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Q2926
#2926
Mathematics → Indefinite Integration → Integration by Substitution — Algebraic, Trigonometric, and Composite Forms
·
Hard
What is
∫
(
cos
x
)
1.5
−
(
sin
x
)
1.5
sin
x
⋅
cos
x
d
x
\int \frac{(\cos x)^{1.5}-(\sin x)^{1.5}}{\sqrt{\sin x\cdot\cos x}}\,dx
∫
s
i
n
x
⋅
c
o
s
x
(
c
o
s
x
)
1.5
−
(
s
i
n
x
)
1.5
d
x
equal to?
Add
NDA Maths strategy
Concept: Trigonometric Substitutions and Identity Reductions
A
sin
x
−
cos
x
+
c
\sqrt{\sin x}-\sqrt{\cos x}+c
sin
x
−
cos
x
+
c
B
sin
x
+
cos
x
+
c
\sqrt{\sin x}+\sqrt{\cos x}+c
sin
x
+
cos
x
+
c
C
2
sin
x
+
2
cos
x
+
c
2\sqrt{\sin x}+2\sqrt{\cos x}+c
2
sin
x
+
2
cos
x
+
c
D
1
2
sin
x
+
1
2
cos
x
+
c
\frac{1}{2}\sqrt{\sin x}+\frac{1}{2}\sqrt{\cos x}+c
2
1
sin
x
+
2
1
cos
x
+
c
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[Q76 · Sep · 2022]
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Q2927
#2927
Mathematics → Differentiation → Differentiation Techniques — Chain Rule, Logarithmic, Composite Functions
·
Moderate
If
y
=
x
x
2
−
16
2
−
8
ln
∣
x
+
x
2
−
16
∣
y = \frac{x\sqrt{x^2-16}}{2} - 8\ln\left|x+\sqrt{x^2-16}\right|
y
=
2
x
x
2
−
16
−
8
ln
x
+
x
2
−
16
, then what is
d
y
d
x
\frac{dy}{dx}
d
x
d
y
equal to?
Add
NDA Maths strategy
A
x
x
2
−
16
x\sqrt{x^2-16}
x
x
2
−
16
B
x
−
x
2
−
16
x-\sqrt{x^2-16}
x
−
x
2
−
16
C
x
2
−
16
\sqrt{x^2-16}
x
2
−
16
D
4
x
2
−
16
4\sqrt{x^2-16}
4
x
2
−
16
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[Q77 · Sep · 2022]
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Q2928
#2928
Mathematics → Differentiation → Differentiation Techniques — Chain Rule, Logarithmic, Composite Functions
·
Moderate
If
y
=
(
x
x
)
x
y = (x^x)^x
y
=
(
x
x
)
x
, then which one of the following is correct?
Add
NDA Maths strategy
A
d
y
d
x
+
x
y
(
1
+
2
ln
x
)
=
0
\frac{dy}{dx}+xy(1+2\ln x)=0
d
x
d
y
+
x
y
(
1
+
2
ln
x
)
=
0
B
d
y
d
x
−
x
y
(
1
+
2
ln
x
)
=
0
\frac{dy}{dx}-xy(1+2\ln x)=0
d
x
d
y
−
x
y
(
1
+
2
ln
x
)
=
0
C
d
y
d
x
−
2
x
y
(
1
+
ln
x
)
=
0
\frac{dy}{dx}-2xy(1+\ln x)=0
d
x
d
y
−
2
x
y
(
1
+
ln
x
)
=
0
D
d
y
d
x
+
2
x
y
(
1
+
ln
x
)
=
0
\frac{dy}{dx}+2xy(1+\ln x)=0
d
x
d
y
+
2
x
y
(
1
+
ln
x
)
=
0
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[Q78 · Sep · 2022]
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Q2929
#2929
Mathematics → Trigonometric Identities → Maximum and Minimum of Trigonometric Expressions
·
Hard
What is the maximum value of
3
(
sin
x
−
cos
x
)
+
4
(
cos
3
x
−
sin
3
x
)
3(\sin x - \cos x) + 4(\cos^3 x - \sin^3 x)
3
(
sin
x
−
cos
x
)
+
4
(
cos
3
x
−
sin
3
x
)
?
Add
NDA Maths strategy
A
1
B
2
\sqrt{2}
2
C
3
\sqrt{3}
3
D
2
2
2
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[Q79 · Sep · 2022]
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Q2930
#2930
Mathematics → Applications of Integration → Area Between Two Curves and Intersection Points
·
Moderate
What is the area of the region (in the first quadrant) bounded by
y
=
1
−
x
2
y = \sqrt{1-x^2}
y
=
1
−
x
2
,
y
=
x
y = x
y
=
x
and
y
=
0
y = 0
y
=
0
?
Add
NDA Maths strategy
A
π
4
\frac{\pi}{4}
4
π
B
π
6
\frac{\pi}{6}
6
π
C
π
8
\frac{\pi}{8}
8
π
D
π
12
\frac{\pi}{12}
12
π
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[Q80 · Sep · 2022]
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Q2931
#2931
Mathematics → Applications of Integration → Area Bounded by a Curve, Lines, and Axes
·
Moderate
What is the area of the region bounded by
x
−
∣
y
∣
=
0
x - |y| = 0
x
−
∣
y
∣
=
0
and
x
−
2
=
0
x - 2 = 0
x
−
2
=
0
?
Add
Lever: Modulus / absolute value behaviour
A
1
B
2
C
4
D
8
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[Q81 · Sep · 2022]
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Q2932
#2932
Mathematics → Trigonometric Identities → Compound Angle Formulas
·
Moderate
If
f
(
α
)
=
sec
2
α
−
1
f(\alpha) = \sqrt{\sec^2\alpha - 1}
f
(
α
)
=
sec
2
α
−
1
, then what is
f
(
α
)
+
f
(
β
)
1
−
f
(
α
)
f
(
β
)
\frac{f(\alpha)+f(\beta)}{1-f(\alpha)f(\beta)}
1
−
f
(
α
)
f
(
β
)
f
(
α
)
+
f
(
β
)
equal to?
Add
Lever: Compound angle: sin/cos/tan(A ± B)
A
f
(
α
−
β
)
f(\alpha-\beta)
f
(
α
−
β
)
B
f
(
α
+
β
)
f(\alpha+\beta)
f
(
α
+
β
)
C
f
(
α
)
f
(
β
)
f(\alpha)f(\beta)
f
(
α
)
f
(
β
)
D
f
(
α
β
)
f(\alpha\beta)
f
(
α
β
)
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[Q82 · Sep · 2022]
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Q2933
#2933
Mathematics → Functions → Domain, Range, and Function Properties
·
Moderate
If
f
(
x
)
=
ln
(
x
+
1
+
x
2
)
f(x) = \ln(x+\sqrt{1+x^2})
f
(
x
)
=
ln
(
x
+
1
+
x
2
)
, then which one of the following is correct?
Add
NDA Maths strategy
Concept: Even and odd functions
A
f
(
x
)
+
f
(
−
x
)
=
0
f(x)+f(-x)=0
f
(
x
)
+
f
(
−
x
)
=
0
B
f
(
x
)
−
f
(
−
x
)
=
0
f(x)-f(-x)=0
f
(
x
)
−
f
(
−
x
)
=
0
C
2
f
(
x
)
=
f
(
−
x
)
2f(x)=f(-x)
2
f
(
x
)
=
f
(
−
x
)
D
f
(
x
)
=
2
f
(
−
x
)
f(x)=2f(-x)
f
(
x
)
=
2
f
(
−
x
)
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[Q83 · Sep · 2022]
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Q2934
#2934
Mathematics → Limits & Continuity → Limit Evaluation Techniques — L'Hôpital, Rationalization, Standard Forms
·
Moderate
What is
lim
x
→
0
x
1
−
cos
4
x
\lim_{x\to 0}\frac{x}{\sqrt{1-\cos4x}}
lim
x
→
0
1
−
c
o
s
4
x
x
equal to?
Add
NDA Maths strategy
A
1
2
2
\frac{1}{2\sqrt{2}}
2
2
1
B
−
1
2
2
-\frac{1}{2\sqrt{2}}
−
2
2
1
C
2
\sqrt{2}
2
D
Limit does not exist
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[Q84 · Sep · 2022]
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Q2935
#2935
Mathematics → Limits & Continuity → Limit Evaluation Techniques — L'Hôpital, Rationalization, Standard Forms
·
Moderate
What is
lim
x
→
π
/
2
4
x
−
2
π
cos
x
\lim_{x\to\pi/2}\frac{4x-2\pi}{\cos x}
lim
x
→
π
/2
c
o
s
x
4
x
−
2
π
equal to?
Add
NDA Maths strategy
A
−
4
-4
−
4
B
−
2
-2
−
2
C
2
2
2
D
4
4
4
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[Q85 · Sep · 2022]
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Q2936
#2936
Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Moderate
If
f
(
x
)
=
x
2
+
x
+
∣
x
∣
x
f(x) = \frac{x^2+x+|x|}{x}
f
(
x
)
=
x
x
2
+
x
+
∣
x
∣
, then what is
lim
x
→
0
f
(
x
)
\lim_{x\to 0} f(x)
lim
x
→
0
f
(
x
)
equal to?
Add
Lever: Modulus / absolute value behaviour
A
0
B
1
C
2
D
lim
x
→
0
f
(
x
)
\lim_{x\to0}f(x)
lim
x
→
0
f
(
x
)
does not exist
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[Q86 · Sep · 2022]
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Q2937
#2937
Mathematics → Limits & Continuity → Limit Evaluation Techniques — L'Hôpital, Rationalization, Standard Forms
·
Easy
What is
lim
h
→
0
sin
2
(
x
+
h
)
−
sin
2
x
h
\lim_{h\to 0}\frac{\sin^2(x+h)-\sin^2x}{h}
lim
h
→
0
h
s
i
n
2
(
x
+
h
)
−
s
i
n
2
x
equal to?
Add
NDA Maths strategy
A
sin
2
x
\sin^2x
sin
2
x
B
cos
2
x
\cos^2x
cos
2
x
C
sin
2
x
\sin2x
sin
2
x
D
cos
2
x
\cos2x
cos
2
x
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[Q87 · Sep · 2022]
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Q2938
#2938
Mathematics → Differentiation → Differentiation Techniques — Chain Rule, Logarithmic, Composite Functions
·
Moderate
Let
f
(
x
)
f(x)
f
(
x
)
be a function such that
f
′
(
x
)
=
g
(
x
)
f'(x) = g(x)
f
′
(
x
)
=
g
(
x
)
and
f
′
′
(
x
)
=
−
f
(
x
)
f''(x) = -f(x)
f
′′
(
x
)
=
−
f
(
x
)
. Let
h
(
x
)
=
{
f
(
x
)
}
2
+
{
g
(
x
)
}
2
h(x) = \{f(x)\}^2+\{g(x)\}^2
h
(
x
)
=
{
f
(
x
)
}
2
+
{
g
(
x
)
}
2
. Consider the following statements: 1.
h
′
(
3
)
=
0
h'(3) = 0
h
′
(
3
)
=
0
2.
h
(
1
)
=
h
(
2
)
h(1) = h(2)
h
(
1
)
=
h
(
2
)
Which of the statements given above is/are correct?
Add
NDA Maths strategy
A
1 only
B
2 only
C
Both 1 and 2
D
Neither 1 nor 2
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[Q88 · Sep · 2022]
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Q2939
#2939
Mathematics → Differentiation → Differentiation Techniques — Chain Rule, Logarithmic, Composite Functions
·
Moderate
If
y
=
ln
2
(
x
2
−
x
+
1
x
2
+
x
+
1
)
y = \ln^2\!\left(\frac{x^2-x+1}{x^2+x+1}\right)
y
=
ln
2
(
x
2
+
x
+
1
x
2
−
x
+
1
)
, then what is
d
y
d
x
\frac{dy}{dx}
d
x
d
y
at
x
=
0
x=0
x
=
0
equal to?
Add
NDA Maths strategy
A
−
2
-2
−
2
B
0
0
0
C
1
1
1
D
2
2
2
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[Q89 · Sep · 2022]
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Q2940
#2940
Mathematics → Differentiation → Differentiation Techniques — Chain Rule, Logarithmic, Composite Functions
·
Hard
If
d
d
x
(
1
+
x
4
+
x
8
1
−
x
2
+
x
4
)
=
a
x
+
b
x
3
\frac{d}{dx}\!\left(\frac{1+x^4+x^8}{1-x^2+x^4}\right) = ax+bx^3
d
x
d
(
1
−
x
2
+
x
4
1
+
x
4
+
x
8
)
=
a
x
+
b
x
3
, then which one of the following is correct?
Add
NDA Maths strategy
A
a
=
b
a=b
a
=
b
B
a
=
2
b
a=2b
a
=
2
b
C
a
+
b
=
0
a+b=0
a
+
b
=
0
D
2
a
=
b
2a=b
2
a
=
b
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[Q90 · Sep · 2022]
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Q2941
#2941
Mathematics → Application of Derivatives → Optimisation — Geometric, Trigonometric, AM-GM
·
Hard
Under which one of the following conditions does the function
f
(
x
)
=
(
p
sec
x
)
2
+
(
q
csc
x
)
2
f(x) = (p\sec x)^2+(q\csc x)^2
f
(
x
)
=
(
p
sec
x
)
2
+
(
q
csc
x
)
2
attain minimum value?
Add
Lever: AM-GM / mean inequalities (incl. x + 1/x ≥ 2)
A
tan
2
x
=
q
p
\tan^2x = \frac{q}{p}
tan
2
x
=
p
q
B
cot
2
x
=
q
p
\cot^2x = \frac{q}{p}
cot
2
x
=
p
q
C
tan
2
x
=
p
q
\tan^2x = pq
tan
2
x
=
pq
D
cot
2
x
=
p
q
\cot^2x = pq
cot
2
x
=
pq
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[Q91 · Sep · 2022]
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Q2942
#2942
Mathematics → Application of Derivatives → Optimisation — Geometric, Trigonometric, AM-GM
·
Easy
Where does the function
f
(
x
)
=
∑
j
=
1
7
(
x
−
j
)
2
f(x) = \sum_{j=1}^{7}(x-j)^2
f
(
x
)
=
∑
j
=
1
7
(
x
−
j
)
2
attain its minimum value?
Add
NDA Maths strategy
A
x
=
3.5
x=3.5
x
=
3.5
B
x
=
4
x=4
x
=
4
C
x
=
4.5
x=4.5
x
=
4.5
D
x
=
5
x=5
x
=
5
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[Q92 · Sep · 2022]
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Q2943
#2943
Mathematics → Application of Derivatives → Monotonicity, Extrema, and Critical Points
·
Moderate
Consider the following statements in respect of the function
f
(
x
)
=
{
∣
x
∣
+
1
,
0
<
∣
x
∣
≤
3
1
,
x
=
0
f(x) = \begin{cases}|x|+1, & 0<|x|\leq3\\1, & x=0\end{cases}
f
(
x
)
=
{
∣
x
∣
+
1
,
1
,
0
<
∣
x
∣
≤
3
x
=
0
: 1. The function attains maximum value only at x=3 2. The function attains local minimum only at x=0 Which of the statements given above is/are correct?
Add
Lever: Modulus / absolute value behaviour
A
1 only
B
2 only
C
Both 1 and 2
D
Neither 1 nor 2
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[Q93 · Sep · 2022]
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Q2944
#2944
Mathematics → Definite Integration → Properties of Definite Integrals — Symmetry, King's, Odd/Even
·
Moderate
What is
∫
0
1
ln
(
1
x
−
1
)
d
x
\int_0^1 \ln\!\left(\frac{1}{x}-1\right)dx
∫
0
1
ln
(
x
1
−
1
)
d
x
equal to?
Add
NDA Maths strategy
A
−
1
-1
−
1
B
0
0
0
C
1
1
1
D
ln
2
\ln 2
ln
2
Tap an option to check your answer.
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[Q94 · Sep · 2022]
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Q2945
#2945
Mathematics → Definite Integration → Fundamental Theorem, Periodic Integrals, and Leibniz Rule
·
Moderate
If
∫
0
π
/
2
(
sin
4
x
+
cos
4
x
)
d
x
=
k
\int_0^{\pi/2}(\sin^4x+\cos^4x)\,dx = k
∫
0
π
/2
(
sin
4
x
+
cos
4
x
)
d
x
=
k
, then what is the value of
∫
0
20
π
(
sin
4
x
+
cos
4
x
)
d
x
\int_0^{20\pi}(\sin^4x+\cos^4x)\,dx
∫
0
20
π
(
sin
4
x
+
cos
4
x
)
d
x
?
Add
NDA Maths strategy
A
k
k
k
B
10
k
10k
10
k
C
20
k
20k
20
k
D
40
k
40k
40
k
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[Q95 · Sep · 2022]
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Q2946
#2946
Mathematics → Definite Integration → Properties of Definite Integrals — Symmetry, King's, Odd/Even
·
Moderate
What is
∫
−
π
/
2
π
/
2
(
e
cos
x
sin
x
+
e
sin
x
cos
x
)
d
x
\int_{-\pi/2}^{\pi/2}(e^{\cos x}\sin x+e^{\sin x}\cos x)\,dx
∫
−
π
/2
π
/2
(
e
c
o
s
x
sin
x
+
e
s
i
n
x
cos
x
)
d
x
equal to?
Add
NDA Maths strategy
A
e
2
−
1
e
\frac{e^2-1}{e}
e
e
2
−
1
B
e
2
+
1
e
\frac{e^2+1}{e}
e
e
2
+
1
C
1
−
e
2
e
\frac{1-e^2}{e}
e
1
−
e
2
D
0
0
0
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[Q96 · Sep · 2022]
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Q2947
#2947
Mathematics → Applications of Integration → Area Between Two Curves and Intersection Points
·
Hard
What is the area of the region enclosed in the first quadrant by
x
2
+
y
2
=
π
2
x^2+y^2=\pi^2
x
2
+
y
2
=
π
2
,
y
−
sin
x
y-\sin x
y
−
sin
x
and
x
=
0
x=0
x
=
0
?
Add
NDA Maths strategy
A
π
3
4
−
1
\frac{\pi^3}{4}-1
4
π
3
−
1
B
π
3
4
−
2
\frac{\pi^3}{4}-2
4
π
3
−
2
C
π
2
2
−
1
\frac{\pi^2}{2}-1
2
π
2
−
1
D
π
2
4
−
2
\frac{\pi^2}{4}-2
4
π
2
−
2
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[Q97 · Sep · 2022]
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Q2948
#2948
Mathematics → Differential Equations → Order, Degree, and Solutions of ODE
·
Moderate
Consider the following statements: 1. The degree of the differential equation
d
y
d
x
+
cos
(
d
y
d
x
)
=
0
\frac{dy}{dx}+\cos\!\left(\frac{dy}{dx}\right)=0
d
x
d
y
+
cos
(
d
x
d
y
)
=
0
is 1. 2. The order of the differential equation
(
d
2
y
d
x
2
)
3
+
cos
(
d
y
d
x
)
=
0
\left(\frac{d^2y}{dx^2}\right)^3+\cos\!\left(\frac{dy}{dx}\right)=0
(
d
x
2
d
2
y
)
3
+
cos
(
d
x
d
y
)
=
0
is 2. Which of the statements given above is/are correct?
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NDA Maths strategy
A
1 only
B
2 only
C
Both 1 and 2
D
Neither 1 nor 2
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[Q98 · Sep · 2022]
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Q2949
#2949
Mathematics → Differential Equations → Formation of ODE from Curves and General Solutions
·
Moderate
What is the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis?
Add
NDA Maths strategy
A
x
d
y
d
x
+
2
y
=
0
x\frac{dy}{dx}+2y=0
x
d
x
d
y
+
2
y
=
0
B
x
d
y
d
x
−
2
y
=
0
x\frac{dy}{dx}-2y=0
x
d
x
d
y
−
2
y
=
0
C
y
d
x
d
y
+
2
x
=
0
y\frac{dx}{dy}+2x=0
y
d
y
d
x
+
2
x
=
0
D
y
d
x
d
y
−
2
x
=
0
y\frac{dx}{dy}-2x=0
y
d
y
d
x
−
2
x
=
0
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[Q99 · Sep · 2022]
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Q2950
#2950
Mathematics → Differential Equations → Solving and Verifying ODEs — Separable, IVP, and Applications
·
Moderate
What is the solution of the differential equation
(
d
y
−
d
x
)
+
cos
x
(
d
y
+
d
x
)
=
0
(dy-dx)+\cos x(dy+dx)=0
(
d
y
−
d
x
)
+
cos
x
(
d
y
+
d
x
)
=
0
?
Add
NDA Maths strategy
A
y
=
tan
(
x
2
)
−
x
+
c
y=\tan\!\left(\frac{x}{2}\right)-x+c
y
=
tan
(
2
x
)
−
x
+
c
B
y
=
1
2
tan
(
x
2
)
−
x
+
c
y=\frac{1}{2}\tan\!\left(\frac{x}{2}\right)-x+c
y
=
2
1
tan
(
2
x
)
−
x
+
c
C
y
=
2
tan
(
x
2
)
−
x
+
c
y=2\tan\!\left(\frac{x}{2}\right)-x+c
y
=
2
tan
(
2
x
)
−
x
+
c
D
y
=
tan
(
x
2
)
−
2
x
+
c
y=\tan\!\left(\frac{x}{2}\right)-2x+c
y
=
tan
(
2
x
)
−
2
x
+
c
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[Q100 · Sep · 2022]
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