1) Solve for x: cos(tan⁻¹x)= sin(cot⁻¹3/4)
A) 1/4 B) 3/4. C) 5/4 D) 6/5
2) Evaluate: lim ₓ→₀{sin x -x)/x³}
A) 1/6 B) -1/6. C) 5/6 D) -5/6
3) For what value of x is the given matrix 2x+4 4
x+ 5 3 a singular matrix ?
A) 3 B) 4. C) 5 D) 6
4) If y= xʸ then value of x dy/dx
A) y/(1-y logx)
B) y²/(1- y logx).
C) -y²/(1- y logx)
D) -y/(1- y logx)
5) using properties of determinants evaluate
1+ a 1 1
1 1+ b 1
1 1 1+ c
A) abc
B) 1/a + 1/b +1/c
C) abc(1/a + 1/b +1/c)
D) abc(1 + 1/a + 1/b +1/c).
6) 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].
A) (π,0) B) (π,2) C) (π, -2). D) (-π,0)
7) 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
A) 4,4 B) 4,-4. C) -4,4 D) -4,-4
8) If cos⁻¹x/a + cos⁻¹y/b = k, then the value of x²/k + y²/b² - sin²k is
A) 2xy B) 2xy/ab C) 2xy cos k D) (2xycos k)/ab.
9) If y= (x+ √(1+x²)))ⁿ, then find the value of (1+x²) d²y/dx² + x dy/dx
A) ny B) n²y. C) ny² D) n²y²
10) Find the equation of the tangent to the curve y= x² - 2x+7 which is parallel to the line 2x- y+9= 0
A) y-2x-3= 0. B) y+2x-3= 0
C) y-2x+3= 0 D) y+2x+3= 0
OR
Find the equation of the tangent to the curve y= x² - 2x+7 which is parpenducular to the line 15x- 5y -13 = 0
A) 36y-12x-227= 0.
B) y+2x-3= 0
C) y-2x+3= 0
D) 36y+12x -227= 0.
11) Find the interval in which the function f given by f(x)= sin x - cos x, 0 ≤ x ≤ 2π is
A) strictly increasing in (0,3π/4).
B) strictly decreasing in (3π/4,π/4)
C) Increasing at (0, π)
D) Decreasing at (0,-π)
12) Let f: N --> N be a function defined as f(x)= 4x² + 12x+ 15. Find f⁻¹(31)
A) 1. B) 4 C) -4 D) -1
13) The area of a right-angled triangle of given hypotenuse is maximum, when the triangle is
A) isosceles triangle.
B) equilateral triangle
C) scalene triangle
D) Isosceles right angled triangle
OR
All the rectangle inscribed in a given fixed circle, the ____ has the maximum area.
A) triangle B) square.
C) equilateral triangle
D) parallelogram
14) the fixed cost of a product is ₹ 18000 and the variable cost per unit is ₹550. If the demand function is p(x)= 400 - 150x, find the breakable values.
A) 8, 10 B) 8, 15. C) 10,15 D) 1,15
15) The cost function for a commodity is C(x)= ₹(200 + 20x - x²/2) find the marginal cost (MC). calculate also the marginal cost when x= 4 and interpret it.
A) 20+x, 10 B) 20- x, 10
C) 20+x, 16 D) 20-x, 16
16) 2 sin⁻¹x = ?
A) sin⁻¹(x √(1+x²))
B) sin⁻¹(2x √(1+x²))
C) sin⁻¹(x √(1- x²))
D) sin⁻¹(2x √(1- x²)).
17) R is a relation in N x N defined by (a, b) R(c, d) if and only if ad= bc. Then R is
A) equivalence relation.
B) Reflexive relation
C) Transitive relation
D) Identity relation
18) find the value of K if A²= 8A + KI, where A= 1 0
-1 7
A) 7 B) -7 C) O D) I
19) If y= log{√(x+1)+√(x-1)}/{√(x+1) - √(x-1)}, then dy/dx is..
A) 1/(x²-1) B) 1/√(x²-1).
C) 1/√(x² +1) D) 1/(x²-1)
20) If x y z
x² y² z² = 0
yz zx xy then find the value of (yz + zx + xy)
A) y- z B) z - x C) x - y D) 0
21) If y= x sin 2x then find the value of x² d²y/dx² - 2x dy/dx is
A) 0 B) 2y C) 4x²y D) -(2y+ 4x²y)
22) The value of tan⁻¹√x is
A) cos⁻¹{(1-x)/(1+x)}
B) cos⁻¹{(1+ x)/(1-x)}
C) 1/2 cos⁻¹{(1-x)/(1+x)}.
D) 1/2 cos⁻¹{(1+ x)/(1-x)}
23) The value of c of Lagrange's mean value theorem if
f(x)= x(x-1)(x-2); a= 0, b= 1/2, i.e., for every x belongs to [0,1/2]
A) 0.24. B) 2.4 C) 24 D) 42
24) 3x - 2, 0< x ≤ 1
If f(x)=2x² - x, 1< x ≤ 2
5x -4, x> 2 then
A) Differentiable at x= 2, continuous at x= 2
B) not Differentiable at x= 2, but continuous at x= 2
C) Differentiable at x=2 , but not continuous at x= 2
D) Neither Differentiable nor continuous at x= 2
25) The curves 2x= y² and 2xy= k cut at right angles if k²=?
A) 2 B) 4 C) 6 D) 8.
26) If f(x)= (4x+3)/(6x-4), x≠ 2/3, then (f o f)(x)= ? Also find f⁻¹
A) x for all , f⁻¹= Intwger
B) x for all x= 2/3, f⁻¹= N
C) x for all x≠2/3, f⁻¹= f.
D) x for all irrational, f⁻¹= R
27) sum of the surface area of a rectangular parpalloid with sides x, 2x and x/3 and a sphere is given to be constant. Then the sum of their volume is minimum if x equal to
A) three times the radius of the sphere.
B) 2 times the radius of the sphere.
C) equal to the radius of the sphere
D) 4 times the radius of the sphere
28) the cost of producing x items per day is given in Rupees as C(x)= 2000 + 100√x. if each item can be sold for ₹10, then the break even point is .
A) 400 B) 500 C) 600 D) 700
29) the marginal cost C is given to be a constant multiple of number of units (x) produced. Find the total cost and the average cost function if the fixed cost is ₹1000 and cost of producing 30 unit is ₹2800
A) 2x²+1000 , 2x+ 1000/x.
B) 2x+1000/x , 2x²+ 1000
C) 2x²+1000x , 2x+ 1000
D) 2x+1000 , 2+ 1000/x
30) For a certain establishment, the total revenue function R and the total cost function C are given by R= 83x - 4x² - 21 and C= x³ - 12x² + 48x + 11, where x= output. Obtain the output for with the profit is maximum.
A) 5 B) 6 V) 7. D) 8
31) the cost function of a Firm is given by C = x³/3 - 5x² + 30x + 10 where C is the total cost for x items. Determine x at which the marginal cost is minimum.
A) 5. B) 6 V) 7 D) 8
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