Limit, Continuity and Differentiability — practice questions

385 curated practice questions from Limit, Continuity and Differentiability (Mathematics JEE Main, NTA Exams) — 132 hard, 206 easy, 47 medium. Every question includes the final answer free; step-by-step worked solutions with a free account.

• If y= _ u(x) ^ v(x) f(t) d t let us define d y d x in a different manner as d y d x =v^ (x) f^ 2 (v(x))-u^ (x) f^ 2…
• _ x 0 e^ 1 x -1 e^ 1 x +1 =
• Let y(x) be a function of x satisfying the relation (x+y)=2 x y , then y^ (x) at x=0 is equal to
• If φ (x) = f (x) + f (2a – x) and f ’’ (x) > 0, a > 0, 0 < x < 2a then
• Let [t] denote the greatest integer ≤ t and _ x 0 x [ 4 x ]=A . Then the function, f(x)= [x^ 2 ] ( x) is discontinuous…
• The number of points with integral coordinates where tangent exists in the curve y = sin –1 2x 1-x^ 2 is
• The least value of the function (x)= _ 5 / 4 ^ x (35 i n t+4 t) d t in the interval [ 5 4 , 4 3 ] is ...
• If a^ 2 +b^ 2 +c^ 2 =1 where a, b, c ∈ R, the the maximum value of (4 a-3 b)^ 2 +(5 b-4 c)^ 2 +(3 c-5 a)^ 2 is
• If a_ n n+1 + a_ 1 n + a_ 2 n-1 + + a_ n-1 2 +a_ n =0 where a_ n , a_ 1 , a_ 2 , a_ n-1 , a_ n R , then the equation a_…
• Given below are two statements. STATEMENT -1: If f(0)=0, f^ (x) = (x+ 1+x^ 2 ) . then f (x) is positive for all x ∈ R …
• The length of subtangent to the curve x 2 y 2 = a 4 at the point (–a, a) is
• Find the value of a for which the function f (x) = x 2 - 2ax + 6 is strictly increasing when x > 0.
• The angle between the tangents to the curve y 2 = 2ax at the points where x = a 2 , is
• If a particle moving along a line follows the law s= 1+t , then the acceleration is proportional to
• If [.] denotes greatest integer function less than or equal to x, then the value of _ x 1 1- x +[ x -1]+[1- x ] is ....
• Evaluate, _ x / 4 x- x ( 4 -x )( x+ x) .
• The two curves y 2 = 4x and x 2 + y 2 – 6x + 1 = 0 at the point (1, 2)
• The maximum value of x 3 – 3x in the intveral [0, 2], is
• Consider the function f (x) = max x 2 , (1 – x) 2 , 2x (1 – x) where 0 ≤ x ≤ 1 . Let RMVT is applicable for f (x) on…
• The point in the interval [0, 2π], where f (x) = e x sin x has maximum slope, is
• The function f( x )= x x is decreasing in the interval
• The value of _ x 0 [ a x - x a ] is ...
• Let f and g be differential functions on R such that fog is the identity function. If for some a,b R, g'(a) = 5 and…
• The value of _ x 2 z^ x +z^ 3-x -6 z^ -x -z^ 1-x is :
• _ x b x-a - b-a x^ 2 -b^ 2
• _ x 0 x-x+ x^ 3 6 x^ 5 =
• If f^ (x)=g(x) and g^ (x)=-f(x) V x and f(2)=4=f^ (2) , then f^ 2 (19)+g^ 2 (19) is
• Differentiate the function y= [x^ 2 +1 ] 2 x w. r. t. x.
• Suppose P ( x )= a _ a + a _ 1 x + a _ 2 x ^ 2 + .+ a _ n x ^ n and |P(x)| |e^ x-1 -1 | x 0 Then, | a _ 1 +2 a _ 2 +3 a…
• Differentiate w.r.t. x : z^ x +x^ 2 +z x+ _ 2 x
• Let f(x)=2 ^ 3 x-3 ^ 2 x+12 x+5,0 x 2 . Then, f (x) is
• The point on x^ 2 =2 y . which is closest to the point (0, 5), will be
• If y= [x+ x^ 2 -1 ]^ 15 + [x- x^ 2 -1 ]^ 15 , then (x^ 2 -1 ) d^ 2 y d x^ 2 +x d y d x is equal to:
• The value of _ x 0 [ x^ 2 x x ] . where [.] denotes greatest integer function.
• The length of the perpendicular from the origin to the normal of curve x = a (cosθ + θ sin θ), y = a (sin θ – θ cos θ)…
• Assertion : The minimum distance of the fixed point (0, y 0 ), where 0 y _ 0 1 2 , from the curve y = x 2 is y 0 …
• Let f (x) = _ 1 ^ x e ^ x (x – 1) (x – 2) dx. Then, f decreases in the interval
• Tangent of acute angle between the curves y = |x 2 –1| and y= 7-x^ 2 at their points of intersection is
• Given below are two statements. One is labelled as Assertion and the other is labelled as Reason. STATEMENT - 1…
• Area of the greatest rectangle that can be inscribed in the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1 is
• The value of p and q for which the function f(x)= array ll (p+1) x+ x x & , x 0 array . is continuous for all x in R, is
• _ x 1 ( 1 x^ 1-x ) equals
• The largest area of a rectangle which has one side on the x-axis and the two vertices on the curve y = e^ -x^ 2 is
• If f(x)= array l x, when x is rational 1-x, when x is irrational, then array .
• The difference between the greatest and least values of the function, f (x) = cos x + 1 2 cos 2x – 1 3 cos 3x is :
• The function f(x)= array cc 1 4^ x -1 ; & x 0 0 & x=0 array . is continuous
• .|f f( x )= a | s i n x |+b e ^ | x | +c | x |^ 3 and if f (x) is differentiable at x = 0, then
• The equation of the tangent to the curve (1 + x 2 ) y = 2 –x, where it crosses the x-axis, is
• The value of _ x 3 ( _ a x-3 x+6 -3 ) is
• On the curve x^ 3 =12 y , find the interval at which the abscissa changes at a faster rate than the ordinate.
• Let y = a x + bx be curve, (2x – y) + λ (2x + y – 4) = 0 be family of lines. If curve has slope - 1 2 at (9, 0) then a…
• If y=f(x) is a curve and if there exists two points A (x_ 1 , f (x_ 1 ) ) and B (x_ 2 , f (x_ 2 ) ) on it such that f^…
• The set of points where the function f (x) = x |x| is differentiable is
• If f(x)= | array ccc x & a & b x & a & b x & a & b array | ,where 0 then the equation f ´ (x) = 0 has, in the interval…
• A function y=f(x) has a second order derivative f^ (x)=6(x-1) . If its graph passes through the point (2,1) and at that…
• Let f:[0,2] R be a twice differentiable function such that f^ (x)>0 , for all x (0,2) . If (x)=f(x)+f(2-x) , then ϕ is:
• If A > 0, B > 0 and A + B = 3 , then the maximum value of tan A tan B is
• The shortest distance between the point ( 3 2 , 0 ) and the curve y= x ,(x>0) , is-
• The angle between the tangents at those points on the curve y = (x + 1) (x – 3) where it meets x-axis, is
• A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm…
• If y = A cos nx + B sin nx, then d ^ 2 y dx ^ 2 =
• If the line ax + by + c = 0 is a tangent to the curve xy = 4, then the possible answer is
• Let II and be the distinct roots of a ^ 2 + b + c = 0 , then _ x 1- [a x^ 2 +b x+c ] (x-a)^ 2 is equal to :
• Limit _ n (n+2)!+(n+1)! (n+3)! , n ∈ N =
• The position of a point in time ‘t’ is given by x = a + bt–ct 2 , y = at + bt 2 . Its acceleration at time ‘t’ is
• The function f (x) = maximum x(2-x) , 2-x is non-differentiable at x equal to :
• The least area of a circle circumscribing any right triangle of area S is :
• The length of the perpendicular from the origin to the normal of curve x = a (cos θ + θ sin θ), y = a (sin θ – θ cos θ)…
• Evaluate _ x 1 1- 2 x 2 x .
• The equation e x - x - 1 = 0 has, apart from x = 0
• Let f (x) = [ array ll x^ 3 -x^ 2 +10 x-5, & x 1 -2 x+ _ 2 (b^ 2 -2 ), & x>1 array . the set of values of b for which f…
• The equation of the tangent to the curve y= 9-2 x^ 2 at the point where the ordinate and the abscissa are equal, is
• If a particle is moving such that the velocity acquired is proportional to the square root of the distance covered…
• Let f (x) = x m/n for x ∈ R where m and n are integers, m even and n odd and 0 < m < n. Then
• If f(x)= array cc 3^ x , & -1 x 1 4-x, & 1<x 4 array , . then f (x) is
• Consider the function f (x) = max x 2 , (1 – x) 2 , 2x (1 – x) where 0 ≤ x ≤ 1 . The interval in which f (x) is…
• Evaluate _ x (x+2) ^ -1 (x+2)- [x ^ -1 x ]
• The tangent to the curve 5x 2 + y 2 = 1 at ( 1 3 ,- 2 3 ) passes through the point
• The function f (x) = 2x 3 – 3x 2 – 12x + 4 has
• _ 4 ( - ) (1- 2 ) is
• If m is the minimum value of k for which the function f(x)=x k x-x^ 2 is increasing in the interval [0, 3] and M is the…
• If y = x n , then the ratio of relative errors in y and x is
• If y= [ ^ -1 1 1+x+x^ 2 + ^ -1 1 x^ 2 +3 x+3 + . .+ ^ -1 1 x^ 2 +5 x+7 + . upton terms ] then y ^ (0) equals
• If y= [x+ 1+x^ 2 ]^ n , then [1+x^ 2 ] d^ 2 y d x^ 2 +x d y d x is
• The function f(x)= ^ -1 ( x+ x) is an increasing function in :
• Let f(x)= array ll -1, & -2 x<0 x^ 2 -1, & 0 x 2 array . and g(x) = |f(x)| + f(|x|). Then, in the interval (-2, 2), g…
• On which of the following intervals are the function x 100 + sin x - 1 decreasing?
• Let f (2) = 4 and f’ (2) = 4. Then, _ x 2 x f(2)-2 f(x) x-2 is given by
• If y = cos –1 ( 2 x-3 x 13 ) ; 0<x< 2 , then dy dx is
• Differentiate a ^ x x x w.r.t. x.
• Differentiate ^ -1 ( 1+ x ^ 2 + 1- x ^ 2 1+ x ^ 2 - 1- x ^ 2 ) w.f.t. x .
• The point on the curve xy 2 = 1, which is nearest to the origin is
• The set of points where the function f (x) = [x] + |1 – x|, –1 ≤ x ≤ 3 where [.] denotes the greatest integer function…
• Let the parabolas y = x (c – x) and y = – x 2 – ax + b touch each other at the point (1, 0), then
• The function f (x) = 2x 3 – 15x 2 + 36x + 4 has local maxima at
• If x = 1-t^ 2 1+t^ 2 and y = 2 t 1+ t ^ 2 , then dy dx is equal to
• _ x 4 x (4 x) ^ 2 x ^ 2 (2 x) is equal to:
• For a real number y, let [y] denotes the greatest integer less than or equal to y. Then, the function f(x)= ( [x- ])…
• If y= _ u(x) ^ v(x) f(t) d t let us define d y d x in a different manner as d y d x =v^ (x) f^ 2 (v(x))-u^ (x) f^ 2…
• The angle of intersection of curves, y = [|sin x| + |cos x|] and x 2 + y 2 = 5, where [.] denotes greatest integral…
• If a, b, c are real, then f(x)= | array ccc x+a^ 2 & a b & a c a b & x+b^ 2 & b c a c & b c & x+c^ 2 array | is…
• If y = 2 x . 3 2x–1 , then dy dx is equal to
• For the curve x m + n = a m – n y 2n , where a is a positive constant and m, n are positive integers
• The normal to the curve x=a( + ) , y=a( - ) at any point 'θ' is such that
• If x is very large, the value of (x^ 2 +16 ) - (x^ 2 +9 ) is
• Derivative of x 6 + 6 x with respect to x is
• Differentiate ^ -1 1+x^ 2 -1 x w.r.t. ^ -1 x .
• Let f be a differentiable function with f (2) = 3 and f ′(2) = 5, and let g be the function defined by g(x) = x f (x)…
• _ x 0 2^ x -1 (1+x)^ 1 2 -1 =
• The value of _ x 0 e^ x + (1+x)-(1-x)^ -2 x^ 2 is equal to
• Assertion : The equation 3x 2 + 4ax + b = 0 has at least one root in (0, 1), if 3 + 4a = 0 . Reason : f (x) = 3x 2 +…
• For |x| 2 + ... to ∞, then dy dx equal to
• _ n 1 2 + 1 2^ 2 + 1 2^ 3 + + 1 2^ n equals
• The value of _ x x( x)^ 3 1+x+x^ 2 is .....
• The sum of tangent and sub-tangent at any point of the curve y = a log (x 2 – a 2 ) varies as
• The derivative of ^ -1 ( x- x x+ x ) , with respect to x 2 , where x (0, 2 ) is:
• If f(x)= array cc [x] [x] , & [x] 0 0, & ,[x]=0 array . , where [.] denotes the greatest integer function, then _ x 0…
• Limit _ x 0^ + (1+ ^ 2 x )^ 5 / x =
• Let f (x) = array cl |x^ 3 +x^ 2 +3 x+ x |(3+ 1 / x), & x 0 0 & x=0 array . then number of points (where f (x) attains…
• Assertion : Among all the rectangles of given perimeter, the square has the largest area. Also among all the rectangles…
• If f ( 9) = 9 and f '(9) = 4 then _ x 9 f(x) -3 x -3 is equal to
• _ x 1^ + x^ 2 -1 + x-1 x^ 2 -1
• If the function f(x)=2 x^ 2 -k x+5 is increasing on [1, 2], then k lies in the interval.
• Let f(x)= 1- x 4 x- , x 4 , x [0, 2 ] . If f(x) is continuous in [0, 2 ] , then f ( 4 ) is
• If _ x (1+ a x + b x^ 2 )^ 2 x =e^ 2 then the values of a and b are
• _ x 0 e c x x is equal to
• _ x 1 2 x^ 3 -1 ^ 6 2 x is equal to
• The function f (x) = x 2 (x –2) 2
• The function f(x)= ^ -1 ( x+ x) is a/an :
• If f (x) = x 1+ x x , x (0, 2 ), then
• The left hand derivative of f(x)=[x] ( x) at x=k, k is an integer is
• _ x 3 x^ 2 -3 x^ 2 +3 3 x-12
• If f ( x )= array cc x (3 e ^ 1 / x +4 ) 2- e ^ 1 / x ; & x 0 0 ; & x =0 array . then
• The interval in which the function x 3 increases less rapidly than 6x 2 + 15x + 5 is :
• If the polynomial equation a n x n + a n–1 x n–1 + .... + a 2 x 2 + a 1 x + a 0 = 0, n positive integer, has two…
• If y= x+e^ x then d^ 2 x d y^ 2 is equal to
• Let p= _ x 1^ + (1+ ^ 2 x )^ 1 2 x then log p is equal to:
• Assertion : If g (x) is a differentiable function g (1) ≠ 0, g (–1) ≠ 0 and Rolles theorem is not applicable to f(x)=…
• The lines y = - 3 2 x and y=- 2 5 x intersect the curve 3x 2 + 4xy + 5y 2 – 4 = 0 at the points P and Q respectively…
• An extremum of the function, f(x)= 2-x cos π (x +3) + 1 ^ 2 sin π (x + 3) 0 < x < 4 occurs at :
• A tangent to the curve y = 1 – x 2 is drawn so that the abscissa x 0 of the point of tangency belongs to the interval…
• The function f(x)= (1+a x)- (1-b x) x is not defined at x=0 . The value which should be assigned to f, so that it is…
• Two tangents to the graph of function f(x)= 17 [1+x^ 2 ] intersect at right angles at a certain point on the y-axis…
• Assertion : Let f : [0, ∞)→[0, ∞) and g : [0, ∞)→[0, ∞) be non-increasing and non-decreasing functions respectively and…
• Consider a function f( x )= ( - 1 - x ) (4 – 3x 2 ) where ‘α’ is a positive parameter Least possible value of the…
• The curve y – e xy + x = 0 has a vertical tangent at
• The maximum value of the function y = x (x –1) 2 , 0 ≤ x ≤ 2 is
• Let f (x) = array lll -x^ 2 , & for & x y = f (x) is
• On the interval [0,1] , the function x^ 25 (1-x)^ 75 takes its maximum value at the point
• If x y = e x–y , then dy dx is equal to
• The diagram shows the graph of the derivative of a function f (x) for 0 f (0) = 0 . Which of the following could be…
• The value of _ n ( x 2 ) ( x 4 ) ( x 8 ) ( x 2^ n ) is
• The minimum value of x x is attained (where x is positive real number) when x is equal to :
• If x = 3 tan t and y = 3 sec t, then the value of d^ 2 y d x^ 2 at t= 4 , is:
• Differentiate x x- x x x- x w.r.t x.
• The greatest and the least value of the function, f (x) = 1-2 x + x ^ 2 - 1+2 x + x ^ 2 , x ∈ (−∞, ∞) are
• Given below are two statements. STATEMENT - 1: f(x)= x^ 3 3 + a x^ 2 2 +x+5 has positive point of maxima for a…
• If f(x)= _ x ^ x^ 2 1 ( t)^ 2 d t, x 0, x 1 then f (x) is monotonically
• x 0 Limit |x|^ x =
• The derivative of e^ x^ 3 with respect to l og x is
• If f (x) = 2 tan –1 x + cos –1 ( 1-x^ 2 1+x^ 2 ), then
• If y=x+ 1 x+ 1 x+ 1 x+ , then find dy dx
• If f_ n (x)=e^ f_ n-1 |x| for all n ∈ N and f_ n (x)=x , then d d x f_ n (x) is equal to:
• The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius…
• Let f: R R be a function defined by f( x )= Min 1+ x _ 1 | x |+1 . Then which of the following is true ?
• _ x 0 x^ 2 +1 -1 x^ 2 +9 -3 is equal to
• Let f ´ (sin x) f ´´ (sin x) > 0 x (0, 2 ) Now consider a function g (x) = f (sin x) + f (cos x) The set of critical…
• For x>1 , if (2 x)^ 2 y =4 e^ 2 x-2 y , then (1+ _ e 2 x )^ 2 d y d x is equal to:
• The triangle formed by the tangent to the parabola y = x 2 at the point with abscissa x 1 , the y-axis and the straight…
• The value of parameter a so that the line (3 – a) x + ay + (a 2 – 1) = 0 is normal to the curve xy = 1, may lie in the…
• Let [.] denotes the greatest integer function and f( x )= [ ^ 2 x ] , then
• If y=f(x) is a curve and if there exists two points A (x_ 1 , f (x_ 1 ) ) and B (x_ 2 , f (x_ 2 ) ) on it such that f^…
• If f (x) = x 3 – x 2 + 100x + 1001, then
• If f(x) is differentiable in the interval [2, 5] where f(2)= 1 5 and f(5)= 1 2 , then there exist a number c, 2 < c < 5…
• If f(1) = 1, f'(1) = 3, then the derivative of f(f(f(x))) + (f(x)) 2 at x = 1 is
• Let f(a)=g(a)=k and their n th derivatives f n (a), g n (a) exist and are not equal for some n. Further, if _ x a f(a)…
• _ h 0 2 [ 3 ( 6 +h )- ( 6 +h ) ] 3 h( 3 )
• The function f(x)= (1+x)- 2 x 2+x is
• A function f such that f ´(a) = f ´´ (a) = ... f 2n (a) = 0 and f has a local maximum value b at x = a, if f (x) is
• If f (x) = l og |x|, x ≠ 0 then f ′(x) equals
• Assertion : Let f (x) = 5 – 4 (x – 2) 2/3 , then at x = 2 the function f (x) attains neither least value nor greatest…
• Let f: R R be such that f(1)=3 and f^ (1)=6 . Then _ x 0 ( f(1+x) f(1) )^ 1 / x equals
• If f(x)= array c 1- 4 x x^ 2 , x 0 array . then at x = 0
• The value of _ x 0 ( (3 x+a)-3 (2 x+a)+3 (x+a)- a x^ 3 ) :
• If y=e^ ^ -1 x and u = l og x, then dy du is
• Assertion : The ratio of length of tangent to length of normal is directly proportional to the ordinate of the point of…
• Let f (x) = (x – 1) 4 (x – 2) n , n ∈ N. then f (x) has
• If then d^ 2 y d x^ 2 + x d y d x +y ^ 2 x is equal to
• The value of _ x 0 x x- (1+x) x^ 2 is
• A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a…
• The set of values of 'a' for which the function f(x)=x^ 2 +a x+1 is an increasing function on [1,2] are :
• Let f(x)=x ^ -1 (- |x|), x [- 2 , 2 ] , then which of the following is true?
• Let (h, k) be a fixed point, where h > 0, k > 0 . A straight line passing through this point cuts the positive…
• If f(x) = _ x^ 2 ^ x^ 2 +1 e^ -t^ 2 d t_ 1 then the function f(x) decreases in
• If t, n, t´, n´ are the lengths of tangent, normal, subtangent & subnormal at a point P (x 1 , y 1 ) on any curve y = f…
• If px 2 + qx + r = 0, p, q, r ∈ R has no real zero and the line y + 2 = 0 is tangent to f (x) = px 2 + qx + r then
• The minimum value of x ^ 2 + 1 1+ x ^ 2 is at:
• The sub-tangent at any point of the curve x m y n = a m + n varies as
• _ x 0 ( 1+5 x^ 2 1+3 x^ 2 )^ 1 / x^ 2 is equal to
• The tangent to the curve x 2 + y 2 = 25 is parallel to the line 3x – 4y = 7 at the point
• The function f(x)= ^ -1 x+x increases in the interval
• If f be a polynomial then, the second derivative of f (e x ) is
• The points on the curve x^ 2 =2 y , which are closest to the point (0, 5) are
• f(x)= array c x+a 2 x, 0 x< 4 2 x x+b, 4 x< 2 a 2 x-b x, 2 x array . is continuous in [0, ] , then
• f(x)= _ n (x-1)^ 2 n -1 (x-1)^ 2 n +1 is discontinuous at
• Let y = a x + bx be curve, (2x – y) + λ (2x + y – 4) = 0 be family of lines. A line of the family cutting positive…
• Assertion : Tangent drawn at the point (0, 1) to the curve y = x 3 – 3x + 1 meets the curve thrice at one point only…
• _ x 0 1 x x(21) is equal to
• The minimum value of 2^ ( x ^ 2 -3 )^ 3 +27 is
• The two curves x 3 – 3xy 2 + 2 = 0 and 3x 2 y – y 3 – 2 = 0
• The points of extrema of f(x)= _ 0 ^ x t t d t in the domain x>0 are :
• Assertion : Shortest distance between | x | + | y | = 2 & x 2 + y 2 = 16 is 4- 2 Reason : Shortest distance between the…
• If _ x ( x^ 2 -1 x+1 -a x-b )=2 , then find the value of b .
• The height of a cylinder is equal to the radius. If an error of α % is made in the height, then percentage error in its…
• If at any point on a curve the sub-tangent and sub-normal are equal, then the length of the normal is equal to
• The minimum value of a tan 2 x + b cot 2 x equals the maximum value of a sin 2 θ + b cos 2 θ where a > b > 0, when
• Consider a function f( x )= ( - 1 - x ) (4 – 3x 2 ) where ‘α’ is a positive parameter Absolute difference between local…
• _ x 0 [ x ^ -1 x-x^ 2 x^ 6 ] equals
• _ x 0 [ (y^ 2 -4 y+11 ) x x ] is equal to [where [.] denotes the greatest integer function]
• Gas is being pumped into a spherical balloon at the rate of 30 ft 3 /min. Then, the rate at which the radius increases…
• How many real solutions does the equation x 7 + 14x 5 + 16x 3 + 30x – 560 = 0 have ?
• Find the shortest distance between xy = 9 and x 2 +y 2 = 1 .
• Differentiate the following function ^ -1 [x 1-x + x 1-x^ 2 ]
• Find the least value of 'a' for which the function f(x)=x^ 2 +a x+1 is increasing on [1,2] .
• For the function f(x)= array cc |x-3|, & x 1 x^ 2 4 - 3 x 2 + 13 4 , & x<1 array . which of the following is incorrect ?
• The angle of intersection of the curve y = x 2 & 6y = 7 – x 3 at (1, 1) is
• _ x 1 1-x^ -2 / 3 1-x^ -1 / 3 is equal to
• The curves x 3 + p xy 2 = –2 and 3x 2 y – y 3 = 2 are orthogonal for
• For the function f(x)= ^ -1 x+ ^ -1 x^ 2 which of the following statements is/are true
• The two tangents to the curve ax 2 + 2hxy + by 2 = 1, a > 0 at the points where it crosses x-axis, are
• If f(x)= array cc |x+2| ^ -1 (x+2) ; & x -2 =2 ; & x=-2 array . then f (x) is
• The function f: R- 0 R is given by f(x)= 1 x - 2 e^ 2 x -1 can be made continuous at x=0 by defining f(0) as
• _ x (1- 4 x-1 )^ 3 x-1 is equal to :
• Given below are two statements. One is labelled as Assertion and the other is labelled as Reason. STATEMENT - 1…
• If y=x^ x^ x^ x^ , then x (1 – y l og x) dy dx
• The value of _ x 7 2- x-3 x^ 2 -49 is
• Assertion : for any triangle ABC sin ( A+B+C 3 ) A+ B+ C 3 Reason : y = sin x is concave downward for x ∈ (0, π].
• Limit _ x 0 e^ x -e^ x x-x is equal to
• If y = a cos ( l og x) + b sin ( l og x) where a, b are parameters, the x 2 y′′ + xy′ is equal to
• Evaluate _ x 1 [ (1+x)- 2 x 3 4^ x-1 -3 x ] (7+x)^ 1 / 3 -(1+3 x)^ 1 / 2 x . .
• For what values of x is the rate of increase of x 3 – 5x 2 + 5x + 8 is twice the rate of increase of x ?
• A line is drawn through the point (1, 2) to meet the coordinate axes at P and Q such that it forms a ∆ OPQ, where O is…
• Evaluate _ x a x a-a x x-a .
• The length of the longest interval, in which the function 3 sin x – 4 sin 3 x is increasing, is
• Limit _ x 1 (1-x+[x-1]+[1-x]) = where [x] denotes greatest integer function
• _ x 2 x^ 3 -8 x^ 2 -4 is equal to
• Let h(x)=f(x)-[f(x)]^ 2 +[f(x)]^ 3 for every real number x, then ...
• _ x 0 x^ 2 ( 1 x ) x is equal to ____
• If x+y + y-x = cthen d^ 2 y d x^ 2 is
• The set of all values of the parameters a for which the points of local minimum of the function y = 1 + a 2 x – x 3…
• If y = (1 + x 2 ) tan –1 x – x, then dy dx is equal to
• If f (9) = 9 and f ‘ (9) = 1, then _ x 9 3- f(x) 3- x is equal to
• _ x 1 [x^ 2 -1 ] x-1 is equal to
• The function f (x) = 2x 2 – log | x | monotonically decreases for
• Differentiate ^ -1 1- x 1+ x w.r.t x
• If P (x) = a 0 + a 1 x 2 + a 2 x 4 + .... + a n x 2n be a polynomial in x ∈ R with 0 1 2 n , then P(x) has
• A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is…
• If y=f(x) is a curve and if there exists two points A (x_ 1 , f (x_ 1 ) ) and B (x_ 2 , f (x_ 2 ) ) on it such that f^…
• The rate of change of the surface area of a sphere of radius r, when the radius is increasing at the rate of 2 cm/s is…
• If x=e^ y+e^ y+2+0 , x>0, then d y d x is
• For x R, f(x)=| 2- x| and g(x)=f(f(x)) then :
• Let f (x) = (1+b 2 ) x 2 + 2bx + 1 and m (b ) the minimum value of f (x) for a given b. As b varies, the range of m (b…
• Let g(x) = - f(-1) 2 x 2 (x – 1) – f (0) (x 2 – 1) + f(1) 2 x 2 (x + 1) – f ′ (0) x (x – 1) (x + 1) where f is a thrice…
• The difference between the greatest and the least value of the function (x)= _ 1 ^ x (t+1) d t on [2,3] is
• If y=a x+b x^ 2 +x has its extremum values at x =-1 and x=z_ d then
• If f( a )=2_ 1 f^ ( a )=1_ 1 g( a )=-1_ 1 g^ ( a )=2 , then the value of _ x a g(x) f(a)-g(a) f(x) x-a is
• The interval in which the function f(x)= _ 0 ^ x ( t t+z - 1 t ) dt will be non-increasing is
• Let y = a x + bx be curve, (2x – y) + λ (2x + y – 4) = 0 be family of lines. Two perpendicular chords of curve y 2 – 4x…
• If f : [–1, 1] → R is a continuously differentiable function such that f (1) > f (–1) and | f ′(y)| < 1 for all y ∈…
• The set of points where the function f (x) = |x–1| e x is differentiable is
• Let f(x)= x a^ 2 +x^ 2 - d-x b^ 2 +(d-x)^ 2 , x F , where a, b and d are non-zero real constants. Then:
• If x log e (log e x) - x 2 + y 2 = 4 (y > 0), then d y d x at x = e is equal to:
• If x = a cos 4 θ, y = a sin 4 θ, then dy dx at θ = 3 4 is
• The set of all points, where the function f( x )= x 1+| x | is differentiable, is
• If y = (1 + x) (1 + x 2 ) (1 + x 4 ) ... (1+x^ 2^ n ) , then dy dx at x = 0 is
• For x (0, ^ -1 5 2 ) , the function f (x) = cot –1 ( 2 x+ 5 x 7 )
• If y = f ( 2 x-1 x^ 2 +1 ) and f ′ (x) = sin x 2 , then dy dx is equal to
• Consider the function f (x) = max x 2 , (1 – x) 2 , 2x (1 – x) where 0 ≤ x ≤ 1 . The interval in which f (x) is…
• Let f(x)= array cc (x-1) ( 1 x-1 ), & if x 1 0, & if x=1 array . Then, which one of the following is true ?
• The value of _ x 1 (1-x) ( x 2 ) will be
• The value of _ x (x+1)(3 x+4) x^ 2 (x-8) is equal to
• For the parabola y 2 = 4ax, the ratio of the sub-tangent to the abscissa is
• Limit _ n 5^ n+1 +3^ n -2^ 2 n 5^ n +2^ n +3^ 2 n+3 =
• If the function f (x) increases in the interval (a, b) then the function φ (x) = [ f (x)] 2 .
• The value of _ x [ x+ x+ x - x ] is
• If y= _ u(x) ^ v(x) f(t) d t let us define d y d x in a different manner as d y d x =v^ (x) f^ 2 (v(x))-u^ (x) f^ 2…
• A wire of length 25m is to be cut into two pieces one of the wires is to be made into square and other into a circle…
• Evaluate _ h a (a+h)^ 2 (a+h)-a^ 2 a h
• _ x 0 ( a^ x +b^ x 2 )^ 1 / x equals to :
• Let f ´ (sin x) f ´´ (sin x) > 0 x (0, 2 ) Now consider a function g (x) = f (sin x) + f (cos x) g (x) increase if x…
• If ax + b x c for all positive x, where a, b, c > 0, then
• Let f (x) = [n + p sin x], x ∈ (0, π), n ∈ I, p is a prime number and [x] denotes the greatest integer less than or…
• Consider the following statements S and R : S : Both sin x and cos x are decreasing functions in the interval ( 2 , ) R…
• If f( x )= a ^ a ^ | x | sgn x ; g( x )= a ^ [ a ^ | x | sgn x ] for a > 1 and x ∈ R, where & [ ] denote the fractional…
• _ x 4 _ 2 ^ ^ 2 x f(t) d t x^ 2 - ^ 2 16 equals
• Which of the following functions have finite number of points of discontinuity ([.] represents greatest integer…
• If y = l og a x + l og x a + l og x x + l og a a, then dy dx is equal to
• The function f (x) = (1+x) cot x is not defined at x = 0. The value of f (0) so that f (x) becomes continuous at x = 0…
• The sub-normal at any point of the curve x 2 y 2 = a 2 (x 2 – a 2 ) varies as
• The value of array l _ x 0 ( 1+ x 1+ x )^ casec x i5... array
• If the ratio of base radius and height of a cone is 1 : 2 and percentage error in radius is λ %, then the error in its…
• _ x 3 ([x-3]+[3-x]-x), where [.] denotes the greatest integer function, is equal to
• For the curve y = 3 sin cos , x=e^ , 0 , the tangent is parallel to x-axis when is:
• If y= x+ x+ x+ then dy dx is equal to
• If a _ 1 =1 and a_ n+1 = 4+3 a_ n 3+2 a_ n , n 1 and if _ n a _ n+1 = a, then the value of a is :
• The local minimum of the function f(x)= x 2 + 2 x is
• _ x -1 [x] + |x| , where [.] denotes the greatest integer function,
• If y 2 = ax 2 + bx + c where a, b, c are constants, then y 3 d ^ 2 y dx ^ 2 , is equal to
• If u= ( x^ 2 +y^ 2 x+y ) , then the value of x L x +y L y is:
• For function f ( x )= n x x which of the following statements are true.
• Assertion : The tangent at x = 1 to the curve y = x 3 – x 2 – x + 2 again meets the curve at x = – 2 . Reason : When a…
• If f(x)= x^ 2 2-2 x ; g(x)= x^ 2 6 x-6 x where 0 < x < 1, then :
• The function f( x )= _ 1 ^ x 2( t -1)( t -2)^ 3 +3( t -1)^ 2 ( t -2)^ 2 dt attains its local maximum value at x =
• _ x 0 ( 4 -x ) ^ 1 / x is equal to
• Let f ´ (sin x) f ´´ (sin x) > 0 x (0, 2 ) Now consider a function g (x) = f (sin x) + f (cos x) g (x) decreases if x…
• If the tangent at each point of the curve y = 2 3 x 3 – 2ax 2 + 2x + 5 makes an acute angle with the positive direction…
• The value of _ x 1 (2-x)^ x 2 is equal to...
• Assertion : The largest term in the sequence a_ n = n^ 2 n^ 3 +200 , n N: (400)^ 2 / 3 600 . Reason : f(x)= x^ 2 x^ 3…
• _ x 2 1+ 2+x - 3 x-2
• A curve passes through the point (2, 0) and the slope of the tangent at any point (x, y) is x 2 – 2x for all values of…
• If f (x) = x 3 + 4x 2 + λx + 1 is a strictly decreasing function of x in the largest possible interval [-2,- 2 3 ] then
• Which of the following pair (s) of curves is/are orthogonal.
• The function f(x)= ^ 4 x+ ^ 4 x is increasing if
• The maximum area of the rectangle that can be inscribed in a circle of radius r, is
• If f(x)=x+ x 1+x + x (1+x)^ 2 + to ∞, then at x = 0, f (x)
• Let x = sin ( ln t) and y = cos ( ln t) then d ^ 2 y dx ^ 2 is
• The abscissa of a point on the curve xy = (a + x) 2 , the tangent at which cuts off equal intercepts on the coordinate…
• If (x – a) 2n (x –b) 2m +1 , where m and n are positive integers and a > b, is the derivative of a function f , then
• The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius = 3 is:
• Slope of tangent to the curve y = 2e x sin ( 4 - x 2 ) cos ( 4 - x 2 ), where 0 ≤ x ≤ 2π is minimum at x =
• A function is defined as follows : f(x)= array cc x^ 3 ; & x^ 2 <1 x ; & x^ 2 1 array . The function is
• If a, b, c, d are positive real numbers then _ n (1+ 1 a+b n )^ 1+d n is equal to...
• If x = a sin θ and y = b cos θ, then d ^ 2 y dx ^ 2 is equal to
• _ x 0 3^ x -1 x+1 -1 is equal to
• Let f (x) be the function given by f(x)=3 x^ 5 -5 x^ 3 +21 x+3 x+4 x+5 . Then,
• The normal to the curve x=a(1+ ) y= asin at 'θ' always passes through the fixed point
• The function y=a(1- x), a>0 is maximum when x is equal to :
• The intercepts on x-axis made by tangents to the curve, y = _ 0 ^ x | t | dt , x R , which are parallel to the line y =…
• The value of _ x 0 1 x^ 3 _ 0 ^ x t (1+t) t^ 4 +4 d t i 5 ,
• The points on the curve y = x 1- x ^ 2 –1 < x < 1 at which the tangent line is vertical are
• A truck is to be driven 300 km on a highway at a constant speed of x kmph. Speed rules of the highway required that 30…
• _ x / 4 ^ 3 x- x (x+ 4 ) is:
• Let g´ (x) > 0 and f ’ (x) < 0, x ∈ R, then
• The value of _ x 2 ( ( x^ 3 -4 x x^ 3 -8 )^ -1 - ( x+ 2 x x-2 - 2 x - 2 )^ -1 ) is ..
• Given below are two statements. One is labelled as Assertion and the other is labelled as Reason. STATEMENT - 1…
• The value of _ x 0 _ 0 ^ x_ 0 ^ 2 ^ 2 t d t x x is
• The greatest value of f( x )=( x +1)^ 1 / 3 -( x -1)^ 1 / 3 on [0, 1] is
• Consider a function f( x )= ( - 1 - x ) (4 – 3x 2 ) where ‘α’ is a positive parameter Number of points of extrema of f…
• _ n 4^ 1 / n -1 3^ 1 / n -1 is equal to
• _ x ( x-1 x+1 )^ x+2 is equal to
• The value of _ x x- x x+ ^ 2 x is
• If y = l og cos x sin x, then dy dx is equal to
• The derivative of l og 10 x with respect to x 2 is
• Let f(x)=x^ ( 1 x ) . when x ≠ 0 and f (x) = 0, when x = 0. Then f (x) will be differentiable at x = 0 if :
• The point on the curve y-e^ x y +x=0 at which curve have vertical tangent is
• A spherical balloon is filled with 4500 cubic meters of helium gas. If a leak in the balloon causes the gas to escape…
• If the function f(x)= array ll a| -x|+1, & x 5 b|x- |+3, & x>5 array . is continuous at x = 5, then value of (a-b) is
• The function ‘ f ’ is defined by f (x) = x p (1 –x) q for all x ∈ R, where p,q are positive integers, has a local…
• A curve is represented parametrically by the equation x = t + e at and y = –t + e at when t ∈ R and a > 0 . If the…
• _ x 0 3^ x -2^ x 4^ x -3^ x is equal to
• The maximum slope of the curve y = –x 3 + 3x 2 + 9x – 27 is
• If a<0, the function f(x)=e^ a x +e^ -a x is a monotonically decreasing function. Then x is given by
• If f (x) is a differentiable function and φ (x) is twice differentiable function and α and β are roots of the equation…
• The solution set of f^ (x)>g^ (x) where f(x)= ( 1 2 ) 5^ 2 x+1 and g(x)=5^ x +4 x 5 is
• x and y are the sides of two squares such that y=x-x^ 2 . Find the rate of change of area of the second square w.r.t…
• If α, β are the roots of x 2 – ax + b = 0, then _ x e^ x^ 2 -a x+b -1 x- is
• If y, z>0 and y+z=c , then minimum value of [ (1+ 1 y ) (1+ 1 z ) ]^ 1 / 2 is equal to
• Assertion : If f (x) is increasing function with concavity upwards, then concavity of f –1 (x) is also upwards. Reason…
• If f(x)= 1 2 x-1 , where [∙] denotes the greatest integer function, then on the interval [0, ]
• If y = f (x) φ (x) , then dy dx is
• The function y = 2 x-1 x-2 (x ≠ 2) with codomain = R – 2
• _ n (0.2)^ _ 5 ( 1 4 + 1 8 + 1 16 + ton terms ) is equal to
• _ x x ( 4 x ) ( 4 x )
• The derivative of sin 2 x with respect to cos 2 x is
• _ x 1 ( x -1)(2 x-3) 2 x^ 2 +x-3 is equal to
• If for x (0, 1 4 ) , the derivative of ^ -1 ( 6 x x 1-9 x^ 3 ) is x g(x) then g(x) equals
• The greatest value of f (x) = (x +1) 1/3 – (x –1) 1/3 on [0, 1] is :
• _ n a ^ n +b^ n a ^ n -b^ n where a>b>1 is equal to
• Differentiate w. r. t. x : 2 ^ -1 x+ 1 x^ 2
• The two curves y = 3 x and y = 5 x intersect at an angle
• If _ x 0 (3+x)- (3-x) x =k, the value of k is
• _ x (2+x)^ 40 (4+x)^ 5 (2-x)^ 45
• The product of the lengths of subtangent and subnormal at any point of a curve is
• Find the fraction which exceeds its n th power by the greatest possible number (n > 1).
• The angle between the tangent at any point P and the line joining P to the origin, where P is a point on the curve l…

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