SET - 1
PART - A (Marks : 10)
1) Choose the correct alternatives:
i) If A={1, 2, 3, 4} and Iₐ be the identity relation on A, then
A) (1,2) belongs to Iₐ
B) (2,2) belongs to Iₐ
C) (2,1) belongs to Iₐ
D) (3,4) belongs to Iₐ
ii) The principle value of tan⁻¹(-√3) is
A) π/3 B) π/4 C) - π/ D) - π/3
iii) If A is an invertible Matrix of order 3 and |A|=5, then the value of |adj A| is equal to
A) 20 B) 21 C) 24 D) 25
iv) If f(x)= log(tan(x/2)), then the value of f'(x) is
A) sin x B) 1/(sin x/2 cos x/2)
C) - cosec x D) Cosec x
v) The value of ∫ |cos x| dx at (π, 0) is
A) 0 B) 1 C) 2 D) none
vi) The area (in square unit) bounded by the curve y =sin x, x-axis and the two ordinates x=π, x= 2π is
A) 1 B) -1 C) -2 D) 2
vii) Angle between the straight lines (x -5)/7 = (y+2)/-5 = z/1 and x/1 = y/2 = z/3 is
A) π/4 B) π/3 C) π/2 D) π
viii) the value of m for which the straight line 3x -2y+z + 3= 0= 4x - 3 + 4z +1 is parallel to the plane 2x - y + mz - 2= 0 is
A) - 2 B) 8 C) -18 D) 11
ix) If the odds against an event are 4 : 5 then the probability the occurrence of the event is
A) 5/9 B) 4/9 C) 4/5 D) 1/9
x) the variance of a binomial distribution with parameters n and p is
A) n/4 B) ≤ n/4 C) > n²/4 D) ≤ n²/4
Part - B (Marks : 70)
2) On the set Q⁺ of all positive rational numbers if the binary operation * is defined by a*b = ab/4 for all a, b belongs to Q⁺, find the identity elements in Q⁺. (2)
3) solve: tan⁻¹(cot x) + cot⁻¹(tan x)= π/4. (2)
4) prove by property
y+ z z y
z z+x x = 4xyz
y x x+ y (2)
OR
If the matrix A= 2 5 and AB=-13 8
1 3 - 8 5 find B.
5) find: lim ₓ→₀ (mᵃˣ - nᵇˣ)/x. (2)
6) if x= t log t, y= (log t)/t, find dy/dx when t= 1. (2)
7) find: ∫(1 - 1/x²) ₑ(x + 1/x) dx. (2)
OR
solve: dy/dx = 1 - x + y - xy
8) Show that the function f(x)= x³/3 - 6x² + 20x - 5 has neither a maximum or minimum value. (2)
9) The radius of a circular plate increases at the rate of 0.002 cm/s. How fast is the area changing when radius is 14 cm? (2)
10) Find the acute angle between the z-axis and the straight line joining the points (3,2,3) and (-3,-1,5). (2)
OR
Prove that the equation of the plane which passes through the point (2,- 3, 5) and which is parallel to the yz- plane is x= 2.
11) Show that the probability that exactly one of the events A and B occurs is P(A)+ P(B) - 2P(AB). (2)
OR
If X is a discrete randomal variable and ' a ' is a constant, show that, E(x - bar x)= 0.
12) Show that the relation R. on the set A={x belongs to z : 0≤ x ≤ 12}, given by R{(a,b): |a - b| is a multiple of 4 and a, b belongs to A} is an equivalence relation on A. (2)
OR
Show that tan⁻¹(1/√3 tan(x/2)) = 1/2 cos⁻¹{(1+ 2 cos x)/(2+ cos x)}.
13) If the matrix
A= 1 x -2
2 2 4
0 0 2 and A² + 2I = 3A. Find x; here I is the unit matrix of order 3. (3)
OR
solve bx matrix method:
2/x - 3/y + 3/z = 10, 1/x + 1/y + 1/z = 10, 3/x - 1/y + 2/z = 13.
14) Prove the determinant. (3)
1 a a²- bc
1 b b²- ca = 0
1 c c² - ab
OR
(a²+ b²)/c c c
a (b²+ c²)/a a = 4abc
b b (c²+ a²)/b
15) If f(x)= tan⁻¹{x/(1+ 20x²)} show that f(x)= {5/(1+ 25x²)} - 4/(1+ 16x²). (3)
OR
Let y= (sin⁻¹x)²+ (cos⁻¹x), show that, (1- x²) d²y/dx² - x dy/dx = 4
16) Evaluate ∫ dx/√(2x³/3 - x² + 1/3). (3)
OR
Evaluate ∫ x²/(x⁴ +1) dx.
17) Solve: (1- x²) dy/dx - xy = x², given y= 2 when x = 0. (4)
OR
Solve: (6x + 9y -7) dx= (2x + 3y -6) dy.
18) AB = 2i - 4j + 5k and BC= i - 2j - 3k in parallelogram ABCD, find a unit vector in direction parallel to the diagonal AC of the parallelogram. (4)
19) prove ∫ log(sin x) dx at (π/2, 0) = π/2 log(1/2). (4)
20) Urn A contains 1 white, 2 black and 3 red balls; Urn B contains two white, one black and one red ball; urn C contains 4 white, 5 black and 3 red balls. One urn is chosen at random and two balls are drawn. These happen to the one white and one red. what is the probability that they come from urn A ? (4)
21) the probability distribution of a random variable X is as follows:
X: 0 1 2 3 4
P(x) 0.20 0.25 0.35 0.14 0.06 Find the probability distribution of the variable Y where y= x² + 5. (5)
22) A company manufacturers two types of toys A and B. Type A requires 5 minutes each for cutting and 10 minutes each for assembling. Type B requires 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. He earns a profit of ₹50 each on type A and ₹ 60 each on type B. formulate the above problem as a LPP to maximize the profit. (5)
OR
Solve graphically:
Minimize Z= 3x+ 5y
Subject to the constraints x+3y≥ 3, x+ y ≥ 2, x, y≥ 0
23) A cylindrical tin can, open at the top, of a given capacity, has to be constructed. Show that the amount of the ti required will be least if the height of the can is equal to its radius. (5)
24) Show that lines r= (i+ j+ k)+ t(i - j + k) and r= (3i - k)+ s(4j - 16k) intersect and find the position vector of their point of their point of their point of intersection. (5)
OR
Find the equation of the plane passing through the point (-1, 1,1) and (1,-1,1) and is perpendicular to the plane x+ 2y + 2z = 5.
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