {x} denotes the fractional part of x then the value of $\left\{\frac{2^{2003}}{17}\right\}$ is

1. 

2. 

3. 

4. 

Q.

Find the sum of 10 terms if the nth term of a sequence is $3^n$- 2.

1. ( - 1) - 20

2. ( - 1) - 20

3. ( - 1) - 20

4. ( - 1) - 30

Q.

If α, β, γ and δ are the roots of $x^4$ + 2 $x^3$ + 3 $x^2$ + 4x + 5 = 0. Find the equation whose roots are $\frac{1}{\alpha }$, $\frac{1}{\beta }$, $\frac{1}{\gamma }$, $\frac{1}{\delta }$

1. x⁴ - 2x³ + 3x² - 4x + 5 = 0

2. x⁴ + 3x³ + 2x² + 4x + 5 = 0

3. 5x⁴ + 4x³ + 3x² + 2x + 1 = 0

4. 5x⁴ - 4x³ + 3x² - 2x + 1 = 0

Q. Which term of the sequence $2,1,\frac{1}{2},\frac{1}{4},\dots$ is $\frac{1}{128}$

1. $9$
2. $8$
3. $10$
4. $7$

Q.

If $\omega$ is a complex cube root of unity , find the equation , whose roots are 2 , 2$\omega$ , 2${\omega }^2$

1. - + x -8 = 0

2. + 8 =0

3. - 8 + 8x -8 = 0

4. - 8 = 0

Q. In a geometric sequence, the first term is a and the common ratio is r. If $S_n$ denotes the sum to n, terms and $U_n={\sum }_{n=1}^n{S}_n$
then $r{S}_n+(1-r)U_n=$

1. na
2. n
3. None of these
4. $n^{2}a$

Q. Find the ${20}^{th}$ and $n^{th}$ terms of the G.P. $\frac{5}{2},\frac{5}{4},\frac{5}{8},...$

Q.

If $a^2,b^2,c^2$ are in AP, prove that

$\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}$ are in AP.

Q.

List all the elements of the following sets:

(i) A = {x : x is an odd natural number}

(ii) B = {x : x is an integer, 1/2 < x < 9/2}

(iii) C = {x : x is an integer, $x^2\le 4$}

(iv) D = {x : x is a letter in the word "LOYAL"}

(v) E = {x : x is a month of a year not having 31 days}

(vi) F = {x : x is a consonant in the English alphabet which precedes K}

Q. Show that the products of the corresponding terms of the sequences form a G.P, and find the common ratio.

Q.

If the cube roots of unity are 1 , $\omega$ , ${\omega }^2$ then the roots of the equation ${(x-1)}^3$ + 8 = 0 are

1. -1 , -1 , -1

2. -1 , 1+ 2 , 1 +

3. -1 , 1-2 , 1-

4. -1 , -1 + 2 , - 1-

Q.

If one root of the quadratic equation $a{x}^2$ + bx + c = 0 is equal to the n^th power of the other root, then the value of $(a{c}^n{)}^{\frac{1}{n+1}}$ + $(a^{n}c{)}^{\frac{1}{n+1}}$

1. b

2. -b

3. 

4. 

Q.

Given a G.P. with a = 729 and $7^{th}$ term 64. determine $S_7$

Q.

Consider two quadratic expressions f(x)=$a{x}^2+bx+c$ and g(x)=$a{x}^2+px+q$ (a, b, c, p, q R, b≠p)such that their discriminents are equal. If f(x)=g(x) has a root x=a then.

1. α will be A.M. of the roots of f(x) = 0 and g(x) = 0

2. α will be A.M of the roots f(x) = 0

3. α will be A.M. of the roots of g(x) = 0

4. α will be A.M. of the roots of f(x) = 0 or g(x) = 0

Q.

Find the sum to indicated number of terms in each of the geometric progressions in $x^3,x^5,x^7,.....nterms(ifx\ne \pm 1).$

Q. If (p+q)th term of G.P. is m and (p-q)th term is n, then pth term will be

1. $mn$
   
2. $(m-n)$
   
3. $\left(\frac{m}{n}\right)$
   
4. $\sqrt{mn}$

Q.

Find the non real roots of the cubic equation $x^3-6x-9=0$

1. 

2. 

3. 

4.