The number of solutions of the equation $2x + 3tan x = pi$ $2x + 3\tan x = \pi$, $xin[-2pi, 2pi] - left\pmfracpi2, pmfrac3pi2right\$ is[cite: 612, 613]:

A. 6
B. 5
C. 4
D. 3

Solution & Explanation

### Related Formula Intersection method for transcendental configurations:

f(x) = g(x)

Plot both curves separately to observe distinct intersection markers inside target domain spans.

Trigonometric Equations diagram for Q60 - JEE Main 2025 Morning

### Core Logic Rearrange terms to group equations into known standard graphing profiles [cite: 1321]:

3tan x = pi - 2x implies tan x = fracpi3 - frac2x3

[cite: 1321] Plot the linear equation line

y = fracpi3 - frac2x3

along with multiple period tracks of the trigonometric function

y = tan x

within the interval

[-2pi, 2pi]

[cite: 613, 1321]. ### Step 1: Point analysis across branches The line has a negative slope and passes through

(0, pi/3)

and

(3pi/2, 0)

. Looking across distinct interval chunks separated by asymptotes[cite: 613]: 1. Branch 1

left(-2pi, -frac3pi2right)

: 1 intersection 2. Branch 2

left(-frac3pi2, -fracpi2right)

: 1 intersection 3. Branch 3

left(-fracpi2, fracpi2right)

: 1 intersection near origin 4. Branch 4

left(fracpi2, frac3pi2right)

: 1 intersection 5. Branch 5

left(frac3pi2, 2piright)

: 1 intersection Counting all distinct points across valid domains gives 5 solutions total[cite: 1337]. ### Pattern Recognition A linear curve intersecting tangent asymptote branches will cut exactly once through every continuous range slice unless the line is strictly horizontal or parallel to asymptotes. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Trigonometric Functions

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Trigonometric Functions

More Trigonometric Functions Previous-Year Questions — Page 2

Q70 2025 Trigonometric Identities

10sin^4theta + 15cos^4theta = 6

, then the value of

frac27csc^6theta + 8sec^6theta16sec^8theta

is:

A. $frac25$ $\frac{2}{5}$
B. $frac34$ $\frac{3}{4}$
C. $frac35$ $\frac{3}{5}$
D. $frac15$ $\frac{1}{5}$

Solution

### Related Formula Trigonometric identity conversion:

cos^2theta = 1 - sin^2theta

### Core Logic Let

sin^2theta = t

. Substitute this into the given equation:

10t^2 + 15(1 - t)^2 = 6 implies 10t^2 + 15(1 - 2t + t^2) = 6

25t^2 - 30t + 9 = 0 implies (5t - 3)^2 = 0 implies t = frac35

Thus,

sin^2theta = frac35

and

cos^2theta = frac25

. ### Step 1: Simplify Target Expression Find individual terms from inverse relations:

csc^2theta = frac53 implies csc^6theta = frac12527

sec^2theta = frac52 implies sec^6theta = frac1258

sec^8theta = left(frac52right)^4 = frac62516

Substitute values into expression:

textNumerator = 27left(frac12527right) + 8left(frac1258right) = 125 + 125 = 250

textDenominator = 16left(frac62516right) = 625

### Step 2: Conclusion

textValue = frac250625 = frac25

### Pattern Recognition Equations structured as

Asin^4theta + Bcos^4theta = C

often yield perfect square trinomial combinations. Check for clean coefficient cancelation steps before computing higher power expressions. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Trigonometric Functions

Q55 2025 Trigonometric Equations and Solutions

Let

A=left\xin(0,pi)-left\fracpi2right\:log_(2/pi)|sin x|+log_(2/pi)|cos x|=2right\

and

B=left\xge0:sqrtx(sqrtx-4)-3|sqrtx-2|+6=0right\.

Then

n(Acup B)

is equal to:

A. $4$ $4$
B. $2$ $2$
C. $8$ $8$
D. $6$ $6$

Solution

### Related Formula Logarithmic addition property:

log(a) + log(b) = log(ab)

. Double \angle sine formula:

2sin x cos x = sin 2x

. ### Step 1: Simplify Set A Combine the logarithmic elements:

log_(2/pi) left(|sin x| cdot |cos x|right) = 2

|sin x cos x| = left(frac2piright)^2 = frac4pi^2

|2sin x cos x| = frac8pi^2 Rightarrow |sin 2x| = frac8pi^2

Since

pi^2 approx 9.87

frac8pi^2 approx 0.81

, which is less than 1. Plotting

|sin 2x| = frac8pi^2

over the specified range

x in (0, pi)

yields exactly 4 real intersection points[cite: 3314, 3906].

Trigonometric Equations graph for Q55 - JEE Main 2025 Evening

Hence,

n(A) = 4

. ### Step 2: Simplify Set B Let

sqrtx = t

where

t ge 0

. The equation becomes[cite: 3316, 3910]:

t(t-4) - 3|t-2| + 6 = 0

**Case I:** If

t < 2

t^2 - 4t - 3(-(t-2)) + 6 = 0 Rightarrow t^2 - 4t + 3t - 6 + 6 = 0 Rightarrow t^2 - t = 0

t = 0, 1 Rightarrow x = 0, 1

[cite: 3912, 3913] **Case II:** If

t > 2

t^2 - 4t - 3(t-2) + 6 = 0 Rightarrow t^2 - 4t - 3t + 6 + 6 = 0 Rightarrow t^2 - 7t + 12 = 0

[cite: 3915, 3916]

(t-3)(t-4) = 0 Rightarrow t = 3, 4 Rightarrow x = 9, 16

[cite: 3917, 3918] Hence, set

B = \0, 1, 9, 16\

, giving

n(B) = 4

. ### Step 3: Calculate Union Since all elements of set A are non-integral angles in

(0, pi)

and elements of set B are pure integers, the sets are completely disjoint (

A cap B = emptyset

n(A cup B) = n(A) + n(B) = 4 + 4 = 8

### Pattern Recognition Always separate functions into disjoint numeric domains (e.g., angles vs. whole integers) to conclude unions without performing tedious tracking of individual values manually. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Trigonometric Functions Class 11 Mathematics: Sets

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The number of solutions of the equation $2x + 3tan x = pi$ $2x + 3\tan x = \pi$, $xin[-2pi, 2pi] - left\pmfracpi2, pmfrac3pi2right\$ is[cite: 612, 613]:

Solution & Explanation

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The number of solutions of the equation 2x + 3tan x = pi$2x + 3\tan x = \pi$, xin[-2pi, 2pi] - left\pmfracpi2, pmfrac3pi2right\$x\in[-2\pi, 2\pi] - \left\{\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}\right\}$ is[cite: 612, 613]:

Solution & Explanation

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The number of solutions of the equation $2x + 3tan x = pi$ $2x + 3\tan x = \pi$, $xin[-2pi, 2pi] - left\pmfracpi2, pmfrac3pi2right\$ is[cite: 612, 613]: