Page No 11.16:
Question 1:
Find the second order derivatives of each of the following functions:
(i) x3 + tan x
(ii) sin (log x)
(iii) log (sin x)
(iv) ex sin 5x
(v) e6x cos 3x
(vi) x3 log x
(vii) tan−1x
(viii) x cos x
(ix) log (log x)
Answer:
(i) We have,
(ii) We have,
(iii) We have,
(iv) We have,
(v) We have,
(vi) We have,
(vii) We have,
(viii) We have,
(ix) We have,
Page No 11.16:
Question 2:
If y = e−x cos x, show that .
Answer:
Here,
Hence proved.
Page No 11.16:
Question 3:
If y = x + tan x, show that .
Answer:
Here,
Hence proved.
Page No 11.16:
Question 4:
If y = x3 log x, prove that .
Answer:
Here,
Hence proved.
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Question 5:
If y = log (sin x), prove that .
Answer:
Here,
Hence proved.
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Question 6:
If y = 2 sin x + 3 cos x, show that .
Answer:
Here,
Hence proved.
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Question 7:
If , show that .
Answer:
Here,
Hence proved.
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Question 8:
If x = a sec θ, y = b tan θ, prove that .
Answer:
Here,
Hence proved.
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Question 9:
If x = a (cos θ + θ sin θ), y = a (sin θ – θ cos θ), prove that
Answer:
We have,
From (i) and (ii), we have
Hence proved.
Page No 11.16:
Question 10:
If y = ex cos x, prove that .
Answer:
Here,
Hence proved.
Page No 11.16:
Question 11:
If x = a cos θ, y = b sin θ, show that .
Answer:
Here,
Hence proved.
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Question 12:
If x = a (1 − cos3 θ), y = a sin3 θ, prove that .
Answer:
Here,
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Question 13:
If x = a (θ + sin θ), y = a (1 + cos θ), prove that .
Answer:
Here,Hence proved.
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Question 14:
If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find .
Answer:
Here,
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Question 15:
If x = a(1 − cos θ), y = a(θ + sin θ), prove that .
Answer:
Here,
Hence proved.
Page No 11.17:
Question 16:
If x = a (1 + cos θ), y = a(θ + sin θ), prove that .
Answer:
Here,
Page No 11.17:
Question 17:
If x = cos θ, y = sin3 θ, prove that .
Answer:
Here,
Hence proved.
Page No 11.17:
Question 18:
If y = sin (sin x), prove that .
Answer:
Here,
Hence proved.
Page No 11.17:
Question 19:
If x = sin t, y = sin pt, prove that .
Answer:
Here,
.
Hence proved.
Page No 11.17:
Question 20:
If y = (sin−1x)2, prove that (1 − x2) .
Answer:
Here,
Hence proved.
Page No 11.17:
Question 21:
If , prove that (1 + x2)y2 + (2x − 1)y1 = 0.
Answer:
Here,
Hence proved.
Page No 11.17:
Question 22:
If y = 3 cos (log x) + 4 sin (log x), prove that x2y2 + xy1 + y = 0.
Answer:
Here,
Hence proved.
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Question 23:
If , show that .
Answer:
Given,
To prove:
Proof:
We have,
...(i)
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Question 24:
If , show that (1 − x2)y2 − xy1 − a2y = 0.
Answer:
Here,
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Question 25:
If log y = tan−1x, show that (1 + x2)y2 + (2x − 1) y1 = 0
Answer:
Here,
Hence proved.
Page No 11.17:
Question 26:
If y = tan−1x, show that .
Answer:
Here,
Hence proved.
Page No 11.17:
Question 27:
If , show that .
Answer:
Here,
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Question 28:
If y = (tan−1x)2, then prove that (1 + x2)2y2 + 2x(1 + x2)y1 = 2.
Answer:
Here,
Hence proved.
Page No 11.17:
Question 29:
If y = cot x show that .
Answer:
Here,
Hence proved.
Page No 11.17:
Question 30:
Find , where .
Answer:
Here,
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Question 31:
If y = ae2x + be−x, show that, .
Answer:
Here,
Hence proved.
Page No 11.17:
Question 32:
If y = ex (sin x + cos x) prove that .
Answer:
Here,
Hence proved.
Page No 11.17:
Question 33:
If y = cos−1x, find in terms of y alone.
Answer:
Here,
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Question 34:
If , prove that .
Answer:
Here,
Hence proved.
Page No 11.17:
Question 35:
If y = 500 e7x + 600 e−7x, show that .
Answer:
Here,
Page No 11.17:
Question 36:
If x = 2 cos t − cos 2t, y = 2 sin t − sin 2t, find .
Answer:
Here,
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Question 37:
If x = 4z2 + 5, y = 6z2 + 7z + 3, find .
Answer:
Here,
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Question 38:
If y log (1 + cos x), prove that
Answer:
Here,
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Question 39:
If y = sin (log x), prove that .
Answer:
Here,
Page No 11.18:
Question 40:
If y = 3 e2x + 2 e3x, prove that
Answer:
Here,
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Question 41:
If y = (cot−1x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2.
Answer:
Here,
Hence proved.
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Question 42:
If y = cosec−1x, x >1, then show that .
Answer:
Here,
Hence proved.
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Question 43:
Answer:
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Question 44:
Answer:
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Question 45:
Answer:
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Question 46:
Answer:
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Question 47:
Answer:
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Question 48:
If find
Answer:
We have,
Also,
Now,
So,
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Question 49:
Answer:
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Question 50:
Answer:
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Question 51:
Answer:
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Question 52:
Disclaimer: There is a misprint in the question. It must be
instead of
.
Answer:
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Question 53:
Disclaimer: There is a misprint in the question,
must be written instead of
Answer:
Page No 11.22:
Question 1:
If x = a cos nt − b sin nt, then is
(a) n2x
(b) −n2x
(c) −nx
(d) nx
Answer:
(b) −n2x
Here,
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Question 2:
If x = at2, y = 2 at, then
(a)
(b)
(c)
(d)
Answer:
(d)
Here,
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Question 3:
If y = axn+1 + bx−n, then
(a) n (n − 1)y
(b) n (n + 1)y
(c) ny
(d) n2y
Answer:
(b) n(n+1)y
Here,
Page No 11.22:
Question 4:
(a) 2
20 (cos 2
x − 2
20 cos 4
x)
(b) 2
20 (cos 2
x + 2
20 cos 4
x)
(c) 2
20 (sin 2
x + 2
20 sin 4
x)
(d) 2
20 (sin 2
x − 2
20 sin 4
x)
Answer:
(b) 220(cos2x + 220cos4x)
Here,
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Question 5:
If x = t2, y = t3, then
(a) 3/2
(b) 3/4t
(c) 3/2t
(d) 3t/2
Answer:
(b) 3/4t
Here,
Page No 11.22:
Question 6:
If y = a + bx2, a, b arbitrary constants, then
(a)
(b)
(c)
(d)
Answer:
(b)
Here,
Page No 11.22:
Question 7:
If f(x) = (cos x + i sin x) (cos 2x + i sin 2x) (cos 3x + i sin 3x) ...... (cos nx + i sin nx) and f(1) = 1, then f'' (1) is equal to
(a)
(b)
(c)
(d) none of these
Answer:
(c)
Here,
Page No 11.22:
Question 8:
If y = a sin mx + b cos mx, then is equal to
(a) −m2y
(b) m2y
(c) −my
(d) my
Answer:
(a) −m2y
Here,
Page No 11.22:
Question 9:
If , then (1 − x)2f '' (x) − xf(x) =
(a) 1
(b) −1
(c) 0
(d) none of these
Answer:
(a) 1
Here,
DISCLAIMER : In the question instead of (1 − x)2f '' (x) − xf(x)
it should be (1 − x)2f ' (x) − xf(x)
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Question 10:
If , then
(a) 2
(b) 1
(c) 0
(d) −1
Answer:
(c) 0
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Question 11:
Let f(x) be a polynomial. Then, the second order derivative of f(ex) is
(a) f'' (ex) e2x + f'(ex) ex
(b) f'' (ex) ex + f' (ex)
(c) f'' (ex) e2x + f'' (ex) ex
(d) f'' (ex)
Answer:
(a) f''(ex)e2x + f'(ex)ex
Since f(x) is a polynomial,
Page No 11.23:
Question 12:
If y = a cos (logex) + b sin (logex), then x2y2 + xy1 =
(a) 0
(b) y
(c) −y
(d) none of these
Answer:
(c) −y
Here,
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Question 13:
If x = 2 at, y = at2, where a is a constant, then is
(a) 1/2a
(b) 1
(c) 2a
(d) none of these
Answer:
(a) 1/2a
Here,
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Question 14:
If x = f(t) and y = g(t), then is equal to
(a)
(b)
(c)
(d)
Answer:
(a)
Here,
x = f(t) and y = g(t)
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Question 15:
If y = sin (m sin−1x), then (1 − x2) y2 − xy1 is equal to
(a) m2y
(b) my
(c) −m2y
(d) none of these
Answer:
(c)−m2y
Here,
Page No 11.23:
Question 16:
If y = (sin−1x)2, then (1 − x2)y2 is equal to
(a) xy1 + 2
(b) xy1 − 2
(c) −xy1+2
(d) none of these
Answer:
(a) xy1 + 2
Here,
Page No 11.23:
Question 17:
If y = etanx, then (cos2x)y2 =
(a) (1 − sin 2x) y1
(b) −(1 + sin 2x)y1
(c) (1 + sin 2x)y1
(d) none of these
Answer:
(c) (1 + sin 2x)y1
Here,
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Question 18:
If , then
(a)
(b)
(c)
(d)
Answer:
Disclaimer: The question given in the book is wrong.
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Question 19:
If , then (2xy1 + y)y3 =
(a) 3(xy2 + y1)y2
(b) 3(xy1 + y2)y2
(c) 3(xy2 + y1)y1
(d) none of these
Answer:
(a) 3(xy2 + y1)y2
Here,
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Question 20:
If , then x3y2 =
(a) (xy1 − y)2
(b) (1 + y)2
(c)
(d) none of these
Answer:
(a) (xy1 − y)2
Here,
Page No 11.23:
Question 21:
If x = f(t) cos t − f' (t) sin t and y = f(t) sin t + f'(t) cos t, then
(a) f(t) − f''(t)
(b) {f(t) − f'' (t)}2
(c) {f(t) + f''(t)}2
(d) none of these
Answer:
(c){f(t) + f''(t)}2
Here,
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Question 22:
If
Answer:
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Question 23:
If , then the value of ar, 0 < r ≤ n, is equal to
(a)
(b)
(c)
(d) none of these
Answer:
(c)
According to the given equation,
Page No 11.23:
Question 24:
If y = xn−1 log x then x2y2 + (3 − 2n) xy1 is equal to
(a) −(n − 1)2y
(b) (n − 1)2y
(c) −n2y
(d) n2y
Answer:
(a) −(n − 1)2y
Here,
Page No 11.23:
Question 25:
If xy − logey = 1 satisfies the equation , then λ =
(a) −3
(b) 1
(c) 3
(d) none of these
Answer:
(c) 3
Here,
Page No 11.24:
Question 26:
If y2 = ax2 + bx + c, then is
(a) a constant
(b) a function of x only
(c) a function of y only
(d) a function of x and y
Answer:
(a) a constant
Here,
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Question 27:
If then is equal to
(a) 25 y (b) 5 y (c) –25 y (d) 15 y
Answer:
Given .
Differentiating both sides w.r.t. x, we get
Hence, the correct answer is option (a).
Page No 11.24:
Question 28:
If then equals
(a)
Answer:
Given:
Rewriting the given equation
Differentiating both sides of the above equation w.r.t. x, we get
Differentiating again w.r.t. x, we get
Hence, the correct answer is option (d).
Page No 11.24:
Question 1:
If y = t10 + 1 and x = t8 + 1, then = ___________________.
Answer:
Given,
y =
t10 + 1 and
x =
t8 + 1.
Differentiating both sides with respect to
t, we get
Differentiating both sides with respect to
t, we get
Differentiating both sides with respect to
x, we get
If
y =
t10 + 1 and
x =
t8 + 1, then
=
.
Page No 11.24:
Question 2:
If x = a sin θ and y = b cos θ, then = ______________________.
Answer:
Given,
and
.
Differentiating both sides with respect to
θ, we get
Differentiating both sides with respect to
θ, we get
Differentiating both sides with respect to
x, we get
If
x = a sin θ and
y =
b cos θ, then
=
.
Page No 11.24:
Question 3:
If y = x + ex, then = _____________________.
Answer:
Differentiating both sides with respect to
x, we get
Again differentiating both sides with respect to
x, we get
If
y = x +
ex, then
=
.
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Question 4:
If
Answer:
Differentiating both sides with respect to
x, we get
Again differentiating both sides with respect to
x, we get
If
Page No 11.24:
Question 5:
If y = x + ex , then = ______________.
Answer:
Differentiating both sides with respect to
x, we get
Differentiating both sides with respect to
y, we get
If
y = x + ex , then
=
.
Page No 11.24:
Question 6:
If y = loge x, then = _____________ .
Answer:
Given:
Now,
Thus, the value of is
Page No 11.24:
Question 1:
If y = a xn + 1 + bx−n and , then write the value of λ.
Answer:
Here,
Page No 11.24:
Question 2:
If x = a cos nt − b sin nt and , then find the value of λ.
Answer:
Here,
Page No 11.24:
Question 3:
If x = t2 and y = t3, find.
Answer:
Here,
Page No 11.25:
Question 4:
If x = 2at, y = at2, where a is a constant, then find .
Answer:
Here,
Page No 11.25:
Question 5:
If x = f(t) and y = g(t), then write the value of .
Answer:
Here.
x = f(t) and y = g(t)
Page No 11.25:
Question 6:
If .....to ∞, then write in terms of y.
Answer:
Here,
Page No 11.25:
Question 7:
If y = x + ex, find .
Answer:
Here,
Page No 11.25:
Question 8:
If y = |x − x2|, then find .
Answer:
Here,
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Question 9:
If , find .
Answer:
Here,
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