১.
If the sum of first n terms of a G.P. is Sn = 7(2n - 1), then the common ratio is:
ব্যাখ্যা
This matches the form:
৪৯তম বিসিএস ⎯ গণিত [৫৫১]
মোট প্রশ্ন২৯
সিলেবাস
Exam - 2
Topics:
1. Summation of finite algebraic and trigonometric series.
2. Polynomials and their roots, Honer’s scheme (synthetic division), Descartes rule of signs.
3. Relation between roots and coefficients. Symmetric functions of roots.
[Source: Class - 2 and Relevant Books]
৪৯তম বিসিএস ⎯ গণিত [৫৫১]
৪৯তম বিসিএস ⎯ গণিত [৫৫১] · তারিখ অনির্ধারিত · ২৯ প্রশ্ন
২.
Evaluate:
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৩.
A series where each term cancels with a part of the next term is called:
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A series where each term cancels with a part of the next term is called telescoping series.
৪.
The sum of the series 5 + 8 + 11 +.........+ 50 is-
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here a=5, d=3, l=50
৫.
For geometric progression is true when
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৬.
The sum 2 + 4 + 6 +.................+ 2n equals:
ব্যাখ্যা
Sn= (n/2) [2a + (n - 1)d] = n(n + 1)
৭.
sin10°+ sin20°+.............+ sin100° equals:
ব্যাখ্যা
৮.
the nth term of tan- 1 (1/3) + tan- 1 (2/9) + ....
ব্যাখ্যা
৯.
A polynomial has 4 sign changes in P(x). The number of positive real roots can be:
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For positive real roots:
From Descarte’s rule of sign change
From Descarte’s rule of sign change
Count the number of sign changes in which is 4
Each sign change corresponds to:
• a possible real root, or
• Fewer roots than the count by a multiple of 2.
• Fewer roots than the count by a multiple of 2.
১০.
For P(x) = x3- 6x2+ 11x - 6, number of sign changes:
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Here number of sign change is = 3
+x3 to - 6x2, then - 6x2 to + 11x, then +11x to - 6
+x3 to - 6x2, then - 6x2 to + 11x, then +11x to - 6
১১.
After synthetic division of x3- 3x2+ 3x - 1 by x - 1, the quotient is:
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x3- 3x2 + 3x - 1
= x2(x - 1) - 2x(x-1) + 1(x - 1)
= (x - 1)(x2- 2x + 1)
So (x2 - 2x + 1) is the quotient
= x2(x - 1) - 2x(x-1) + 1(x - 1)
= (x - 1)(x2- 2x + 1)
So (x2 - 2x + 1) is the quotient
১২.
Synthetic division is most efficient when divisor is:
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Synthetic division is most efficient when divisor is linear
১৩.
In synthetic division, if the remainder is 0, then:
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In synthetic division, if the remainder is 0, then the divisor is a factor
১৪.
A polynomial equation of degree 5 can have how many complex roots at most?
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A polynomial equation of degree n can have how many complex roots at most n.
১৫.
The polynomial P(x) = x2+ 4x + 4 has:
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Discriminant, D = 42 - 4.1.4 = 0
One real repeated root
১৬.
The remainder when P(x) = x3+ 2x2- x + 5 is divided by x - 1 is:
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Remainder = P(1) = 13 + 2.12 - 1 + 5 = 7
১৭.
The degree of (x2+ 1)(x - 2 + x3) is:
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Degree is sum of highest degrees: 2 + 3 = 5
১৮.
If α, β are roots of 3x2 - 2x + 1 = 0, the equation whose roots are α2, β2 is:
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১৯.
If 2x2+ 2x - k is a full square, then what is the value of k?
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Since 2x2 + 2x - k is a square then discriminant of 2x2 + 2x - k = 0 is = 0
So, 22- 4.2.(- k) = 0 or, k = - (1/2)
২০.
What is the maximum value of 5 + 3x - x2?
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২১.
For what value of p, x2+ px + 1 = 0 has complex roots?
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Here x2 + px + 1=0
Now for complex roots,
D = 0
Or, P2- 4.1.1<0
Or, p2 < 4
Or, - 2
২২.
If the roots 3x2- 10x + k = 0 has roots a and 1/a , what is the value of k?
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২৩.
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২৪.
Which one of the following is the quadratic equation having root 5 + 3i?
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If one root is, a = 5 + 3i, other root is b=5 - 3i
So the equation is, x2-(a + b)x + ab=0
Or, x2- (5 + 3i + 5 - 3i)x + (5 + 3i)(5 - 3i) = 0
Or, x2- 10x + 34 = 0
So the equation is, x2-(a + b)x + ab=0
Or, x2- (5 + 3i + 5 - 3i)x + (5 + 3i)(5 - 3i) = 0
Or, x2- 10x + 34 = 0
২৫.
If one root of a quadratic equation is, 1/(2 - √5) find the other one-
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২৬.
Evaluate:
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২৭.
In a telescoping series, which terms usually remain?
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By the definition, a telescoping series is one where consecutive terms cancel out, leaving only a few terms at the beginning and end.
২৮.
What is the nth term of the series
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২৯.
Result of dividing x2 - 3x + 2 by x - 1:
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Now, no, remainder left.
Else we can use synthetic division method.