Two Dimensional Geometry jee mains questions | JEE Main Two Dimensional Geometry Previous Year Question

Q 31 :
Let the range of the function $f (x) = 6 + 16 \cos x \cdot \cos (\frac{π}{3} - x) \cdot \cos (\frac{π}{3} + x) \cdot \sin 3 x \cdot \cos 6 x, x \in R$ be $[α, β]$ . Then the distance of the point $(α, β)$ from the line 3x + 4y + 12 = 0 is: [2025]

8
11
9
10

1

2

3

4

(2)

$f (x) = 6 + 16 (\frac{1}{4} \cos 3 x) \sin 3 x \cdot \cos 6 x$

$[∵ \cos θ \cos (\frac{π}{3} - θ) \cos (\frac{π}{3} + θ) = \frac{1}{4} \cos 3 θ]$

$= 6 + 4 \cos 3 x \sin 3 x \cos 6 x = 6 + \sin 12 x$

$∵$ Range of sin x = [–1, 1]

$∴$ Range of f(x) is [5, 7]

$\Rightarrow (α, β) \equiv (5, 7)$

$∴$ Distance of point (5, 7) from the line 3x + 4y + 12 = 0

$=$ $| \frac{15 + 28 + 12}{5} |$ $= 11$

Q 32 :
Let the lines 3x – 4y – $α$ = 0, 8x –11y – 33 = 0 and 2x – 3y + $λ$ = 0 be concurrent. If the image of the point (1, 2) in the line 2x – 3y + $λ$ = 0 is $(\frac{57}{13}, \frac{- 40}{13})$ , then $| α λ |$ is equal to [2025]

101
113
84
91

1

2

3

4

(4)

As the three lines are concurrent,

$| \begin{matrix} 3 & - 4 & - α \\ 8 & - 11 & - 33 \\ 2 & - 3 & λ \end{matrix} |$ $= 0$

$\Rightarrow 3 (- 11 λ - 99) + 4 (8 λ + 66) - α (- 24 + 22) = 0$

$\Rightarrow - 33 λ - 297 + 32 λ + 264 + 2 α = 0$

$\Rightarrow - λ - 33 + 2 α = 0$ ...(i)

As image, (1, 2) w.r.t. 2x – 3y + $λ$ = 0 is $(\frac{57}{13}, \frac{- 40}{13})$

$\frac{\frac{57}{13} - 1}{2} = \frac{\frac{- 40}{13} - 2}{- 3} = - 2 (\frac{2 - 6 + λ}{4 + 9})$

$\Rightarrow \frac{22}{13} = \frac{- 2 (- 4 + λ)}{13} \Rightarrow λ = - 7$

Substitute $λ$ = –7 in (i), we get

$7 - 33 + 2 α = 0 \Rightarrow a = 13$

$∴ | α λ | = | 13 \times (- 7) | = 91$

Q 33 :
Two equal sides of an isosceles triangle are along – x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is: [2025]

$- 2 \sqrt{10}$
12
–6
6

1

2

3

4

(4)

Slope of given lines are $m_{1} = \frac{1}{2}$ and $m_{2} = - 1$ .

Since, AB = AC, then $∠$ ABC = $∠$ ACB

Now, angle between AB and BC = angle between AC and BC

$\frac{m - \frac{1}{2}}{1 + \frac{1}{2} (m)} = \frac{- 1 - m}{1 - m}$

$\Rightarrow \frac{2 m - 1}{2 + m} = \frac{m + 1}{m - 1}$

$\Rightarrow 2 m^{2} - 3 m + 1 = m^{2} + 3 m + 2$

$\Rightarrow m^{2} - 6 m - 1 = 0$

Sum of roots = 6

$∴$ Required sum is 6.

Q 34 :
Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then ${(Q R)}^{2}$ is equal to _________. [2025]

0%

(28)

Let $∠ P R N = θ$

$\Rightarrow P R = \cos e c θ and P Q = 4 \sec (30 ° + θ)$

For equilateral $△$ PQR.

Let PR = PQ

$\Rightarrow cosec θ = 4 \sec (θ + 30 °)$

$\Rightarrow \cos (θ + 30 °) = 4 \sin θ$

$\Rightarrow \frac{\sqrt{3}}{2} \cos θ - \frac{1}{2} \sin θ = 4 \sin θ$

$\Rightarrow \tan θ = \frac{1}{3 \sqrt{3}}$

$\Rightarrow cosec θ = \sqrt{28}$

Now, $Q R^{2} = d^{2} = {cosec}^{2} θ = 28$ .

Q 35 :
If $α = 1 + \sum_{r = 1}^{6} {(- 3)}^{r - 1} {}^{12}C_{2 r - 1}$ , then the distance of the point $(12, \sqrt{3})$ from the line $α x - \sqrt{3} y + 1 = 0$ is _________. [2025]

0%

(5)

Given, $α = 1 + \sum_{r = 1}^{6} {(- 3)}^{r - 1} {}^{12}C_{2 r - 1}$

$= 1 + \sum_{r = 1}^{6} {}^{12}C_{2 r - 1} \frac{{(\sqrt{3} i)}^{2 r - 1}}{\sqrt{3} i}$

Let $t = \sqrt{3} i$ , then we have

$α = 1 + \frac{1}{\sqrt{3} i} [{}^{12}C_{1} t + {}^{12}C_{3} t^{3} + . . . + {}^{12}C_{11} t^{11}]$

$= 1 + \frac{1}{\sqrt{3} i} [\frac{{(1 + \sqrt{3} i)}^{12} - {(1 - \sqrt{3} i)}^{12}}{2}]$

$= 1 + \frac{1}{\sqrt{3} i} [\frac{{(- 2 ω^{2})}^{12} - {(2 ω)}^{12}}{2}] = 1$ [ $∵ ω^{3} = 0$ ]

$∴$ Distance of $(12, \sqrt{3})$ from $x - \sqrt{3} y + 1 = 0$ is

$| \frac{12 - \sqrt{3} (\sqrt{3}) + 1}{\sqrt{{(1)}^{2} + {(- \sqrt{3})}^{2}}} |$ $= \frac{12 - 3 + 1}{\sqrt{1 + 3}} = 5$ .

Topic Question Set

Q 31 :
Let the range of the function $f (x) = 6 + 16 \cos x \cdot \cos (\frac{π}{3} - x) \cdot \cos (\frac{π}{3} + x) \cdot \sin 3 x \cdot \cos 6 x, x \in R$ be $[α, β]$ . Then the distance of the point $(α, β)$ from the line 3x + 4y + 12 = 0 is: [2025]

Q 32 :
Let the lines 3x – 4y – $α$ = 0, 8x –11y – 33 = 0 and 2x – 3y + $λ$ = 0 be concurrent. If the image of the point (1, 2) in the line 2x – 3y + $λ$ = 0 is $(\frac{57}{13}, \frac{- 40}{13})$ , then $| α λ |$ is equal to [2025]

Q 33 :
Two equal sides of an isosceles triangle are along – x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is: [2025]

Q 34 :
Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then ${(Q R)}^{2}$ is equal to _________. [2025]

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Q 35 :
If $α = 1 + \sum_{r = 1}^{6} {(- 3)}^{r - 1} {}^{12}C_{2 r - 1}$ , then the distance of the point $(12, \sqrt{3})$ from the line $α x - \sqrt{3} y + 1 = 0$ is _________. [2025]

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Topic Question Set

Q 31 : Let the range of the function f(x)=6+16 cos x·cos(π3–x)·cos(π3+x)·sin 3x·cos 6x, x∈R be [α, β]. Then the distance of the point (α, β) from the line 3x + 4y + 12 = 0 is: [2025]

Q 32 : Let the lines 3x – 4y – α= 0, 8x –11y – 33 = 0 and 2x – 3y + λ = 0 be concurrent. If the image of the point (1, 2) in the line 2x – 3y + λ = 0 is (5713,–4013), then |αλ| is equal to [2025]

Q 33 : Two equal sides of an isosceles triangle are along – x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is: [2025]

Q 34 : Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then (QR)2 is equal to _________. [2025]

0%

Q 35 : If α=1+∑r=16(–3)r–1 C2r–112, then the distance of the point (12,3) from the line αx–3y+1=0 is _________. [2025]

0%

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Q 31 :
Let the range of the function $f (x) = 6 + 16 \cos x \cdot \cos (\frac{π}{3} - x) \cdot \cos (\frac{π}{3} + x) \cdot \sin 3 x \cdot \cos 6 x, x \in R$ be $[α, β]$ . Then the distance of the point $(α, β)$ from the line 3x + 4y + 12 = 0 is: [2025]

Q 32 :
Let the lines 3x – 4y – $α$ = 0, 8x –11y – 33 = 0 and 2x – 3y + $λ$ = 0 be concurrent. If the image of the point (1, 2) in the line 2x – 3y + $λ$ = 0 is $(\frac{57}{13}, \frac{- 40}{13})$ , then $| α λ |$ is equal to [2025]

Q 33 :
Two equal sides of an isosceles triangle are along – x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is: [2025]

Q 34 :
Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then ${(Q R)}^{2}$ is equal to _________. [2025]

Q 35 :
If $α = 1 + \sum_{r = 1}^{6} {(- 3)}^{r - 1} {}^{12}C_{2 r - 1}$ , then the distance of the point $(12, \sqrt{3})$ from the line $α x - \sqrt{3} y + 1 = 0$ is _________. [2025]