1) 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
2) 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)
3) 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)
4) 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
5) 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
6) 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.
7) If f(x)= x |x| then dy/dx is
A) 2x B) -2x C) 2|x| D) none
8) 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
9) 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
10) sin cos⁻¹(x) w.r.t. cos⁻¹x is
A) cosx B) - x C) x D) none
11) 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
12) find the value of K if A²= 8A + KI, where A= 1 0
-1 7
A) 7 B) -7 C) O D) I
13) 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)
14) 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)
15) 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
16) 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
17) The curves 2x= y² and 2xy= k cut at right angles if k²=?
A) 2 B) 4 C) 6 D) 8.
18) 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
19) 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
20) 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
21) 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
