Set - I
2) Prove that the relation R in the set A={ 1, 2, 3, 4, 5} given R={(a, b): |a - b| is even}, is an equivalence relation.
3) Let f: N-->N be defined by
(n + 1 )/2 , if n is odd
f(x)= n/2, if n is even
Find whether the function f is bisective. f is not bijective, f is not one-one
1) let * be binary operation on the set of rational numbers given as a* b = (2a- b)², a, b ∈ Q. Find
A) 3*5. 1
B) 5*3. 49
C) Is 3*5=5*3? No
1) Show that the relation S in the set R of real numbers, defined as A={(a, b) ∈ R and a ≤ b³} is neither reflexive, nor symmetric, nor transitive.
2) Let f: R---> R be defined as f(x)= 10x+7. Find the function g: R---> R such that g o f = f o g = Iᴿ⁺ . (x-7)/10
1) Find the value of
A) cot(π/2 - 2 cot⁻¹√3). √3
B) tan⁻¹{2sin(2 cos⁻¹(√3/2))}. π/3
C) tan[2 tan⁻¹(1/2)- cot⁻¹3]. 9/13
2) Solve:
A) sin{cot⁻¹(x+1)}= cos(tan⁻¹x) -1/2
B) tan⁻¹{(x-2)/(x-4)} + tan⁻¹{(x+2)/(x+4)}= π/4. ±√2
C) cos⁻¹x + sin⁻¹(x/2)=π/6. 1
3) Prove:
A) tan⁻¹(3/4) + tan⁻¹(3/5) - tan⁻¹(8/19)=π/4.
B) sin⁻¹(8/17) + sin⁻¹(3/5) = tan⁻¹(77/36)
C) tan⁻¹(1/4) + tan⁻¹(2/9) =1/2 sin⁻¹(4/5)
4) Find the principal value of tan⁻¹√3 - sec⁻¹(-2). -π/3
5) If cos⁻¹x + cos⁻¹y + cos⁻¹z=π then show that x² + y²+ z²+ 2xyz = 1.
1) A= 3 1
7 5 find x and y so that A² + xI = yA. Hence, find A⁻¹. 8,8 & 5/8 -1/8
-7/8 3/8
2) Find the adjoint of 1 0 -1
3 4 5
0 -6 -7 and Hence find A⁻¹.
2 6 4 & 1/10 3/10 1/5
21 -7. -8. 21/20 -7/20 -2/5
18 6 4 -9/10 3/10 1/5
3) solve by Martin Rule:
2/x +3/y. +10/z = 4 ; 4/x -6/y +5/z. = 1; 6/x + 9/y - 20/z = 2. 2,3,5
4) Find A⁻¹ if A= 0 1 2
1 2 3
3 1 1 Hence, solve the system equation y+ 2x +8= 0, x+ 2y +3z +14= 0, 3x+ y +z +8= 0.
1/2 -1/2 1/2
-4 3 -1
5/2 -3/2 1 and -1,- 2, -3
5) Solve: x+ y+ z = 9, 2x+ 5y +7z = 52, 2x+ y -z = 0. 1, 3, 5
6) If A=1 3 B= y 0 & C= 5 6
0 x 1 2 1 8 With the relation 2A + B = C, then find the value of (x+y). 6
7) Write a 2 x 2 Matrix which is both symmetric and skew-symmetric. 0 0
0 0
8) Given A= 0 2b -2
3 1 3
3a 3 -1 is symmetric, find the value of a and b. 2/3,3/2
9) A= 1 3 -2 1 -2 -3
-3 0 -2 -2 4 7
2 1 0 find A⁻¹. -3 5 9
10) If A= -5 1 3 & B= 1 1 2
7 1 -5 3 2 1
1 -1 1 2 1 3 find AB. Then solve the following equations by Matrix method. x+ y+ 2z = 1, 3x+ 2y +z = 7, 2x+ y +3z = 2. 2, 1, -1
1) without expanding prove :
A) a+b b+ c c+a
c a b = 0
1 1 1
2) a b c
a² b² c²
b+ c c+a a+ b
= (a- b)(b - c)(c- a) (a+b + c)
3) 1+ a²- b² 2ab -2b
2ab 1- a²+ b² 2a
2b -2a 1- a² - b²
= (1+ a²+ b²)³
4) - a² ab ac
ba - b² bc = 4a²b²c²
ca cb - c²
5) 1+ a² ab ac
ba b²+1 bc = 1+ a²+ b²+ c²
ca cb c²+1
6) a b - c c+ b
a+ c b c- a
a- b b+ a c
= (a+ b+ c)(a²+ b²+c²)
1) If the function f(x) given by:
2ax + b, if x > 1
f(x) = 11, if x= 1
5ax -2b, if x< 1 is continuous at x= 1, then find the value of a and b. 3,2
2) Find the value of k, so that the function f Defined by:
k cos x/(π - 2b), if x ≠ π/2
f(x) = 3, if x= π/2
is continuous at x=π/3. 6
3) Find the value of k, so that the function f Defined by:
kx + 1, if x ≤ π
f(x) = cosx, if x> π
5ax -2b, if x< 1 is continuous at x =π. -2/π
1) Find whether the following function is differentiable at x= 1 and x =2 or not
x , x < 1
f(x) = 2- x, 1≤ x ≤ 2
-2+3x -x², x> 2. f(x) is not differentiable at x=1, f(x) is differentiable at x= 2
2) For what value for k for the function defined by f(x) = k(x²+2), if x≤0
4x +6, if x> 0 is continuous at x= 0 ? Hence check the differentiability of f(x) at x= 0. k= 3, f is not differentiable at x= 0
1) DIFFERENTIATE THE FOLLOWING:
A) tan⁻¹√{(1+ cosx)/(1- cosx)}. -1/2
B) tan⁻¹(secx + tanx). 1/2
C) tan⁻¹√{(a- x)/(a+x)}. -a/2√(a²- x²)
D) log (secx + tanx). secx
E) sin⁻¹(x³). 3x²/√(1- x⁶)
F) log tan(π/4 + x/2). Secx
G) sin⁻¹{(2x/(1+ x²)} w.r.t. tan⁻¹x. 2
H) If tan⁻¹{√(1- x²)/x} w.r.t. cos⁻¹(2x√(1- x²). -1/2
2) If y= [x +√(x²+ a²]ⁿ then show dy/dx= ny/√(x² + a²).
3) If y= log [x +√(x²+ a²] then show (x²+ a²) d²y/dx²+ x dy/dx = 0.
4) If xʸ = eˣ⁻ʸ, show dy/dx = log x/(1+ log x)².
5) If y= (sin⁻¹x)², show (1- x²) d²y/dx² - x dy/dx - 2 = 0.
6) If eʸ(x +1)= 1, show that d²y/dx² = (dy/dx)².
7) If √(1- x²) + √(1- y²)= a(x - y), show that dy/dx =√{(1- y²)/(1- x²)}.
8) If y= [x +√(1+ x²]ⁿ then show (1+ x²)d²y/dx²+ x dy/dx = n²y.
9) If xᵐ yⁿ = (x+y)ᵐ⁺ⁿ, then show dy/dx = y/x.
10) If y= xˣ then show d²y/dx²- 1/y (dy/dx)²- y/x = 0.
11) If x√(1+ y)+ y√(1+ x)=0(x≠y), then show dy/dx = -1/(1+ x)².
1) lim ₓ→₀ (sinx - x)/x³. -1/6
2) lim ₓ→₀ (x - tan x)/x³. -1/3
3) lim ₓ→₀ (1 - tan x)/cos 2x. 1
4) lim ₓ→₀ (1 - logx - x)/(1- 2x+x²). -1/2
5) lim ₓ→₀ (x - tan⁻¹x)/(x - sinx). 2
6) lim ₓ→π/2 (tan5x)/tanx. 1/5
7) lim ₓ→₀ (cosecx - 1/x) 0
8) lim ₓ→π/2 (x tanx - π/2 secx). -1
9) lim ₓ→π/2. (Cosx log tanx) 0
10) lim ₓ→₀ (1+ sinx)ᶜᵒᵗˣ e
1) Verify Rolle's theorem for the following:
A) f(x)= x²- 5x +6 on [2,3]. c=2.5
B) f(x)= log{(x²+ ab)/x(a+ b)} on (a,b), where 0 < a < b. c=√(ab)
C) f(x)= sin x - sin 2x on [0,π]. c= 32°32' , 126° 23'
2) Verify Lagrange 's mean value theorem for the following:
A) f(x)= (x-3)(x-6)(x-9) on [3,5]. 4.8
B) y=√(x-2) on [2,3], where the tangent is parallel to the chord joining the end points of the curve. (9/4,1/2)
1) The side of an equilateral triangle is increasing at the rate of 2cm/s. At what rate is its area increasing, when the side of the triangle is 20 cm? 20√3 cm²/s
2) Find intervals in which the function f(x)= 3x⁴- 4x³- 12x²+5 is
A) strictly increasing. (-1,0)U(2, ∞)
B) strictly decreasing. (-∞,-1) & (0,2)
3) Find the intervals in which the function f(x)= (x-1)(x+2)² is
A) increasing. (-∞,-2)U(0,∞)
B) decreasing. (-2,0)
4) Find intervals in which the function f(x)= sinx+cos x, 0≤ x≤ 2π is
A) increasing. (0,π/4) &(5π/4,2π)
B) decreasing. (π/4,5π/4)
1) Using differentials, find the approximate value of √49.5. 7.035
2) The area of a circle of radius r increases at the rate of 5 cm²/s.
A) Find the rate at which the radius increases. 5/2πr cm/s
B) Also, find the value of this rate when the circumference is 10 cm. 0.5 cm/s
1) Find the slope of the tangent to the curve y= 3x²- 4x at the point whose x-coordinate is 2. 8
2) Find the points on the curve y= x³- 11x²+5 at which the equation of the tangent is y= x -11. (2,-9) and (-2,19)
3) Find the equation of the tangent to the curve 3= x²+ 3y which is parallel to the line y - 4x +5= 0. 4x - y +13= 0
4) Find the equation of the normal to the curve y= x³+ 2x+6, which is parallel to the line 14y +x +4= 0. x + 14y= 254
5) Find the equation of the tangent and normal to the curve x= 1- cos k, y= k - sin k at k=π/4. 4√2 x +(8- 4√2)y =π(2 -√2)
1) An open box with a square base is to made out of a given quantity of cupboard of area C² square units. Show that the maximum volume of the box is C³/6√3 cubic units.
2) If the sum of lengths of the hypotenuse and a side of a right angled triangle is given. Show that the area of a triangle is maximum, when the angle between them is π/3.
3) If the length of three sides of a trapezium other than the base is 10cm each then find the area of the trapezium when it is maximum. 75√3cm²
4) Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius r cm. 4πr³/3√3
5) Show that the semivertical angle of a right circular cone of maximum volume and given slant height is tan⁻¹√2.
6) Find the point on the curve y²= 2x which is nearest to the point (1,4). (2,2)
7) Three numbers are given whose sum is 180 and the ratio of the first two is 1:2. If the product of the numbers is greatest, find the number. 40,80,60
8) Prove that the Area of a right angled triangle of a given hypotenuse is maximum when the triangle is isosceles.
9) If the sum of the length of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum when the angle between them is π/3.
10) A closed right circular cylinder has volume 539/2 cubic units. Find the radius and the height of the cylinder, so that the total surface area is minimum. 7/2, 7
11) Find the maximum volume of the cylinder which can be inscribed in a sphere of 3√3cm. radius. (Leave the answer in terms of π). 108π cm³
12) Find the maximum and minimum values of:
A) -(x -1)²+ 10. 10, nil
B) sin 2x +5. 6,4
C) sinx - cos x. .√2, -√2
1) ∫ tan(2 log x)/x dx. 1- 1/2 log|cos(2logx)|
2) ∫ cosecx/log(tan(x/2)) dx. Log|log(tan(x/2))|
3) ∫ cos x/(sinx √sinx) dx. 2log|√sinx +1|
4) ∫e²ˣ/(2+ eˣ) dx. eˣ - 2 log(2+ eˣ)
5) ∫secx log|secx + tan x| dx. 1/2 (log|secx + tanx|)²
6) ∫x³ log x dx. x⁴/16 (4 log x -1)
7) ∫x tan⁻¹(x²) dx. x²/2 tan⁻¹x² 1/4 log(1+ x⁴)
8) ∫ (sin⁻¹x)² dx. x(sin⁻¹x)² + 2√(1- x²) sin⁻¹x - 2x
9) ∫ (cos⁻¹x)/x². (cos⁻¹x)/x + log|1+ √(1- x²)/x |
10) ∫ (x -1)eˣ/(x+1)³ dx. eˣ/(x+1)²
11) ∫ logx/(1+ log x)² dx. x/(1+ log x)
12) ∫ eˣ{(2+ sin2x)/(1+ cos2x)} dx. eˣ tan x
13) ∫ eˣ{(1- sin x)/(1- cosx)} dx. - eˣcot(x/2)
14) (sin⁻¹x)² dx. x(sin⁻¹x)² - 2{-√(1-x²) ∫sin⁻¹x + x}
15) ∫ x/(x²+ x +1) dx. 1/2 log|x²+ x+1| - 1/√3 tan⁻¹{(3x+1)√3}
16) ∫ (x⁴+1)/(x²+1) dx. x³/3 - x + 2 tan⁻¹x
17) ∫ x²/(x⁴+1)dx. 1/2√2 tan⁻¹{(x²-1/x√2} + 1/4√2 log|(x²- x√2+1)/(x²+ x√2+1)|
18) ∫ dx/(sin⁴x+cos⁴x)dx. 1/√2 tan⁻¹{(tan²x -1)/√2 tanx}
19) ) ∫ x²/(x⁴²- 4x+3)dx. x + 9/2 log|x -3| - 1/2 log|x-1|
20) ) ∫ (x²-5x-1)/(x⁴+x²+1)dx. 1/2 log|{(x²-x+1)/(x²+x+1)} - 5/√3 tan⁻¹{(2x²+1/√3}
21) ) ∫ (4x+5)/(2x²+ x-3)dx. 2 √(2x²+ x -3) + 2√2 log|(x+ 1/4 + √(x²+ x/2- 3/2)|
22) ¹₀∫ sin⁻¹{2x/(1+ x²)} dx. π/2 - log2
23) ∫ (3 sinx + 4 cos x)/(sinx + cos x) dx at (π/2,0). 7π/4
24) ¹₀∫ x tan⁻¹x dx. π/4 -1/2
25) ²₁∫ 2/(4x²-1) dx. 1/2 Log (9/5)
26) ∫ 4x sinx/(1+ cos x) dx at (π, 0). π²
27) ³₁∫ [|x -1| + |x -2|+ |x -3|] dx. 5
28) ∫ x sin x/(1+ cos²x) dx at (π,0). π²/4
29) Prove: x tan x/(secx + tanx) dx at π,0) =π/2 (π -2)
30) ²₁∫ (x²+5x) dx. 59/6
31) ³₋₃∫|x+2| dx. 13
32) ∫sin2x log tanx dx at (π/2,0). 0
33) ∫ log tan x dx at (π/2,0). 0
34) ∫ √sinx/(√sinx + √cosx) at π/2,0). π/4
Solve:
1) cos dy/dx + cos 2x = cos 3x. y= sin2x - 2sin x - x + log|secx + tan x|
2) log(dy/dx)= 2x - 3y. 2e³ʸ= 3e²ˣ + c₂ where c₂ = 6c₁, c₂ is an arbitrary constant.
3) 2 dy/dx = y/x + y²/x². 2 log |(y-x)/y| = log |x|
4) x dy/dx - y = √(x²+ y²). y+ √(x²+ y²) = C x²
5) dy/dx + sin(x + y)= sin (x - y). Log|cosy - cot y| = -2 sinx
6) x dy/dx = y - x tan(y/x). x sin(y/x) = C
7) Find the particular solution of the differential equation (3xy + y²)dx + (x²+ xy) dy = 0, from x= 1 and y= 1. y² x² + 2yx³ = 3.
8) y dx -(x + 2y²) dy = 0. x/y = 2y+ c
9) y log x dy/dx + y = 2/x log x, x> 0. y log x = -2/x (1+ cosx) + c, x> 0
10) (1+ y + x²y) dx + (x + x³) dy= 0. xy = - tan⁻¹x + c
11) cos²x dy/dx + y = tan x. yeᵗᵃⁿˣ = e ᵗᵃⁿˣ (tanx -1)+ c
12) (x+1) dy/dx - y = e³ˣ(x +1)². y= 1/3 (x+1)e³ˣ + (x+1) c
13) dy/dx + y(1- 2x))x² = 1. y= x²(1+ e¹/ˣ.
7) x dy/dx + y = x cos x + sin x given y(π/2)= 1
1) Find the probability that a leap year, selected at random, will contain 53 Sundays. 2/7
2) A word consists of 9 letters of 4 consonants and 5 vowels. Three letters are chosen at random. Find the probability that more than one vowel is chosen. 25/42
3) If two balls are drawn from a bag containing three red balls and 4 blue balls , find the probability that
A) they are of the same colour. 3/7
B) they one of different colours. 4/7
4) Two horses are considered for a race. The probability of selection of the first horse is 1/4 and that of the second is 1/3. What is the probability that
A) both of them will be selected? 1/12
B) only one of them will be selected? 5/12
C) none of them will be selected? 1/2
5) Bag A contains 6 red and 5 blue balls and another bag B contains 5 red and 8 blue balls. A ball is drawn from bag A without seeing its colour and it is put into the bag B. Then a ball is drawn from bag B at random. Find the probability that the ball drawn is blue in colour. 93/154
6) On answering a question of a multiple choice test, a student either knows the answer or guesses. Let 3/5 be the probability that he knows the answer and 2/5 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/3, what is the probability that the students knows the answer, given that he answered it correctly? 9/11
7) Bag A contains 3 white and 4 black balls and bag B contains 4 white and 5 black balls. One ball is transferred from bag A to bag B and then a ball is drawn at random from bag B. If the ball so drawn is white, find the probability that the transferred ball is black. 16/31
8) A man is known to speak the truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six. 3/8
9) Suppose that 5% of men and 0.25% of women have grey hair. A grey- haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females. 20/21
10) Find the mean and variance for the following probability distribution:
X: 0 1 2 3
P(x): 1/8 3/8 3/8 1/8 3/2,3/4
11) Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Find the mean and standard deviation of the number of kings. 2/13, 0.373
12) If the mean and variance of a binomial distribution are 9 and 6 respectively, find the distribution. (2/3 + 1/3)²⁷
13) Find the binomial distribution when the sum of its mean and variance for five trials is 4.8. (1/5+ 4/5)⁵
14) A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die. 625/3x6⁵
1) Find a unit vector perpendicular to each of the vector a+ b and a - b where a= 3i+ 2j+ 2k and b= i+ 2j - 2k. 2i/3 -2j/3 -k/3
2) The vectors -2i+ 4j +4j and -4i-2k represent the diagonal BD and AC of a parallelogram ABCD. Find the area of the parallelogram. 6√5
3) For any three vectors a, b, c, prove that [a+ b, b+ c c+ a]= 2[a b c]
4) Prove by vector method that in any triangle ABC, a/sinA = b/sinB = c/sinC.
5) Find the length and the foot of the perpendicular drawn from the point (2,-1,5) to the line (x -11)/10 =(y+2)/-4 = (z+8)/-11. (1,2,3), √14
6) Find the equation of the line passing through the point (-1,3,-2) and perpendicular to the lines x/1= y/2 = z/3 and (x+2)/-3= (y-1)/2 = (z+1)/5. (x+1)/2 = (y-3)/-7 = (z+2)/4.
7) Find the shortest distance between the lines (x-1)/2= (y-2)/3 = (z-3)/4 and (x-2)/3 = (y-4)/4 = (z-5)/5. 1/√6
8) Find the shortest distance between the lines (x-8)/3= (y+9)/-16 = (z-10)/7. (x-15)/3 = (y- 29)/8 = (5-z)/5. Also, find the equation of this line. 14, (x-5)/2 = (y-7)/3 = (z-3)/6.
9) Find the equation of the plane passing through the intersection of the planes 2x+ 3y - z+1= 0 and x+ y - 2z+3= 0 and perpendicular to the plane 3x - y - 2z-4= 0. 7x+ 13y +4z-9= 0
10) Find the equation of the plane passing through the intersection of the planes 2x+ y +z-1= 0 and 2x+ 3y - z+4= 0 and parallel to the x-axis y -3z+6= 0
11) Find the distance of the point (1,-2,3) from the plane x- y +z-5= 0 and x+ y - 2z+3= 0 measured parallel to the line (x+1)/2 = (y+3)/3 = (z+1)/6. 1 unit
12) Find the equation of the plane through (1, 2, 3) and perpendicular to the plane x+ y + 2z- 3= 0 and 3x+ 2y +z= 4. 3x - 5y + z + 4= 0
13) Find the equation of the plane passing through (2,1, 0) and (5,0,1) and (4,1,1). x + y - 2z -3 = 0
1) Draw a rough sketch of the curve y²+1= x, x≤ 2. Find the area enclosed by the curve and the line x=2. 4/3
2) Find the area enclosed by the curve y²= x and the line y+ x=2. 9/2
3) Find the area enclosed by the curve y= x² and x=y². 1/3
4) Draw a rough sketch of the curve y= (x-1)² and y=|x -1|. Hence find the area enclosed by the curve. 1/3
5) Find the area of the region enclosed by the curve 4y - y²= x and the y-axis. 32/3
6) Find the area enclosed by the curve y= 2x - x² and x=y. 1/6
7) Find the area of the region in the first quadrant enclosed by the x-axis, the line y= x and the circle x²+ y²= 32. 4π
1) A television manufacturer finds that total cost for the production and marketing of x television sets is
C(x)= 300 x²+ 4200x+ 13500
Each product is sold for₹8400. Determine the break even points. 5,9
2) The average cost function associated with producing and marketing x units of an item is given by AC= 2x -11+ 50/x.
A) Find the total cost function and the marginal function. 2x²-11x+ 50, 4x-11
B) The range of values of the output x, for which AC is increasing. x> 5
3) The average cost function AC for a commodity is given by AC= x +5+ 36/x in terms of output x.
A) Find the total cost function and the marginal function. x²+5x+6, 2x+5
B) The range of values of the output x, for which AC is increasing. x> 6
4) Given that the total cost function AC for a commodity is given by C(x)= x³/3+ 3x²- 7x+ 6 in terms of output x.
A) Find the marginal cost(MC). x²+6x-7
B) find the average cost(AC). x²/3 + 3x -7+ 16/x
C) prove that marginal average cost (MAC) = x(MC)- C(x)/x².
5) The marginal cost function MC for a product is given by MC =2/√(2x+9) and the fixed cost is ₹14. Find the average cost for 8 units of output. ₹9/4
6) Find the total revenue and demand function if the marginal revenue function (MR)= 100- 9x². 100x - 3x³, p= 100- 3x²
1) Out of the following two regression lines, find the line of regression of y on x: 3x+12y=7; 3x+ 9y=46. 3x+12y=7
2) The equation of two lines of regression are 4x+3y= -7 and 3x+4y +8=0.
A) find the mean values x and y. -4/7, -11/7
B) find the regression coefficients bᵧₓ and bₓᵧ. -3/4, -3/4
C) Find the correlation coefficient between x and y. -3/4
3) The regression lines are represented by 4x+10y=9 and 6x+3y=4. Find the regression line of x on y and y on x. 4x+10y=9 and 6x+3y=4
4) Find the equation of the lines of regression for the data:
X: 1 2 3 4 5
Y: 7 6 5 4 3
And hence find an estimate of the variable y for x= 3.5 from the appropriate line of regression. Y on x is x+y=8, x on y is x+y=8, y=4.5
1) A mill owner buys two types of machines A and B for his mills. Machine A occupies 1000 sq.m of area and requires 12 men to operate, while Machine B occupies 1200 sq m of area and requires 8 men to operate. The owner has 7600 sq.m of area available and 72 men to operate the machines. If machine A produces 50 units and machine B produces 8 units daily, how many machine of each type should be buy to maximize the daily output? Use linear programming to find the situation. A=4; B= 3
2) A new cereal formed of a mixture of bran and rice, contains atleast 88g of protein and atleast 36mg of iron. Knowing that bran contains 89g of protein and 40mg of iron per kilogram and that rice contains 100g of protein and 30mg of iron per kilogram, find the minimum cost of producing a kilogram of this new cereal if bran costs ₹88 per kg and rice costs ₹25 per kg. ₹26.8 at the point (0.6,0.4)
SHORT ANSWER QUESTIONS
Set -2
1) Solve: cos(sin⁻¹x)= 1/7. ±4√3/7
2) ∫ sin³x cos²x dx. -1/3 cos³x + 1/5 cos⁵x + c
3) Use Matrix solve: 2x+3y= 10; x+ 6y= 4. 16/3,-2/9
4) Find dy/dx: y+ siny = x². x sec²(y/2)
5) Let f : X--> Y be a function defined on a relation R on X given by R={(a,b): f(a)= f(b). Show that R is an equivalence relation on X.
6) If f: R---> R is defined by f(x)= (3 - x)¹⁾³, then find f o f(x). x
7) If P(A)= 7/13, P(B)= 9/13 and P(A∩ B)= 4/13, find P(A/B). 4/9
8) Consider f: R ₊ -->[9,∞) given by f(x)= 5x²+ 6x -9. Prove that f is invertible with f⁻¹(y)= {√(54+5y) -3}/5.
9) The mean number of success of a binomial distribution (p+q)ⁿ is 240 where p is the probability of success. The standard deviation is 12. Calculate the values of n, p and q. 2/5,3/5,600
10) Find the value of tan(2tan⁻¹(1/5)). 5/12
11) The vector 2i+ j - k is perpendicular to i+ 4j - Mk, if M is equal to
A) 0 B) -1 C) -2 D) 2 -2
12) Solve the determinant x² x 1
0 2 1= 28
3 1 4
13) Find the Inverse of -2 5
3 4
14) Using Matrix, solve : 5x+7y =-2; 4x+6y =- 3. 9/2,-7/2
15) The probability density function y of a continuous variable x is given by y= k/x, 0≤ x ≤ 2 and y = 0 for all other values of x. Calculate the value of k and the probability that x < 1. 2, 1/4
16) Solve: cos⁻¹x+ sin⁻¹(x/2)= π/6. ±1
17) dy/dx of cos⁻¹√{(1- cosx)/2}. -1/2
19) If f is an invertible function defined as f(x)= (3x-4)/5, then find f⁻¹(x). (5x+4)/3
20) Find inverse of A= 1 2
3 7
21) The binary operation * R x R --> R is defined as a * b= 2a+ b. Find (2*3)*4. 18
22) Find the probability of throwing a total of 3 or 5 or 11 with two dice. 2/9
23) Find the magnitude of the vector ax b if a= 3i+ 4j and b= 5j +12k. 15√17
24) ¹₀∫ dx/(2x -3). -1/2 log 3
25) dy/dx of √{(1- cos2x)/(1+ cos 2x)}. sec²x
26) A company wants to lunch a new product. It invested ₹37500 as fixed cost and ₹200 per unit as the variable cost of production. The revenue function for the sale of x units is given by 4825x - 125x². Find the break even points. 12, 25
27) In a simultaneously throw of two dice, find the probability of getting a total of 7. 1/6
28) Does the Lagrange 's mean value theorem apply to f(x)= x¹⁾³ ; -1≤ x≤1? N
29) ∫ x²/(1- x) dx. -x²/2+ x + log|1- x|
30) Form the differential equation y= aeᵇˣ, where a and b are arbitrary constants. y(d²y/dx²)= (dy/dx)²
31) Find the value of M so that the vectors 2i+ 3j+ 4k and 4i - 2j - Mk may be perpendicular to each other. -1/2
32) If sin⁻¹x + sin⁻¹y = 2π/3 then find the value of cos⁻¹x + cos ⁻¹y. .π/3
33) Find the differential equations of the family of curves y= Aeˣ + Be⁻ˣ, where A and B are arbitrary constants. d²y/dx²= y
34) Find the value of p for which the vectors a= 3i+ 2j+ 9k and b= i+ pj + 3k are parallel. 2/3
35) You roll a pair of dice and record the sum of numbers on the top of these dice. Find the probability that the score is atleast 10. 1/6
36) A card is drawn at random from a well shuffled pack of 52 cards. Find the probability that it is neither an Ace nor a king.
37) ∫ 8/{(x+2)(x²+4)} dx log|x+2| - 1/2 log|x²+4)| + tan⁻¹(x/2)
38) Find the Cartesian equations of the line which passes through the point (-2,4,-5) and is parallel to the line (x-3)/3 = (y-4)/5 = (z+8)/6. (x+2)/3 = (y-4)/5 = (z+5)/6.
39) If a=3i+ j +2k and b= 2i - 2j+ 4k, find the magnitude of a x b. 8√3
40) if y= tan⁻¹[{√(1+ x²) - 1}/x] then find dy/dx. 1/2(1+ x²)
41) Determine the binomial distribution whose mean is 9 and whose standard deviations is 3/2. Find the probability of obtaining one success at the most.
43) if f(x)= 27x³ and g(x)= x¹⁾³, then find g o f(x). 3x
44) If f: R---> R defined by f(x)= (3x+5)/2 is an invertible function, then find f⁻¹(x). (2x-5)/3
45) What is the probability that a number selected from the numbers 1,2,3,4,...25 is a prime number? 9/25
46) ∫ x/{(x-2)(x -1)} dx. log|(x+2)²/(x-1)|
47) ᵅ₀∫ √x/{√x +√(a- x)} dx. 9/2
48) ∫ sec²x/cosec²x dx. tanx - x
49) Find the differential equations for y= a cos(x+ b), where a, b are parameter. d²y/dx² + y= 0
51) Find the area between the curve y= 2x² + 3x, the x-axis and the ordinate x= 3. 63)2
52) Without expanding the determinant, show that
1 a b+ c
1 b c+a = 0
1 c a+ b
54) ²₀ ∫ √x/{√x+ √(3- x)} dx. 1/2
55) Determine the order and degree of the differential equations d²y/dx² = cos 3x+ sin 3x. Also, state whether it is linear or bon-linear. 2, 1, linear
56) The regression equation of y on x is 3x - 5y =13 and the regression equation of x on y is 2x - y= 7. Estimate the value of x when y= 10. 8.5
57) If eˣ + eʸ = eˣ⁺ʸ prove that dy/dx= eʸ⁻ˣ = 0.
58) Find K if ᴷ₀∫ dx/(2+ 8x²)=π/16. 1/2
59) If y= sin⁻¹√{(1+ x²)/2} , find dy/dx. x/√(1- x⁴)
60) Using differentials, find the approximate value of √49.5. 7.035
61) Differentiate xˣ. xˣ(log x +1)
62) Find the local maxima and minima for the function x³- 12x. 2 at -16 and-2 16
63) ∫ dx/(1+ cosx). Cosecx - cot x
64) ∫ √(1+ cos 2x) dx at (π/2,0). √2
65) ¹₀∫ 2x/(1+ x²) dx. log 2
66) Solve: 1/sin⁻¹x dy/dx = 1. y= x sin⁻¹x + √(1- x²)
67) If y= x + tan x, prove cos²x d²y/dx² - 2y + 2x = 0.
68) lim ₓ→₃ (x⁴-81)/(x-3). 108
69) ∫ x sin⁻¹x/√(1- x²) dx. -√(1- x²) sin⁻¹x + x
70) If a= 3i + j + 2k and b= 2i- 2j + 4k, find the magnitude of a xb. 8√3
71) ∫ sin³x dx at (π/4,-π/4). 0
72) ⁵₂∫(3x²-5). 102
73) Solve: dy/dx= e³ˣ⁻²ʸ+ x²e⁻²ʸ. e²ʸ/2= e³ˣ/3 + x³/3.
74) find dy/dx if y tan x - y² cosx + 2x = 0. (y sec²x+ y² sinx+2)/(2y cosx - tanx)
75) A bag contains 5 white and 3 black balls and 4 balls are successively drawn out and not replaced. Find the probability that they are alternately of different colours. 1/7
76) Out of the following two regression equation lines, find the line of regression of y on x; 3x+ 12y = 9 ; 3y+ 9x = 46. 3x+ 12y = 9
77) Find the are enclosed between the line 3x - 2y +12= 0 and the parabola y = 3 x²/4. 27 units
78) Solve: dy/dx + 2y tan x = sinx. y= cos x + C cos²x.
79) Find a unit vector perpendicular to the plane a and b, where a= i - 2j + 3k and b= i + 2j - k. ±1/√3(-i + j+k)
80) ₁²∫ 2/(4x²-1) dx. 1/2 log(9/5)
81) If P(A)= 0.4, P(B)= 0.3 and P(A∩B)= 0.2, find
A)P(A/B). 2/3
B) P(B/A). 1/2
82) ∫ x/(1+ sinx) dx at (π,0). π
83) solve: 7x = 6y - 8 and 12y = 9+ 5x by Martin rule. -7/9,23/54
84) If A= 5 3
-1 -2 find A⁻¹.
85) Verify Rolle's theorem for the function f(x)= sinx in the interval (π/4, 3π)4).
86) ²₋₂∫ |x +1| dx. 5
87) Find the particular solution of the differential equations (x-y)(dx+ dy)= dx - dy, given that y=-1, when x=0. Log|x -y| = x + y+1
88) The probability of an event X occurring is 0.5 and that of Y is 0.3. if X and Y are mutually exclusive events, find the probability of neither X nor Y occuring. 0.2
89) Ten cards numbered 1 to 10 are placed in a box, and one card is drawn randomly. If it is known that the number on the card drawn is more than 3, what is the probability that it is an even number? 4/7
90) find the inverse Matrix of 1 2
3 7
91) The probability of solving a problem by three students A, B and C are 1/2, 1/3, and 1/4 respectively. The probability that the problem will be solved is:
a) 1/4 B) 1/2 C) 3/4. D) 1/3. 3/4
92) Find M if the vectors a= i+ 3j + k, b= 2i- j - k, and c= Mi+ 3k are coplanar. 7
93) The function f(x)= x⁴- 62x² + kx +9 attains its maximum value on the interval [0,2] at x=1. Find the value of k. 120
94) If y= 2 sinx + 3cosx, find the value of d²y/dx² + y. 0
95) Find the area of a parallelogram whose adjacent dides are given by a= 3i- 6j + 2k and b=2 i+ j -2k. 5√17
96) Using Martin rule, solve the following system of equation: 6x+ y - 3z=5, x+ 3y - 2z=5, 2x+ y +4z=8. 1,2,1
97) Without expanding prove the determinant. x+y y+z z+ x
z x y =0
1 1 1
98) If A= x 4 1 B= 2 1 2 & C= x
1 0 2 4
0 2 -4 -1
And the relation ABC= 0, find x. -2, or -1
99) dy/dx of cos⁻¹{(1- x²)/(1+ x²)}. 2/(1+ x²)
100) ∫ dx/√(4x²-9). 2 log|x + √(x² - 9/4)|
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