JEE Coordinate Geometry PYQs with Solutions PDF, Download

Let $$a_{1},\dfrac{a_{2}}{2},\dfrac{a_{3}}{2^{2}},....,\dfrac{a_{10}}{2^{9}}$$ be a G.P. of common ratio $$\dfrac{1}{\sqrt{2}}$$. If $$a_{1}+a_{2}+....+a_{10}=62$$, then $$a_{1}$$ is equal to:

Let $$a_{1},a_{2},a_{3},...$$ be a G.P. of increasing positive terms such that $$a_{2}.a_{3}.a_{4}=64\text{ and }a_{1}+a_{3}+a_{5}=\frac{813}{7}.\text{ Then }a_{3}+a_{5}+a_{7}$$ is equal to :

Let $$a_{1}=1$$ and for $$n\geq1,a_{n+1}=\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}(n+1)^2}$$. Then $$\left|\sum_{ n=1}^{ \infty}\left( a_n - \frac{2}{n^2}\right)\right|$$ is equal to ______.

Let f and g be functions satisfying f(x+ y) =f(x)f(y), f (1) =7 and g(x+ y) = g(xy), g(1) =1, for all $$x,y \epsilon N$$. If $$\sum_{x=1}^n \left(\frac{f(x)}{g(x)}\right) = 19607$$, then n is equal to:

Suppose $$a, b, c$$ are in A.P. and $$a^2, 2b^2, c^2$$ are in G.P. If $$a < b < c$$ and $$a + b + c = 1$$, then $$9(a^{2}+b^{2}+c^{2})$$ is equal to _____________.

Let $$\overrightarrow{AB} = 2\widehat{i}+4\widehat{j}-5\widehat{k}$$ and $$ \overrightarrow{AD} = \widehat{i}+2\widehat{j}+\lambda\widehat{k}, \lambda\text{ }\epsilon \text{ } R$$. Let the projection of the vector $$ \overrightarrow{v}=\widehat{i}+\widehat{j}+\widehat{k}$$ on the disgonal $$\overrightarrow{AC}$$ of the parallelogram ABCD be of length one unit. If $$\alpha> \beta$$, be the roots of the equation $$\lambda^{2}x^{2}-6\lambda x+5=0$$, then $$2\alpha-\beta$$ is equal to

Let $$\overrightarrow{r}=2\widehat{i}+\widehat{j}-2\widehat{k}, \overrightarrow{b}=\widehat{i}+\widehat{j}\text{ and }\overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}$$. Let $$\overrightarrow{d}$$ be a vector such that $$|\overrightarrow{d}-\overrightarrow{a}|=\sqrt{11},|\overrightarrow{c}\times \overrightarrow{d}|=3$$ and the angle between $$\overrightarrow{c}\text{ and }\overrightarrow{d}$$ is $$\frac{\pi}{4}$$. Then $$\overrightarrow{a}. \overrightarrow{d}$$ is equal to

Let $$\overrightarrow{a}= 2\widehat{i}-\widehat{j}-\widehat{k}, \overrightarrow{b}=\widehat{i}+ 3\widehat{j}-\widehat{k}$$ and $$\overrightarrow{c} = 2\widehat{i}+\widehat{j}+3\widehat{k}.$$ Let $$\overrightarrow{\nu}$$ be the vector in the plane of the vectors $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$, such that the length of its projection on the vector $$\overrightarrow{C}$$ is $$\frac{1}{\sqrt{14}}$$. Then $$\mid \overrightarrow{\nu} \mid$$ is euqal to

Let $$\overrightarrow{a}= 2\widehat{i}-5\widehat{j}+5\widehat{k}$$ and $$\overrightarrow{b}= \widehat{i}-\widehat{j}+3\widehat{k}$$. If $$\overrightarrow{C}$$ is a vector such that $$2(\overrightarrow{a}\times\overrightarrow{c})+3(\overrightarrow{b}\times\overrightarrow{c})= \overrightarrow{0}$$ and $$(\overrightarrow{a}-\overrightarrow{b})\cdot\overrightarrow{c}=-97,$$ then $$\mid \overrightarrow{c}\times\widehat{k} \mid^{2}$$ is equal to

For three unit vectors $$\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$$ satisfying $$|\overrightarrow{a}-\overrightarrow{b}|^{2}+|\overrightarrow{b}-\overrightarrow{c}|^{2}+|\overrightarrow{c}-\overrightarrow{a}|^{2}=9$$ and $$|2\overrightarrow{a}+k\overrightarrow{b}+k\overrightarrow{c}|+3$$. the positive value of k is

Let a point A lie between the parallel lines $$L_{1}\text{ and }L_{2}$$ such that its distances from $$L_{1}\text{ and }L_{2}$$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC, where the points B and C lie on the lines $$L_{1}\text{ and }L_{2}$$, respectively, is:

Among the statements
(S1) : If A(5, -1) and B(-2, 3) are two vertices of a triangle, whose orthocentre is (0, 0), then its third vertex is (- 4,- 7) and
(S2) : If positive numbers 2a, b, c are three consecutive terms of an A.P., then the lines ax + by + c = 0 are concurrent at (2,-2),

A rectangle is formed by the lines x= O, y = O, x=3 and y = 4. Let the line L be perpendicular to 3x +y + 6 = 0 and divide the area of the rectangle into two equal parts. Then the distance of the point $$\left(\frac{1}{2},-5\right)$$ from the line L is equal to :

LetA(l, 0), B(2, -1) and $$C\left(\frac{7}{3}, \frac{4}{3}\right)$$ be three points. If the equation of the bisector of the angle ABC is $$\alpha x+\beta y=5$$, then the value of $$\alpha^{2} +\beta^{2}$$ is

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______.

Let $$f(t)=\int_{}^{}\left(\frac{1-\sin(\log_{e}{t})}{1-\cos(\log_{e}{t})}\right)dt,t > 1$$.
If $$f(e^{\pi/2})=-e^{\pi/2}\text{ and }f(e^{\pi/4})=\alpha e^{\pi/4}$$, then $$\alpha$$ equals

Let $$I(x)=\int\frac{3dx}{\left(4x+6\right)\left(\sqrt{4x^{2}}+8x+3\right)}$$ and $$I(0)=\frac{{\sqrt{3}}}{4}+20.$$
If $$I\left( \frac{1}{2} \right)=\frac{a\sqrt{2}}{b}+c, \text { Where a,b,c } \in N,gcd(a,b)=1, \text{ a+b+c is equal to}$$

The area of the region, inside the ellipse $$x^{2}+4y^{2}=4$$ and outside the region bounded by the curves y=|x|-1 and y=1-|x|, is:

The value of the integral $$\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1+\sqrt[3]{\tan2x}}$$ is:

Let $$ f: [1 , \infty ) \rightarrow R$$ be a differentiable function. If $$6 \int_{1}^{x} f(t)dt=3x f(x)+ x^{3}-4$$ for all $$x\geq 1$$ then the value of $$f(2)-f(3)$$ is

If the domain of the function f(x) = $$\sin^{-1}\frac{1}{x^{2}-2x-2}$$, is $$\left[-\infty, \alpha\right] \cup \left[\beta,\gamma\right]\cup \left[\delta,\infty\right],$$ then $$\alpha+\beta+\gamma+\delta$$ is equal to

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $$16\left(\left(\sec^{-1}x\right)^{2}\left(\cosec^{-1}x\right)^{2}\right) \text{is :} $$

$$ \text{If for some } \alpha,\beta;\; \alpha\le\beta,\; \alpha+\beta=8$$ and  $$\sec^2(\tan^{-1}\alpha)+\cosec^2(\cot^{-1}\beta)=36,$$ $$\alpha^2+\beta^2$$  is:_______

Let $$\alpha$$ and $$\beta$$ respectively be the maximum and the minimum values of the function $$f(\theta)=4\left(\sin^4\left(\frac{7\pi}{2}-\theta\right)+\sin^4(11\pi+\theta)\right)-2\left(\sin^6\left(\frac{3\pi}{2}-\theta\right)+\sin^6(9\pi-\theta)\right),\ \ \theta\in\ R$$. Then $$\alpha+2\beta$$ is equal to:

The least value of $$(\cos^{2} \theta- 6\sin \theta \cos \theta + 3\sin^{2} \theta +2)$$ is

The sum of all values of $$\theta \in [0,2\pi]$$ satisfying $$2\sin^{2}\theta =\cos2\theta \text{ and }2\cos^{2}\theta =3\sin\theta$$ is

The number of solutions of the equation $$4\sin^2 x - 4\cos^3 x + 9 - 4\cos x = 0$$; $$x \in [-2\pi, 2\pi]$$ is:

For a triangle ABC, let $$\overrightarrow{p} = \overrightarrow{BC}, \overrightarrow{q}= \overrightarrow{CA}$$ and $$\overrightarrow{r} = \overrightarrow{BA}$$. If $$|\overrightarrow{p}| = 2\sqrt{3}, |\overrightarrow{q}|=2$$ and $$\cos\theta = \frac{1}{\sqrt{3}}$$ where $$\theta$$ is the angle between $$\overrightarrow{p}$$ and $$\overrightarrow{q}$$, then $$|\overrightarrow{p} \times \left(\overrightarrow{q}-\overrightarrow{3r}\right)|^2 +3|\overrightarrow{r}|^2$$ is equal to :

Let L be the line $$\frac{x+1}{2}=\frac{y+1}{3}=\frac{z+3}{6}$$ and let S be the set of all points (a, b, c) on L, whose distance from the line $$\frac{x+1}{2}=\frac{y+1}{3}=\frac{z-9}{0}$$a long the line L is 7. Then $$\sum_{(a,b,c)\in S} (a+b+c) $$ is equal to :

Let $$ P(\alpha,\beta, \gamma)$$ be the point on the line $$\frac{x-1}{2}=\frac{y+1}{-3}=z$$ at a distance $$4\sqrt{14}$$ from the point (1, -1, 0) and nearer to the origin. Then the shortest di stance, between the Lines $$\frac{x-\alpha}{1}=\frac{y-\beta}{2}=\frac{z-\gamma}{3}$$ and $$\frac{x+5}{2}= \frac{y-10}{1}=\frac{z-3}{1}$$, is equal to

If the distances of the point (1 , 2, a) from the line $$\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}$$ along the lines $$L_{1}:\frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b}$$ and $$L_{2}:\frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c}$$ are equal, then a + b + c is equal to

Let a line pass through two distinct points P(−2,−1, 3) and , and be parallel to the vector $$3\widehat{i}+2\widehat{j}+2\widehat{k}$$. If the distance of the point Q from the point R(1, 3, 3) is 5 , then the square of the area of △PQR is equal to :

The distance of the line $$\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}$$ from the point (1, 4, 0) along the line $$\frac{x}{1}=\frac{y-2}{2}=\frac{z+3}{3}$$ is :

Let  $$P$$  be the image of the point  $$Q(7,-2,5)$$ in the line  $$L:\;\frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4},$$ and  $$R(5,p,q)$$  be a point on $$L.$$ Then the square of the area of  $$\triangle PQR$$ is  $$\underline{\hspace{2cm}}.$$

Let a straight line L pass through the point P(2,-1,3) and be perpendicular to the lines $$\frac{x-1}{2}=\frac{y+1}{1}=\frac{z-3}{-2}$$ and $$\frac{x-3}{1}=\frac{y-2}{3}=\frac{z+2}{4}.$$ If the line L intersects the yz-plane at the point Q , then the distance between the points P and Q is :

For the matrices $$A=\begin{bmatrix}3  -4 \\1  -1 \end {bmatrix}$$ and $$B=\begin{bmatrix}-29  49 \\-13  18 \end{bmatrix}$$, if  $$\left(A^{15} + B \right) \begin{bmatrix}x \\y\end{bmatrix} = \begin{bmatrix}0 \\0 \end{bmatrix}$$, then among the following which one is true ?

Let |A|=6, Where A is a $$3\times3$$ matrix. If $$|adj(3adj(A^{2}\cdot adj(2A)))|=2^{m}\cdot3^{n},m,n\epsilon N$$, then m+n is equal to:

If $$A=\begin{bmatrix}2 & 3 \\3 & 5 \end{bmatrix}$$, then the determinant of the matrix $$ (A^{2025}-3A^{2024}+ A^{2023})$$ is

The number of $$3\times 2$$ matrices A, which can be formed using the elements of the set {-2, -1 , 0, 1, 2} such that the sum of all the diagonal elements of $$A^{T}A$$ is 5, is_____

Let $$f(x)=\int_{}^{} \frac{7x^{10}+9x^{8}}{(1+x^{2}+2x^{9})^{2}}dx, x>0, \lim_{x \rightarrow 0}f(x)=0$$ and $$f(1)=\frac{1}{4.}$$ If $$A= \begin{bmatrix}0 & 0 & 1 \\ \frac{1}{4} & f'(1) & 1 \\ \alpha^{2} & 4 & 1 \end{bmatrix}$$ and B = adj(adj A) be such that |B| = 81 , then $$\alpha^{2}$$ is equal to

Let n be the number obtained on rolling a fair die. If the probability that the system
x - ny + z = 6
x + (n - 2)y + (n + 1)z = 8
(n - 1)y + z = 1
has a unique solution is $$\frac{k}{6}$$, then the sum of k and all possible values of n is:

Among the statements :
I: If $$ \begin{vmatrix}1 & \cos\alpha & \cos\beta \\\mathbf{\cos\alpha} & 1 & \mathbf{\cos\gamma} \\\mathbf{\cos\beta} & \mathbf{\cos\gamma} & 1\end{vmatrix}=\begin{vmatrix}0 & \mathbf{\cos\alpha}&\mathbf{\cos\beta} \\\mathbf{\cos\alpha} & 0 & \mathbf{\cos\gamma} \\\mathbf{\cos\beta} & \mathbf{\cos\gamma} & 0\end{vmatrix}$$, then $$\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=\frac{3}{2}$$, and

II: $$\begin{vmatrix}x^{2}+x & x+1 & x-2 \\2x^{2}+3x-1 & 3x & 3x-3 \\x^{2}+2x+3 & 2x-1 & 2x-1\end{vmatrix} = px + q$$, then $$p^{2}=196q^{2}$$

The system of linear equations
$$x + y + z = 6$$
$$2x + 5y + az =36$$
$$x + 2y + 3z = b$$

Let M and m respectively be the maximum and the minimum value of
$$f(x) =\begin{vmatrix}\mathbf{1+\sin^{2}x} & \mathbf{\cos^{2}x} & \mathbf{4\sin 4x} \\\mathbf{\sin^{2}x} &\mathbf{1+\cos^{2}x} & \mathbf{4\sin 4x} \\\mathbf{\sin^{2}x} &\mathbf{\cos^{2}x} & \mathbf{1+4\sin 4x}\end{vmatrix}$$, $$x \in R$$ then $$M^{4}-m^{4}$$ is equal to :

Let $$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix}$$ and $$|2A|^3 = 2^{21}$$ where $$\alpha, \beta \in Z$$, Then a value of $$\alpha$$ is

Let $$f: R \rightarrow (0, \infty)$$ be a twice differentiable function such that f(3) = 18, f'(3) = 0 and f" (3) = 4. Then $$\lim_{x \rightarrow 1}\left(\log_{a}\left(\frac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^{2}}}\right)$$ ls equal to :

Lety = y (x) be a differentiable function in the interval $$(0, \infty)$$ such that y(l) = 2, and $$\lim_{t \rightarrow x} \left( \frac{t^{2}y(x)-x^{2}y(t)}{x-t} \right) = 3$$ for each x > 0. Then 2){2) is equal to

If $$\lim_{x \rightarrow \infty}((\frac{e}{1-e})(\frac{1}{e}-\frac{x}{1+x}))^{x}=\alpha$$ then the value of $$\frac{\log_{e}^{\alpha}}{1+\log_{e}^{\alpha}}$$ equals :

$$\lim_{x\to 0}\cosec x \left( \sqrt{2\cos^2 x+3\cos x} - \sqrt{\cos^2 x+\sin x+4} \right)$$ is:

If $$\lim_{x\to 1}\frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}} = \frac{m\sqrt{5}}{n(2n)^{2/3}}$$, where gcd(m, n) = 1, then $$8m + 12n$$ is equal to ______.