B.Sc. Part 1 Mathematics
DIFFERENTIAL CALCULCUS
Ques1: Define limit of a function by giving one example. [Purvanchal 2012; Bundel. 2006; Kanpur B.Sc. 1992; Agra 1992]
Ques2: Define limit of a function. What are infinite limits and limits at infinity ? Make it clear. [Kanpur B.Sc. 2001]
Ques3: A function f on R is defined by
f(x) = { 1, when x is rational
-1, when x is irrational.
Show that f is discontinuous at every point of R. [Agra 1992]
f(x) = { 1, when x is rational
-1, when x is irrational.
Show that f is discontinuous at every point of R. [Agra 1992]
Ques4: Discuss the continuity of f(x) = [x] at x = 0. [Rohil. 2008]
Ques5: Define continuity of a function at a point by giving one example. [Bundel. 2010; Kanpur B.A. 1994, B.Sc. 1992, 2011; Purvanchal 2010, 12]
Ques6: Discuss the continuity of the function f at x = 1/2, where
0, for x = 0
1-x, for 0<x<1/2
f(x) ={ 1/2, for x = 1/2
3-2x, for 1/2 < x < 1
1, for x = 1. [Kashi 2011]
0, for x = 0
1-x, for 0<x<1/2
f(x) ={ 1/2, for x = 1/2
3-2x, for 1/2 < x < 1
1, for x = 1. [Kashi 2011]
Ques7: Describe various kinds of discontinuity by giving an example of each kind. [Purvanchal 2008; Avadh 2014; Kanpur B.Sc. 1994]
Ques8: Obtain points of discontinuity and their types for the function f defined by :
0, for x = 0
1/2 - x, for 0 < x < 1/2
f(x) ={ 1/2, for x = 1/2
3/2 - x, for 1/2 <x< 1
1, for x =1.
Also draw the graph of the function. [Rohil. 2010; Avadh 2010, 13]
0, for x = 0
1/2 - x, for 0 < x < 1/2
f(x) ={ 1/2, for x = 1/2
3/2 - x, for 1/2 <x< 1
1, for x =1.
Also draw the graph of the function. [Rohil. 2010; Avadh 2010, 13]
Ques9: Show that f(x) = x|x| is differentiable at x = 0. [Kanpur B.Sc. 2005]
Ques10: If function f(x) is continuous, but not differentiable at x = 0, then show that xf(x) is differentiable at x = 0. [Kanpur B.Sc. 2006]
Ques11: Prove that the derivative of an even function is always an odd function. [Kanpur B.Sc. 2010]
Ques12: Draw the graph of the function
y = |x - 1| + |x - 2| in the interval [0, 3] and discuss the continuity and differentiability of the function in this interval. [Meerut 2007, 09]
y = |x - 1| + |x - 2| in the interval [0, 3] and discuss the continuity and differentiability of the function in this interval. [Meerut 2007, 09]
Ques13: Let f(x + y) = f(x).f(y), for all x and y. If f'(0) = 1, show that
f'(x) = f(x), for all x. [Kanpur B.Sc. 1989]
f'(x) = f(x), for all x. [Kanpur B.Sc. 1989]
Ques14: Define differentiability of a function at a point by giving suitable examples. Show that the function f(x) = |x| is not differentiable at x = 0. [Kanpur B.Sc. 1990, 92, 94; Avadh 1993, 96; Bundel. 1993; Garhwal B.Sc. 1991]
Ques15: Give an example of a continuous function which is not differentiable. [Kanpur B.Sc. 1994, 2001, 05]
Ques16: Define differentiability and continuity of a function at a point. Give an example of function which is continuous at the origin but is not differentiable. [Purvanchal 2011, 13]
Ques17: Define differentiability of a function. Prove that the function y = sin x differentiable. [Kanpur B.A. 1993]
Ques18: Test for continuity and differentiability at x = 1/2 of the following function : f(x) = 0 for 0<x<1/2, f(x) = 1 for x = 1/2, f(x) = 3 for 1/2<x. [Kanpur B.Sc. 1993]
Ques19: Sketch the function y = |x - 2| in the interval (-1, 3). Is this function (a) continuous (b) differentiable at x = 2 ? [Roorkee 1984]
Ques20: Show that f(x) = |x - 1|, 0 < x < 2, is not derivable at x = 1 but continuous. [Kanpur B.Sc. 1993, 94]
Ques21: Show that the function f(x) = |x| + |x - 1| is continuous but not differentiable at x = 0 and x = 1. [Bundel. 2013; Kashi 2010; Meerut 2005, 08]
Ques22: Show that the function f defined by
f(x) = |x-1| + 2 |x-2| + 3 |x-3|
is continuous but not differentiable at the points 1, 2 and 3. [Bundel. 2009; Purvanchal 2008]
Ques23: Discuss the applicability of Rolle's theorem to the following functions :
f(x) = |x| in [-1, 1]. [Kanpur B.Sc. 1994]
Ques24: Verify Rolle's theorem for the following function :
f(x) = (x-1)(x-4) in [1, 4]. [Kanpur B.A. 1992]
Ques25: Find ' c ' of Lagrange's M.V.T. if
f(x) = x(x-1)(x-2); a = 0, b = 1/2. [Kanpur B.Sc. 1995]
Ques26: Apply Lagrange's M.V.T. to show that
x/1+x < log (1+x) < x, x > 0. [Bundel. 1991, 2011]
Ques27: Find c in Cauchy's M.V.T. if
f(x) = x(x-1)(x-2), g(x) = x(x - 2)(x -3) defined in [0, 1/2]. State the conditions clearly. [Rohil. 2013]
Ques28: If in Cauchy's mean value theorem, we write
f(x) = 1/√x and g(x) = √x, then c is the geometric mean between a and b. [Rohil. 2014]
Ques29: Find 'c' of the mean value theorem when :
1) f(x) = log x in [1, e]. [Kanpur B.Sc. 2003]
2) f(x) = (x-1)(x-2)(x-3) in [0, 4]. [Rohil. 2014; Kanpur B.Sc. 1994]
Ques30: State Lagrange's form of Mean Value Theorem, and examine if it holds for the function f(x) = |x| in the interval [-1, 1]. [Kanpur B.Sc. 2010]
Ques31: Check the hypothesis and applicability of Cauchy's M.V.T. for the function :
f(x) = sin x and g(x) = cos x in [- π / 2, 0 ]. [Meerut 2013]
Ques32: Assuming that sin x can be expanded in a series in ascending powers of x, obtain the expansion giving the n th term. [Kanpur B.Sc. 2000; Avadh 1995]
Ques33: Expand the following by Maclaurin's theorem :
1) tan x. [Kanpur B.Sc. 2014; Agra 1981; Avadh 1995]
2) log sec x. [Bundel. 1993, 2011, 13; Agra 2009, 13; Purvanchal 1990; Avadh 1996, 2002; Rohil. 1991, 2013]
3) log (1+ sin x). [Kanpur B.A. 1993; Meerut 2007; Avadh 1997]
4) cos x. [Kanpur B.Sc. 2006; Avadh 1995]
5) sec x. [Bundel. 1990]
6) log (1+x). [Agra 2005, 10; Meerut 2003,11]
Ques34: Expand the following function in ascending powers of x :
log (1 + tan x). [Kanpur B.Sc. 2003]
Ques35: Expand sin x in powers of x - 1\2 π. [Kanpur B.Sc. 1998, B.A. 1994; Bundel. 1993; Lucknow 2005; Purvanchal 1991; Kashi 2010, 12; Avadh 2008; Agra 2006; Meerut 2005; Rohil. 2006, 10]
Ques36: Expand the following in the powers of the quantity indicated.
1) cos x in powers of x - 1/4 π. [Kanpur B.Sc. 2002; Avadh 2014]
2) sin x in powers of x - 1/4 π. [Avadh 1993]
Ques37: Expand log sin x in powers of x - a. [Meerut 2001, 06, 13; Kanpur B.Sc. 1995]
Ques38: Expand log sin x in powers of x - 2. [Kanpur B.Sc.1992; Avadh 2007; Purvanchal 2013; Garhwal 2000; Kamuan 2001; Meerut 1990, 91]
Ques39: Find the first three terms in the expansion of log sec x in ascending powers of x. [Purvanchal 1990; Rohil. 1991]
Ques40: If x = u(1+v) and y = v(1+u), find the Jacobian of x, y with respect to u, v. [Meerut 2013; Kanpur B.Sc. 1994; Avadh 2002]
Ques41: If x + y + z = u, y + z = uv, z = uvw, find the value of the Jacobian of x, y, z with regard to u, v, w. [Kanpur B.Sc. 1990, 99; Rohil. 2005]
Ques42: Verify the chain rule for the Jacobians if
x = u, y = u tan v, z = w. [Kanpur B.Sc. 2014]
Ques43: Show that the functions
X = x + 3y + 2z, Y = 3x + 4y - 2z, Z = 11x + 18y - 2z are not independent and find a relation between them. [Kanpur B.Sc. 1992, B.A. 1991]
Ques44: If u = x + 3y + 2z, v = 3x + 4y - 2z and w = 7x + 11y - 2z, show that u, v, w are connected by a functional relation and find it. [Kanpur B.Sc. 1991]
Ques45: Find the maximum value of u, where
u = sin x sin y sin (x+y). [Kanpur B.A. 1992, 93, B.Sc. 1992; Avadh 1999; Agra 2002; Rohil. 2013]
Ques46: Discuss maxima / minima of the function
u = sin x + sin y + sin (x + y). [Avadh 2000; Purvanchal 2009, 11]
Ques47: Find a point within a triangle such that the sum of the squares of its distances from the three vertices is a minimum. [Kanpur B.Sc. 2010; Kumaun 2008]
Ques48: A rectangular box, open at the top, is to have a given capacity. Find the dimensions of the box so that the material required for the construction of the box be minimum. [Kanpur B.Sc. 2014]
Ques49: Find the conditions of optimality of a function of two variables. [Kanpur B.Sc. 1992]
Ques50: Find the maximum values of the following function:
u = xy(1-x-y). [Kanpur B.Sc. 2003]
Ques51: Find the equation of the tangent at the point t to the cycloid:
x = a(t + sin t), y = a(1 - cos t). [Agra 2014]
Ques52: Find ds/dx for the following curve:
y = a log sec (x/a). [Bundel. 2013]
Ques53: Find the radius of curvature at any point (x,y) of the curve :
y = c log sec (x/c). [Purvanchal 2009; Kanpur B.Sc. 2007; Avadh 1997]
Ques54: Prove that the radius of curvature at any point 't' of the cycloid x = a(t + sin t), y = a(1 - cos t) is 4a cos 1/2 t. [Kanpur B.Sc. 2000, B.A. 1994, 97; Bundel. 2012, 14; Avadh 1990; Kumaun 2002; Garhwal 2003; Agra 2011]
Ques55: Find the radius of curvature at any point 't' of the curve
x = a(cos t + log tan 1/2 t), y = a sin t. [Kanpur B.Sc. 1991,2013]
Ques56: Find the radius of curvature of the following curve:
y = 4 sin x - sin 2x at the point where x = π / 2. [Purvanchal 2012]
Ques57: Find the radius of curvature at the vertex of the cycloid:
x = a(t + sin t), y = a(1 - cos t). [Bundel. 2007]
Ques58: In the curve y = a log sec (x/a), prove that the chord of curvature parallel to the axis of y is of constant length. [Rohil. 2009; Avadh 1995, 99, 2000]
Ques59: Show that the chord of curvature through the pole for the curve
p = f(r) is 2f(r)/f ' (r). [Avadh 2001]
Ques60: Show that the chord of curvature through the focus of a parabola is four times the focal distance of the point and the chord of curvature parallel to the axis (x-axis) has the same length. [Avadh 1994]
Ques61: Find the envelope for the following curve:
y = mx + a/m, the parameter being m. [Kanpur B.Sc. 2005; Avadh 1991, 97]
Ques62: Find the envelope of the straight line
x/a + y/b = 1, where the parameters a and b are connected by the following relation in which c is a constant :
a + b = c. [Garhwal 2001, 03; Avadh 2002]
Ques63: Find the envelope of straight lines of given length which slides with its extremities on two fixed straight lines at right angles. [Purvanchal 2009]
Ques64: A straight line of given length slides with its extremities on two fixed straight lines at right angles. Find the envelope of the circles drawn on the sliding line as diameter. [Purvanchal 2008]
Ques65: Prove that the evolute of the tractix
x = a ( cos t + log tan 1/2t ), y = a sin t, is the catenary y = a cosh (x/a). [Kanpur 1994; Avadh 1991, 95, 2000]
Ques66: Show that the evolute of an equiangular spiral is an equal equiangular spiral. [Avadh 1994]
Ques67: Find asymptotes of the curve y = 2/x-3. [Kanpur B.Sc. 2005]
Ques68: Examine y = cos x for concavity and convexity in the range (0, 2π). [Kanpur B.Sc. 2014]
Ques69: Find the points of inflexion of the curve x = log (y/x). [Rohil.2014; Purvanchal 2011]
Ques70: Trace the curve :
x = (y-1)(y-2)(y-3). [Kanpur B.Sc. 2009; Avadh 1997; Kashi 2010]
Ques22: Show that the function f defined by
f(x) = |x-1| + 2 |x-2| + 3 |x-3|
is continuous but not differentiable at the points 1, 2 and 3. [Bundel. 2009; Purvanchal 2008]
Ques23: Discuss the applicability of Rolle's theorem to the following functions :
f(x) = |x| in [-1, 1]. [Kanpur B.Sc. 1994]
Ques24: Verify Rolle's theorem for the following function :
f(x) = (x-1)(x-4) in [1, 4]. [Kanpur B.A. 1992]
Ques25: Find ' c ' of Lagrange's M.V.T. if
f(x) = x(x-1)(x-2); a = 0, b = 1/2. [Kanpur B.Sc. 1995]
Ques26: Apply Lagrange's M.V.T. to show that
x/1+x < log (1+x) < x, x > 0. [Bundel. 1991, 2011]
Ques27: Find c in Cauchy's M.V.T. if
f(x) = x(x-1)(x-2), g(x) = x(x - 2)(x -3) defined in [0, 1/2]. State the conditions clearly. [Rohil. 2013]
Ques28: If in Cauchy's mean value theorem, we write
f(x) = 1/√x and g(x) = √x, then c is the geometric mean between a and b. [Rohil. 2014]
Ques29: Find 'c' of the mean value theorem when :
1) f(x) = log x in [1, e]. [Kanpur B.Sc. 2003]
2) f(x) = (x-1)(x-2)(x-3) in [0, 4]. [Rohil. 2014; Kanpur B.Sc. 1994]
Ques30: State Lagrange's form of Mean Value Theorem, and examine if it holds for the function f(x) = |x| in the interval [-1, 1]. [Kanpur B.Sc. 2010]
Ques31: Check the hypothesis and applicability of Cauchy's M.V.T. for the function :
f(x) = sin x and g(x) = cos x in [- π / 2, 0 ]. [Meerut 2013]
Ques32: Assuming that sin x can be expanded in a series in ascending powers of x, obtain the expansion giving the n th term. [Kanpur B.Sc. 2000; Avadh 1995]
Ques33: Expand the following by Maclaurin's theorem :
1) tan x. [Kanpur B.Sc. 2014; Agra 1981; Avadh 1995]
2) log sec x. [Bundel. 1993, 2011, 13; Agra 2009, 13; Purvanchal 1990; Avadh 1996, 2002; Rohil. 1991, 2013]
3) log (1+ sin x). [Kanpur B.A. 1993; Meerut 2007; Avadh 1997]
4) cos x. [Kanpur B.Sc. 2006; Avadh 1995]
5) sec x. [Bundel. 1990]
6) log (1+x). [Agra 2005, 10; Meerut 2003,11]
Ques34: Expand the following function in ascending powers of x :
log (1 + tan x). [Kanpur B.Sc. 2003]
Ques35: Expand sin x in powers of x - 1\2 π. [Kanpur B.Sc. 1998, B.A. 1994; Bundel. 1993; Lucknow 2005; Purvanchal 1991; Kashi 2010, 12; Avadh 2008; Agra 2006; Meerut 2005; Rohil. 2006, 10]
Ques36: Expand the following in the powers of the quantity indicated.
1) cos x in powers of x - 1/4 π. [Kanpur B.Sc. 2002; Avadh 2014]
2) sin x in powers of x - 1/4 π. [Avadh 1993]
Ques37: Expand log sin x in powers of x - a. [Meerut 2001, 06, 13; Kanpur B.Sc. 1995]
Ques38: Expand log sin x in powers of x - 2. [Kanpur B.Sc.1992; Avadh 2007; Purvanchal 2013; Garhwal 2000; Kamuan 2001; Meerut 1990, 91]
Ques39: Find the first three terms in the expansion of log sec x in ascending powers of x. [Purvanchal 1990; Rohil. 1991]
Ques40: If x = u(1+v) and y = v(1+u), find the Jacobian of x, y with respect to u, v. [Meerut 2013; Kanpur B.Sc. 1994; Avadh 2002]
Ques41: If x + y + z = u, y + z = uv, z = uvw, find the value of the Jacobian of x, y, z with regard to u, v, w. [Kanpur B.Sc. 1990, 99; Rohil. 2005]
Ques42: Verify the chain rule for the Jacobians if
x = u, y = u tan v, z = w. [Kanpur B.Sc. 2014]
Ques43: Show that the functions
X = x + 3y + 2z, Y = 3x + 4y - 2z, Z = 11x + 18y - 2z are not independent and find a relation between them. [Kanpur B.Sc. 1992, B.A. 1991]
Ques44: If u = x + 3y + 2z, v = 3x + 4y - 2z and w = 7x + 11y - 2z, show that u, v, w are connected by a functional relation and find it. [Kanpur B.Sc. 1991]
Ques45: Find the maximum value of u, where
u = sin x sin y sin (x+y). [Kanpur B.A. 1992, 93, B.Sc. 1992; Avadh 1999; Agra 2002; Rohil. 2013]
Ques46: Discuss maxima / minima of the function
u = sin x + sin y + sin (x + y). [Avadh 2000; Purvanchal 2009, 11]
Ques47: Find a point within a triangle such that the sum of the squares of its distances from the three vertices is a minimum. [Kanpur B.Sc. 2010; Kumaun 2008]
Ques48: A rectangular box, open at the top, is to have a given capacity. Find the dimensions of the box so that the material required for the construction of the box be minimum. [Kanpur B.Sc. 2014]
Ques49: Find the conditions of optimality of a function of two variables. [Kanpur B.Sc. 1992]
Ques50: Find the maximum values of the following function:
u = xy(1-x-y). [Kanpur B.Sc. 2003]
Ques51: Find the equation of the tangent at the point t to the cycloid:
x = a(t + sin t), y = a(1 - cos t). [Agra 2014]
Ques52: Find ds/dx for the following curve:
y = a log sec (x/a). [Bundel. 2013]
Ques53: Find the radius of curvature at any point (x,y) of the curve :
y = c log sec (x/c). [Purvanchal 2009; Kanpur B.Sc. 2007; Avadh 1997]
Ques54: Prove that the radius of curvature at any point 't' of the cycloid x = a(t + sin t), y = a(1 - cos t) is 4a cos 1/2 t. [Kanpur B.Sc. 2000, B.A. 1994, 97; Bundel. 2012, 14; Avadh 1990; Kumaun 2002; Garhwal 2003; Agra 2011]
Ques55: Find the radius of curvature at any point 't' of the curve
x = a(cos t + log tan 1/2 t), y = a sin t. [Kanpur B.Sc. 1991,2013]
Ques56: Find the radius of curvature of the following curve:
y = 4 sin x - sin 2x at the point where x = π / 2. [Purvanchal 2012]
Ques57: Find the radius of curvature at the vertex of the cycloid:
x = a(t + sin t), y = a(1 - cos t). [Bundel. 2007]
Ques58: In the curve y = a log sec (x/a), prove that the chord of curvature parallel to the axis of y is of constant length. [Rohil. 2009; Avadh 1995, 99, 2000]
Ques59: Show that the chord of curvature through the pole for the curve
p = f(r) is 2f(r)/f ' (r). [Avadh 2001]
Ques60: Show that the chord of curvature through the focus of a parabola is four times the focal distance of the point and the chord of curvature parallel to the axis (x-axis) has the same length. [Avadh 1994]
Ques61: Find the envelope for the following curve:
y = mx + a/m, the parameter being m. [Kanpur B.Sc. 2005; Avadh 1991, 97]
Ques62: Find the envelope of the straight line
x/a + y/b = 1, where the parameters a and b are connected by the following relation in which c is a constant :
a + b = c. [Garhwal 2001, 03; Avadh 2002]
Ques63: Find the envelope of straight lines of given length which slides with its extremities on two fixed straight lines at right angles. [Purvanchal 2009]
Ques64: A straight line of given length slides with its extremities on two fixed straight lines at right angles. Find the envelope of the circles drawn on the sliding line as diameter. [Purvanchal 2008]
Ques65: Prove that the evolute of the tractix
x = a ( cos t + log tan 1/2t ), y = a sin t, is the catenary y = a cosh (x/a). [Kanpur 1994; Avadh 1991, 95, 2000]
Ques66: Show that the evolute of an equiangular spiral is an equal equiangular spiral. [Avadh 1994]
Ques67: Find asymptotes of the curve y = 2/x-3. [Kanpur B.Sc. 2005]
Ques68: Examine y = cos x for concavity and convexity in the range (0, 2π). [Kanpur B.Sc. 2014]
Ques69: Find the points of inflexion of the curve x = log (y/x). [Rohil.2014; Purvanchal 2011]
Ques70: Trace the curve :
x = (y-1)(y-2)(y-3). [Kanpur B.Sc. 2009; Avadh 1997; Kashi 2010]
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