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ESSAY ANSWER QUESTIONS (7 MARKS)
DE MOIVRE’S THEOREM
11 7 4 9 5 4
1. Find all the roots of the equation i) x - x + x - 1 = 0 ii) x - x + x - 1 = 0.
2. If cos + cos + cos = 0 = sin + sin + sin then show that
i) cos 3 + cos 3 + cos 3 = 3cos ()
ii) sin 3 + sin 3 + sin 3 = 3sin ()
3
2 2 2 2 2 2
3. If cos + cos + cos sin + sin + sin , Prove that cos + cos + cos = = sin + sin + sin
2
n
2n 2n n + 1
4. If n is an integer then show that (1 + i) + (1 - i) = 2 cos .
2
n2
n
2
n n
2 cos
5. If n is a positive integer, show that (1 + i) + (1 - i) =
4
n
2 n n n+1
6. If are the roots of the equation x - 2x + 4 = 0 then for any n N show that + = 2 cos
3
1 1 1
1 Q
2 2 1
n n 2n
(piQ) (PiQ) 2(P Q ) .cos tan
7. If n is a positive integer, show that
n P
n
n
n n + 1 n
8. If n is an integer then show that (1 + cos + i sin ) + 1 cos isin = 2 cos cos
2 2
8
3
1sin icos
8 8
9. Show that one value of is - 1.
1sin icos
8 8
2n
z 1
2n1
10. If n is an integer and z = cis , , then show that = i tan n
2n
2
z 1
n 2 n
11. If (1+x) = a + a x+ a x +.......+a x , then show that
0 1 2 n
n
n
2
2 cos
i) a - a + a -......... = .
0 2 4
4
n
n
2
2 sin
ii) a - a + a -......... = .
1 3 5
4
12. State and prove de-moivres theorem for an integral index
THEORY OF EQUATIONS
3 2
1. Solve : 4x - 24x + 23x + 18 = 0, given that the roots are in A.P.
3 2
2. Solve the equation x - 7x + 14x - 8 = 0, given that the roots are in geometric progression.
3 2
3. Solve the equation 15x - 23x + 9x - 1 = 0, given that the roots of are in H.P.
3 2
4. Solve 18x + 81x + 121x + 60 = 0 given that a root is equal to half the sum of the remaining roots.
4 3 2
5. Solve the equation x - 2x + 4x + 6x - 21 = 0, the sum of two roots being zero.
4 3 2
6. Solve the equation x - 5x + 5x + 5x - 6 = 0, the product of two roots being 3.
4 3 2
7. Solve the equation x + 4x - 2x - 12x + 9 = 0, if it has a pair of equal roots.
4 3 2
8. Find the roots of x - 16x + 86x - 176x +105 = 0.
4 3 2
9. Solve 6x - 35x + 62x - 35x +6 = 0.
5 4 3 2
10. Solve : 2x + x - 12x - 12x + x + 2 = 0
6 5 4 2
11. Solve the equation : 6x - 25x + 31x - 31x + 25x - 6 = 0.
3 2
12. Solve x - 9x + 14x + 24 = 0 given that two of the roots are in the ratio 3:2.
5 4 3 2
13. Find the repeated roots of the equation x - 3x - 5x + 27x - 32x + 12 = 0.
4 3 2
14. Solve the equation x + 2x - 5x + 6x + 2 = 0, given that one root of it is 1 + i.
5 3 2
15. Find the algebraic equation of degree 5 whose roots are the translates of the roots of x + 4x -x +11=0 by -3.
4 3 2
16. Transform x + 4x + 2x - 4x - 2 = 0 into another equation in which the coefficient of second highest power
of x is zero and find the transformed equation.
3 2
17. If the roots of the equation x + 3px + 3qx + r = 0
3
i) are in Arithmetic progression, then show that 2p - 3pq + r = 0.
3 3 3
ii) Are in G.P then show that p r = q . (iii) Are in H.P show that 2q = 4(3pq-r).
BINOMIAL THEOREM
n + 1
(1 + x) - 1
C C C
1 2 2 n n
1 Prove that C + x + x + ..... + x = .
0 (n + 1) x
2 3 n + 1
n
C C C
3 5 2 - 1
1
Deduce that + + + ....... = .
2 4 6 n + 1
2n
2 Prove that C C + C C + C C + .... + C C = C .
0 r 1 r + 1 2 r + 2 n - r n n + r
2n
2 2 2 2 2n
Deduce that i) C + C + C + ...... + C = C . ii) C C C C C C ......C C C
0 1 1 2 2 3 n1 n n 1
0 1 2 n n
nd rd th n
3. If the 2 , 3 and 4 terms in the expansion of (a + x) are respectively 240, 720, 1080, find a, x, n.
n
4. a) If (7 + 4 ) = I + f where I and n are positive integers and 0 < f < 1 then show that (i) I is an odd
3
positive integer (ii) (I + f) (1- f) = 1.
bx bx
find the relation between a and b, where a and b are real numbers.
n
6. i) If the coefficients of rth, (r + 1)th, (r + 2)nd terms in the expansion of (1 + x) are in A.P, then show that
2 2
n - (4r + 1) n + 4r - 2 = 0.
9 10 11 n 2
ii) If the coefficients of x , x , x in the expansion of (1 + x) are in A.P. then prove that n - 41n + 398 = 0.
n
7. If the coefficients of 4 consecutive terms in the expansion of (1 + x) are a , a , a , a respectively, then
1 2 3 4
a a 2a
1 3 2
show that + .
a + a a + a a + a
1 2 3 4 2 3
8. State and prove “Binomial theorem for a positive integral index n.”
2
n n 2
C n(n+1) (n+2)
3
r
r
9. If n is a positive integer, prove that .
n
C 12
r=1
r-1
n
10. If P and Q are the sum of odd terms and the sum of even terms respectively in the expansion of (x + a) , then prove that
2 2 2 2 n 2n 2n
i) P - Q = (x - a ) ii) 4PQ = (x + a) - (x - a) .
n
n1
11. Prove that (C + C ) (C + C ) (C + C )..............(C + C ) = C C ...........C .
0 1 1 2 2 3 n-1 n 0 1 n
n!
BINOMIAL THEOREM SERIES PROBLEMS
3 3. 5 3.5.7
+ + +.......
1. Find the sum of infinite series
4 4.8 4.8.12
4 4.6 4.6.8
...............
2. If t = then prove that 9t = 16.
5 5.10 5.10.15
1 1.3 1.3.5
2
3. If x = ..............., then find the value of 3x + 6x.
5 5.10 5.10.15
1.3 1.3.5 1.3.5.7
2
4. If x = + ..............., then prove that 9x + 24x = 11.
3.6 3.6.9 3.6.9.12
3.5 3.5.7 3.5.7.9
...............
5. Find the sum of the series
5.10 5.10.15 5.10.15.20
3 3.5 3.5.7
6. Find sum of the infinite series - + -..... .
4.8 4.8.12 4.8.12.16
5 5.7 5.7.9
2
7. If x = + ..............., then find the value of x + 4x.
2 3
(2!)3
(3!) 3 (4!) 3
7 1 1.3 1 1.3.5 1
1+ ...............
8. Find the sum of the series .
2 4 6
5 1.2 1.2.3
10 10 10
2 3
2 1 2.5 2.5.8
1 1
9. Find the sum of the infinite series 1 + . + + +........ .
3 2 3.6 3.6.9
2 2
x x(x-1) x(x-1) (x-2)
10 Show that for any non zero rational number x, 1+ + ..................
2 2.4 2.4.6
x x(x+1) x(x+1) (x+2)
= 1+ + ..................
3 3.6 3.6.9
.
MEASURES OF DISPERSION
1. Calculate the mean deviation about the mean for the following data
Class interval 2 5 7 8 10 35
Frequency 6 8 10 6 8 2
2. Find the mean deviation from the mean of the following data, using the step deviation method
Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70
No. of Students 6 5 8 15 7 6 3
x 6 9 3 12 15 13 21 22
i
f 4 5 3 2 5 4 4 3
i
4. Find the mean deviation from the median of the following data.
Age (Years) 20-25 25-30 30-35 35-40 40-45 45-50 50-55 55-60
No. of workers (f) 120 125 175 160 150 140 100 30
i
5. Calculate the variance and standard deviation for the discrete frequency distribution
x 4 8 11 17 20 24 32
i
f 3 5 9 5 4 3 1
i
6. Calculate the variance and standard deviation of the following continuous frequency distribution
Class interval 30-40 40-50 50-60 60-70 70-80 80-90 90-100
Frequency 3 7 12 15 8 3 2
7. The following tables gives the daily wages of workers in a factory. Compute the standard deviation and the
coefficient of variation of the wages of the workers
Wages 125-175 175-225 225-275 275-325 325-375 375-425 425-475 475-525 525-575
No. of Workers 2 22 19 14 3 4 6 1 1
8. The scores of two cricketers A and B in 10 innings are given below. Find who is a better run getter and
who is a more consistent player
Scores of A : x 40 25 19 80 38 8 67 121 66 76
i
Scores of B : y 28 70 31 0 14 111 66 31 25 4
i
9. The mean of 5 observations is 4.4. Their variance is 8.24. If three of the observations are 1, 2 and 6. Find
the other two observations.
PROBABILITY
1 State and explain the axioms that define ‘Probability function’. Prove addition theorem on probability.
i.e. P (E E ) = P(E ) + P(E ) - P(E E ).
1 2 1 2 1 2
2 A, B, C are three horses in a race. The probability of A to win the race is twice that of B, and probability of B is twice that
of C. What are the probabilities of A, B, C to win the race. Also find the probability that A loses in the race.
3. A, B, C are 3 newspapers from a city, 20% of the population read A, 16% read B, 14% read C, 8% read both
A and B, 5% read both A and C, 4% read both B and C and 2% read all the three. Find the percentage of the
population who read atleast one newspaper and find the percentage of the population who read the newspaper
A only.
4. The probabilities of three events A, B, C are such that P(A) = 0.3, P(B) = 0.4, P(C) = 0.8, P(A B) = 0.08,
P(A C) = 0.28, P(ABC) = 0.09 and P(ABC) > 0.75. Show that P(BC) lies in the interval [0.23, 0.48]
1+3p 1-p 1-2p
, ,
5. The probabilities of three mutually exclusive events are respectively, given as .
3 4 2
1 1
p
Prove that .
3 2
6. A, B, C are aiming to shoot a balloon. A will succed 4 times out of 5 attempts. The chance of B to shoot the balloon is
3 out of 4 and that C is 2 out of 3. If the three aim the balloon simultaneously, then find the probability that atleast two
of them hit the balloon.
7. In a shooting test the probability of A, B, C hitting the targets are 1/2, 2/3 and 3/4 respectively. If all of them fire at the
same target, find the probability that i) only one of them hits the target, ii) at least one of them hits the target.
1 1
8. If E , E , E are three independent events such that P(E E E ) = , P(E E E ) = ,
1 2 3 3
1 2
1 2 3 4 8
1
P(E E E ) = , then find P(E ), P(E ), P(E ).
1 2 3
4 1 2 3
9. Define conditional event and Conditional Probability. There are 3 black and 4 white balls in one bag; 4
black and 3 white balls in the second bag. A die is rolled and the first bag is selected if it is 1 or 3, and the
second bag for the rest. Find the probability of drawing a black ball from the selected bag.
10. State and prove Baye’s theorem.
11. Three boxes numbered I, II, III contain 1 white, 2 black and 3 red balls; 2 white, 1 black and 1 red ball;
4 white, 5 black and 3 red balls respectively. One box is randomly selected and a ball is drawn from it.
If the ball is red then find the probability that it is from box II.
12. Three boxes B , B , B contain balls with different colours as follows:
1 2 3
White Black Red
B 2 1 2
1
B 3 2 4
2
B 4 3 2
3
A die is thrown. If 1 or 2 turns up on the dice, box B is selected; if 3 or 4 turns up B is selected; if 5 or 6
1 2
turns up, then B is selected. If a box is selected like this, a ball is drawn from that box. If the ball is red,
probaility that the student is a girl.
RANDOM VARIABLE AND DISTRIBUTIONS
1 The probability distribution of a random variable X is given below:
X = x 1 2 3 4 5
i
P(X=x) k 2k 3k 4k 5k
i
Find the value of k and the mean, varianceof X.
2. X = x -2 -1 0 1 2 3
P(X = x) 0.1 k 0.2 2k 0.3 k
is the probability distribution of a random variable X. Find the value of K and the variance of X.
3. A random variable X has the following probability distribution.
X = x 0 1 2 3 4 5 6 7
2 2 2
P(X = x) 0 k 2k 2k 3k k 2k 7k + k
Find (i) k (ii) The mean (iii) P(0 < X < 5)
4. A cubical die is thrown. Find the mean and variance of X, giving the number on the face that shows up.
3 2
5. The range of a random variable X is {0, 1, 2}. Given that P(X = 0) = 3C , P(X = 1) = 4C - 10C , P(X = 2) = 5C - 1.
Find (i) the value of C (ii) P(X < 1) (iii) P(1 < X < 2) (iv) P(0 < X < 3).
6. One in nine ships is likely to be wrecked when they set on sail. When 6 ships are set on sail, find the probability for :
i) atleast one will arrive safely ii) exactly three will arrive safely
7. If the mean and variance of a binomial variate X are 2.4 and 1.44 respectively, find P(1 < X < 4).
8. In the experiment of tossing a coin n times, if the variable X denotes the number of heads and P(X = 4),
P(X = 5), P(X = 6) are in A.P, then find n.
5
9. If the difference between the mean and variance of binomial variate is then, find the probability for the
9
event of 2 successes when the experiment is conducted 5 times.
k
C
10. The range of a random variable X is {1, 2, 3,..........} and P(X = k) = ; k = 1, 2, 3,............. Find the
k!
value of c and P(0 < x < 3).
(K + 1)C
11. If X is a random variable with the probability distribution P(X = K) = (K = 0, 1, 2, ....), then find C.
K
2
2
12. If X : S R is a discrete random variable with range {x , x , x ,.........}; is mean and is variance of X
1 2 3
2 2 2
then prove that + = X P(X = X ).
r r
SHORT ANSWER QUESTIONS (4 MARKS)
COMPLEX NUMBERS
1. Show that the points in the Argand diagram represented by the complex numbers 2 + 2i, - 2 - 2i, -2 + 2 are
3 3i
the vertices of an equilateral triangle.
2. Show that the four points in the Argand plane represented by the complex numbers 2 + i, 4 + 3i, 2 + 5i, 3i are the
vertices of a square.
-3 1 7
+ i, 1+i
3. Show that the points in the Argand plane represented by the complex numbers -2 + 7i, are
4-3i,
2 2 2
the vertices of rhombus.
3 2
4. If z = 3 - 5i, then show that z - 10z + 58z + 136 = 0.
x y
1/3 2 2
5. If (x - iy) = a - ib, then show that = 4(a - b ).
a b
6. If z = x+iy and if the point P in the Argand plane represents z, find the locus of z satisfying the equation |z - 2 - 3i| = 5.
zi
7. If the point P denotes the complex number z = x + iy in the argand plane and if is a purely imaginary number,,
z1
find the locus of P.
z 2
8. If the amplitude of , find its locus.
z6i 2
z 4
9. Determine the locus of z, z 2i, such that Re = 0.
z2i
32isin
10. Find the real values of in order that is a
12isin
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