Limit, Continuity and Differentiability — practice questions

233 curated practice questions from Limit, Continuity and Differentiability (Mathematics JEE Advanced, NTA Exams) — 140 medium, 93 hard. Every question includes the final answer free; step-by-step worked solutions with a free account.

• The function f(x)= _ -1 ^ x t (e^ t -1 )(t-1)(t-2)^ 3 (t-3)^ 5 d t has a local minimum at x =
• If P ( x ) is a polynomial of degree less than or equal to 2 and S is the set of all such polynomials so that P (0) =…
• Let f(x)= array cc 0, & x x
• If _ x ( x^ 2 +x+1 x+1 -a x-b )=4, then
• Let f(x)= x x^ 2 , x>0 Let x 1 x 2 x 3 ..... x n f and y 1 y 2 y 3 y n f . Then which of the following options is/are…
• Let y= array l ( x+ 2 x+ 3 x)^ 2 +( x+ 2 x+ 3 x)^ 2 array then which of the following is correct ?
• Let f(x)=x^ n , n ~W . The number of values of n for which f^ (p+q)=f^ (p)+f^ (q) is valid for all +ve p & q is
• A curve is represented parametrically by the equations x = f ( t )= a ^ ( b ^ t ) and y = g ( t )= b ^ - ( a ^ t ) a, b…
• The function f ( x ) = sin 4 x + cos 4 x increases if
• Let f : R → R , g : R → R and h : R → R be differentiable functions such that f ( x ) = x 3 + 3 x + 2, g ( f ( x )) = x…
• A curve is represented parametrically by the equations x = f ( t )= a ^ ( b ^ t ) and y = g ( t )= b ^ - ( a ^ t ) a, b…
• If 0 1 1+x + 2 x 1+x^ 2 + 4 x^ 3 1+x^ 4 + 8 x^ 7 1+x^ 8 + =
• The value of _ x 0 ( [ 100 x x ]+ [ 99 x x ] ), where [.] represents greatest integral function is
• _ x 4 _ 2 ^ ^ 2 x f(t) d t x^ 2 - ^ 2 16 equals
• Let b be a non-zero real number. Suppose f : ℝ → ℝ is a differentiable function such that f (0) =1 . If the derivative…
• Let f(x)= e^ x (x-1)(x-2) d x then f decreases in the interval,
• For a real number y , let [ y ] denote the greatest integer less than or equal to y . Then the function f(x)= [(x- )]…
• If x 2 + y 2 = t + 1 t and x 4 + y 4 = t 2 + 1 t ^ 2 , then dy dx is equal to
• Let f (x) be a polynomial of degree 3 such that f (3) = 1, f ′ (3) = – 1, f ‘’ (3) = 0 and f ‘’’(3) = 12 . Then the…
• The function f(x)= ( +x) (e+x) is
• Let f : R → R be any function. Define g : R → R by g ( x ) = | f ( x )| for all x . Then g is
• Let f : (0, 1) → be the function defined as where n ∈ . Let g : (0, 1) → be a function such that _ x^ 2 ^ x 1-t t d t x…
• Let g(x)= (x-1)^ n ^ m (x-1) ; 0 x m and n are integers, m ≠ 0, n > 0, and let p be the left hand derivative of | x –…
• If and g(x)= x an x where , then in this interval
• If f (x) = array ll x (3 e^ 1 / x +4 ) 2-e^ 1 / x , & x 0 0 & , x=0 array . , then f (x) is
• Consider two functions f (x) = _ n ( x n )^ n and g(x) = –x 4b where b = _ x ( x^ 2 +x+1 - x^ 2 +1 ) Then. Number of…
• AP is a diameter of a unit circle with centre at O. Let AC be an arc of this circle, which subtends angle θ radian at…
• Let f (x) = ( x ) ( x ) ( x ) for all real x, where ( x ), ( x ) and ( x ) are differentiable functions of x. If f ‘(2)…
• f (x), g(x), h(x) are functions having non-zero derivatives. The derivative of f (x) w.r.t g(x) is α(x) and derivative…
• Let f : (0, π) → R be a twice differentiable function such that _ t x f(x) t-f(t) x t-x = ^ 2 x for all x ∈ (0, π). If…
• Limit _ x -1 2- 2 x x^ 2 -|x| =
• Let f(x)= 1-x(1+|1-x|) |1-x| ( 1 1-x ) for x 1 . Then
• Suppose f and g are functions having second derivatives f” and g” everywhere, if f (x) . g (x) = 1 for all x and f’ and…
• Let f : R → R be a function defined by f ( x ) = max x , x 3 . The set of all points where f ( x ) is not…
• For each t ∈ R , let [ t ] be the greatest integer less than or equal to t . Then _ x 0^ + x ( [ 1 x ]+ [ 2 x ]+ .+ […
• AP is a diameter of a unit circle with centre at O. Let AC be an arc of this circle, which subtends angle θ radian at…
• Let f : [0, 1] → [0, 1] be the function defined by Consider the square region S = [0, 1] × [0, 1]. Let G = ( x , y ) ∈…
• If y 2 = P (x), a polynomial of degree n 3, then 2 d d x (y^ 3 d^ 2 y d x^ 2 )
• Let f : R → (0, 1) be a continuous function. Then, which of the following function(s) has (have) the value zero at some…
• 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 the line ax + by + c = 0 is a normal to the curve xy = 1, then
• If f ( x ) = x n , then the value of f(1)- f^ (1) 1! + f^ (1) 2! - f^ (1) 3! + + (-1)^ n f^ n (1) n! is
• A curve is represented parametrically by the equations x = f ( t )= a ^ ( b ^ t ) and y = g ( t )= b ^ - ( a ^ t ) a, b…
• For any positive integer n , define f n : (0, ∞) → R as for all x ∈ (0, ∞) Here, the inverse trigonometric function tan…
• _ n 1 1-n^ 2 + 2 1-n^ 2 + + n 1-n^ 2 is equal to
• The smallest positive root of the equation, tan x – x = 0 lies in
• The function f ( x ) = max (1 – x ), (1 + x ), 2 , x ∈(–∞ , ∞), is
• If _ l = | array lll x & ~b & ~b a & x & ~b a & a & x array | and _ 2 = | array ll x & ~b a & x array | are given, then
• If x + | y |= 2 y , then y as a function of x is
• The function f ( x ) = 1 + |sin x | is
• Let U(x) and V(x) are differentiable functions such that U(x) V(x) =7 . If U ^ ( x ) V ^ ( x ) = p and ( V ( x ) U ( x…
• If sin y = x sin (a + y), then dy dx is
• Let and be functions defined by f ( x ) = [ x 2 – 3] and g ( x ) = | x | f ( x ) + |4 x – 7| f ( x ), where [ y ]…
• If y 2 = P ( x ), a polynomial of degree 3, then 2 d d x (y^ 3 d^ 2 y d x^ 2 ) equals
• Where [ x ] denotes the greatest integer less than or equal to x then _ x 0 f(x) equals
• f is defined in [–5, 5] as f(x)= array l x, if x is rational -x, if x is irrational array . Then
• Let S be the set of all twice differentiable functions f from to such that for all x ∈(–1,1). For f ∈ S , let X f be…
• Let f(x)= array cc e^ x , & 0 x 1 2-e^ x-1 , & 1 and g(x)= _ 0 ^ x f(t) d t, x [1,3] then g ( x ) has
• Let f be the differentiable for ∀ x . If f (1) = –2 and f ′( x ) ≥ 2 for [1, 6], then
• Let f(x)= array c |x| ; for 0<|x| 2 1 ; for x=0 array , then at x=0, f . has
• If y = ( ax + b cx + d ), then 2 dy dx ~d ^ 3 y dx ^ 3 is equal to
• Let f : R → R be a function. We say that f has PROPERTY 1: if _ h 0 f(h)-f(0) |h| exists and is finite, and PROPERTY 2…
• If y = a x+b x^ 2 +c , where a, b, c are constants then (2xy’ + y) y’’’ is equal to
• If f is a differentiable function satisfying = 0 for all n ≥ 1, n ∈ I , then
• The value of a for which _ x 0 (e^ x -1 )^ 4 ( x^ 2 a^ 2 ) _ e 1+ x^ 2 2 =8, is
• Let for all x > 0 . Then
• The value of _ x 0 (( x)^ 1 / x +(1 / x)^ x ) , where x > 0 is
• Let h ( x ) = f ( x ) – ( f ( x )) 2 + ( f ( x )) 3 for every real number x , Then
• Let denote the set of all real numbers. Let f : → be defined by f(x)= array cl 6 x+ x 2 x+ x & ext if x eq 0, 7 3 & ext…
• If f (x) = | array ccc x^ n & n! & 2 x & n 2 & 4 x & n 2 & 8 array | then the value of d ^ n dx ^ n [f( x )]_ x =0 is
• Consider all rectangles lying in the region (x, y) R R: 0 x 2 and 0 y 2 (2 x) and having one side on the x -axis. The…
• If f ( x ) = min 1, x 2 , x 3 , then
• If f : R → R is a function defined by f ( x ) = [ x ] cos ( 2 x-1 2 ) , where [ x ] denotes the greatest integer…
• 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 at x = 0 so that…
• Let f (x) = sin x, g (x) = 2x and h (x) = cos x. If φ (x) = [go (f h)] (x), then φ “ ( 4 ) is equal to
• The total number of local maxima and local minima of the function f(x)= array ll (2+x)^ 3 , & -3<x -1 x^ 2 / 3 , &…
• For positive integer n , define + 48-3 n-3 n^ 2 12 n+3 n^ 2 + + 25 n-7 n^ 2 7 n^ 2 Then, the value of _ n f(n) is equal…
• Consider the functions defined implicitly by the equation y 3 – 3 y + x = 0 on various intervals in the real line. If x…
• _ x 0 x^ 4 -x^ 4 x^ 4 +x^ 20 x^ 4 (e^ 2 x^ 4 -1-2 x^ 4 ) is equal to
• If f r (x), g r (x), h r (x), r = 1, 2, 3 are polynomials in x such that f r (a ) = g r (a )= h r (a), r = 1, 2, 3 and…
• _ n 1 n _ r=1 ^ 2 n r n^ 2 +r^ 2
• Let h ( x ) = min x , x 2 for every real number of x . Then
• Let f : R → R be any function. Defined g : R → R by g ( x ) = | f ( x )| for all x . Then g is
• Let f , g : [–1, 2] → R be continuous functions which are twice differentiable on the interval (–1, 2). Let the values…
• Let f : R → R be a differentiable function and f (1) = 4. Then the value of _ x 1 1 x-1 _ 4 ^ f(x) 2 t d t is
• Let f ( x ) be a non-constant twice differentiable function defined on (–∞, ∞) such that f ( x ) = f (1 – x ) and f^ (…
• If f (x) = |x –1| and g (x) = f [ f f (x) ], then for x >2, g ’ (x) is equal to
• The function defined by f ( x ) = ( x + 2) e – x is
• For x R, _ x ( x-3 x+2 )^ x =
• The derivative of f (tan x) w.r.t. g (sec x) at x = 4 , where f ’(1) = 2 and g ’ ( 2 ) = 4, is
• Let S be the set of all (α, β) ∈ × such that _ x (x^ 2 ) ( _ e x )^ ( 1 x^ 2 ) x^ ( _ e (1+x) )^ =0 Then which of the…
• Limit _ x 0 (4^ x -1 )^ 3 ( x p ) (1+ x^ 2 3 ) =
• Let [ x ] be the greatest integer less than or equals to x . Then, at which of the following point(s) the function f (…
• If _ x 0 ((a-n) n x- x) n x x^ 2 =0, where n is non zero real number, then a is equal to
• If α, β are the roots of the quadratic equation ax 2 + bx + c = 0 then, Limit _ x 1- (a x^ 2 +b x+c ) (x- )^ 2 =
• Let f : [1, ∞) → be a differentiable function such that and 3 _ 1 ^ x f(t) d t=x f(x)- x^ 3 3 x [1, ) . Let e denote…
• The value of _ x 2 [1^ 1 / ^ 2 x +2^ 1 / ^ 2 x + +n^ 1 / ^ 2 x ]^ ^ 2 x is
• If f ( x ) = then, on the interval [0, π], where [ x ] denotes the greatest integer function of x .
• If x is a real number in [0, 1] then the value of _ m _ n [1 + cos 2m (n!πx)] is given by, where [x] represents…
• _ x 2 x-( x)^ x 1- x+ x equals
• If F(x) = f (x) g (x) and f ’ (x) g ’ (x) = c, then
• If f (n + 1) = 1 2 f( n )+ 9 f( n ) , n ∈ N and f (n) > 0 for all n ∈ N then _ n f (n) is equal to
• If a continuous function f defined on the real line R ; assumes positive and negative values in R then the equation f (…
• Let a ∈ R and let f : R → R be given by f ( x ) = x 5 – 5 x + a . Then
• AP is a diameter of a unit circle with centre at O. Let AC be an arc of this circle, which subtends angle θ radian at…
• Column–I Column–II (A) Lim _ x x 8 x 8 x = (P) 8 (B) Lim _ x 0 [- ^ 2 ] x^ 2 - [- ^ 2 ] x^ 2 ^ 2 (x) = (Q) 2 (C) Lim _…
• Let f(x)= _ 1 ^ x 2-t^ 2 d t , then the real roots of the equation x 2 – f ′( x ) = 0 are
• The function f ( x ) = ( x 2 – 1)| x 2 – 3 x + 2|+ cos(| x |) is not differentiable at
• On the interval [0, 1] the function x 25 (1 – x ) 75 takes its maximum value at the point
• The limit of sequence 2 , 2 2 , 2 2 2 ....... is
• Let f :  →  and g :  →  be two non-constant differentiable functions. If f ′( x ) = ( e ( f ( x ) − g ( x )) ) g ′(…
• Limit _ y 0 ( Limit _ x (x (1+ a y x ) )- (x (1+ by x ) ) y )=
• If f(x)= _ 0 ^ x e^ t^ 2 (t-2)(t-3) d t for all x ∈ (0, ∞), then
• If _ x 0 ( a^ x +b^ x +c^ x 3 )^ / x , (a, b, c, λ > 0) is equal to
• If _ x 0 x(1+a x)-b x x^ 3 =1, then
• A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open…
• For every integer n , let a n and b n be real numbers. Let function f : R → R be given by f(x)= array l a_ n + x, for x…
• Let f : [ a , b ] → [1, ∞) be a continuous function and let g : R → R be defined as g(x)= array llc 0 & if & x b array…
• If f ( x ) is differentiable and strictly increasing function, then the value of is
• Suppose f(x) = e ax + e bx , where a ≠ b, and that f ”(x) – 2f ’ (x) –15f(x) = 0 for all x. Then the product ab is
• For x ∈ R, _ x ( x-3 x+2 )^ x is equal to
• The left hand derivative of f ( x ) = [ x ] sin(π x ) at k , k an integer is
• For a ∈ R (the set of all real numbers), a ≠ –1, _ n (1^ a +2^ a + +n^ a ) (n+1)^ a-1 [(n a+1)+(n a+2)+ +(n a+n)] = 1…
• Let f be a non-negative function defined on the interval [0, 1]. If _ 0 ^ x 1- (f^ (t) )^ 2 d t= _ 0 ^ x f(t) d t, 0 x…
• If , then f ( x ) is
• If f (x) = x^ 2 -10 x+25 , then the derivative of f (x) on the interval [0, 7] is
• The value of _ x a^ - a^ 2 -x^ 2 ( 2 a-x a+x ) is
• Let f : [0, 1] → R (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f (0) =…
• Let f be a real-valued function defined on the interval (0, ∞) by f(x)= x+ _ 0 ^ x 1+ t d t Then which of the following…
• Let L= _ x 0 a- a^ 2 -x^ 2 - x^ 2 4 x^ 4 , a>0 . If L is finite, then
• Let f(x)= 1-x^ n+1 1-x and g (x) = 1 – 2 x + 3 x^ 2 - .........+ (–1) n n +1 x ^ n . Then the constant term in f’(x) ×…
• Let [ x ] denote the greatest integer less than (or) equal to x . If f ( x ) = [ x sin π x ] then f ( x ) is
• The domain of the derivative of the function f(x)= array ll ^ -1 x, & if |x| 1 1 2 (|x|-1), & if |x|>1 array . is
• If f ( x ) = x α log x and f (0) = 0, then the value of α for which Rolle’s theorem can be applied in [0,1] is
• If f (x) = l og | 2x |, x ≠ 0, then f ′ (x) is equal to
• If f(x)= 1 [ x] where [.] denotes the greatest function, then
• _ n 1 n (1+e^ 1 / n +e^ 2 / n + +e^ n-1 n ) is equal to
• If y = e tan x , then cos 2 x d ^ 2 y dx ^ 2 =
• x 0 Limit ^ -1 ( x)
• People living at Mars, instead of the usual definition of derivative D f(x), define a new kind of derivative, D*f(x) by…
• For the function f(x)=x 1 x , x 1
• The number of points in (–∞, ∞), for which x 2 – x sin x – cos x = 0, is
• Suppose the function f(x) – f(2x) has the derivative 5 at x = 1 and derivative 7 at x = 2 . The derivative of the…
• Let f : (0, 1) → be the function defined as f ( x ) = [4 x] (x- 1 4 )^ 2 (x- 1 2 ), where [ x ] denotes the greatest…
• Let f (x) = a [x] + b e |x| + c |x| 2 , where a, b and c are real constants. where [x] denotes greatest integer f (x)…
• Given that for each a ∈ (0, 1), _ h 0^ + _ h ^ 1-h t^ -a (1-t)^ a-1 d t exists. Let this limit be g ( a ). In addition…
• If f : R → R is a differentiable function such that ( x ) > 2 f ( x ) for all x ∈ R and f (0) = 1, then
• _ x 0 x 2 x-2 x x (1- 2 x)^ 2 is
• Let denote the set of all real numbers. Define the function f: R R by f(x)= array cc 2-2 x^ 2 -x^ 2 1 x & ext if x eq 0…
• Which of the following functions are continuous on (0, π ) ?
• Let f & g be differentiable functions satisfying g ’ (a ) = 2, g (a ) = b & fog = I (Identity function). Then f ‘(b )…
• A non zero polynomial with real coefficients has the property that f (x) = f’(x) . f’’ (x). The leading coefficient of…
• The function f(x)= array ll |x-3| & , x 1 x^ 2 4 - 3 x 2 + 13 4 & , x<1 array . .
• Limit _ x 5 x^ 2 -9 x+20 x-[x] where [x] is the greatest integer not greater than x
• If f : R → R is a twice differentiable function such that f ′′( x ) > 0 for all x ∈ R , and f ( 1 2 )= 1 2 , f(1)=1 …
• If f (x) = | x–3 | and φ (x) = (fof) (x), then for x > 10, φ’ (x) is equal to
• The set of points where is differentiable, is
• Let f: [ 1 2 , 1 ] R (the set of all real numbers) be a positive, non-constant and differentiable function such that f…
• If Lim _ x 0 x^ 3 a+x (b x- x) =1, a > 0, then a + 2b is equal to
• If f(x)=A ( x 2 )+B, f^ ( 1 2 )= 2 , then constants A and B are
• Let f (x) be a polynomial function of second degree. If f (1) = f (–1) and a, b, c are in AP, then f ′ (a ), f ′ (b )…
• _ x 0 ^ 2 x x^ 2 equals
• The values of a, b and c such that _ x 0 a e^ x -b x+c e^ -x x x =2 are
• The function f (x) = ( 2 x)^ ^ 2 2 x is not defined at x = π/4 . The value of f (π/4) so that f is continuous at x =…
• Let [⋅] denote the greatest integer function and f ( x ) = [tan 2 x ], then
• Let y = x 3 – 8x + 7 and x = f (t). If dy dt = 2 and x = 3 at t = 0, then dx dt at t = 0 is given by
• Let a , b ∈ R and f : R → R be defined by f ( x ) = a cos(| x 3 – x |) + b | x | sin(| x 3 + x |). Then f is
• Suppose we define the definite integral using the following formula _ a ^ b f(x) d x= for more accurate result for When…
• Consider the following S and R . S : Both sin x , cos x are decreasing functions in the interval ( 2 , ) R : If a…
• If G(x)=- 25-x^ 2 then _ x 1 ( G(x)-G(1) x-1 )=
• where [.] denotes greatest integer function, is equal to
• If y = f (x) is an odd differentiable function defined on (–∞, ∞) such that f ′ (3) = –2, then f ′ (–3) equals
• If y = | array ccc x & x & x x & - x & x x & 1 & 1 array | , then dy dx is equal to
• If [x] denotes the greatest integer ≤ x, then Limit _ n 1 n^ 4 ( [1^ 3 x ]+ [2^ 3 x ]+ + [n^ 3 x ] ) equals
• If f(x)= _ x^ 2 ^ x^ 2 +1 e^ -t^ 2 d t then f ( x ) increases in
• _ h 0 2 [ 3 ( 6 +h )- ( 6 +h ) ] 3 h( 3 h- h) is equal to
• Let f : R R be a function such that and f ′ (0) = 3 . Then
• Let g : R → R be a differentiable function with g (0) = 0, g ′(0) = 0 and g ′(1) ≠ 0 . Let f(x)= array cc x |x| g(x), &…
• For every twice differentiable function f: R [-2,2] with ( f (0)) 2 + ( f ′(0)) 2 = 85, which of the following…
• If _ x 0 (cos x + a sin bx) 1/x = e 2 , then the values of a and b are
• Let p ( x ) = a 0 + a 1 x 2 + a 2 x 4 + - - - - - a n x 2 n be a polynomial in a real variable x with 0 a 0 a 1 a 2 - …
• If the normal to the curve y = f(x) at the point (3, 4) makes an angle 3π/4 with the positive x -axis, then f ′(3) =
• Function f : R → R satisfies the functional equation f(x-y)= f(x) f(y) If f’(0) = p and f’ (a ) = q, then f’ (–a) is
• 112. Let f ( x ) = (1 – x ) 2 sin 2 x + x 2 for all x ∈ R , and let for all x ∈ (1, ∞). Which of the following is true ?
• Consider the polynomial: f ( x ) = 1 + 2 x + 3 x 2 + 4 x 3 . Let s be the sum of all distinct real roots of f ( x ) and…
• Which of following functions differentiable at x = 0?
• by appropriately matching the information given in the three columns of the following table. Let f ( x ) = x + log e x…
• Let f: [- 2 , 2 ] R be a continuous function such that f(0)=1 and _ 0 ^ 3 f(t) d t=0 Then which of the following…
• Limit _ x 2 (e^ x-2 -1 ) (x-1) =
• If y = a ln x + bx 2 + x has its extremum values at x = –1 and x = 2, then
• If f(x)= aligned 3 x^ 2 +12 x-1, & -1 x 2 37-x, & 2<x 3 aligned .
• Limit _ x 3 (x^ 3 +27 ) (x-2) (x^ 2 -9 ) =
• Assertion : If a and b are positive and [x] denotes greatest integer _ x 0^ + x a [ ~b x ]= b a Reason : _ x x x =0…
• Consider the function f : (– ∞, ∞) → (– ∞, ∞) defined by Let g(x)= _ 0 ^ e^ x f^ (t) 1+t^ 2 d t Which of the following…
• If f (x) = cos 2 x + cos 2 (x+ 3 ) –cos x cos (x+ 3 ) then f ’(x) is
• There exists a function f ( x ) satisfying f (0) = 1, f ′(0) = –1, f ( x ) > 0 for all x and
• Let x 0 be the real number such that e x 0 + x 0 = 0 . For a given real number α, define g(x)= 3 x e^ x +3 x- e^ x - x…
• The function f (x) = max. (1 – x), (1 + x), 2 , x ∈ (–∞, ∞), is
• Let [x] denotes the greatest integer less than or equal to x. If f (x) = [x sin π x], then f (x) is
• Let f(x)=e^ x -e^ -x -2 x- 2 3 x^ 3 , then the least value of n for which | d^ n f(x) d x^ n |_ x=0 is non-zero
• Let f : [0, 2] → R be a function which is continuous on [0, 2] and is differentiable on (0, 2) with f (0) = 1 . Let…
• Let g ( x ) = x f ( x ), where f(x)= array cl x 1 x , & x 0 0, & x=0 array . At x=0 .
• Let f (x) = a + b |x| + c |x| 4 , where a, b and c are real constants. Then f (x) is differentiable at x = 0 if
• If F(x)= (f ( x 2 ) )^ 2 + (g ( x 2 ) )^ 2 where f ′′ ( x ) = – f ( x ) and g ( x ) = f ′ ( x ) and given that F (5) =…
• If ax 2 + 2hxy + by 2 = 1, then d ^ 2 y dx ^ 2 is equal to
• Let f : ℝ → ℝ and g : ℝ → ℝ be functions satisfying f ( x + y ) = f ( x ) + f ( y ) + f ( x ) f ( y ) and f ( x ) = xg…
• Let the function f : ℝ → ℝ be defined by f ( x ) = x 3 − x 2 + ( x − 1)sin x and let g : ℝ → ℝ be an arbitrary…
• _ h 0 (a+3 h)-3 (a+2 h)+3 (a+h)- a h^ 3 is
• Let f ( x ) be a continuously differentiable function on the interval (0, ∞) such that f (1) = 2 and _ t x t^ 10…
• Let f : R → (0, ∞) and g : R → R be twice differentiable functions such that f ′′ and g ′′ are continuous functions on…
• The integer n , for which _ x 0 ( x-1) ( x-e^ x ) x^ n is a finite non-zero number is
• _ h 0 f (2 h+2+h^ 2 )-f(2) f (h-h^ 2 +1 )-f(1) , given that f ′(2) = 6 and f ′(1) = 4
• Let f : (0, ∞) → R and F(x)= _ 0 ^ x f(t) d t If , then f (4) equals
• f ( x ) is cubic polynomial with f (2) = 18 and f (1) = –1. Also f ( x ) has local maxima at x = –1 and f ( x ) has…
• _ n 1^ p +2^ p +3^ p + +n^ p n^ p+1 is
• The value of Limit _ x 0 ( x)- x x^ 4 is equal to
• If f (a) = 2, f ‘ (a) = 1, g (a) = –1, g ’ (a) = 2, then _ x a g( x ) f( a )-g( a ) f( x ) x - a is equal to
• If f (x) = sin ( 2 [x]-x^ 5 ), 1 is equal to
• If f is an even function such that _ h 0^ + f( ~h )-f(0) h has some finite non-zero value, then
• If f ( a ) = 2, f ′ ( a ) = 1, g ( a ) = –1, g ′ ( a ) = 2 then the value of
• Let f : R → R be given by f(x)= array ll x^ 5 +5 x^ 4 +10 x^ 3 +10 x^ 2 +3 x+1, & x Then which of the following options…
• For every pair of continuous function f , g : [0, 1] → R such that max f ( x ) : x ∈ [0, 1] = max g ( x ) : x ∈ [0, 1]…
• Consider two functions f (x) = _ n ( x n )^ n and g(x) = –x 4b where b = _ x ( x^ 2 +x+1 - x^ 2 +1 ) Then. g (x) is
• If y = ksinpx, then the value of the determinant is equal to
• Consider two functions f (x) = _ n ( x n )^ n and g(x) = –x 4b where b = _ x ( x^ 2 +x+1 - x^ 2 +1 ) Then. f (x) is
• Let f and g be real valued functions defined on interval (– 1, 1) such that g ″( x ) is continuous, g (0) ≠ 0, g ′(0) =…
• Let f(x)= array cl x^ 2 | x |, & x 0 0, & x=0 array , x R, . then f is
• If ( y )^ ( x / 2) + 3 2 Sec –1 (2x) + 2 x tan (log (x+2)) = 0 then dy/dx at x = –1 is
• Let f (x) = 2/(x+1) and g(x) = 3x. It is given that (fog) (x 0 ) = (gof) (x 0 ). Then (gof)’ (x 0 ) equals
• _ x 1 1- 2(x-1) x-1
• A curve y = f ( x ) passes through (1, 1) and tangent at P ( x , y ) cuts the x -axis and y axis at A and B…
• The value of Lim _ x 0 x- x ^ -1 x- ^ -1 x is
• The value of _ x 0 1 x^ 3 _ 0 ^ x t (1+t) t^ 4 +4 d t is

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