Full Marks- 90
SECTION A
1) (any 5) 1x5= 5
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) 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
2)
i) solve the determinants
x² x 1 (2)
0 2 1 = 28
3 1 4
ii) Solve the equation for x:
cos(tan⁻¹x)= sin(cot⁻¹3/4). (1)
iii) using Matrix rule, solve the system of equations: 5x + 7y = -2, and 4x+ 6y = -3. (2)
iv) Evaluate: lim ₓ→₀{(sin x - x)/x³}. (1)
v) Given that the events A and B are such that P(A)=1/2, P(B)= p, P(A U B)= 3/5. Find p if A and B are
a) mutually exclusive
b) Independent. (2)
vi) Evaluate: ∫ (x sin x)/(1 + sin x) dx at (π, 0). (2)
vii) (1)
For what value of x is the given matrix 2x + 4 4
x + 5 3 a singular Matrix
viii) if y= xʸ, prove that x dy/dx = y²/(1- y log x). (2)
ix) The probability of A, B, C solving a problem 1/2, 1/3, 1/4 respectively. Find the probability that the problem will be solved. (2)
x) If f(x)= 27x³ and g(x)= ³√x, then find gof(x). (1)
3) Using properties of determinant, Prove that 1 + a 1 1
1 1+ b 1
1 1 1+ c = abc(1+ 1/a + 1/b + 1/c) (3)
4) Apply Rolle's theorem to find a point (or points) on the curve y= - 1+ cos x where the tangent is parallel to the x-axis in [0, 2π]. (3)
5) If the following function is differentiable at x= 2, then find the values of a and b.
OR
If the following function is differentiable at x= 2, then find the values of a and b.
f(x)= x², if x ≤ 2
ax+ b, if x > 2 (3)
6) if cos⁻¹(x/a) + cos⁻¹(y/b) = k , prove that x²/a² - (2xy cos k)/ab + y²/b² = sin²k. (3)
7) if y= {x+ √(1+ x²)}ⁿ , then show that (1+ x²) d²y/dx² + x dy/dx = n²y. (3)
8) Two balls are drawn one after another (without replacement) from a bag containing 2 white, 3 red and 5 blue balls. What is the probability that at least one ball is red ? (3)
9) Evaluate ∫ (3x+1)/√(5 - 2x - x²) dx. (3)
OR
evaluate ∫ √x/{√x + √(a - x)} dx
10) Find the equation of the tangent to the curve y= x² - 2a + 7 which is
a) a parallel to the line 2x - y +9= 0
b) Perpendicular to the line 5y - 15x= 13. (3)
11) Find the interval in which the function f given by f(x)= sin x - cos x, 0 ≤ x ≤ 2π is strictly increasing or strictly decreasing. (2)
12) solve: dy/dx + y/x = x². (2)
13) Solve: dy/dx + 1 = eˣ⁺ʸ. (2)
14) Let f: N --> N be a function defined as f(x)= 4x² + 12x + 15, show that f: N --> S is invertible (where S is the range of f). Find the inverse of f and hence find f⁻¹(31) and f⁻¹(87). (3)
15) Using matrix, solve the following equations: 5x+ 3y + z= 1z, 2x+ y+ 3z= 19, x+ 2y+ 4z= 25. (3)
OR
If A= 1 -1 0
2 5 3
0 2 1 find A⁻¹.
16) Prove that the area of a right angled triangle of given hypotenuse is maximum, when the triangle is isosceles. (3)
OR
Show that of all rectangle inscribed in a given fixed circle, the square has the maximum area.
17) Evaluate: ∫ x² sin⁻¹x dx. (2)
18) ∫ x dx/(x²+ 4x +3). (2)
19) A pair of dice is thrown 4 times. If getting a double is considered a success, find the probability distribution of the number of success and show that its mean is 2/3. (3)
20) One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from the first bag to the second and then a ball is drawn from 2nd. Find the probability that the ball drawn is white. (3)
SECTION - B
21) if a= 2i + j + 3k and b= 3i + 5j - 2k represent two adjacent sides of a triangle, then find the area of the triangle. (2)
22) If (8,2,0), B(4,6,-7), C(-3, 1, 2) and D(-9,-2,4) are four given points, then find the angle between AB and CD. (2)
23) Find the vector and Cartesian equation of the plane which bisects the line joining the points (3,-2,1) and (1,4,-3) at right angles. (2)
24) If a= 7i- 2j + 3k, b= i - j + 2k, c= 2i + 8j are three vectors then find a.(b x c) and (a x b). c. (4)
25) find the equation of the plane passing through the line of intersection of the planes x+ 2y+ 3z - 5= 0 and 3x - 2y - z + 1= 0 and cutting off equal intercepts on the OX and OZ axes. (4)
OR
Find the co-ordinate of the points on the line (x -1)/2 = (y+2)/3 = (z - 3)/6 which are at a distance of 3 units from the point (1,-2,3).
26) Find the area of the region included between the parabola y= 3x²/4 and the line 3x- 2y + 12= 0. (4)
27) Find the area bounded by the line y= x, The x-axis and the ordinate8 x= 0 and x= 4. (2)
SECTION- C
28) The fixed cost of a product is ₹ 18000 and the variable cost per unit is ₹550. if the demand function is p(x)= 4000 - 150x, find the break even values. (2)
29) Given x + 4y = 4 and 3x + y= 16/3 are regression lines. Find the line of regression of x on y. (2)
30) The cost function for a commodity is C(x)= ₹(200 +20x - x²/2)
a) find the marginal cost(MC).
b) calculate the marginal cost when x= 4. (2)
31) Two regression lines represented by 2x+ 3y - 10= 0 and 4x + y - 5 = 0. find the line of Regression of y on x. (4)
OR
Fit a straight line to the following data. Treating y as the dependent variable.
X: 1 2 3 4 5
Y: 7 6 5 4 3 Hence, estimate the value of y when x= 3.5
32) the marginal cost function of a firm is MC= 33 log x. Find the total cost function when the cost of producing one unit is ₹11. (4)
OR
If the marginal cost of a commodity is equal to half its average cost, show that fixed cost is zero. If the cost of producing 9 units of the commodity is ₹60, find the cost function.
33) A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of the first machine is 12 hours and that of the second machine is 9 hours per day. Each unit of product A requires 3 hours in both machines and each unit of product B requires 2 hours on the first machine and 1 hour on the second machine. Each unit of product A is sold at profit of ₹7 and that of B at a profit of ₹4. find graphically the production level per day for maximum profit. 6
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