Question -1. 1x 10= 10
i) Find the GM between a³b and ab³
A) ab B) ab²C) a²b D) a²b²
ii) Value of (i+ i²+ i³+ i⁴)/(1+ i) is
A) 0 B) 0i C) o+0i D) 1
iii) If (A xB)={(3,2),(3,4),(5,2),(5,4)} find B
A) |2,4| B) [2,4] C) (2,4) D) {2,4}
iv) (A∩B) U(A - B) is
A) A B) B C) A' D) (AB)'
v) In how many ways can the letters of the word BANANA be arranged?
A) 60 B) 72 C) 144 D) 210
vi) Two dice are thrown simultaneously. The probability that the number on both the dice are same
A) 1/6 B) 1/18 C) 1/36 D) none
vii) If cosx= -1/2, what is the general value of x.
A) 30 B) 60 C) 90 D) none
viii) If the distance between the points (-3,3) and (4,y) be 5√2 units, find the value of y.
A) 3 B) 4 C) 5 D) 6 units
ix) Find the equation of straight line whose x-intercept and y-intercept are 3 and - 4 respectively.
A) x/3 + y/4 = 1 B) x/3 - y/4 = 1
C) - x/3 + y/4 = 1 D) x/3 + y/4 = -1
x) The middle term of (x/y - y/x)¹⁰ is
A) 5th B) 6th C) 7th D) 4th
Question 2. 2x10= 20
i) Find the value of sin 105°
ii) Find the nth term of the GP 12, 4, 4/3, 4/9
iii) Find the modulus of (1+ 3i)/(2- i)
iv) The mean deviation of 7,8, 4, 13, 9, 5, 16
v) Find the value of cos(-1170).
vi) Find the radius and centre of the circle x²+ y² = 36 is
vii) Find the number of arrangement of word MONDAY.
OR
Find the number of arrangement of x²y³z⁴
viii) Find the coefficient of x⁷ in the expansion of (x² + 1/x)¹¹
ix) Find the value of sin 75 + cos 75
x) Find the area of the triangle whose vertex are (3,2,(4,-2),(-4,7) respectively.
Question 3. 3x10= 30
i) Find the eccentricity of an ellipse whose latus rectum is one half of its minor axis.
OR
Find the equation of the hyperbola whose vertices are are (0,±3) and the foci are (0,±5)
ii) Find the square root of 5 - 12i
iii) a) lim ₓ→₁ (x²-1)/(x +1).
b) lim ₓ→₂ (x³ +1)/(x²+1+ 3x)
iv) How many numbers of four digits can be formed with the digits 1, 2, 3, 4, 5, 6, 7 .
OR
Prove cos 3π/32 = 1/2 √[2+√{2+ √(2+ √2)}].
v) A die is thrown and at the same time a card is drawn from a pack of 52 playing cards. Find the probability of getting 5 and Ace of hearts.
vi) find dy/dx: y=√{(1- cos 2x)/(1+ cos 2x)).
vii) if the three consecutive vertices of a parallelogram be A(3,4-1), B(7,10,-3) and C(5,-2,7). Find the fourth vertex D.
OR
Prove cos²(π - a/2) - cos²(π/8 + a/2)= 1/√2 sin a.
viii) √3 sin10 + sin 20 = cos 50.
OR
Prove cos 9 cos 27 cos 63 cos 81 = 1/16
ix) Prove that the straight lines is concurrent x+ y+5= 0, x- y+1= 0, 3x - y+ 7= 0.
x) If 2 cos a= x+ 1/x, then show that 2 cos 3a = x³ + 1/x³.
OR
A 4 digited number is written by the digits 1,2,3 and 4 and where no digits is repeated in any number. Find the probability that the number is
A) odd
B) mutiple of 4.
Question 4. 4x5= 20
i) The midpoints of the sides of a triangle are (1,5,-1),(0,4,-2) and (2,3,4) find its vertices
OR
Find the fourth term from the end of (x⁴ + 1/x³)¹⁵ .
ii) In how many can a committee of 6 persons be formed taken atleast 3 gentlemen and 2 ladies from 10 gentlemen and 7 ladies, where two particular ladies refuse to serve in the same committee together.
iii) Prove Cos²A + cos²(A- 120) + cos²(A+ 120) = 3/2.
OR
If n be any real integer, find the value of cosec {nπ/2 +(-1)ⁿ π/6}.
iv) In an examination, 56% of the candidates failed in English and 48% failed in science. If 18% failed in both English and science, find the percentage of those who passed in both the subjects.
OR
Which term in the expansion of (2x² - 1/x)¹² is independent of x ? Find the value of that term.
v) The coordinates of the points A, B and C are (6,3),(-3,5) and (4,-2) respectively and that of the point P(x,y). Show that the ratio of the area of PBC and ABC is |(x+ y -2)/7|.
Or
Find the equation of the straight line which passes through the point of intersection of the straight lines y- 2x +2= 0 and y - 3x +5= 0 and is at a distance of 7/√2 units from the origin.
Question 5. (20 marks)
i) If the standard deviations of the numbers 2, 3, 2x, 11 is 35, find the possible value of x. (2)
OR
Find the standard deviation of
X: 10 15 18 20 25
F: 3 2 5 8 2
ii) Find the mean deviation of:
Class: 20-30 30-40 40-50 50-60
F: 3 7 12 8 (3)
iii) Find the combined standard deviations of 10 numbers when mean of x and y are 20, 30 respectively. And standard deviations of x and y are 3 and 4 respectively. (4)
iv) Two dice are thrown simultaneously. Find the probability of getting
A) a doublet
B) a multiple of 3 as the sum
C) multiple of 3 and 5
D) multiple of 3 or 5. (4)
v) Find the derivative from the first principle. √(4-x). (3)
vi) Find the equation of a circle concentric with the circle 2x²+ 2y² - 6x + 8y +1= 0. (4)
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