Condition of Concurrency of 3 Straight Lines

Q.

The direction cosines of the line joining the points $\left(4,3,-5\right)$ and $\left(-2,1,-8\right)$ are

1. $\left(\frac{6}{7}\right),\left(\frac{2}{7}\right),\left(\frac{3}{7}\right)$

2. $\left(\frac{2}{7}\right),\left(\frac{3}{7}\right),\left(\frac{-6}{7}\right)$

3. $\left(\frac{6}{7}\right),\left(\frac{3}{7}\right),\left(\frac{2}{7}\right)$

4. None of these

Q. If the system of linear equations
$2x+y-z=3$
$x-y-z=\alpha$
$3x+3y+\beta z=3$
has infinitely many solutions, then $(\alpha +\beta -\alpha \beta )$ is equal to

Q. If the following system of linear equations
$2x+y+z=5x-y+z=3x+y+az=b$
has no solution, then :

1. $a=-\frac{1}{3},b\ne \frac{7}{3}$
2. $a\ne -\frac{1}{3},b=\frac{7}{3}$
3. $a\ne \frac{1}{3},b=\frac{7}{3}$
4. $a=\frac{1}{3},b\ne \frac{7}{3}$

Q. PQ is the double ordinate of the parabola y²=4ax. Then the locus of its point of trisection is

Q. The system of linear equations
$\lambda x+2y+2z=5$
$2\lambda x+3y+5z=8$
$4x+\lambda y+6z=10$ has:

1. no solution when $\lambda =2$
2. infinitely many solutions when $\lambda =2$
3. no solution when $\lambda =8$
4. a unique solution when $\lambda =-8$

Q. The value of $k\in R,$ for which the following system of linear equations
$3x–y+4z=3,$
$x+2y–3z=–2,$
$6x+5y+kz=–3,$
has infinitely many solutions, is:

1. $-3$
2. $-5$
3. $5$
4. $3$

Q. Drawn from origin are 2 perpendicular lines forming an isosceles triangle together with the straight line 2x + y = a , then area of this triangle is

Q. If $|z+\frac{6}{z}|=5,$ then greatest value of $\left|z\right|$ is equal to

1. 5
2. 6
3. 1
4. 3

Q.

Points $(-2,4,7)$, $(3,-6,-8)$ and $(1,-2,-2)$ are:

1. Collinear

2. Vertices of an equilateral triangle

3. Vertices of an isosceles triangle

4. None of the above

Q. The value(s) of $\lambda$ for which the system of equations $x+y-3=0,(1+\lambda )x+(2+\lambda )y-8=0$ and $x-(1+\lambda )y+(2+\lambda )=0$
is consistent, is:

1. $\frac{5}{3}$
2. $-\frac{5}{3}$
3. $1$
4. $-1$

Q.

A point $P$ moves in such a way that the ratio of its distance from two coplanar points is always a fixed number $(\ne 1)$. Then, its locus is a

1. Parabola

2. Circle

3. Hyperbola

4. Pair of straight lines

Q. In a quadrilateral $ABCD$, $\overrightarrow{AC}$ is the bisector the angle between $\overrightarrow{AB}$ and $\overrightarrow{AD}$ which is $\frac{2\pi }{3}$. If $15|\overrightarrow{AC}|=3|\overrightarrow{AB}|=5|\overrightarrow{AD}|$, then cosine of the angle between $\overrightarrow{BA}$ and $\overrightarrow{DC}$ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

1. $-\frac{1}{\sqrt{7}}$
2. $\frac{2}{\sqrt{7}}$
3. $\frac{1}{\sqrt{7}}$
4. $-\frac{2}{\sqrt{7}}$

Q. Let $\lambda and\alpha$ be real. Find the set of all values of $\lambda$ for which the system of linear equations
$\lambda x+\left(sin\alpha \right)y+\left(cos\alpha \right)z=0,x+\left(cos\alpha \right)y+\left(sin\alpha \right)z=0and-x+\left(sin\alpha \right)y-\left(cos\alpha \right)z=0$
has a non - trivial solution.
For $\lambda =1$, the values of $\alpha$ are ___.
($n$ belongs to integers)

1. $\alpha =n\pi$
2. $\alpha =n\frac{\pi }{2}$
3. $n\pi +\frac{\pi }{4}$
4. $n\frac{\pi }{2}+\frac{\pi }{4}$

Q. The number of non-trivial solutions of the system $x-y+z=0,x+2y-z=0,2x+y+3z=0$ is

1. $0$
2. $1$
3. $2$
4. infinite

Q.

Find the equation of the line joining the points $(-1,3)$ and $(4,-2)$.

Q. Using properties of determinants, prove that $\left|\begin{array}{ccc}{a}^2+2a& 2a+1& 1\\ 2a+1& a+2& 1\\ 3& 3& 1\end{array}\right|={(a-1)}^3$

Q. Solution of the sysytem of equations, $x+2y+z=7,x+3z=11,2x-3y=1$, is $(x,y,z)$ then $x+y-z$ is equal to:

1. $-1$
2. $0$
3. $1$
4. $6$

Q. $\mathrm{If}\mathrm{the}\mathrm{lines}{a}_{1}x+b_{1}y+1=0,a_{2}x+b_{2}y+1=0\mathrm{and}{a}_{3}x+b_{3}y+1=0\mathrm{are}\mathrm{concurrent},\mathrm{then}\mathrm{the}\mathrm{points}(a_1,b_1),(a_2,b_2)\mathrm{and}(a_3,b_3)\mathrm{will}\mathrm{be}\mathrm{collinear}.$

1. False
2. True

Q.

PM and PN are the perpendiculars from any point P on the rectangular hyperbola $xy=c^2$ to the asymptotes.

If the locus of the mid point of MN is a conic, then its eccentricity is

1. 

2. 

3. 

4. 

Q. If value of $\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|$ is A then value of $\left|\begin{array}{ccc}d& e& f\\ a& b& c\\ g& h& i\end{array}\right|$ is equal to

1. A
2. -A
3. \N
4. 2A

Q.

The area between x = y² and x = 4 is divided into two equal parts by the line x = a, find the value of a.

Q. 29. Find the area of a triangle formed by a plane 2x-3y+4z=12on axes is

Q. Three lines are drawn from the origin $O$ with direction cosines proportional to $(1,-1,1),(2,-3,0)$ and $(1,0,3),$ Then three lines are

1. not coplanar
2. coplanar
3. perpendicular to each other
4. coincident

Q. Area of a square ABCD is 36 and side AB is parallel to the X-axis. Vertices A, B and C lie on the graphs of $y=lo{g}_{a}x,y=2lo{g}_{a}x$ and $y=3lo{g}_{a}x$ respectively. Then a =

1. $3^{\frac{1}{6}}$
2. $\sqrt{3}$
   
3. $6^{\frac{1}{3}}$
   
4. $\sqrt{6}$

Q. Show that the points $P(a,b+c),Q(b,c+a)$ and $R(c,a+b)$ are collinear.

Q.

Area under the circle x² + y² = 16 is

1. 4 π sq units

2. π sq units

3. 13 π sq units

4. 16 π sq units

Q. Find the locus of a point which is such that two of normals drawn from it to the parabola y^2=4ax are at right angles.

Q.

Find k, if the straight lines y-3kx+4=0, (2k-1)x-(8k-1) y-6=0

Q. If P, Q and R are any three points on the curve whose equation is xy=c² then prove that the orthocentre of the triangle PQR also lies on that curve. (wirhout using parabola.)

Q. The number of value of k, for which the system the system of equation (k + 1)x + 8y = 4k $\Rightarrow$ kx + (k + 3)y = 3k - 1 has no solution, is

1. infinite
2. 1
3. 2
4. 3