Continuity and Differentiability

CBSE Class 12 Mathematics · 123 board questions from this chapter; 122 listed below, 2023–2026

• Differential of e^e^x with respect to x is:2026
• If f(x)= {(x^2-4 x-5)/(x+1), x ≠-1; k, x=-1} is continuous at x=-1, then the value of k is:2026
• If f(x)= {(sin x)/x+ cos x, x ≠ 0; k, x=0} is continuous at x=0, then the value of k is:2026
• 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:2026
• The greatest integer function, f(x)=[x], 0<x<3 is not differentiable at how many points?2026
• 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)= {x^2 sin 1/x, x ≠ 0; k(x+1), x=0} is a continuous function, is:2026
• If e^x+y=3 x, then dy/dx is2026
• Check whether function f(x) defined as f(x)={ (|x-3|)/(2(x-3)), x<3; (x-6)/6, x ≥ 3 is continuous at x=3. or not? OR If…2026
• If x=t+1/t and y=t-1/t, find dy/dx at t=2.2026
• Show that the function f(x)= {(cos x)/(-x+(π)/2), x ≠ (π)/2; 1, x=(π)/2} is continuous at x=(π)/2. OR Find whether the function…2026
• 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=0.2026
• 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)2026
• 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^2.2026
• If ( sin x)^y=y^cos x, then find dy/dx.2026
• 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…2026
• 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…2026
• If f(x)= {(1- sin^3 x)/(3 cos^2 x), for x ≠ (π)/2; k, for x=(π)/2} 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)= {(sin^2 a x)/(x^2), x ≠ 0; 1, x=0} is continuous at x=0, then the value of a is:2025
• If f(x)= {(log (1+a x)+ log (1-b x))/x, for x ≠ 0; k, for x=0} is continuous at x=0, then the value of k is:2025
• If tan^-1(x^2-y^2)=a, where ' a ' is a constant, then (d y)/(d x) is:2025
• If f(x)= {1, if x ≤ 3; ax+b, if 3<x<5; 7, if 5 ≤ x} is continuous in R, then the values of a and b are:2025
• If f(x)= {3 x-2, 0<x ≤ 1; 2 x^2+ax, 1<x<2} is continuous for x ∈(0,2), then a is equal to:2025
• If y=a cos ( log x)+b sin ( log x), then x^2 y_2+xy_1 is:2025
• If f(x)= {3 a x-b, x>1; 11, x=1; -5 a x-2 b, x<1} 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:2025
• The function f defined by f(x)= {x, if x ≤ 1; 5, if x>1} is not continuous at:2025
• If A denotes the set of continuous functions and B denotes set of differentiable functions, then which of the following depicts…2025
• Assertion: f(x)= {x sin 1/x, x ≠ 0; 0, x=0} is continuous at x=0. Reason (R): When x → 0, sin 1/x is a finite value between -1…2025
• Assertion: f(x)= {3 x-8, x ≤ 5; 2 k, x>5} is continuous at x=5 for k=5/2. Reason (R): For a function f to be continuous at x=a,…2025
• 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.2025
• 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).2025
• 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=0.2025
• Differentiate ((5^x)/(x^5)) with respect to x. OR If -2 x^2-5 x y+y^3=76, then find (d y)/(d x).2025
• If x=e^(x/y), then prove that dy/(d x)=(x-y)/(x log x). OR If f(x)= {2 x-3,-3 ≤ x ≤-2; x+1,-2<x ≤ 0} Check the differentiability…2025
• 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,…2025
• Find k so that f(x)= {(x^2-2 x-3)/(x+1), x ≠-1; k, x=-1} is continuous at x=-1. OR Check the differentiability of function…2025
• 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…2025
• Differentiate tan^-1 (√(1-x^2))/x 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+1))/2.2025
• 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…2025
• 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 y)/(d…2025
• If y= cos^-1(e^x), then (d y)/(d x) is:2024
• If x=at, y=a/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)= {x^2+3, x ≠ 0; 1, x=0}?2024
• The number of points of discontinuity of f(x)= {|x|+3, if x ≤-3; -2 x, if -3<x<3; 6 x+2, if x ≥ 3} is:2024
• 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:2024
• The value of constant c that makes the function f defined by f(x)= {x^2-c^2, if x<4; c x+20, if x ≥ 4} continuous for all real…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)= {(√(4+x)-2)/x, x ≠ 0; k, x=0}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:2024
• 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…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=0.2024
• If x=e^x / y, prove that dy/dx=(log x-1)/(( log x)^2) OR Check the differentiability of f(x)= {x^2+1, 0 ≤ x<1; 3-x, 1 ≤ x ≤ 2} at…2024
• Verify whether the function f defined by f(x)= {x sin (1/x), x ≠ 0; 0, x=0} is continuous at x=0 or not. OR Check for…2024
• 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).2024
• 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…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: d/(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)).2024
• 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…2024
• 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.2024
• 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)= {a x+b, x>-1; b x^2-3, x…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/(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/(d x)[f(x)]=a x+b and f(0)=0, then f(x) is equal to2023
• The function f(x)=|x| is2023
• The derivative of x^2 x w.r.t. x is2023
• The function f(x)=[x], where [x] denotes the greatest integer less than or equal to x, is continuous at2023
• 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:2023
• If x=A cos 4 t+B sin 4 t, then (d^2 x)/(d t^2) is equal to2023
• The function f(x)=x|x|, x ∈ R is differentiable2023
• If tan ((x+y)/(x-y))=k, then dy/dx is equal to2023
• The value of k for which f(x)= {3 x+5, x ≥ 2; kx^2, x<2} is a continuous function, is:2023
• The value of k for which function f(x)= {x^2, x ≥ 0; kx, x<0} is differentiable at x=0 is:2023
• If y=(cos x- sin x)/(cos x+ sin x), then (d y)/(d x) is:2023
• 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)={ (1- cos x)/(2 x^2), if x ≠ 0; k, if x=0 is continuous at x=0.. OR If…2023
• If (x^2+y^2)^2=x y, then find (d y)/(d x).2023
• If f(x)= {x^2, if x ≥ 1; x, if x<1}, then show that f is not differentiable at x=1. OR Find the value(s) of ' λ ', if the…2023
• If y=(x+√(x^2-1))^2, then show that (x^2-1)((d y)/(d x))^2=4 y^2.2023
• 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).2023
• If y=√(a x+b), prove that y((d^2 y)/(d x^2))+(dy/dx)^2=0. OR If f(x)= {ax+b; 0<x ≤ 1; 2 x^2-x; 1<x<2} is a differentiable…2023
• 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…2023
• Let f(x) be a real valued function. Then its - Left Hand Derivative (L.H.D.): Lf^′(a)= lim_h → 0 (f(a-h)-f(a))/-h - Right Hand…2023
• 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…2023

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