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Find the value of $b$ so that the lines $\frac{x-1}{2}=\frac{y-b}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ are intersecting lines. Also, find the point of intersection of these given lines.

35. (b) OR: Find the angle between the lines $2x=3y=-z$ and $6x=-y=-4z$.

35. (a) Show that the following lines do not intersect each other : $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5};\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{-2}$

25. (b) OR: The equations of a line are $5x-3=15y+7=3-10z$. Write the direction cosines of the line and find the coordinates of a point through which it passes.

Find the vector and the Cartesian equations of a line passing through the point (1,2,-4) and parallel to the line joining the points A(3,3,-5) and B(1,0,-11). Hence, find the distance between the two lines. OR Find the equations of the line passing through the points A(1,2,3) and B(3,5,9). Hence, find the coordinates of the points on this line which are at a distance of 14 units from point B.

Find the area of the region bounded by the curves x^{2}=y, y=x+2 and x-axis, using integration.

Find the area of the following region using integration: {(x,y): y² ≤ 2x and y ≥ x-4}

Sketch the region bounded by the lines 2x+y=8, y=2, y=4 and the y-axis. Hence, obtain its area using integration.

If $y=(x+\sqrt{x^{2}-1})^{2}$;, then show that $(x^{2}-1)(\frac{dy}{dx})^{2}=4y^{2}.$

(a) Differentiate $\text{sec}^{-1}\left(\frac{1}{\sqrt{1-x^2}}\right)$ w.r.t. $\sin^{-1}\left(2x\sqrt{1-x^2}\right)$.
OR
(b) If $y = \tan x + \sec x$, then prove that $\frac{d^2y}{dx^2} = \frac{\cos x}{(1-\sin x)^2}$.

In $\Delta ABC$, $\vec{AB}=\hat{i}+\hat{j}+2\hat{k}$ and $\vec{AC}=3\hat{i}-\hat{j}+4\hat{k}$. If D is mid-point of BC, then vector $\vec{AD}$ is equal to :

The number of corner points of the feasible region determined by the constraints x-y\ge0, 2y\le x+2, x\ge0, y\ge0 is:

The feasible region of a linear programming problem is shown in the figure below: ... Which of the following are the possible constraints?

18. The probability that A speaks the truth is $\frac{4}{5}$ and that of B speaking the truth is $\frac{3}{4}$. The probability that they contradict each other in stating the same fact is :

∫ from -1 to 1 [|x-2| / (x-2)] dx, x≠2 is equal to

If f(x)=a(x-cos\~x) is strictly decreasing in R, then 'a' belongs to

Four friends Abhay, Bina, Chhaya and Devesh were asked to simplify \(4~AB+3(AB+BA)-4~BA,\) where A and B are both matrices of order \(2\times2\). It is known that \(A\ne B\ne I\) and \(A^{-1}\ne B\). Their answers are given as:
Abhay: \(6 AB\),
Bina : \(7 AB-BA\),
Chhaya: \(8 AB\),
Devesh: \(7 BA - AB\).
Who answered it correctly?

\(If~A=[\begin{matrix}2&1\\ -4&-2\end{matrix}].\) then the value of \(I-A+A^{2}-A^{3}+...is\):

Find the matrix \(A^{2}\), where \(A=[a_{ij}]\) is a \(2\times2\) matrix whose elements are given by \(a_{ij}=\) maximum (i, j) - minimum (i, j):

If \(P(A)=\frac{1}{7}\), \(P(B)=\frac{5}{7}\) and \(P(A\cap B)=\frac{4}{7},\) then \(P(\overline{A}|B)\) is: