Continuity and Differentiability

CBSE Class 12 Mathematics · every board question from this chapter, 2023–2026 · 62 of them repeated across years

• Differential of e^e^x with respect to x is:2026asked 3×
• If f(x)=. is continuous at x=-1, then the value of k is:2026asked 3×
• If f(x)=. is continuous at x=0, then the value of k is:2026asked 3×
• If e^-x+e^-y=2, then (dy)/(dx) is2026
• Derivative of cos^-1(( sin x+ cos x)/(√2)),-(π)/4<x<(π)/4 with respect to x is:2026asked 3×
• The greatest integer function, f(x)=[x], 0<x<3 is not differentiable at how many points?2026asked 3×
• If x+y=x y, then (dy)/(dx) is2026
• If 2 cos^-1 x = y, then dy/dx is:2026
• The value of k for which the function f(x)= casesx^2 sin 1/x, & x ≠ 0 k(x+1), & x=0 cases is a continuous function, is:2026asked 3×
• If e^x+y=3 x, then (dy)/(dx) is2026
• Check whether function f(x) defined as f(x)= is continuous at x=3. or not? OR If √3(x^2+y^2)=4 x y, then find (dy)/(dx) at (1/2,…2026asked 3×
• If x=t+1/(t) and y=t-1/(t), find (dy)/(dx) at t=2.2026
• Show that the function f(x)=. is continuous at x=(π)/2. OR Find whether the function f(x)=. at x=2 is differentiable or not.2026asked 3×
• Differentiate x^x with respect to x log x. OR If y=P cos ux+Q sin ux, show that (d^2 y)/(dx^2)+u^2 y=0.2026asked 3×
• If x=a^2 sin^3 t, y=b cos^3 t, then find (dy)/(d x) at t=(π)/4.2026
• If x=e^ sin^-1 t, y=e^ cos^-1 t, find (dy)/(dx) at t=1/(√2)2026asked 2×
• If x=e^t+1/t and y=e^t-1/t, then find (dy)/(dx) at t=-2.2026
• If x= sin t- cos t, y= sin t cos t, find (dy)/(d x) at t=(π)/4.2026
• If xy=e^x-y, then find (dy)/(dx). OR Differentiate tan^-1( √1+x^2-√1-x^2√1+x^2+√1-x^2) with respect to cos^-1 x^2.2026asked 3×
• If ( sin x)^y=y^ cos x, then find (dy)/(dx).2026asked 3×
• If x=3 sin t- sin 3 t, y=3 cos t- cos 3 t find (dy)/(d x) and prove that (d^2 y)/(d x^2)=(-cosec^3 2 t · cosec t)/32026
• If y √x^2+1= log √x^2+1-x, show that (x^2+1) (dy)/(dx)+xy+1=0. OR Find the differential of x^ cot x+(2 x^2-3)/(2 x^2-x+2) with…2026asked 3×
• If x= cos t, y= cos mt, prove that (1-x^2) (d^2 y)/(d x^2)-x (dy)/(d x)+m^2 y=0.2026
• If x=a( sin t-t cos t) and y=b( cos t+t sin t), then find (d y)/(d x) and (d^2 y)/(d x^2).2026
• Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in…2026asked 3×
• If f(x)=. is continuous at x=(π)/2, then the value of k is:2025
• If y= sin^-1 x, then (1-x^2) (d^2 y)/(d x^2) is equal to:2025
• If y= log 2 x(√2 x), then (d y)/(d x) is equal to:2025
• If f(x)=. is continuous at x=0, then the value of a is:2025asked 4×
• If f(x)=. is continuous at x=0, then the value of k is:2025asked 3×
• If tan^-1(x^2-y^2)=a, where ' a ' is a constant, then (d y)/(d x) is:2025asked 3×
• If f(x)=. is continuous in R, then the values of a and b are:2025
• If f(x)=. is continuous for x ∈(0,2), then a is equal to:2025asked 3×
• If y=a cos ( log x)+b sin ( log x), then x^2 y 2+xy 1 is:2025asked 3×
• If f(x)=. is continuous at x=1, then the values of a and b are:2025
• Let f^′(x)=3(x^2+2 x)-4/(x^3)+5, f(1)=0. Then, f(x) is:2025asked 3×
• The function f defined by f(x)= casesx, & if x ≤ 1 5, & if x 1 cases is not continuous at:2025asked 3×
• If A denotes the set of continuous functions and B denotes set of differentiable functions, then which of the following depicts…2025asked 3×
• Assertion: f(x)=. is continuous at x=0. Reason (R): When x → 0, sin 1/(x) is a finite value between -1 and 1.2025asked 3×
• Assertion: f(x)=. is continuous at x=5 for k=5/2. Reason (R): For a function f to be continuous at x=a, lim x → a^- f(x)= lim x →…2025asked 3×
• Differentiate 2^ cos^2 x w. r. t cos^2 x. OR If tan^-1(x^2+y^2)=a^2, then find (dy)/(dx).2025asked 3×
• Differentiate √e^√2 x with respect to e^√2 x for x 0. OR If (x)^y=(y)^x, then find (d y)/(d x).2025asked 3×
• Differentiate sin x√ cos x with respect to x. OR If y=5 cos x-3 sin x, prove that (d^2 y)/(d x^2)+y=0.2025asked 3×
• Differentiate ( 5^xx^5) with respect to x. OR If -2 x^2-5 x y+y^3=76, then find (d y)/(d x).2025asked 3×
• If x=e^x/(y), then prove that (dy)/(d x)=(x-y)/(x log x). OR If f(x)=. Check the differentiability of f(x) at x=-2.2025asked 3×
• Differentiate y= sin^-1(3 x-4 x^3) w.r.t. x, if x ∈[-1/2, 1/2]. OR Differentiate y= cos^-1((1-x^2)/(1+x^2)) with respect to x,…2025asked 3×
• Find k so that f(x)=. is continuous at x=-1. OR Check the differentiability of function f(x)=x x at x=0.2025asked 2×
• Differentiate y=√ log sin ((x^3)/3-1) with respect to x.2025
• A spherical medicine ball when dropped in water dissolves in such a way that the rate of decrease of volume at any instant is…2025
• Differentiate log (x^x+cosec^2 x) with respect to x.2025
• Show that the derivative of tan^-1( sec x+ tan x),[-(π)/2<x<(π)/2] with respect to x is equal to 1/2.2025
• If y= log (√x+1/(√x))^2, then show that x(x+1)^2 y 2+(x+1)^2 y 1=2. OR If x √1+y+y √1+x=0,-1<x<1, x ≠ y, then prove that…2025asked 3×
• Differentiate tan^-1 √1-x^2x w. r. t. cos^-1(2 x √1-x^2), x ∈(1/(√2), 1) OR Find (dy)/(dx), if y=x^ tan x+ √x^2+12.2025
• If √1-x^2+√1-y^2=a(x-y), then prove that (dy)/(dx)=√(1-y^2)/(1-x^2). OR If x=a( cos θ+ log tan (θ)/2) and y= sin θ, then find…2025asked 2×
• For a positive constant 'a', differentiate a^t+1/(t) with respect to (t+1/(t))^a, where t is a non-zero real number. OR Find (d…2025asked 3×
• If y= cos^-1(e^x), then (d y)/(d x) is:2024
• If x=at, y= /(t), then (dy)/(dx) is:2024
• The derivative of 5^x w.r.t. e^x is:2024
• If y= sin^-1 x, then (d^2 y)/(d x^2) is:2024
• Which of the following statements is true for the function f(x)=.?2024asked 2×
• The number of points of discontinuity of f(x)=. is:2024asked 3×
• If e^x^2 y=c, then (dy)/(dx) is:2024
• Derivative of e^2 x with respect to e^x, is:2024
• Derivative of x^2 with respect to x^3, is:2024
• The derivative of sin (x^2) w.r.t. x, at x=√π is:2024asked 2×
• The value of constant c that makes the function f defined by f(x)= casesx^2-c^2, & if x<4 c x+20, & if x ≥ 4 cases continuous for…2024
• If x e^y=1, then the value of (dy)/(dx) at x=1 is:2024
• The function f(x)= x + x-2 is2024
• For what value of k, the function given below is continuous at x=0? f(x)=.2024
• If sin (x y)=1, then (dy)/(d x) is equal to:2024
• d/(d x)[ cos ( log x+e^x)] at x=1 is:2024
• Derivative of e^ sin^2 x with respect to cos x is:2024asked 3×
• The derivative of 2^x w.r.t. 3^x is:2024
• The number of points, where f(x)=[x], 0<x<3 ([•] denotes greatest integer function) is not differentiable is:2024
• Check whether the function f(x)=x^2 x is differentiable at x=0 or not. OR If y=√ tan √x, prove that √x (d y)/(d x)=(1+y^4)/(4 y).2024
• Check the differentiability of function f(x)=[x] at x=-3, where [·] denotes greatest integer function. OR If x^1 / 3+y^1 / 3=1,…2024
• If f(x)= tan 2 x, then find the value of f^′(x) at x=(π)/3. OR If y=cosec( cot^-1 x), then prove that √1+x^2 (dy)/(dx)-x=0.2024asked 2×
• If x=e^x / y, prove that (dy)/(dx)=( log x-1)/(( log x)^2) OR Check the differentiability of f(x)=. at x=1.2024asked 3×
• Verify whether the function f defined by f(x)=. is continuous at x=0 or not. OR Check for differentiability of the function f…2024asked 2×
• If y= cos^3( sec^2 2 t), find (dy)/(dt). OR If x^y=e^x-y, prove that (dy)/(dx)=( log x)/((1+ log x)^2).2024asked 3×
• If y=√ cos x+y, prove that (d y)/(d x)=( sin x)/(1-2 y). OR Show that the function f(x)= x ^3 is differentiable at all points of…2024
• Verify whether the function f defined by is continuous at x=0 or not. OR Check for differentiability of the function f defined by…2024
• Check the differentiability of f(x)= cos x at x=(π)/2. OR If y=A sin 2 x+B cos 2 x and (d^2 y)/(d x^2)-k y=0, find the value of k.2024
• If x=e^ cos 3 t and y=e^ sin 3 t, prove that (dy)/(dx)=-(y log x)/(x log y). OR Show that: /(dx)( x )=(x)/( x ), x ≠ 02024
• Find (dy)/(dx), if ( cos x)^y=( cos y)^x. OR If √1-x^2+√1-y^2=a(x-y), prove that (d y)/(d x)=√(1-y^2)/(1-x^2).2024asked 3×
• Given that y=( sin x)^x · x^ sin x+a^x, find (dy)/(d x).2024
• Given that x^y+y^x=a^b, where a and b are positive constants, find (d y)/(d x).2024
• Find (dy)/(dx), if y=( cos x)^x+ cos^-1 √x is given.2024
• If x cos (p+y)+ cos p sin (p+y)=0, prove that cos p (d y)/(d x)=- cos^2(p+y), where p is a constant. OR Find the value of a and b…2024asked 3×
• If y=( tan^-1 x)^2, show that (x^2+1)^2 ( d^2 y)/(d x^2)+2 x(x^2+1) (dy)/(dx)=2.2024
• If √1-x^2+√1-y^2=a(x-y), prove that (dy)/(dx)=√(1-y^2)/(1-x^2). OR If y=( tan x)^x, then find (dy)/(dx).2024asked 3×
• If x=a sin^3 θ, y=b cos^3 θ, then find (d^2 y)/(d x^2) at θ=(π)/4.2024
• Find the values of a and b so that the following function is differentiable for all values of x: f(x)= casesa x+b, & x -1 b…2024
• If y=( log x)^2, prove that x^2 y^′ ′+xy^′=2. OR If y= sin ( tan^-1 e^x), then find (dy)/(dx) at x=0.2024
• If x^30 y^20=(x+y)^50, prove that (dy)/(dx)=(y)/(x). OR Find (d y)/(d x), if 5^x+5^y=5^x+y.2024
• If /(d x) f(x)=2 x+3/x and f(1)=1, then f(x) is2023
• If f(x)=2 x +3 sin x +6, then the right hand derivative of f(x) at x=0 is:2023
• The function f(x)= x -x is:2023
• If /(d x)[f(x)]=a x+b and f(0)=0, then f(x) is equal to2023asked 2×
• The function f(x)= x is2023asked 2×
• The derivative of x^2 x w.r.t. x is2023asked 2×
• The function f(x)=[x], where [x] denotes the greatest integer less than or equal to x, is continuous at2023asked 3×
• If y= log ( sin e^x), then (d y)/(d x) is:2023
• If y= sin^2(x^3), then (d y)/(d x) is equal to:2023asked 2×
• If x=A cos 4 t+B sin 4 t, then (d^2 x)/(d t^2) is equal to2023asked 2×
• The function f(x)=x x, x ∈ R is differentiable2023
• If tan ((x+y)/(x-y))=k, then (dy)/(dx) is equal to2023asked 3×
• The value of k for which f(x)=. is a continuous function, is:2023
• The value of k for which function f(x)=. is differentiable at x=0 is:2023asked 3×
• If y=( cos x- sin x)/( cos x+ sin x), then (d y)/(d x) is:2023asked 3×
• If xy=e^x-y, then show that (dy)/(dx)=(y(x-1))/(x(y+1)).2023
• Find the value of k for which the function f given as f(x)= is continuous at x=0.. OR If x=a cos t and y=b sin t, then find (d^2…2023
• If (x^2+y^2)^2=x y, then find (d y)/(d x).2023asked 2×
• If f(x)=., then show that f is not differentiable at x=1. OR Find the value(s) of ' λ ', if the function f(x)=.2023asked 2×
• If y=(x+√x^2-1)^2, then show that (x^2-1)((d y)/(d x))^2=4 y^2.2023asked 2×
• A particle moves along the curve 3 y=a x^3+1 such that at a point with x -coordinate 1, y -coordinate is changing twice as fast…2023
• If y=x^1/(x), then find (dy)/(dx) at x=1. OR If x=a sin 2 t, y=a( cos 2 t+ log tan t), then find (d y)/(d x).2023asked 3×
• If y=√a x+b, prove that y((d^2 y)/(d x^2))+((dy)/(dx))^2=0. OR If f(x)=. is a differentiable function in (0, 2), then find the…2023asked 3×
• If x=√a^ tan^-1 t, y=√a^ cot^-1 t, then show that x (dy)/(dx)+y=0.2023
• Differentiate sec^-1( 1√1-x^2) w.r.t. sin^-1(2 x √1-x^2). OR If y= tan x+ sec x, then prove that (d^2 y)/(d x^2)=( cos x)/((1-…2023asked 3×
• Let f(x) be a real valued function. Then its - Left Hand Derivative (L.H.D.): Lf^′ = lim h → 0 (f(a-h)-f )/-h - Right Hand…2023asked 3×
• The equation of the path traced by a roller-coaster is given by the polynomial f(x)=a(x+9)(x+1)(x-3). If the roller-coaster…2023asked 3×

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