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 Multiple Choice QuestionsMultiple Choice Questions

1.

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.

increment space equals space 1 half straight p subscript 1 straight a space equals space 1 half straight p subscript 2 straight b space equals space 1 half straight p subscript 3 straight b
straight p subscript 1 comma space straight p subscript 2 comma space straight p subscript 3 space are space in space straight H. straight P.
rightwards double arrow space fraction numerator 2 increment over denominator straight a end fraction comma fraction numerator 2 increment over denominator straight b end fraction comma fraction numerator 2 increment over denominator straight c end fraction space are space in space straight H. straight P.
rightwards double arrow space 1 over straight a comma 1 over straight b comma 1 over straight c space are space in space straight H. straight P
rightwards double arrow straight a comma straight b comma space straight c space are space in space straight A. straight P
rightwards double arrow space sin space straight A comma space sin space straight B comma space sin space straight C space are space in space straight A. straight P.
322 Views

2.

If a1, a2, a3 , ....,an , .... are in G.P., then the value of the determinant open square brackets table row cell log space straight a subscript straight n end cell cell log subscript straight n plus 1 end subscript end cell cell log subscript straight n plus 2 end subscript end cell row cell log space straight a subscript straight n plus 3 end subscript end cell cell log space straight a subscript straight n plus 4 end subscript end cell cell log space straight a subscript straight n plus 5 end subscript end cell row cell log space straight a subscript straight n plus 6 end subscript end cell cell log space straight a subscript straight n plus 7 end subscript end cell cell log space straight a subscript straight n plus 8 end subscript end cell end table close square brackets is

  • 0

  • -2

  • 1

  • 2


A.

0

262 Views

3.

Suppose four distinct positive numbers a1, a2, a3, a4 are in G.P. Let b1 = a1, b2 = b1 + a2, b3 = b2 + a3 and b4 = b3 + a4.
Statement-I: The numbers b1, b2, b3, b4are neither in A.P. nor in G.P. Statement-II: The numbers b1, b2, b3, b4 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.

a1 = 1
a2 = 2
a3 = 4
a4 = 8

So, b1 = 1
b2 = 1 + 2 = 3
b3 = 3 + 4 = 7
b4 = 7 + 8 = 15
The numbers b1, b2, b3 , b4 are not in G.P. and A.P. Statement-I is correct but Statement-II wrong.



233 Views

4.

The sum of series fraction numerator 1 over denominator 2 space factorial end fraction space plus fraction numerator 1 over denominator 4 factorial end fraction space plus fraction numerator 1 over denominator 6 space factorial end fraction space plus space.... space is

  • fraction numerator left parenthesis straight e squared minus 1 right parenthesis over denominator 2 end fraction
  • fraction numerator left parenthesis straight e minus 1 right parenthesis squared over denominator 2 straight e end fraction
  • fraction numerator left parenthesis straight e squared minus 1 right parenthesis over denominator 2 straight e end fraction
  • fraction numerator left parenthesis straight e minus 1 right parenthesis squared over denominator straight e end fraction

B.

fraction numerator left parenthesis straight e minus 1 right parenthesis squared over denominator 2 straight e end fraction
straight e space equals space 1 space plus space fraction numerator 1 over denominator 1 factorial end fraction space plus fraction numerator 1 over denominator 2 factorial end fraction space plus fraction numerator 1 over denominator 3 factorial end fraction space plus space fraction numerator 1 over denominator 4 factorial end fraction space plus space....... infinity space.... space left parenthesis straight i right parenthesis
straight e to the power of negative 1 end exponent space equals 1 space minus space fraction numerator 1 over denominator 1 factorial end fraction space plus fraction numerator 1 over denominator 2 factorial end fraction space minus fraction numerator 1 over denominator 3 factorial end fraction space plus space fraction numerator 1 over denominator 4 factorial end fraction space minus space....... infinity space.... space left parenthesis ii right parenthesis
on space adding space equ space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis
straight e space plus straight e to the power of negative 1 end exponent space space equals space 2 space plus space fraction numerator 2 over denominator 2 space factorial end fraction space plus fraction numerator 2 over denominator 4 factorial end fraction space plus........ space infinity
fraction numerator straight e squared plus 1 over denominator straight e end fraction minus 2 space equals space fraction numerator 2 over denominator 2 factorial end fraction space plus fraction numerator 2 over denominator 4 factorial end fraction space plus....... infinity
fraction numerator straight e squared space plus 1 minus 2 straight e over denominator straight e end fraction space equals space 2 space open square brackets fraction numerator 1 over denominator 2 factorial end fraction fraction numerator 1 over denominator 4 space factorial end fraction space plus..... infinity close square brackets
fraction numerator left parenthesis straight e minus 1 right parenthesis squared over denominator 2 straight e end fraction space equals space fraction numerator 1 over denominator 2 factorial end fraction space plus fraction numerator 1 over denominator 4 factorial end fraction space plus space...... infinity
120 Views

5.

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

  • x2 + 18x +16 = 0

  • x2-18x-16 = 0

  • x2+18x-16 =0

  • x2-18x +16 =0


D.

x2-18x +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)]

211 Views

6.

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, Tm = 1/n  and Tn = 1/m, then a-b equals

  • 0

  • 1

  • 1/mn

  • 1 over straight m plus 1 over straight n

A.

0

Given space that comma space straight T subscript straight m space equals space 1 over straight n
rightwards double arrow space straight a space plus left parenthesis straight m minus 1 right parenthesis. straight d space equals 1 over straight n space..... space left parenthesis straight i right parenthesis
and space straight T subscript straight n space equals 1 over straight m
rightwards double arrow space straight a space plus space left parenthesis straight n minus 1 right parenthesis straight d space equals space 1 over straight m space..... space left parenthesis ii right parenthesis
On space solving space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis space we space get
straight a space equals 1 over mn space and space straight d space equals space 1 over mn
so comma space straight a minus straight d space equals space 1 over mn minus 1 over mn space equals space 0
254 Views

7.

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.

174 Views

8.

The sum of the first n terms of the series 1 squared space plus space 2.2 squared space plus space 3 squared plus space 2.4 squared space plus space 5 squared space plus space 2.6 squared plus... space is space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis squared over denominator 2 end fraction when n is even. When n is odd the sum is

  • fraction numerator 3 straight n space left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction
  • fraction numerator straight n squared space left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction
  • fraction numerator straight n squared space left parenthesis straight n plus 1 right parenthesis over denominator 4 end fraction
  • open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis squared over denominator 2 end fraction close square brackets squared

B.

fraction numerator straight n squared space left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction

19. The sum of n terms of given series = fraction numerator straight n left parenthesis straight n plus 1 right parenthesis squared over denominator 2 end fractionif n is even.
Let n is odd i.e. n = 2m + 1
Then, S2m+1 = S2m + (2m + 1)th term

fraction numerator left parenthesis straight n minus 1 right parenthesis straight n squared over denominator 2 end fraction space plus straight n to the power of th space term
space equals space fraction numerator left parenthesis straight n minus 1 right parenthesis straight n squared over denominator 2 end fraction space plus space straight n squared
left parenthesis because space straight n space space is space odd space equals space 2 straight m plus 1 right parenthesis
space equals space straight n squared open square brackets fraction numerator straight n minus 1 plus 2 over denominator 2 end fraction close square brackets space equals space fraction numerator left parenthesis straight n plus 1 right parenthesis straight n squared over denominator 2 end fraction

178 Views

9.

If straight x space equals space sum from straight n equals space 0 to infinity of straight a to the power of straight n comma space straight y space equals space sum from straight n equals 0 to infinity of straight b to the power of straight n comma space sum from straight n equals 0 to infinity of straight c to the power of straight n 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.

straight x space equals sum from straight n equals 0 to infinity of space straight a to the power of straight n space equals space fraction numerator 1 over denominator 1 minus straight a end fraction space space space space space space space space space straight a equals space 1 minus 1 over straight x
straight y space equals sum from straight n equals 0 to infinity of space straight b to the power of straight n space space space equals fraction numerator 1 over denominator 1 minus straight b end fraction space space space space space space space space straight b equals space space 1 minus 1 over straight y
straight z space equals space sum from straight n equals space 0 space to infinity of straight c to the power of straight n space equals space fraction numerator 1 over denominator 1 minus straight c end fraction space space space space space space space space space straight c space equals space 1 minus 1 over straight z
straight a comma straight b comma space straight c space are space in space straight A. straight P.
2 straight b space equals space straight a plus straight c
2 open parentheses 1 minus 1 over straight y close parentheses space equals 1 minus 1 over straight x plus 1 minus 1 over straight y
2 over straight y space equals space 1 over straight x plus 1 over straight z
rightwards double arrow space straight x comma straight y comma straight z space are space in space straight H. straight P.
117 Views

10.

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

  • m2 – m(4r – 1) + 4r2 – 2 = 0

  • m2 – m(4r+1) + 4r2 + 2 = 0

  • m2 – m(4r + 1) + 4r2 – 2 = 0

  • m2 – m(4r – 1) + 4r2 + 2 = 0


C.

m2 – m(4r + 1) + 4r2 – 2 = 0

Given space to the power of straight m straight C subscript straight r minus 1 end subscript space comma space to the power of straight m straight C subscript straight r space plus straight C presuperscript straight m subscript straight r plus 1 end subscript space are space in space straight A. straight P.
2 to the power of space straight m end exponent straight C subscript 2 space equals space fraction numerator blank to the power of straight m straight C subscript straight r minus 1 end subscript over denominator straight C presuperscript straight m subscript straight r end fraction space plus fraction numerator straight C presuperscript straight m subscript straight r plus 1 end subscript over denominator straight C presuperscript straight m subscript straight r end fraction
space equals space fraction numerator straight r over denominator straight m minus straight r plus 1 end fraction space plus fraction numerator straight m minus straight r over denominator straight r plus 1 end fraction
rightwards double arrow space straight m squared minus straight m space left parenthesis 4 straight r plus 1 right parenthesis space plus 4 straight r squared space minus 2 space equals space 0 space
197 Views