CONTINUITY & DIFFERENTIABLE
1) Show that f(x)= x³ is continuous at x= 2.
2) Show that f(x)=[ x] is not continuous at x= n, where n is any integer.
3) Show that f(x)={ x, if x is an integer
0, if x is not integer
is discontinuous at each integral value of x.
4) Show that f(x)={x/|x|, when x ≠ 0,
1, when x= 0
is discontinuous at x= 0.
5) If f(x)={ (x²-1)/(x -1) for x≠ 1
2 for x= 1
Show that f(x) is continuous at x= 1.
6) Determine the value of k for which the function
f(x)={ (sin5x)/3x, if x≠ 0
k, if x= 0
is continuous at x=0. 5/3
7) Show that the function is continuous at x= 0
f(x)={ x sin(1/x), when x ≠ 0
0, when x = 0
8) Let f(x)={ (sinx)/x + cosx, when x ≠ 0
2, when x= 0
Show that f(x) is continuous at x= 0.
9) Show that the function is discontinuous at x= 0
f(x)={ (sin²ax)/x, when x≠ 0
1, when x= 0
Redefine the function in such a way that it becomes continuous at x= a.
10) Is the function, f(x)= (3x + 4 tanx)/x continuous at x= 0 ? If not, how many the function be defined to make it continuous at this point ? No,
11) Discuss the continuity of the function
f(x)={ 3x -2, when x≤ 0
x +1, when x> 0 at x= 0.
12) Discuss the continuity of the function
f(x)={ - x, when x≤ 0
x, when 0< x ≤1
2-x, when 1< x<2
1, when x> 2
at each of the point x= 0,1,2. Y:0, y:1, n:2
13) Show that the function
f(x)={ 2x, if x< 2
2, if x= 2
x², if x> 2
has a removable discontinuity at x= 2.
CONTINUOUS FUNCTIONS
1) Let f(x)={ x if x≥ 1
x² if x< 1
Is a continuous function? Why ?
2) Prove that f(x)= |x | is a continuous function.
3) Discuss the continuity of the function.
f(x)= {2x -1, if x< 0;
2x +1, if x≥ 0.
4) Discuss the continuity of the function
f(x)= { (sinx)/x, if x < 0;
(x+1), if x≥ 0.
5) Discuss the continuity of the function
f(x)={ x/|x|, if x ≠ 0;
0, if x= 0.
6) Locate the point of discontinuity of the function
f(x)= { (x⁴-16)/(x -2), if x≠ 2
16, if x= 2.
7) Determine the value of k so that the function
f(x)={ kx², if x< 2;
3, if x> 2 is continuous.
8) Let f(x)={ 1, if x≤ 3;
ax+ b, if 3<x<5;
7, if 5≤ x.
Find the values of a and b so that f(x) is continuous.
9) Show that the function f(x)= √(x⁴+3) is continuous at each point.
10) Show that the function f(x)= |sinx + cosx| is continuous at x=π.
DIFFERENTIABLE
DIFFERENTIATION
DIFFERENTIATION
Raw- 1
1) ₑtan⁻¹{(y- x²)/x²} then dy/dx is
a) 2x[1+ tan(logx)]+ x sec²(logx)
b) x[1+ tan(logx)]+ sec²(logx)
c) 2x[1+ tan(logx)]+ x² sec²(logx)
d) 2x[1+ tan(logx)]+ sec²(logx)
2) If x= a cos⁴θ, y= a sin⁴θ then dy/dx at θ= 3π/4 is
a) 0 b) 1 c) -1 d) -2
3) d/dx(xˣ) is
a) xˣ(1- logx)
b) xˣ logx
c) xˣ⁺¹(1+ logx)
d) xˣ(1+ logx)
4) The differential coefficient of ₑx³ w.r.t. logx is
a) ₑx³ b) 3x³ ₑx³ c) 3x² ₑx³ d) 3x² ₑx³ + 3x²
5) The second derivative of a sin³ t w.r.t. a cos³t at t=π/4 is
a) 2 b) 1/12a c) 4√2/3a d) 0
6) The derivative of the function f(x)= 3|x +2| at the point, x= -3 is
a) -3 b) 3 c) 0 d) does not exist
7) If y= √[x + √{x + √(x+....... ∞ Then the value of dy/dx is
a) x/(2y -1)
b) 2/(2y -1)
c) 1/(2y -1)
d) x/(y -1)
8) If y= sinx + eˣ, then the value of d²x/dy² is
a) (sinx - eˣ)/(cosx + eˣ)
b) 1/(eˣ - sinx)
c) (sinx - eˣ)/(cosx + eˣ)²
d) (sinx + eˣ)/(cosy + eˣ)³
9) If sin⁻¹x + sin⁻¹y =π/2, then dy/dx is
a) x/y b) -x/y c) y/x d) - y/x
10) If dx/dy = u and d²x/dy² = c, then d²y/dx² is
a) - v/u² b) v/u² c) -v/u³ d) v/u³
Raw- 2
1) If y= √sin√x then dy/dx is
a) 1/(2√sin√x)
b) (√cos√x)/2x
c) 1/(2√cos√x)
d) (√cos√x)/(4√x √sin√x)
2) If f(x)= cos⁻¹[(1- (logx)²)/(1+ (logx)²)], then the value of f'(e) is
a) 2/e b) 1/e c) 1 d) 1/e²
3) If y= a cos mx - b sin mx, then d²y/dx² is
a) - m²y b) m²y c) -my d) my
4) If y= ₓeˣ, then dy/dx is
a) y(logx + eˣ)
b) ylogx( 1/2 + eˣ)
c) yeˣ(logx + 1/x)
d) yeˣ(logx + x)
5) If 2ˣ + 2ʸ= 2ˣ⁺ʸ, then dy/dx at x= y= 1 is
a) 0 b) -1 c) 1 d) 2
6) If x= sin⁻¹t, y= log(1- t²), 0≤ t < 1, then the value of d²y/dx² at t= 1/3 is
a) -9/4 b) -9/8 c) 9/4 d) 9/8
7) If y= {x + √(1+ x²)}ⁿ, then (1+ x²) d²y/dx² + x dy/dx is
a) -y b) n²y c) - n²y d) 2n²y
8) If siny + ₑ- x cosy = e Then the value of dy/dx at (1,π) is
a) 0 b) 1 c) e d) -1
9) If x= 2 cos t + cos2t and y= 2 sin t - sin2t, then dy/dx at t=π/4 is
a) -(√2+1) b) √2 c) (√2- 1) d) 1- √2
10) If logx = z, then x² d²y/dx² is
a) d²y/dz² b) d²y/dz² + dy/dz d) d²y/dz² - 2dy/dz
Raw - 3
1) The value of x, at which the first derivative of the function (x + 1/x) w.r.t. x is 3/4, are
a) ±1/2 b) ±2/√3 c) ±√3/2 d) ±2
2) If f(x)= sin3x cos4x then f"(π/2) is
a) 24 b) 25 c) -25 d) -24
3) If y= x + x²+ x³+ .....∞ where |x|< 1, then for |y|< 1 the value of dx/dy is
a) 1- 2y + 3y² -......∞
b) y + y² + y²+.....∞
c) 1- y + y² -y³+.....∞
d) 1+ 2y + 3y² +......∞
4) The derivative of sec⁻¹{1/(2x²-1)} w.r.t √(1- x²) at x = 1/2 is
a) 2 b) 4 c) 1 d) -2
5) The derivative of log₅(log₇x)(x > 7) is
a) 1/xlogₑx
b) 1/(xlogₑ5logₑ7)
c) 1/(xlogₑ5logₑ7log₇x) d) none
6) If 2y = (x - a) √(2ax - x²)+ a² sin⁻¹{(x - a)/a}, then the value of dy/dx is
a) √(ax - x²) b) √(x²- ax) c) √(x²- 2ax) d) √(2ax - x²)
7) If y= x³ then the value of d²y/dx²/{1+ (dy/dx)²}³/² at the point (1,1) is
a) 3/5√10 b) 5/3√10 c) 4/3√10 d) 3/4√10
8) If x= 1/z, y= f(x) and d²y/dx² = kz³ dy/dz + z⁴d²y/dz², then find k
a) -1 b) 1 c) 2 d) -2
9) If xy = ax²+ b/x, then the value of x² d²y/dx²+ 2x dy/dx is
a) y b) - y c) -2y.d) 2y
Paper 4
1) The derivative of (secx + tanx)(secx - tanx) is
a) 2secx(secx + tanx)
b) 2sec¹x(secx + tanx)²
c) 2secx(secx + tanx)²
d) secx(secx + tanx)
2) If y= sinx° and z= log₁₀x, then dy/dx is
a) x°cosx°/log₁₀e
b) xcosx°/logₑ10
c) xcosx°/log₁₀e
d) x°cosx°/logₑ10
3) The value of d/dx[tan⁻¹{√x(3- x)/(1- 3x)}] is
a) 3/2(1-x)√x
b) 3/2(1+ x)√x
c) 2/(1 + x)√x
d) 3/(1 + x)√x
4) If x= sin t and y= cos pt, then which of the following is true?
a) (1- x²)y₂ + xy₁ + p²y=0
b) (1- x²)y₂ + xy₁ - p²y=0
c) (1 + x²)y₂ + xy₁ + p²y=0
d) (1- x²)y₂ - xy₁ + p²y=0
5) If y= √ₓ√x√ˣ....∞ and x dy/dx = f(y)/(2- y logx), then the value of f(y) is
a) y log y b) log y c) 2y d) y²
6) If x= eᵗ sin t and y= eᵗ cos t, then the value of (x + y)² d²y/dx² - 2x dy/dx id
a) 2y b) -2y c) 4y d) -4y
7) If y= f(x)² and f'(x)= √(3x²+1) then the value of dy/dx at x= 2 is
a) 4√13 b) 2√13 c) 28 d) 14
8) If x= secθ - cosθ, y= secⁿθ - cosⁿθ and (x²+ 4)(dy/dx)²= k(y²+4) then the value of k is
a) n² b) 2n c) - n² d) -2n
9) If y= 4ax, then the value of d²y/dx². dy/dx is
a) 2a/y³ b) - 2a/y³ c) - a/y³ d) a/y³
Paper 5
1) If f(x)= logₓ (logₑx) then f'(e) is
a) 0 b) e c) 1/e d) 1/e²
2) If y= (cos⁻¹x)², then the value of (1- x²)y₂ - xy₁ is
a) 4 b) 2 c) y d) 2y
3) If y= sin(x²), z= ₑy¹ and t= √z, then the value of dt/dx is
a) xyz/t b) 2xyz/t c) (2xyz cos(x²))/t d) (-xyz cos(x²))/t
4) If y= aˣ b²ˣ⁻¹, then the value of d²y/dx² is
a) y(log ab²)² b) y(log ab²) c) y(log a½b)² d) y²(log ab²)
5) If y= √[sinx +√{sinx + √sinx+........∞, then dy/dx is
a) - cosx/(2y -1)
b) sinx/(1- 2y)
c) - sinx/(1- 2y)
d) cosx/(2y -1)
6) If y= logₐx + logₓa + logₓx + logₐa, then the value of dy/dx is
a) (logx)/x + x/loga
b) 1/x(loga) - log a/x(logx)²
c) 1/x(logx)
d) 1/x + xloga
7) If y= ₑ{(1/2) log(1+ tan²x)}, then dy/dx is
a) ₑ{log(1+ tan²x)} b) sec²x c) (1/2) aec²x d) secx tanx
8) If (x + y)ᵐ⁺ⁿ= xᵐ yⁿ, then the value of dy/dx is
a) x/y b) xy c) y/x d) - x/y
9) If x+ y = eˣ⁻ʸ then d²y/dx² is
a) 4(x +y)/(x + y +1)³
b) 2(x +y)/(x + y +1)³
c) 4(x +y)/(x + y +1)²
d) 2(x +y)/(x + y +1)²
10) If x= 2 cosθ - cos2θ and y= 2 sinθ - sin2θ, then the value of d²y/dx² at θ =π/2 is
a) -2 b) -3/2 c) 3/2 d) 0
Test paper - 6
1) If y= sin(πeˣʸ/6), then the value of dy/dx at x= 0 is
a) √3/24 b) √3π/24 c) √3/12 d) √3/12
2) If y= log(tan x/2)+ sin⁻¹(cosx) then dy/dx is
a) secx +1 b) cosecx +1 c) cosecx -1 d) secx -1
3) If y= 1/(1+ x + x²+ x³) then d²y/dx² at x= 0 is
a) 0 b) 1 c) -1 d) 2
4) If y= sinx log(tan x/2) then d²y/dx²+ y is
a) - cotx b) tan x c) cot x d) - tan x
5) If t is a parameter and x= t²+ 2t, y= t³- 3t, then d²y/dx² at t = 1 is
a) -3/8 b) 3/8 c) -3/4 d) 3/4
6) d/dx[sin²cot⁻¹√{(1- x)/1+ x)}] is equal to
a) -1/2 b) -1 c) 1 d) 1/2
7) If x= sin⁻¹(3t - 4t³) and y= cos⁻¹√(1- t²), then dy/dx is
a) 1/3 b) 1/2 c) 2 d) 3/2
8) The derivative of sin²x w.r.t cos²x is
a) tan²x b) tanx c) - tanx d) none
9) If y= cosec⁻¹{(x +1)/(x -1)}+ cos⁻¹{(x - 1)/(x + 1)} then dy/dx
a) 1 b) π c) 0 d) π/2
10) If y= tan⁻¹[{√(1+ x²)-1}/x] and tan⁻¹{2x/(1- x²)}, then dy/dx is
a) 1/8 b) 1/4 c) 1/2 d) 1
11) If f(x)= 1/(1- x), then the derivatives of the composite function f[f{f(x)}] is equal to
a) 1 b) 1/2 c) 0 d) 2
Paper - 7
1) if eʸ + xy= e² then d²y/dx² at x= 0 is
a) 0 b) 1 c) 1/e² d) 1/e
2) Derivative of sin⁻¹x w.r.t cos⁻¹√(1- x²) is
a) 1/(1- x²) b) cos⁻¹x c) 1 d) none
3) If x= a(θ+ sinθ), y= a(1- cosθ), then the value of dy/dx is
a) tanθ b) tan θ/2 c) cotθ d) cot θ/2
4) If y= 2 sin⁻¹{(x -2)/√6} - √(2+ 4x - x³), the, the value of dy/dx at x= 2 is
a) -2/√6 b) 1/√6 c) 2/√6 d) √6/2
5) If y= sec(tan⁻¹x), then dy/dx is
a) x/√(1+ x²)
b) - x/√(1+ x²)
c) x/√(1- x²) d) none
6) If y= (x sin⁻¹x)/√(1- x²)+ log√(1- x²), then the value of d²y/dx² at x= 0 is
a) 0 b) 1/2 c) 1 d) 2
7) Observe the following statements:
i) f(x)= ax⁴¹+ bx⁻⁴⁰=> f"(x)/f(x)= 1640x⁻²
ii) d/dx {2x/(1- x²)= 1/(1+ x²)
Which of the following is correct?
a) I is true but ii is false
b) both i and ii are true
c) neither I nor ii is true
d) I is false but ii is true
PAPER- 8
1) If x√(1- y²)+ y√(1- x²)= k, then dy/dx at x= 0 is
a) √(1- k²) b) - √(1- k²) c) k d) - k
2) If x= a cotθ and y= 1/(x²+ a²), then the value of d²y/dx² at θ= π/6 is
a) 1/4a⁴ b) - 1/4a⁴ c) 2/a⁴ d) - 2/a⁴
3) If y= (sin⁻¹x)²+ (cos⁻¹x)², then which of the following is correct?
a) (1- x²)y₂ + xy₁ +4=0
b) (1+ x²)y₂ - xy₁ -4=0
c) (1- x²)y₂ - xy₁ -4=0
d) (1- x²)y₂ - xy₁ +4=0
4) y= ₑmsin⁻¹x, then the value of [d²y/dx²]ₓ₌₀ is
a) m²ₑmx b) - m²ₑmx c) m² d) - m²
5) If x√(1+ y)+ y√(1+ x)= 0 Then dy/dx is
a) 1/(1+ x²) b) - 1/(1+ x²) c) 1/(1+ x)² d) -1/(1+ x)²
6) If x²+ y²= t + 1/t and x⁴+ y⁴= t²+ 1/t², then the value of - x³ydy/dx is
a) 1 b) 1/2 c) 1/3 d) 1/4
7) If sin y = x sin(a + y), then dy/dx is
a) sin(a+ y)
b) {sin²(a+ y)}/sin a
c) {sin²(a+ y)}/cosa
d) {sin²(a+ y)}/sin y
8) If y= tan⁻¹[{√(1+ x²) - √(1- x²)}/{√(1+ x²) - √(1- x²)}], then dy/dx is
a) x²/√(1- x⁴)
b) x²/√(1+ x⁴)
c) x/√(1- x⁴)
d) x/√(1+ x⁴)
9) If x sin y + y cos x=π, then the value of y"(0) is
a) π b) -π c) 1 d) 0
Paper -9
1) The derivative of the function tan⁻¹[{2x √(1- x²)}/(1- 2x²)] w.r.t the function tan⁻¹[{√(1+ x²)-1}/x] at x= 0 is
a) 1 b) 2 c) 4 d) 8
2) If f(x) g(x)= k (a constant) and g"(x)/g'(x) = f"(x)/f'(x), then the value of a is
a) 4 b) -4 c) 2 d) -2
3) If 2x = y¹⁾⁵ + y⁻¹⁾⁵ and (x²-1)y₂ + xy₁= ky, then the value of k is
a) 5 b) -5 c) 25 d) -25
4) If y= 2ˣ. 3²ˣ⁻¹, then the value of d²y/dx² is equal to
a) log2 log3 b) (log18)²y c) (log18)y d) (kog24) y
5) if log(x + y)= 2xy, then the value of y'(0) is
a) 1 b) -1 c) 2 d) 0
6) If y= logⁿx where logⁿ means log log log .... repeated n times, then the value of x logx log²x og³x....logⁿx
ⁿ
₁₀ ₑ⁻¹²₂₁ₓ ˣᵗ ᵗ ²²²²²θ θ θθθθθ ⁿ ²²²²³³³³³ ∞ eˣ⁻ʸ dy/dx
ₑ⁻¹²⁴ θ ˣ ˣ⁺¹³³ˣ ∞ ʸ ˣ⁺ʸ ⁿ
1) If x= √t + 1/√t, then 2t dx/dt + x is equal to
a) √t b) 2 √t c) 3 √t 4) none
2) If (x + 4)y= x then x dy/dx + y(y -1) is equal to
a) 1 b) 2 c) 1/2 d) none
3) If y= xⁿ⁻¹ log x, then x dy/dx + (1- n)y is equal to
a) xⁿ⁻¹ b) xⁿ⁻² c) nxⁿ⁻¹ d) none
4) If y= 6x⁴ + 4x³ - 24ax² - 36x + 21 and dy/dx|ₓ₌₂ = 12 then the value of a is
a) 1 b) 2 c) 3 d) 4
5) If y= 1/(1+ xᑫ⁻ᵖ + xʳ⁻ᵖ) + 1/(xᵖ⁻ᑫ + xʳ⁻ᑫ) + 1/(1+ xᵖ⁻ʳ + xᑫ⁻ʳ) , then the value of dy/dx is equal to
a) xᵖ⁺ᑫ⁺ʳ b) 1/(p+ q+ r) c) 1 d) 0
6) If y= (1+ sinx)/cos x, then cosx dy/dx is
a) sin²x b) 1+ y c) y d) none
7) If x² + y² = 4, then y dy/dx + x is
a) 4 b) 0 c) -1 d) none
8) If y= xˣ (x > 0), then dy/dx is
a) y(1+ logx) b) 1+ logx c) y logx d) x(1+ log y)
9) If y= 2x tan⁻¹x - log(1+ x²), then dy/dx = k tan⁻¹x where the value of k is
a) 1 b) 1/2 c) 1/3 d) 2
10) If y= sinx°, then dy/dx =π/180 cosθ where θ is equal to
a) x b) xᶜ c) x/180 d) πx/180
11) If y= logₑ(logₑx), then dy/dx = k/logₑx where k is equal to
a) x b) 1 c) 1/x d) none
12) Let y= sin⁻¹(cosx). Then
a) dy/dx = -1 where sinx < 0
b) dy/dx = 1 where sinx > 0
c) dy/dx does not exist at x=π
d) dy/dx does not exist anywhere
13) If cosx= (1- t²)/(1+ t²), tan y = (3t - t³)/1- 3t²) (0< t < 1/√3), then dy/dx is equal to
a) 1 b) 1/2 c) 3/2 d) 2
14) If x⁶y² = (x + y)⁸, then dy/dx is
a) x/y b) y/x c) - y/x d) 1
15) If x² + y² = t + 1/t and x⁴ + y⁴ = t² + 1/t², then dy/dx is
a) -1/x³y b) 1/x³y c) x³/y d) - y/x³
16) If y= cos[2 sin⁻¹(cosx)], then dy/dx is
a) 4 sinx cosx b) sin2x c) 2 cos2x d) none
17) If y= log(secx + tanx), then dy/dx is
a) sinx b) cosx c) secx d) cosecx
18) If y= log tan(π/2+ x/2), then dy/dx is
a) secx b) tanx c) cosecx d) none
19) If y= log[log(logx)], then dy/dx is
a) log(logx) b) logx log(logx) c) 1/(x logx log(logx)) d) 1/(logx log(logx))
20) If y= (tanx)ˢᶦⁿˣ, then dy/dx|ₓ₌π/4 is equal to
a) √2 b) 1/√2 c) 2 d) none
ˣʸʸₓ₌₀²³²⁻¹⁷³¹⁰²²⁻¹²²ₓ₌₁⁻¹²²²²⁻¹²²²³²²⁴⁴³ₓ₌₀ₓ₌₀ᵗ²²ₜ₌ₑₑₑ⁻¹²²²²²⁻¹²₂₁⁴²²ₓ₌₀ᵧ₌₁²²²²²³²ₑ⁻¹ₑ⁻¹ˣ²²₅₃₃₂₃₃₃³²³³³ⁿⁿⁿ⁻¹ʸˣ⁻ʸ²²²ᵐⁿᵐ⁺ⁿ⁻¹⁻¹³³∈∈∈∈ ²²²
∈∈∈∈∈
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