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Exam: Maharashtra HSC Class 12
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Set · 4 questions
Q801
#801
Mathematics → Differentiation → Higher Order Derivatives
·
Hard
Add
If
y
=
sin
(
m
cos
−
1
x
)
y = \sin (m \cos^{-1} x)
y
=
sin
(
m
cos
−
1
x
)
then show that
(
1
−
x
2
)
d
2
y
d
x
2
−
x
d
y
d
x
+
m
2
y
=
0
(1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + m^2 y = 0
(
1
−
x
2
)
d
x
2
d
2
y
−
x
d
x
d
y
+
m
2
y
=
0
Show model answer
[QEx 1.5 Q.3 (ix) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q802
#802
Mathematics → Differentiation → Higher Order Derivatives
·
Hard
Add
If
y
=
log
(
log
2
x
)
y = \log (\log 2x)
y
=
lo
g
(
lo
g
2
x
)
, show that
x
y
2
+
y
1
(
1
+
x
y
1
)
=
0
x\, y_2 + y_1 (1 + x\, y_1) = 0
x
y
2
+
y
1
(
1
+
x
y
1
)
=
0
.
Show model answer
[QEx 1.5 Q.3 (x) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q803
#803
Mathematics → Differentiation → Higher Order Derivatives
·
Hard
Add
If
x
2
+
6
x
y
+
y
2
=
10
x^2 + 6xy + y^2 = 10
x
2
+
6
x
y
+
y
2
=
10
, show that
d
2
y
d
x
2
=
80
(
3
x
+
y
)
3
\frac{d^2y}{dx^2} = \frac{80}{(3x + y)^3}
d
x
2
d
2
y
=
(
3
x
+
y
)
3
80
.
Show model answer
[QEx 1.5 Q.3 (xi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q804
#804
Mathematics → Differentiation → Higher Order Derivatives
·
Hard
Add
If
x
=
a
sin
t
−
b
cos
t
x = a \sin t - b \cos t
x
=
a
sin
t
−
b
cos
t
,
y
=
a
cos
t
+
b
sin
t
y = a \cos t + b \sin t
y
=
a
cos
t
+
b
sin
t
, show that
d
2
y
d
x
2
=
−
x
2
+
y
2
y
3
\frac{d^2y}{dx^2} = -\frac{x^2 + y^2}{y^3}
d
x
2
d
2
y
=
−
y
3
x
2
+
y
2
.
Show model answer
[QEx 1.5 Q.3 (xii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Set · 12 questions
Find the
n
th
n^{\text{th}}
n
th
derivative of the following :
Q805
#805
Mathematics → Differentiation → Higher Order Derivatives
·
Moderate
Add
(
a
x
+
b
)
m
(ax + b)^m
(
a
x
+
b
)
m
Show model answer
[QEx 1.5 Q.4 (i) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q806
#806
Mathematics → Differentiation → Higher Order Derivatives
·
Easy
Add
1
x
\frac{1}{x}
x
1
Show model answer
[QEx 1.5 Q.4 (ii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q807
#807
Mathematics → Differentiation → Higher Order Derivatives
·
Easy
Add
e
a
x
+
b
e^{ax + b}
e
a
x
+
b
Show model answer
[QEx 1.5 Q.4 (iii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q808
#808
Mathematics → Differentiation → Higher Order Derivatives
·
Moderate
Add
a
p
x
+
q
a^{px + q}
a
p
x
+
q
Show model answer
[QEx 1.5 Q.4 (iv) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q809
#809
Mathematics → Differentiation → Higher Order Derivatives
·
Moderate
Add
log
(
a
x
+
b
)
\log (ax + b)
lo
g
(
a
x
+
b
)
Show model answer
[QEx 1.5 Q.4 (v) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q810
#810
Mathematics → Differentiation → Higher Order Derivatives
·
Easy
Add
cos
x
\cos x
cos
x
Show model answer
[QEx 1.5 Q.4 (vi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q811
#811
Mathematics → Differentiation → Higher Order Derivatives
·
Moderate
Add
sin
(
a
x
+
b
)
\sin (ax + b)
sin
(
a
x
+
b
)
Show model answer
[QEx 1.5 Q.4 (vii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q812
#812
Mathematics → Differentiation → Higher Order Derivatives
·
Moderate
Add
cos
(
3
−
2
x
)
\cos (3 - 2x)
cos
(
3
−
2
x
)
Show model answer
[QEx 1.5 Q.4 (viii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q813
#813
Mathematics → Differentiation → Higher Order Derivatives
·
Moderate
Add
log
(
2
x
+
3
)
\log (2x + 3)
lo
g
(
2
x
+
3
)
Show model answer
[QEx 1.5 Q.4 (ix) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q814
#814
Mathematics → Differentiation → Higher Order Derivatives
·
Moderate
Add
1
3
x
−
5
\frac{1}{3x - 5}
3
x
−
5
1
Show model answer
[QEx 1.5 Q.4 (x) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q815
#815
Mathematics → Differentiation → Higher Order Derivatives
·
Hard
Add
y
=
e
a
x
⋅
cos
(
b
x
+
c
)
y = e^{ax} \cdot \cos (bx + c)
y
=
e
a
x
⋅
cos
(
b
x
+
c
)
Show model answer
[QEx 1.5 Q.4 (xi) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
Report
Q816
#816
Mathematics → Differentiation → Higher Order Derivatives
·
Hard
Add
y
=
e
8
x
⋅
cos
(
6
x
+
7
)
y = e^{8x} \cdot \cos (6x + 7)
y
=
e
8
x
⋅
cos
(
6
x
+
7
)
Show model answer
[QEx 1.5 Q.4 (xii) · Maharashtra State Board (Class 12) — Differentiation (Balbharati textbook)]
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Q817
#817
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Find the vector equation of the line passing through the point having position vector
3
i
^
+
4
j
^
−
7
k
^
3\hat{i}+4\hat{j}-7\hat{k}
3
i
^
+
4
j
^
−
7
k
^
and parallel to
6
i
^
−
j
^
+
k
^
6\hat{i}-\hat{j}+\hat{k}
6
i
^
−
j
^
+
k
^
.
Show model answer
[QMisc A Q.1 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q818
#818
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Find the vector equation of the line which passes through the point
(
3
,
2
,
1
)
(3, 2, 1)
(
3
,
2
,
1
)
and is parallel to the vector
2
i
^
+
2
j
^
−
3
k
^
2\hat{i}+2\hat{j}-3\hat{k}
2
i
^
+
2
j
^
−
3
k
^
.
Show model answer
[QMisc A Q.2 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q819
#819
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Moderate
Add
Find the Cartesian equations of the line which passes through the point
(
−
2
,
4
,
−
5
)
(-2, 4, -5)
(
−
2
,
4
,
−
5
)
and parallel to the line
x
+
2
3
=
y
−
3
5
=
z
+
5
6
\frac{x+2}{3}=\frac{y-3}{5}=\frac{z+5}{6}
3
x
+
2
=
5
y
−
3
=
6
z
+
5
.
Show model answer
[QMisc A Q.3 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q820
#820
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Obtain the vector equation of the line
x
+
5
3
=
y
+
4
5
=
z
+
5
6
\frac{x+5}{3}=\frac{y+4}{5}=\frac{z+5}{6}
3
x
+
5
=
5
y
+
4
=
6
z
+
5
.
Show model answer
[QMisc A Q.4 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q821
#821
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Moderate
Add
Find the vector equation of the line which passes through the origin and the point
(
5
,
−
2
,
3
)
(5, -2, 3)
(
5
,
−
2
,
3
)
.
Show model answer
[QMisc A Q.5 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q822
#822
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Moderate
Add
Find the Cartesian equations of the line which passes through points
(
3
,
−
2
,
−
5
)
(3, -2, -5)
(
3
,
−
2
,
−
5
)
and
(
3
,
−
2
,
6
)
(3, -2, 6)
(
3
,
−
2
,
6
)
.
Show model answer
[QMisc A Q.6 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q823
#823
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Moderate
Add
Find the Cartesian equations of the line passing through
A
(
3
,
2
,
1
)
A(3, 2, 1)
A
(
3
,
2
,
1
)
and
B
(
1
,
3
,
1
)
B(1, 3, 1)
B
(
1
,
3
,
1
)
.
Show model answer
[QMisc A Q.7 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q824
#824
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Hard
Add
Find the Cartesian equations of the line passing through the point
A
(
1
,
1
,
2
)
A(1, 1, 2)
A
(
1
,
1
,
2
)
and perpendicular to vectors
b
ˉ
=
i
^
+
2
j
^
+
k
^
\bar{b}=\hat{i}+2\hat{j}+\hat{k}
b
ˉ
=
i
^
+
2
j
^
+
k
^
and
c
ˉ
=
3
i
^
+
2
j
^
−
k
^
\bar{c}=3\hat{i}+2\hat{j}-\hat{k}
c
ˉ
=
3
i
^
+
2
j
^
−
k
^
.
Show model answer
[QMisc A Q.8 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q825
#825
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Hard
Add
Find the Cartesian equations of the line which passes through the point
(
2
,
1
,
3
)
(2, 1, 3)
(
2
,
1
,
3
)
and perpendicular to lines
x
−
1
1
=
y
−
2
2
=
z
−
3
3
\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}
1
x
−
1
=
2
y
−
2
=
3
z
−
3
and
x
−
3
=
y
2
=
z
5
\frac{x}{-3}=\frac{y}{2}=\frac{z}{5}
−
3
x
=
2
y
=
5
z
.
Show model answer
[QMisc A Q.9 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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