1) The area(in square unit) of the region bounded by the curve x²= 4y, the line x= 2 and x-axis is -
a) 1 b) 2/3 c) 4/3 d) 8/3
2) Let(a sec θ, b tanθ) and Q(a secα, b tanα) where θ + α=π/2, be two on the hyperbola x²/a²- y²/b²= 1. If (h,k) be the point of intersection of the normals at P and Q, then the value of k is
a) (a²+ b²)/a
b) - (a²+ b²)/a
c) (a²+ b²)/b
d) -(a²+ b²)/b
3) The equation of the tangent to the curve (1+ x²)y = 2- x where it crosses the x-axis is
a) x+ 5y = 2 b) x- 5y = 2 c) 5x - y = 2 d) 5x + y = 2
4) The area (in square unit) bounded by the parabola y²= 4ax and x²= 4ay is
a) 64a²/3 b) 32a²/3 c) 16a²/3 d) 7a²/3
5) Equations of the tangent and the normal drawn at the point (6,0) on the ellipse x²/36 + y²/9 = 1 respectively are
a) x= 6, y=0
b) x+ y= 6, y - x +6=0
c) x= 0, y=3
d) x= - 6, y=0
6) The area (in square unit) of the figure by the curve y = cosx and y = sinx and the ordinates x= 0, x=π/4 is
a) √2 +1 b) √2 - 1 c) 1/√2 d) (√2 -1)/√2
7) The straight line x + y = a will be a tangent to the ellipse x²/9+ y²/16= 1 if the value of a is
a) 8 b) ±10 c) ±5 d) ±6
8) The equation of the tangent to the parabola y²= 8x which is perpendicular to the line x - 3y +8= 0 is
a) 3x + y + 2 = 0
b) 3x - y - 1 = 0
c) 9x - 3y + 2 = 0
d) 9x + 3y + 2 = 0
9) The area (in square unit) bounded by the Parabola y²= 8x and its latus rectum is
a) 16/3 b) 25/3 c) 16√2/3 d) 32/3
10) If the curve y²= 4x and xy= k cut orthogonally, then the value of k² will be
a) 16 b) 32 c) 36 d) 8
11) The area (in square unit) bounded by the curve -3y²= x -9 and the lines x= 0, y= 0 and y= 1 is
a) 8/3 b) 3/8 c) 83
12) If the slope of the normal to the curve x³= 8a²y at P is (-2/3), then the coordinates of P are
a) (2a,a) b) (a,a) c) (2a, -a) d) none
13) If a> 2b > 0, then the positive value of m for which the line y= mx - b √(1+ m²) is a common tangent to the circles x²+ y²= b² and (x - a)²+ y²= b² is
a) 2b/√(a²- 4b²)
b) √(a²- 4b²)/2b
c) 2b/(a- 2b)
d) b/(a- 2b)
14) The area in square unit of the region bounded by the lines y= |x -1| and y= 3- |x| is
a) 6 b) 2 c) 4 d) 3
15) The minimum value of f(x)= x²+ 250/x is
a) 55 b) 25 c) 50 d) 75
16) If f(x)=kx³- 9x²+ 9x +3 is increasing function then
a) k< 3 b) k ≤ 3 c) k> 3 d) k is indeterminate
17) If f(x)= 1/(4x²+ 2x +1), then its maximum value is
a) 2/3 b) 4/3 c) 3/4 d) 1
18) If f(x)= 1/(x +1) - log(1+ x), x> 0, then f(x) is
a) a decreasing function
b) an increasing function
c) neither increasing nor decreasing
d) increasing when x> 1.
19) Let α, β be the roots of x²+ (3- λ)x - λ=0, then the value of λ for which α²+ β² is minimum, is
a) 0 b) 1 c) 3 d) 2
20) The function f(x)=2x³- 3x² -12x +4 has
a) no maxima and minima
b) one maximum and one minimum
c) two maxima
d) two minima
21) The height of the cylinder of maximum volume that can be inscribed in a sphere of radius a, is
a) 3a/2 b) √2 a/3 c) 2a/√3 d) a/√3
22) Maximum value of (logx)/x in [0, ∞) is
a) (log2)/2 b) 0 c) 1/e d) e
23) Let the function f: R--R be defined by f(x)=2x + cosx; then f(x)
a) has maximum value at x=0
b) has minimum value at x=π
c) is a decreasing function
d) is an increasing function
24) The maximum distance from the origin of a point on the curve x= a sin t - b sin(at/b), y= a cos t - b cos(at/b), both a, b> 0, is
a) a- b b) a+ b c) √(a²+ b²) d) √(a²- b²)
25) The velocity v of a particle moving along a straight line is given by a+ bv²= x², where x is the distance of the particle from the origin. Then the acceleration of the particle is
a) x/b b) bx c) x/a d) b/x
26) If the slope of the tangent at (x,y) to a curve passing through the point (2,1) is (x²+ y²)/2xy , then the equation of the curve is
a) 2(x²- y²)= 3x
b) 2(x²- y²)= 3y
c) x(x²- y²)= 6
d) 2px(x² + y²)= 6
27) A particle moves uniform acceleration f along a straight line. If v be it's velocity at time t and s be the distance described during the interval, then
a) s= 2vt - ft²
b) s= vt - ft²/2
c) s= vt/2 - ft²
d) s= vt - ft²/2
28) A particle moving in a straight line traverses a distance x in time t, if t= x²/2 + x, then the retardation of the particle is
a) equal to its velocity
b) constant
c) is equal to the cube of its velocity
d) equal to the square of its velocity
29) If y= 3x²+ 2 and if x changes from 10 to 10.1, then the approximate change in y will be
a) 8 b) 6 c) 5 d) 4
30) The rate of change of surface area of a sphere of radius r when the radius is increasing at the rate of 2 cm/sec is proportional to
a) 1/r² b) r² c) r d) 1/r
Paper - 4
C- TEST PAPER- (1)
Section A
(Multiple Choice Questions) Each question carries 1 mark
1) If A= [aᵢⱼ] is a square metrix of order 2 such that
aᵢⱼ= {1, when i≠ j
0, when i= j then A² is
a) 1 0 b) 1 1 c) 1 1 c) 1 0
1 0 0 0 1 0 0. 1
2) if A and B are invertible square matrices of the same order, then which of the following is not correct ?
a) adjacent A= |A|. A⁻¹
b) det(A⁻¹)= [det(A)]⁻¹
c) (AB)⁻¹= B⁻¹A⁻¹
d) (A+ B)⁻¹= B⁻¹ + A⁻¹
3) if the area of the triangle with vertices (-3,0),(30) and (0,k) is 9 square. units, then the value/s of k will be
a) 9 b) ±3 c) -9 d) 6
4) If f(x)= {kx/|x|, if x< 0
3, if x≥ 0 is continuous at x= 0, then the value of k is
a) -3 b) 0 c) 3 d) any real number
5) The lines r= i+ j - k + λ(2i + 3j - 6k) and r= 2i - j + k + μ(6i + 9j - 18k), where λ and μ are scalars) are
a) coincident b) skew c) intersecting d) parallel
6) The degree of the differential equation
[1+ (dy/dx)²]³⁾² = d²y/dx² is
a) 4 b) 3/2 c) 2 d) not defined
7) The corner points of the bounded feasible region determined by a system of linear constraints are (0,3),(1,1) and (3,0). Let Z= px + qy, where p,q> 0. The condition on p and q, so that the minimum of Z occurs at (3,0) and (1,1) is
a) p= 2q b) p= q/2 c) p= 3q d) p= q
8) ABCD is a rhombus whose diagonals intersect at E. Then EA+ EB + EC + ED equals to
a) 0 b) AD c) 2BD d) 2AD
9) For any integer n, the value of
∫ₑsin²x cos³(2n +1)x dx at (π,0) is
a) -1 b) 0 c) 1 d) 2
10) The value of |A|, if
0 2x -1 √x
A= 1- 2x 0 2√x
-√x -2√x 0
Where x ∈ R⁺, is
a) (2x +1)² b) 0 c) (2x +1)⅔ d) none
11) The feasible region corresponding to the linear constraints of a Linear programming problem is given below.
Which of the following is not a constraint to the given linear programming problem.
a) x + y ≥ 2
b) x + 2y ≤ 10
c) x - y ≥ 1
d) x - y ≤ 1
12) If a= 4i + 6j and b= 3j + 4k, then the vector form of the component of a along b is
a) 18(3i + 4k)/5
b) 18(3j + 4k)/25
c) 18(3j + 4k)/5
d) 18(2i + 4k)/25
13) Given that A a square metrix of order 3 and |A|= -2, then |adjacent(2A)| is equal to
a) -2⁶ b) 4 c) -2⁸ d) 2⁸
14) A problem in Mathematics is given to 3 students whose chances of solving it are 1/2, 1/3, 1/4, respectively. If the events of their solving the problem are independent, then the probability that the problem will be solved, is
a) 1/4 b) 1/3 c) 1/2 d) 3/4
15) The general solution of the differential equation ydx - x dy = 0; (given x, y> 0), is of the form
a) xy= c b) x = cy² c) y= cx d) y= cx²
(Where c is an arbitrary positive constant of integration)
16) The value of λ for which two vectors 2i - j + 2k and 3i + λj + k are perpendicular is,
a) 2 b) 4 c) 6 d) 8
17) The set of all points, where the functions f(x)= x + |x| is differentiable, is
a) (0,∞) b) (-∞,0) c) (-∞,0) U (0,∞) d) (-∞,∞)
18) If the direction cosines of a line are < 1/c, 1/c, 1/c > , then
a) a< c < 1 b) c> 2 c) c=±√2 d) c=±√3
Assertion - Reason Based Questions
In the following questions , a statement of Assertion(A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices .
a) Both A and R are true and R is the correct explanation of A.
b) Both A and R are true but R is not the correct explanation of A
c) A is true but R is false
d) A is false but R is true.
19) Let f(x) be a polynomial function of degree 6 such that
d/dx [f(x)]= (x -1)³(x -3)², then
Assertion (A): f(x) has a minimum at x= 1.
Reason (R): When d/dx [f(x)]< 0.
∀ x ∈ (a - h, a) and d/dx [f(x)]> 0,
∀ x ∈ (a, a+ h), where h is an infinitesimally small positive quantity, then f(x) has a minimum at x= a, provided f(x) is continuous at x= a.
20) Assertion (A): The relation
f: {1,2,3,4} --> {x, y, z, p} defined by
f={(1,x),(2,y),(3,z)} is a bijective function.
Reason (R) The function
f: {1,2,3} ---> {x,y,z,p} such that
f: {(1,x),(2, y),(3, z)} is one-one.
SECTION - B
(This section comprises of very short Answer type questions (VSA) of 2 marks each)
21) Find the value of sin⁻¹[cos(33π/5)]
OR
Find the domain of sin⁻¹(x²-4).
22) Find the interval in which the functions
f: R---> R defined by f(x)= xeˣ, is increasing.
23) If f(x)= 1/(4x²+ 2x +1), x ∈ R, then find the maximum value of f(x).
OR
Find the maximum profit that a company can make, if the profit function is given by P(x)= 72 + 42x - x², where x is the number of units and P is the profit in rupees.
24) Evaluate ¹₋₁ ∫ logₑ {(2- x)/(2+ x)} dx.
25) Check whether the function f: R--->R defined by f(x)= x³+ x, has any critical point/s are not ? If yes, then find the point/s.
SECTION - C
(This section comprises of short Answer type questions (SA) of 3 marks each)
26) Evaluate ∫ (2x²+3)/{x²(x²+9)} dx, x≠ 0.
27) The random variable X has a probability distribution P(X) of the following form, where 'k' is some real number.
k, if X= 0
2k, if X = 1
P(X)=. 3k, if X= 2
0, otherwise
a) Determine the value of k.
b) Find P(X< 2).
c) Find P(X > 2).
28) Evaluate ∫ √{x/(1- x³)} dx, x ∈(0,1),
OR
Evaluate ∫ logₑ(1+ tanx) dx at (π/4,0).
29) Solve the differential equation
yeˣ/ʸ dx = (xeˣ/ʸ + y²) dy, (y≠ 0).
OR
Solve the differential equation
(cos²x)dy/dx + y = tanx; (0≤ x<π/2).
30) Solve the following Linear Programming Problem graphically
Minimize Z= x + 2y
Subject to the constraints , x +2≥ 100, 2x - y≤ 0, 2x + y ≤ 200, x, y ≥ 0.
OR
Solve the following Linear Programming Problem graphically
Maximum Z= - x + 2y.
subject to constraints , x≥3, x+ y ≥ 5, x + 2y ≥ 6, y≥ 0.
31) If (a + bx)eʸ/ˣ = x, then prove that
x d²y/dx²= {a/(a + bx)}².
SECTION - D
(This section section comprises of long answer type questions (LA) of 5 marks each)
32) Make a rough sketch of the region
{(x,y): 0≤ y ≤ x½+1,
0≤ y ≤ x +1, 0≤ x ≤ 2) and find the area of the region , using the method of integration.
33) Let N be the set of all natural numbers and R be a relation on N x N defined by (a, b) R(c,d) <=> ad= bc for all
(a, b),(c,d) ∈ N x N. Show that R is an equivalence relation on N x N. Also, find the equivalence class of (2,6) i.e., [(2,6)].
OR
Show that the function
f: R---> {x ∈ R : -1< x < 1} defined by
f(x)= x/{1+ |x|} , x ∈ R is one-one and onto function.
34) Using the matrix method, solve the following system of linear equations .
2/x + 3/y + 10/z = 4,
4/x - 6/y + 5/z = 1,
6/x + 9/y - 20/z = 2.
35) Find the coordinates of the image of the point (1,6,3) with respect to the line.
r= (j+ 2k) + λ(i + 2j + 3k), where λ is a scalar. Also, find the distance of the image from the y-axis.
OR
An aeroplane is flying along the line
r= λ(i - j + k), where λ is a a scalar and another aeroplane is flying along the line
r= i - j + μ(-2j + k), where μ is a scalar . At what points on the lines should they reach, so that the distance between them is the shortest ? Find the shortest possibly distance between them.
SECTION - E
(This section comprises of 3 case - study/passage -based questions of 4 marks each
36) Read the following passage and answer the questions given below.
In an office 3 employees James, Sofia and Oliver process incoming copies of a certain form. James processes 50% of the forms. Sophia processes 20% and Oliver the remaining 30% of the forms. James has an error rate of 0.06, Sophia has an error rate of 0.04 and Oliver has an error rate of 0.03.
Based on the above information, answer the following questions .
a) Find the probability that Sophia processed the form and committed an error.
b) Find the total probability of committing an error in processing the form .
c) The manager of the company wants to do a quality check. During inspection, he selects a form at random from the days output of process from. If the form selected at random has an error; then find the probability that the form is not processed by James.
OR
Let E be the event of committing an error in processing the form and let E₁, E₂ and E₃ be the events that James, Sophia and and Oliver processed the form.
Find the value of ³ᵢ₌₁∑ P(Eᵢ/E).
37) Read the following passage and answer the questions given below.
Teams A, B, C went for playing a tug of war game. Team A, B, C have attached a rope to a metal ring and is trying to pull the ring into their own area.
Team A pulls with force F₁ = 6i + 0j kN,
Team B pulls with force F₂ = -4i + 4j kN,
Team C pulls with force F₃ = -3i - 3j kN,
a) What is the magnitude of the force of Team A ?
b) Which team will win the game ?
c) Find the magnitude of the resultant force exerted by the teams.
OR
In what direction is the ring getting pulled?
38) Read the following passage and answer the questions given below.
The relation between the height of the plant (y in cm) with respect to its exposure to the sunlight is governed by the following equation y= 4x - x²/2, where x is the number of days exposed to the sunlight, for x≤ 3.
i) Find the rate of growth of the plant with respect to the member of days exposed to the sunlight.
ii) Does the rate of growth of the plant increase or decrease in the first three days ? What will be the height of the plant after 2 days ?
PAPER - III
1) What is the domain of the f(x)= (x²-1)/(x -4) ?
2) For what values of x will the given matrix
-x x 2
2 x -x
x -2 -x be a singular?
3) Evaluate ∫ sinx cos²x(sin²x + cosx) dx at (π/2,-π/2).
5) Evaluate sin(2 tan⁻¹ (1/3))
6) If a b
c -a is a square root of the 2 x 2 identity matrix, then what is the relation between a,b and c ?
8) Find the solution of the equation of determinant
cosx sinx cosx
-sinx cosx sinx = 0
-cosx -sinx cosx
10) Evaluate: ∫ ₂2ˣ 2ˣ dx
12) Let f(x)= x²/(x²+1) for x ≥ 0. Then find f⁻¹(x).
13) Find the probability of drawing a diamond card in each of the two consecutive draws from a well shuffled pack of cards, if the card drawn is not replaced after the first draw.
14) Solve the equation
sin[2cos⁻¹(cot(2tan⁻¹x)]= 0.
Or
Show that
2 tan⁻¹[tan(x/2) tan(π/4 - y/2)]= tan⁻¹{sinx cosx/(siny + cosx)}
15) Find the differential equation of all straight lines which are at fixed distance 'p' from the origin.
16)
17) Evaluate:∫ √(2x²-1)/(1- x²) dx
Or
∫ dx/((x¹⁾² + x¹⁾³).
18) If c= a(sink - k cost) and y= a(cost + k sink) find dy/dx at k=π/4.
20) Find the inverse of A
A= 2 1 3
4 -1 0
-7 2 1
21) 1, if x ≤ 3
If f(x)= ax+ b, if 3< x < 5
7, if 5 ≤ x
Determine the value of a and b so that f(x) is continuous.
Or
Determine the value of a, b, c for which function
(sin(a+1)+ sinx)/x, x< 0
f(x)= c , x=0
(√(x + bx²) - √x)/b√x³, x> 0
may be continuous at x=0
4) If a, b, c are three mutually perpendicular unit vectors then what is the value of|a+ b + c| ?
7) a and b are unit vectors. If a+ b is a unit vector then what is the angle between a and b?
9) Equation of the plane passing through (2,3,-1) and is perpendicular to the vector 3i - 4j + 7k.
11) If a,b,C are vectors such that a.b= a.c, ax b = ax c, a≠ 0, then show that b= c
19) A line with direction ratios < 2,7, -5> is drawn to intersect the lines
(x -5)/3 = (y-7)/-1 = (z+2)/1 and (x +3)/-3 = (y-3)/2 = (z -6)/4
Find the coordinates of the points of intersection.
PAPER - II
1) If A= 3 1
7 5 find x and y so that A²+ xI₂ = yA.
2) Evaluate: tan[2 tan⁻¹(1/5) - π/4].
3) State the reason why the relation
R={(a,b): a≤ b²}
on the set R of real numbers is not reflexive.
4) ∫ (xeˣ)/(x +1)² dx.
5) Deepak rolls two dice and gets a sum more than 9. What is the probability that the number on the first die is even?
6) Y= tan⁻¹{5x/(1- 6x²)}, -1/√6< x <1/√6, then show that dy/dx = 2/(1+ 4x²) + 3/(1+ 9x²).
7) Solve: (y + xy) dx + y(1- y²) dy = 0.
8) Let f: [0,∞)--> R be a function defined by f(x)= 9x²+ 6x -5. Show that f is not invertible. Modify only the codamain of f to make f invertible and then find its inverse.
9) Using properties of determinants, Prove that
b²+ c² ab ac
ba c²+ a² bc= 4a²b²c²
ca cb a²+ b².
10) Show that: tan[sin⁻¹(1/√17) + cos⁻¹(9/√85)]= 1/2.
11) Discuss the continuity of the function
f(x}= 2x -1, x< 1/2
3 -6c, x≥ 1/2
at x = 1/2
12) Find differentiation:
y= tan⁻¹[{√(1+ x²) -1}/x].
13) ∫ (tanx + tan²x)/(1+ tan²x) dx.
14) ³₁∫ (3x² +1) dx.
15) Show that the equation of the normal at any point t on the curve x= 3 cos t - cos³t and y= 3 sin t - sin³ t is 4 (y cos³t - x sin³t)= 3 sin 4t.
16) The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
17) Solve: (tan⁻¹y - x) dy = (1+ y²) dx.
18) Bag I had two and three black balls, Bag II has four red and one black ball and Bag III has three white and two black balls. A bag is selected at random and a ball is drawn at random. What is the probability of drawing a red ball?
19) Using matrix method, solve
x - 2y - 2z -5=0; -x + 3y +4=0; -2x + z - 4=0.
20) Find inverse of
-1 1 2
1 2 3
3 1 1
21) Show that the right circular cone of least curved surface area and given volume has an altitude equal to √2 times the radius of the base.
22) The sum of the surface areas of a cuboid with sides x, 2x and x/3 and a sphere is given to be constant. Show that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
23) Evaluate: ∫ x²/{(x² +4)(x² +9)} dx.
24) Three persons A, B and C apply for a job of manager in a private company. Chances of their selection (A,B and C) are in the ratio 1:2:4. The probability of A, B and C can introduce changes to improve profits of the company are 0,8,0.5,0.3 respectively. If the change does not take place, find the probability that it is due to the appointment of C.
25) The vectors from origin to the points A and B are a= 3i - 6j +2k and b= 2i + j - 2k respectively. Find the area of the triangle OAB.
26) Find the Cartesian equation of the line passing through the points(3,-2,-5) and (1,-4,-7).
27) Show that (a x b)²= a.a a. b
a.b b.b
28) If the vectors ai+ j + k, i + bj + k and i+ j+ ck are coplanar (a,b,c≠1), then show that 1/(1- a) + 1/(1- b) + 1/(1- c)= 1.
29) Find the shortest distance between the lines whose vector equations are
r= (1- t)i + (t - 2)j +(3- 2t)k, and
r= (s +1)i + (2s -1)j - (2s +1)k.
30) Find the equation of the plane passing through the point (1,1,-1) and perpendicular to the planes x +2y + 3z -7=0 and 2x - 3y + 4z = 0.
31) Sketch the graphs of y= x(4- x) and find the area bounded by the curve, x-axis and the lines x=0 and x= 5.
32) The marginal cost of production of a commodity is 30+ 2x. It is known that fixed costs are Rs 120. Find
a) Find the total cost of producing 100 units.
b) Find the cost of increasing output from 100 to 200 units.
33) You are given the following two lines of regression. Find the regression of Y on X and X on Y and satisfy your answer.
34) The cost function for a commodity is C(x)= 200+ 20x - x²/2 (in Rs)
a) Find the marginal cost MC.
b) Calculate the marginal cost when x= 4 and interpret it.
35) Fit a straight line to the following data, treating y as the dependent variable:
x: 14 12 13 14 12
y: 22 23 22 24 24
Hence, predict the value of y when x= 16.
36) You are given the following data:
x y
Arithmetic mean. 36 85
Standard deviation 11 8
37) Given the total cost function for x units of a commodity as C(x)= x³/3 + 3x² - 7x + 16. Find
a) the marginal cost
b) the average cost
c) show that the marginal cost is given by {x MC - C(x)}/x²
38) Given the price of a commodity is fixed at Rs 55 and its cost function is C(x)= 30x + 250.
a) Determine the break even point.
b) What is the profit when 12 items are sold?
39) An aeroplane can carry a maximum of 200 passes. A profit of Rs 1000 is made in each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves atleast 20 seats for executive class. However, atleast four times as many passengers prefer to travel by economy class then by executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit earned?
Paper - 1
1) Without expanding at any stage, find the value of the determinant:
2 x y+ z
∆= 2 y z+ x
2 z x+ y (2)
2) Solve: sin⁻¹cos (sin⁻¹x)=π/3. (2)
3) Find the value of k if
M= 1 2
2 3 and M² - kM - I₂ =0. (2)
4) Evaluate : ∫ sin³⁾²x/(sin³⁾²x + cos³⁾²x) dx at (π/2,0). (2)
5) Find dy/dx, if x= at² and y= 2at. (2)
6) Find the differential equation of the family of curves y= Aeˣ + Be⁻ˣ, where A and B are arbitrary constants. (2)
7) Find the intervals in which the function f(x) is strictly increasing where f(x)= 10 - 6x - 2x². (2)
8) A family has two children. What is the probability that both children are boys given that atleast one of them is a boy? (2)
9) Given that the events A and B are such that P(A)= 1/2 , P(A U B)= 3/5 and P(B)= k. Find k if
a) A and B are mutually exclusive
b) A and B are independent. (2)
10) Ley R⁺ be the set of all positive real numbers and f: R⁺ ---[4,∞): f(x)= x² + 4. Show that inverse of f exists and find f⁻¹. (2)
11) Using properties of determinants Prove
x x² 1+ px³
y y² 1+ py³
z z² 1+ pz³ = (1+ pxyz)(x - y)(y - z)(z - x), where p is any scalar. (3)
12) Prove that tan⁻¹(1/2) = π/4 - (1/2) cos⁻¹(4/5). (3)
13) Show that the function f(x)= |x -1|, x ∈ R, is continuous at x= 1 but not differentiable. (2)
14) If y= ₑa cos⁻¹x, where -1≤ x ≤ 1 then show that: (1- x²)y₂ - xy₁ - a²y = 0. (3)
15) ∫ (6x +7)/√{(x -5)(x -4)} dx. (3)
16) Evaluate : ³₁∫ (x² + x)dx
17) Find the equation of the normal to the curve y= x³+ 2x + 6 which are parallel to the line x + 14y + 4=0. (2)
18) A circle disc of radius 3cm is being heated. Due to expansion, its radius is increasing at the rate of 0.05 cm/s. Find the rate at which its area is increasing when the radius the is 3.2 cm. (2)
19) Solve the following differential equation: x dy/dx + 2y = x² logx. (2)
20) Let X denote the number of hours you study during a randomly selected school day. The probability that X can take the values 'x' has the following form, where 'k' is some unknown constant.
P(X= x) = 0.1 if x= 0
kx if x= 1 or 2
k(5- x), ifx= 3 or 4
0, otherwise
a) Find the value of k.
b) What is the probability that you study
i) atleast two hours?
ii) exactly two hours?
iii) atmost 2 hours? (3)
21) If the matrix
A= 3 -2 3 & B= -1 -5 -1
2 1 -1 -8 -6 9
4 -3 2 -10 1 7
With the relation AB, hence solve the system of equations
3x - 2y + 3z = 8; 2x + y - z = 1; 4x - 3y + 2z = 4. (3)
22) Find inverse of
1 3 -2
-3 0 -1
2 1 0 (3)
23) Show that the altitude of a right circular cone of maximum volume that can be inscribed in a sphere of radius R is 4r/3. (3)
24) An open topped box is to be made by removing equal squares from each corner of a 3m by 8m rectangle sheet of aluminium and folding up the sides. Find the volume of the largest such box. (3)
25) ∫ (3x +5)/(x³- x² - x +1) dx. (3)
26) A, B, C throw a die one after the other in the same order till one of them gets a 6 and wins the game. Find their respective probability of winning if A starts the game. (3)
27) Find the cost of increasing from 100 to 200 units if the marginal cost in Rs per unit is given by the function MC = 0.003 x²- 0.01x + 2.5. (1)
28) Given that the observation are (9,-4),(10,-3),(11,-1), (13,1),(14,3), (15,5),(16,8), find the two lines of regression. Estimate the value of y when x= 13.5. (2)
29) Find the regression coefficients bᵧₓ and bₓᵧ and the two lines of regression for the following data.
X: 2 6 4 7 5
Y: 8 8 5 6 2
Also compute the correlation coefficient. (2)
30) If the demand function is given by x= (600- p)/8, where the price is Rs p per unit and the manufacturer produces x units per week at the total cost of Rs x²+ 78x + 2500, find the value of x for which the profit is maximum. (3)
31) The fixed cost of a new product is Rs 35000 and the variable cost per unit is Rs 500. If the demand function p= 5000 - 100x, find the break even value/s. (3)
32) A toy company manufacturers two types of dolls, A and B. Market test and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demands for the dolls of type B is atmost half of that for dolls of type A. Further, the production level of type A can exceed three timee the production of dolls of other type by atmost 600 units. If the company makes a profit of Rs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximize the profit ? (3)
