i) The domain for which the function f(x)= 3x²- 2x and g(x)= 3(3x -2) are equal, will be
a) {1,2/3} b) {1,3} c) {2/3,3} d) {2/3,0}
ii) The value of tan {π/2 - tan⁻¹(1/3)} is equals to
a) 1/3 b) 3 c) 2/3 d) 3/2
iii) If two rows or two columns of a determinant are identical then value of the determinant is
a) 0 b) 2 c) - 1 d) 1
iv) if f(x)= - f(x), then the value of ᵃ₋ₐ∫ f(x) dx is equals to
a) 2a b) a c) a/2 d) 0
v) If y= tan⁻¹{5 - x).(1+ 5x)}, then value of dy/dx is
a) -1/(1+ x²) b) 1/(1+ x²) c) 5 d) 5/(1+ x²)
vi) If P(A)= 3/7, P(B)= 4/7 and P(A∩B) =2/9, then the value of P(A/B) is equals to
a) 7/18 b) 14/27 c) 5/18 d) 4/9
vii) A coin is tossed 10 times . The probability of getting had six times is
a) ¹⁰C₅. 1/2¹⁰ b) ¹⁰C₃. 1/2¹⁰ c) ¹⁰C₄. 1/2¹⁰ d) ¹⁰C₈. 1/2¹⁰
viii) If xy =1 then the value of d²y/dx² is
a) 0 b) 3 c) -1 d) 2
ix) The integrating factor of the differential equation (x + y+ z) dy/dx =1 is
a) e⁻ʸ b) eˣ c) e⁻ˣ d) eʸ
x) Form the differential equation of the family of curves y= A cos(x + B), where A and B are arbitrary constants.
2)a) Find the maximum value of 1 1 1
1 1+sinx 1
1 1 1 + cosx (1)
b) Solve the equation for x:
cos(tan⁻¹x)= sin(cot⁻¹(3/4)). (1)
c) Find the intervals in which the function f(x)= 3x⁴- 4x³- 12x²+ 5 is strictly decreasing. (1)
4) Given that event 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)
5) Evaluate ∫ (x sinx)/(1+ sinx) dx at (π,0). (2)
6) For what value of x is the given matrix
2x+4 4
x+ 5 3 is a singular matrix ? (2)
7) If y = xʸ, prove that x dy/dx = y²/(1- y logx). (2)
8) The probabilities of A, B, and C solving a problem are 1/2, 1/3 and 1/4 respectively . Find the probability that the problem will be solved. (3)
9) Prove: 1+ a 1 1
1 1+b 1 = abc(1/a+1/b+1/c). (3)
1 1 1+c
10) If the following function is differentiable at x=2, then find the value of a and b.
f(x)= x², if x ≤ 2
ax + b, if x > 2. (3)
OR
Find the value of k. So that the function f defined by
f(x)= kx +1, if x ≤π
cosx, if x > π
is continuous at x =π
11) if cos⁻¹(x/a)+ cos⁻¹(y/b)= K, Show that x²/a² - (2xy cosK)/ab + y²/b² = sin²K. (4)
12) If y= {x + √(1+ x²)}ⁿ, then show that (1+ x²) d²y/dx² + x dy/dx = n²y. (4)
13) 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 atleast one ball is red ? (4)
14) Evaluate ∫ (3x +1)/√(5 - 2x - x²) dx. (4)
OR
Prove ³₁∫ dx/{x²(x +1} = 2/3 + log(2/3).
14) Find the equations of the tangent to the curve y= x²- 2x +7 which is
a) parallel to the line 2x - y +9=0.
b) perpendicular to the line 5y - 15x =13. (4)
OR
Find the intervals in which the function f given by f(x)= sinx - cosx, 0 ≤ x ≤ 2π is strictly increasing or strictly decreasing.
15) a) Solve : dy/dx + y/x = x². (2)
b) Solve : dy/dx + 1 = eˣ⁺ʸ. (2)
16) 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 f⁻¹(31) and f⁻¹(87). (4)
17) Using matrices , solve the following equations: 5x+ 3y + z =16, 2x+ y + 3z =19, x+ 2y + 4z =16pp25. (6)
OR
If A= 1 -1 0
2 5 3
0 2 1 Find A⁻¹.
18) Prove that the area of a right angle triangle of given hypotenuse is maximum, when the triangle is isosceles. (6)
OR
Show that of all the rectangle inscribed in a given fixed circle, the square has the maximum area.
19) Evaluate
a) ∫ x² sin⁻¹x dx. (3)
b) ∫ x/(x²+ 4x +3) dx. (3)
20) A pair of dice is thrown 4 times. If getting a double is considered a success, find the probability distribution of the number of successes and show that its mean is 2/3. (3)
21) a) 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 the second. Find the probability that the ball drawn is white. (3)
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22) i) If a= i+ 3j - k and b = 2i + 6j + μk. If a and b vectors are parallel, then the value of μ is
a) 3 b) -6 c) -3 d) -2 (1)
ii) The angle between the two planes x - y +2z =9 and 2x + y +z =7 is
a) 30° b) 45° c) 60° d) 90° (1)
iii) If a= 2i + j + 3k and b = 3i + 5j - 2k represent two adjacent sides of a triangle, then find the area of the triangle. (1)
iv) If A(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. (1)
23) If a and b are two vectors, then show that
(a x b)²= a.a a.b
a. b b.b (4)
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 plane passing through the line of intersection number 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 coordinates of the points in 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)
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27)a) The fixed cost of a product is Rs18000 and the variable cost per unit is Rs550. If the demand function is p(x)= 4000 - 150x, find the break-even values. (2)
b) Given x + 4y =4 and 3x + y =16/3 are regression lines. Find the line of regression of x and y. (2)
c) The cost function for a commodity is C(x)= Rs(200+20x - x²/2)
i) Find the marginal cost (MC).
ii) Calculate the marginal cost when x= 4 and interprete it. (2)
28) Two regression lines are represented by 2x + 3y -10=0 and 4x + y -5=0. Find the lines 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.
29) The marginal cost function of a firm is MC= 33 logx. Find the total cost function when the cost producing one unit is Rs11. (4)
30) 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 and both machines and each unit 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 Rs7 and that of B at a profit of Rs4. Find graphically the production level per day for maximum profit.
(6)
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