If in a triangle ABC, the altitudes from the vertices A, B, C on opposite sides are in H.P., then sin A, sin B, sin C are in
G.P.
A.P.
Arithmetic − Geometric Progression
H.P.
B.
A.P.
If a_{1}, a_{2}, a_{3} , ....,a_{n} , .... are in G.P., then the value of the determinant is
0
-2
1
2
A.
0
Suppose four distinct positive numbers a_{1}, a_{2}, a3, a_{4} are in G.P. Let b_{1} = a_{1}, b_{2} = b_{1} + a_{2}, b_{3} = b_{2} + a_{3} and b_{4} = b_{3} + a_{4}.
Statement-I: The numbers b_{1}, b_{2}, b_{3}, b_{4}are neither in A.P. nor in G.P. Statement-II: The numbers b_{1}, b_{2}, b_{3}, b_{4} are in H.P.
Both statement-I and statement-II are true but statement-II is not the correct explanation of statement-I
Both statement-I and statement-II are true, and statement-II is correct explanation of Statement-I
Statement-I is true but statement-II is false.
Statement-I is false but statement-II is true
C.
Statement-I is true but statement-II is false.
a_{1} = 1
a_{2} = 2
a_{3} = 4
a_{4} = 8
So, b_{1} = 1
b_{2} = 1 + 2 = 3
b_{3} = 3 + 4 = 7
b_{4} = 7 + 8 = 15
The numbers b_{1}, b_{2}, b_{3} , b_{4} are not in G.P. and A.P. Statement-I is correct but Statement-II wrong.
Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation
x^{2} + 18x +16 = 0
x^{2}-18x-16 = 0
x^{2}+18x-16 =0
x^{2}-18x +16 =0
D.
x^{2}-18x +16 =0
Let α and β be two numbers whose arithmetic mean is 9 and geometric mean is 4.
∴ α + β = 18 ........... (i)
and αβ =16 ........... (ii)
∴ Required equation is x2 - (α + β)x + (αβ) = 0 ⇒ x2 - 18x + 16 = 0 [using equation (i) and equation (ii)]
Let T be the rth term of an A.P. whose first term is a and the common difference is d. If for r some positive integers m, n, m ≠ n, T_{m} = 1/n and T_{n} = 1/m, then a-b equals
0
1
1/mn
A.
0
If the letters of word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number
601
600
602
603
A.
601
Alphabetical order is A, C, H, I, N, S
No. of words starting with A – 5!
No. of words starting with C – 5!
No. of words starting with H – 5!
No. of words starting with I – 5!
No. of words starting with N – 5!
SACHIN – 1 601.
The sum of the first n terms of the series when n is even. When n is odd the sum is
B.
19. The sum of n terms of given series = if n is even.
Let n is odd i.e. n = 2m + 1
Then, S_{2m+1} = S_{2m} + (2m + 1)^{th} term
If where a, b, c are in A.P. and |a| < 1, |b|<1, |c|< 1, then x, y, z are in
G.P.
A.P.
Arithmetic − Geometric Progression
H.P.
D.
H.P.
If the coefficients of rth, (r+ 1)th and (r + 2)th terms in the binomial expansion of (1 + y)^{m} are in A.P., then m and r satisfy the equation
m^{2} – m(4r – 1) + 4r^{2} – 2 = 0
m^{2} – m(4r+1) + 4r^{2} + 2 = 0
m^{2} – m(4r + 1) + 4r^{2} – 2 = 0
m^{2} – m(4r – 1) + 4r^{2} + 2 = 0
C.
m^{2} – m(4r + 1) + 4r^{2} – 2 = 0