Area of Triangle with Coordinates of Vertices Given

Q. The value of k for which the points (2, -3), (k, -1) and (0, 4) are collinear, is .

1. $k=-\frac{10}{7}$
2. $k=\frac{10}{7}$
3. $k=\frac{18}{7}$
4. $k=-\frac{18}{7}$

Q.

If a line makes $\alpha ,\beta ,\gamma ,\delta$ angles with four diagonals of a cube, then the value ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma +{\sin }^2\delta$ is

1. $\frac{4}{3}$

2. $1$

3. $\frac{8}{3}$

4. $\frac{7}{3}$

Q.

Show that the points (a + 5, a - 4), (a - 2, a + 3) and (a, a) do not lie on a straight line for any value of a.

Q.

If $y={\left(\tan x\right)}^{cotx}$, then $\frac{dy}{dx}=$

1. $y\left(\cos e{c}^{2}x\left(1-\log \left(\tan x\right)\right)\right)$

2. $\frac{dy}{dx}=y\left(\cos e{c}^{2}x\left(1+\log \left(\tan x\right)\right)\right)$

3. $\frac{dy}{dx}=y\left(\cos e{c}^{2}x\left(\log \left(\tan x\right)\right)\right)$

4. none of these

Q.

Three distinct points $P(3u^2,2u^3);Q(3v^2,2v^3)andR(3w^2,2w^3)$ are collinear and equation $a{x}^3+b{x}^2+cx+d=0$ has roots u, v and w, then which of the following is true

1. b = 0

2. c = 0

3. d = 0

4. ab = 0

Q. If the points A(m, −1), B(2, 1) and C(4, 5) are collinear, find the value of m.

Q. Tangent at the point $(2\sqrt{2},3)$ to the hyperbola$\frac{x^2}{4}-\frac{y^2}{9}$= 1 meet its asymptotes at A and B, then area of the triangle OAB, O being the origin is

1. 6 sq. units
2. 3 sq. units
3. 12 sq. units
4. 2 sq. units

Q. $OX,OY$ are positive $X$-axis, positive $Y$-axis respectively where $O=(0,0,0)$ . The d.c.'s of the line which bisects $\angle XOY$ are

1. $(1,1,0)$
2. $(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$
   
3. $(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})$
   
4. $(0,1,1)$

Q. If the straight lines $x=1+s,y=-3-\lambda s,z=1+\lambda s$ and $x=\frac{t}{2},y=1+t,z=2-t$, with parameters s and t respectively, are co-planar, then $\lambda$ equals

1. 0
2. -1
3. $\frac{-1}{2}$
4. -2

Q.

The points (-2, 3, 5), (1, 2, 3) and (7, 0, -1) are collinear because.

1. AC + BC = AB

2. AB + AC = BC

3. none of the above

4. AB + BC = AC

Q. Let
$P_1=\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|,$$P_2=\left|\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right|,$$P_3=\left|\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right|,$$P_4=\left|\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right|,$$P_5=\left|\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right|,$$P_6=\left|\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right|,$ and $X=\sum _{k=1}^6{P}_k\left[\begin{array}{ccc}2& 1& 3\\ 1& 0& 2\\ 3& 2& 1\end{array}\right]P_k^T,$ where $P_k^T$ denotes the transpose of the matrix $P_k.$ Then which of the following options is/are correct?

1. $\text{If}X\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=\alpha \left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],\text{then}\alpha =30$
2. $X$ is a symmetric matrix
3. The sum of diagonal entries of $X$ is $18$
4. $X-30I$ is an invertible matrix

Q.

If $A,B,C$ are angles of a triangle, then the value of $\left|\begin{array}{ccc}{e}^{-2iA}& e^{iC}& e^{iB}\\ e^{iC}& e^{-2iB}& e^{iA}\\ e^{iB}& e^{iA}& e^{-2iC}\end{array}\right|$ is

1. $0$
2. $e^{ABC}$
3. $-4$
4. $16$

Q. The equation of a plane bisecting the angle between the plane $2x-y+2z+3=0$ and $3x-2y+6z+8=0$ is/are:

1. $5x-y-4z-3=0$
2. $5x-y-4z-45=0$
3. $23x-13y+32z+5=0$
4. $23x-13y+32z+45=0$

Q. If the $D.R^{\prime }s$ of a line are $(1+\lambda ,1-\lambda ,2)$ and it makes an angle of ${60}^{\circ }$ with $Y$ axis, then the value of $\lambda$ can be

1. $2+\sqrt{5}$
2. $3+\sqrt{5}$
3. $3-\sqrt{5}$
4. $2-\sqrt{5}$

Q. Let $O=(0,0),A=(a,11)$ and $B=(b,37)$ are the vertices of an equilateral triangle $OAB$, then $a$ and $b$ satisfy the relation :

1. $(a^2+b^2)-3ab=138$
2. $(a^2+b^2)-4ab=138$
3. $(a^2+b^2)-ab=124$
4. $(a^2+b^2)+3ab=130$

Q.

Determine the point in the YZ-plane which is equidistant from three points $A(2,0,3),B(0,3,2)$ and $C(0,0,1)$.

Q. Find the values of other five trigonometric functions if $\sin x=\frac{3}{5},x$ lies in second quadrant.

Q. If area of a triangle is $35squnits$ with vertices $(2,-6)$, $(5,4)$ and $(k,4)$, then find the value of $k$.

Q.

Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5)

Q. If the value of determinant $\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 1+\cos A& 1+\cos B& 1+\cos C\\ {\cos }^{2}A+\cos A& {\cos }^{2}B+\cos B& {\cos }^{2}C+\cos C\end{array}\right|$ is zero for $△ABC,$ whose angles are $A,B$ and $C.$ Then $△ABC$ can be

1. Equilateral triangle
2. Isosceles triangle
3. Scalene triangle
4. Right angled triangle, whose one angle is ${60}^{\circ }.$

Q.

The area of the quadrilateral PQRS whose vertices are P (-5, 7), Q (-4, -5), R (-1, -6) and S (4, 5) is

1. 54

2. 17.5

3. 109

4. 72

Q. the area of the triangle formed by the vertices (0, 0), (2, 1) and (5, 3) is

1. 1
2. 2
3. 1/2
4. 1/4

Q. If the area of the triangle with vertices $(-2,0),(0,4)$ and $(0,k)$ is $4$ square units, find the values of $k$ using determinants.

Q.

Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.

Q.

Find the equation of the set of points which are equidistant from the points (1, 2 , 3) and (3, 2, -1)

1. y + 2z = 0

2. x - 2y = 0

3. x - 2z = 0

4. x + 2z = 0

Q. Find the values of other five trigonometric functions if $\cot x=\frac{3}{4},x$ lies in third quadrant.

Q. Inclination of the lines drawn through the point $(1,2)$ and cutting the line $x+y=4$ at a distance of $\frac{\sqrt{6}}{3}$ units from the point $(1,2)$ are

1. $\frac{\pi }{6}$ and $\frac{\pi }{3}$
2. $\frac{\pi }{8}$ and $\frac{3\pi }{8}$
3. $\frac{\pi }{12}$ and $\frac{5\pi }{12}$
4. none of these

Q.

For $x\in (0,\frac{5\mathrm{\pi }}{2})$, define $f\left(x\right)={\int }_0^x\sqrt{t}\sin tdt$​​. Then $f$ has:-

1. Local maximum at $\mathrm{\pi }$ and $2\mathrm{\pi }$

2. Local minimum at $\mathrm{\pi }$ and local maximum at $2\mathrm{\pi }$

3. local maximum at $\pi$ and a local minimum at $2\pi$

4. local minimum at $\pi$ and $2\pi$

Q. Tangent at a point $P_1$ (other than origin) on the curve $y=x^3$ meets the curve again at $P_2.$ The tangent at $P_2$ meets the curve at $P_3$ and so on. Then which of the following is/are correct?

1. Abscissa of $P_1,P_2,\cdots ,P_3$ are in G.P.
   
2. Ordinate of $P_1,P_2,\cdots ,P_3$ are in G.P.
3. Ratio of area of $\Delta P_1{P}_2{P}_3$ and $\Delta P_2{P}_3{P}_4$ is $\frac{1}{8}$
4. Ratio of area of $\Delta P_1{P}_2{P}_3$ and $\Delta P_2{P}_3{P}_4$ is $\frac{1}{16}$

Q. Let two points be $A(1,-1)$ and $B(0,2)$. If a point $P(x^{\prime },y^{\prime })$ be such that the area of $\Delta PAB=5$ sq. units and it lies on the line, $3x+y-4\lambda =0$, then the value of $\lambda$ is:

1. $4$
2. $1$
3. $-3$
4. $3$