FBQ1: When the sequence of partial sums tends to an infinite limit, oscillates either finitely or infinitely the series is said to be ____ Answer: divergent FBQ2: Both Taylor series and Maclaurin series only represent the function f(x) in their interval of ______ Answer: Convergence FBQ3: When functions are expanded at x = a, we have Taylor’s expansion and when functions are expanded at x = 0 then we have _____ expansion Answer: Maclaurin FBQ4: By considering the hypothesis of mean value theorem, Given that f(x) = x2 + 2x +1 a = 1, b =2 Answer: 4 FBQ5: By considering the hypothesis of mean value theorem, Given that fx=x2+2x+1 and a=1, b = 2 find fb=_____ Answer: 9 FBQ6: By considering the hypothesis of mean value theorem, Given that fx=x2+2x+1 and a=1, b = 2 find fıc=_____ Answer: 5 FBQ7: _____ rule is a technique for approximating the definite integral Answer: Trapezoidal FBQ8: _____ rule is an arithmetical rule for estimating the area under a curve where the values of an odd number of ordinates including those at each end. Answer: Simpson’s FBQ9: The trapezoidal rule is also known as ____ rule Answer: Trapezium FBQ10: The ∂2f∂x∂y of the function fx,y=3x2-x3y3+5xy+6y3 evaluate at the points x=1 and y=2 is ___________ Answer: -31 FBQ11: The ∂2f∂y2 of the function fx,y=3x2-x3y3+5xy+6y3 evaluate at the points x=1 and y=2 is ___________ Answer: 60 FBQ12: The limx→2 ⁡x2-2xx2-4 is ___________ Answer: ½ FBQ13: The limx→∞ ⁡xx3+5 is ___________ Answer: 0 FBQ14: If fx=x(x2-x-2) satisfies Mean Value Theorem , the value c is ___________ Answer: 1/3 FBQ15: The exponential form of the function fx=1+x+x22!+x33!+x44!+x55!+⋯ is _______ Answer: exp x FBQ16: Find the limit of $\lim_{(x, y)\rightarrow (2, 1)} x+3y^{2}$ is ________ Answer: 5 FBQ17: Find the limit of $\lim_{(x, y)\rightarrow(2,4)} \frac{x+y}{x-y}$ is ________ Answer: -3 FBQ18: Find limit $\lim_{(x, y, z)\rightarrow (1, 2, 5)} \sqrt(x+y+z)$ is __________ Answer: 3 FBQ19: The coefficient of $$x^{2}$$ in the Taylor series about $$x=0$$ for $$f(x)=e^{-x^{2}}$$ is _____________ Answer: -1 FBQ20: The coefficient of $$x^{3}$$ in the Taylor series about x=0 for f(x)=sin 2x is _______________ Answer: -4/3 FBQ21: Let $f(x)=\frac{\sin x}{1+x^{2}}$ and $$y^{n}$$ denote the $$n^{th}$$ derivative of f(x) at x=0 then the value of $$y^{100}+900y^{98}$$ is _________ Answer: 0 FBQ22: If the first derivative at x=0 of the function $$f(x)=\frac{\cos (x)}{x^{2}-x+1}$$ is ______ Answer: 2 FBQ23: Given $$f(x,y)= 2x^{2}y$$, the value $$\frac{\partial f(x,y)}{\partial x}$$ at x=2 and y=4 is ________ Answer: 24 FBQ23: .Given that the function $$f(x)=\frac{2(x+3)}{x^{2}+x-2}$$ has an absolute maximum on the -2<x<q. The maximum value is ________ Answer: 2 FBQ25: The points of inflection of the function $$f(x)=x^{4}-12x^{3}+6x-9$$ on the interval $$-2\leq x\leq 10$$ are ____________ and _________ Answer: 0, 6 FBQ26: The value of a such that the function $$f(x)=x^{2}+ax+5, when f(2)= 15 is _______ Answer: 3 FBQ27: If x2+y2-2x-6y+5=0, the value d2ydx2 at x=3, y=2 is _____ Answer: 5 FBQ28: If the Mean Value Theorem satisfies fx=x2 on the interval -2, 1 , then the value of c is _____ Answer: -1/5 FBQ29: The minimum value of$$f(x,y)=x^{2}+y^{2}+6x+12$$is _________ Answer: 3 FBQ30: Suppose w=x3yz+xy+z+3 and x=3cos⁡t, y=3sin⁡t and w=2t. The value dwdtt=π2 is _____________ Answer: 7 FBQ31: Let$$f(x)=\frac{e^{x} sin(x^{2})}{x}$$, then the value of the fifth derivative at x=0 is ___________ Answer: 21 FBQ32: Leibniz rule gives the Nth derivative of multiplication of _____ functions Answer: Two FBQ33: Leibniz theorem is applicable if n is a ________ integer Answer: Positive FBQ34: If nth derivative of$$xy_{3}+x^{2}y_{2}+x^{3}y_{0}=0$$then order of its nth differential equation is _________ Answer: n+3 FBQ35: For the function$$f(x)=\frac{sin x}{x^{2}}$$. ________ are the number of points exist in the interval$$[0, 7\pi]$$such that$$f’(c)= 0$$Answer: True FBQ36:$$f(x)=\frac{sin x}{x}$$. ________ are the number of points exist in the interval$$[0, 18\pi]$$such that$$f’(c)= 0$$Answer: 18 FBQ37: For all second degree polynomials with y = ax2 + bx + k, it is seen that the Rolles’ point is at c = 0. Also the value of k is zero. Then the value of b is _____ Answer: 0 FBQ38: For second degree polynomial it is seen that the roots are equal. Then ______ is the relation between the Rolles point c and the root x Answer: c=x FBQ39: Rolle’s Theorem is a special case of ___________ theorem Answer: Mean value FBQ40: The value of$$c$$if$$f(x)=x(x-3)e^{3x}$$, is continuous over interval [0, 3] and differentiable over interval (0, 3)_________ (Answer to 3 decimal) Answer: 2.703 FBQ41: The value of ‘a’ are _____ and ______,if f(x) = ax2+32x+4 is continuous over [-4, 0] and differentiable over (-4, 0) and satisfy the Rolle’s theorem. Hence find the point in interval (-2,0) at which its slope of a tangent is zero Answer: 8, -2 FBQ42: For the function f(x) = x2 – 2x + 1. We have Rolles point at x = 1. The coordinate axes are then rotated by 45 degrees in anticlockwise sense. What is the position of new Rolles point with respect to the transformed coordinate axes___________ Answer: 3/2 FBQ43: If f(a)=f(b) in mean value theorem, then it becomes ________ theorem Answer: Rolle’s FBQ44: Mean Value theorem is applicable to the functions continuous in closed interval [a, b] and ___________ in open interval (a, b) Answer: Differentiable FBQ45: Mean Value theorem is also known as ___________ theorem Answer: Lagrange’s FBQ46: The point c is _________ in the curve f(x) = x3 + x2 + x + 1 in the interval [0, 1] where slope of a tangent to a curve is equals to the slope of a line joining (0,1) Answer: 0.54 FBQ47: _________ is the point c between [2,9] where, the slope of tangent to the function f(x)=1+∛x-1 at point c is equals to the slope of a line joining point (2,f(2)) and (9,f(9)).(Providing given function is continuous and differentiable in given interval). Answer: 4.56 FBQ48: _______ is the point c between [-1,6] where, the slope of tangent to the function f(x) = x2+3x+2 at point c is equals to the slope of a line joining point (-1,f(-1)) and (6,f(6)).(Providing given function is continuous and differentiable in given interval). Answer: 2.5 FBQ49: The necessary condition for the maclaurin expansion to be true for function f(x) is f(x) should be continuous and ______ Answer: Differentiable FBQ50: The limit$$\lim_{(x, y)/rightarrow (0, 0)} \frac{x^{3}-y^{3}}{x-y} is _______ Answer: 0 MCQ1: A single valued function of x is said to be continuous at x=a if Answer: lim fx= f(a) MCQ2: Which of the following is discontinuous at x = 0 Answer: Sin xx MCQ3: A function y = f(x ) is said to be differentiable at a point x = a if Answer: f1(x) exists that point MCQ4: Find the derivative of y = Sin-1x Answer: 11- x2 MCQ5: Suppose u = f(x, y) = x2 + y2, where x = cosh4t and y = 2t + t2. Find the total derivative of u with respect to t Answer: 4sinh8t + 8t + 12t2 + 4t3 MCQ6: If f(u) = Sin u and u = x2+y2  find fx Answer:  Cos U1+x2  MCQ7: If f(u) = Sinu and u = x2+y2  find fy Answer: y Cos Ux2+y2  MCQ8: Partial derivatives are said to be continuous if Answer: MCQ9: Obtain the slope of the tangent at the point (2,3) of the curve 6 x2 + 3xy + x4 + 3y2 = 0 Answer: -65 24 MCQ10: A function f (x, y) of two variables is said to have a local maximum at (a,b) if there exists a rectangular region containing (a,b) such that ____ Answer: f(x, y)≤ f(a, b) MCQ11: The local maxima and minima are called the ____ of (x, y) Answer: extreme MCQ12: To test for critical point if fxxfyy - fxy2< 0 then this gives  Answer: saddle point MCQ13: Obtain the stationary points of f(x, y) = x2+y2 subject to the constraint condition 3x+2y = 6 Answer: 18 13 ,12 13 MCQ14: A function f(x, y) is said to be homogeneous of degree m if Answer: f(kx, ky) = km f(x, y) MCQ15: What is the degree of the function f(x, y) = x3+4xy2- 3y3 Answer: three MCQ16: If x and y are rectangular Cartesian coordinates, u = f(x, y) satisfies laplace’s equation if Answer: ∂2f∂x2  + ∂2f∂y2  = 0 MCQ17: A function f(x, y) is said to have a maximum value of point (x, y) = (a, b) if Answer: f(a+h, b+k)-f(a, b)<0 MCQ18: A function f(x, y) is said to have a minimum value of point (x, y) if Answer: f(a+h, b+k)- f(a, b)>0 MCQ19: If exy+x+y=1, evaluate dy dx  at (0,0) Answer: -1 MCQ20:  If xy + Sin y = 2 find dy dx  Answer: -y x+Cos y MCQ21:  If z= Sin (x+y), x = u2+ v2,  y=2uv. Evaluate dzdu Answer: 2(u+v) Cos(x + u) MCQ22:  With the usual notation a series cannot be convergent unless Answer: limn→∞⁡Un=0 MCQ23:  Let U1+ U2+ .  .  . Un+ .  .  .   be a series of positive terms. If limn→∞⁡Un+1Un>1. Then the series Answer: Diverges MCQ24: As n→∞ of the series 1+12+13+14+ .  .  .  is Answer: divergent MCQ25:  For the series 12+23+34+45+ .   .   . an expression of Un+1 is given by Answer: n+1n+2 MCQ26:  By considering the D’ Alembert test for positive terms if limn→∞⁡Un+1Un=1, then the series is Answer: inconclusive MCQ27: By the comparison test, the series 11P+12P+13P+14P + .  .  . +1nP   ___ if p > 1 Answer: converges MCQ28: Find limn→∞⁡Sin2xx2 Answer: 1 MCQ29: Evaluate limx→0⁡Sinhx-Sinxx3 Answer: 1/3 MCQ30: The Taylor’s series is given by Answer: fx+h= fx+hfıx+h2fıı(x)2!+ .  .  . MCQ31: Find limx→0⁡tan⁡x-xx3  Answer: 1/3 MCQ32: Determine limx→1⁡ x3-2x2+4x-34x2-5x+1 Answer: 1 MCQ33: Find the second order derivatives of the function. fx=x2-cosx at x=π4 Answer: 2+12 MCQ34: Find the third order derivatives of the function. fx=x2-cosx at x=π4  Answer: -12 MCQ35: limx→0⁡tan⁡x-xSin x-x  is Answer: -2 MCQ36: From the Taylor’s expansion of Cos π3+x in ascending powers of x up to the x3 term find fı π3 Answer: -32 MCQ37: From the Taylor’s expansion of Cos π3+x in ascending powers of x up to the x3 term find fıı x Answer: -cosx  MCQ38: From the Taylor’s expansion of Cos π3+x in ascending powers of x up to the x3 term find fıv π3 Answer: ½ MCQ39: From the Taylor’s expansion of Cos π3+x in ascending powers of x up to the x3 term find fıv x Answer: Cos⁡x MCQ40: Suppose fx is a function continuous on a close interval a≤x≤b and differentiable on the open interval a<x<b and if fa= fb= 0, then fıc Answer: 0 MCQ41: From the Maclaurin expansion fx=In(1+x) find fıııx Answer: 21+x3 MCQ42: From the Maclaurin expansion fx=In(1+x) find fıvx Answer: -6(1+x)4 MCQ43: From the Maclaurin expansion fx=In(1+x) find fıı0 Answer: -1 MCQ44: From the Maclaurin expansion fx=In(1+x) find fv0 Answer: 4! MCQ45: Using Simpson’s rule with 6 equally spaced intervals and by considering the integral ∫064+x3dx. Find The number of ordinates Answer: 7 MCQ46: Using Simpson’s rule with 6 equally spaced intervals and by considering the integral ∫064+x3dx. Find ∆x = strip width Answer: 1 MCQ47: Using Simpson’s rule with 6 equally spaced intervals and by considering the integral ∫064+x3dx. Find Area Answer: 22.6square units MCQ48: The two segment trapezoidal rule of integration is exact for integrating at most ____ order of polynomial Answer: first MCQ49: Using trapezoidal rule with five (5) equally spaced intervals and by considering the integral. ∫12  1x dx. Evaluate b-an Answer: 1/5 MCQ50: Using trapezoidal rule with five (5) equally spaced intervals and by considering the integral. ∫12  1x dx, evaluatethe area of the integral Answer: 17532520