# Placeholder

All new and solved questions in Placeholder category

##### Number of equilateral triangles with \$y=sqrt{3}(x-1)+2\$ and \$y=-sqrt{3} x\$ as two of its sides, is(A) 0(B) 1(C) 2(D) none of these
Number of equilateral triangles with \$y=sqrt{3}(x-1)+2\$ and \$y=-sqrt{3} x\$ as two of its sides, is (A) 0 (B) 1 (C) 2 (D) none of these...
##### If the distance of any point \$P(x, y)\$ from the origin is defined as \$d(x, y)=operatorname{Max} .{|x|,|y|}\$ and \$d(x, y)=k\$ (nonzero constant), then the locus of the point \$P\$ is(A) a straight line(B) a circle(C) a parabola(D) none of these
If the distance of any point \$P(x, y)\$ from the origin is defined as \$d(x, y)=operatorname{Max} .{|x|,|y|}\$ and \$d(x, y)=k\$ (nonzero constant), then the locus of the point \$P\$ is (A) a straight line (B) a circle (C) a parabola (D) none of these...
##### If \$a, b, c\$ form an A. P. with common difference \$d(eq 0)\$ and \$x, y, z\$ form a G. P. with common ratio \$r eq 1\$ ), then the area of the triangle with vertices \$(a, x),(b, y)\$ and \$(c, z)\$ is independent of(A) \$b\$(B) \$r\$(C) \$d\$(D) \$x\$
If \$a, b, c\$ form an A. P. with common difference \$d( eq 0)\$ and \$x, y, z\$ form a G. P. with common ratio \$r eq 1\$ ), then the area of the triangle with vertices \$(a, x),(b, y)\$ and \$(c, z)\$ is independent of (A) \$b\$ (B) \$r\$ (C) \$d\$ (D) \$x\$...
##### A line of fixed length 2 units moves so that its ends are on the positive \$x\$-axis and that part of the line \$x+y=\$ 0 which lies in the second quadrant. The locus of the mid-point of the line has the equation(A) \$(x+2 y)^{2}+y^{2}=1\$(B) \$(x-2 y)^{2}+y^{2}=1\$(C) \$(x+2 y)^{2}-y^{2}=1\$(D) none of these
A line of fixed length 2 units moves so that its ends are on the positive \$x\$-axis and that part of the line \$x+y=\$ 0 which lies in the second quadrant. The locus of the mid-point of the line has the equation (A) \$(x+2 y)^{2}+y^{2}=1\$ (B) \$(x-2 y)^{2}+y^{2}=1\$ (C) \$(x+2 y)^{2}-y^{2}=1\$ (D) none of t...
##### A straight line through the origin \$O\$ meets the parallel lines \$4 x+2 y=9\$ and \$2 x+y+6=0\$ at points \$P\$ and \$Q\$, respectively. The point \$O\$ divides the segment \$P Q\$ in the ratio(A) \$1: 2\$(B) \$3: 4\$(C) \$2: 1\$(D) \$4: 3\$
A straight line through the origin \$O\$ meets the parallel lines \$4 x+2 y=9\$ and \$2 x+y+6=0\$ at points \$P\$ and \$Q\$, respectively. The point \$O\$ divides the segment \$P Q\$ in the ratio (A) \$1: 2\$ (B) \$3: 4\$ (C) \$2: 1\$ (D) \$4: 3\$...
##### Let \$O\$ be the origin and let \$A(2,0), B(0,2)\$ be two points. If \$P(x, y)\$ is a point such that \$x y>0\$ and \$x+y<\$ 2 , then(A) \$P\$ lies either inside the triangle \$O A B\$ or in the third quadrant(B) \$P\$ cannot be inside the triangle \$O A B\$(C) \$P\$ lies inside the triangle \$O A B\$(D) none of these
Let \$O\$ be the origin and let \$A(2,0), B(0,2)\$ be two points. If \$P(x, y)\$ is a point such that \$x y>0\$ and \$x+y<\$ 2 , then (A) \$P\$ lies either inside the triangle \$O A B\$ or in the third quadrant (B) \$P\$ cannot be inside the triangle \$O A B\$ (C) \$P\$ lies inside the triangle \$O A B\$ (D) none o...
##### Consider the equation \$y-y_{1}=mleft(x-x_{1}ight)\$. In this equation, if \$m\$ and \$x_{1}\$ are fixed and different lines are drawn for different values of \$y^{1}\$, then,(A) the lines will pass through a single point(B) there will be one possible line only(C) there will be a set of parallel lines(D) none of these
Consider the equation \$y-y_{1}=mleft(x-x_{1} ight)\$. In this equation, if \$m\$ and \$x_{1}\$ are fixed and different lines are drawn for different values of \$y^{1}\$, then, (A) the lines will pass through a single point (B) there will be one possible line only (C) there will be a set of parallel lines (...
##### \$D\$ is a point on \$A C\$ of the triangle with vertices \$A(2,\$,3), \$B(1,-3), C(-4,-7)\$ and \$B D\$ divides \$A B C\$ into two triangles of equal area. The equation of the line drawn through \$B\$ at right angles to \$B D\$ is(A) \$y-2 x+5=0\$(B) \$2 y-x+5=0\$(C) \$y+2 x-5=0\$(D) \$2 y+x-5=0\$
\$D\$ is a point on \$A C\$ of the triangle with vertices \$A(2,\$, 3), \$B(1,-3), C(-4,-7)\$ and \$B D\$ divides \$A B C\$ into two triangles of equal area. The equation of the line drawn through \$B\$ at right angles to \$B D\$ is (A) \$y-2 x+5=0\$ (B) \$2 y-x+5=0\$ (C) \$y+2 x-5=0\$ (D) \$2 y+x-5=0\$...
##### If two points \$A(a, 0)\$ and \$B(-a, 0)\$ are stationary and if \$angle A-angle B=heta\$ in \$Delta A B C\$, the locus of \$C\$ is(A) \$x^{2}+y^{2}+2 x y an heta=a^{2}\$(B) \$x^{2}-y^{2}+2 x y an heta=a^{2}\$(C) \$x^{2}+y^{2}+2 x y cot heta=a^{2}\$(D) \$x^{2}-y^{2}+2 x y cot heta=a^{2}\$
If two points \$A(a, 0)\$ and \$B(-a, 0)\$ are stationary and if \$angle A-angle B= heta\$ in \$Delta A B C\$, the locus of \$C\$ is (A) \$x^{2}+y^{2}+2 x y an heta=a^{2}\$ (B) \$x^{2}-y^{2}+2 x y an heta=a^{2}\$ (C) \$x^{2}+y^{2}+2 x y cot heta=a^{2}\$ (D) \$x^{2}-y^{2}+2 x y cot heta=a^{2}\$...
##### The straight line \$y=x-2\$ rotates about a point where it cuts the \$x\$-axis and becomes perpendicular to the straight line \$a x+b y+c=0 .\$ Then, its equation is(A) \$a x+b y+2 a=0\$(B) \$a x-b y-2 a=0\$(C) \$b y+a y-2 b=0\$(D) \$a y-b x+2 b=0\$
The straight line \$y=x-2\$ rotates about a point where it cuts the \$x\$-axis and becomes perpendicular to the straight line \$a x+b y+c=0 .\$ Then, its equation is (A) \$a x+b y+2 a=0\$ (B) \$a x-b y-2 a=0\$ (C) \$b y+a y-2 b=0\$ (D) \$a y-b x+2 b=0\$...
##### If the point \$Pleft(a^{2}, aight.\$ ) lies in the region corresponding to the acute angle between the lines \$2 y=x\$ and \$4 y=x\$, then(A) \$a in(2,6)\$(B) \$a in(4,6)\$(C) \$a in(2,4)\$(D) none of these
If the point \$Pleft(a^{2}, a ight.\$ ) lies in the region corresponding to the acute angle between the lines \$2 y=x\$ and \$4 y=x\$, then (A) \$a in(2,6)\$ (B) \$a in(4,6)\$ (C) \$a in(2,4)\$ (D) none of these...
##### The point \$(4,1)\$ undergoes the following three successive transformations(A) Reflection about the line \$y=x-1\$(B) Translation through a distance 1 unit along the positive \$x\$-axis(C) Rotation through an angle \$frac{pi}{4}\$ about the origin in the anti-clockwise direction. Then, the coordinates of the final point are(A) \$(4,3)\$(B) \$left(frac{7}{2}, frac{7}{2}ight)\$(C) \$(0,3 sqrt{2})\$(D) \$(3,4)\$
The point \$(4,1)\$ undergoes the following three successive transformations (A) Reflection about the line \$y=x-1\$ (B) Translation through a distance 1 unit along the positive \$x\$-axis (C) Rotation through an angle \$frac{pi}{4}\$ about the origin in the anti-clockwise direction. Then, the coordinates o...
##### A light ray emerging from the point source placed at \$P(2,3)\$ is reflected at point ' \$heta\$ on the \$y\$-axis and then passes through the point \$R(5,10)\$. Coordinates of ' \$Q\$ ' are(A) \$(0,3)\$(B) \$(0,2)\$(C) \$(0,5)\$(D) none of these
A light ray emerging from the point source placed at \$P(2,3)\$ is reflected at point ' \$ heta\$ on the \$y\$-axis and then passes through the point \$R(5,10)\$. Coordinates of ' \$Q\$ ' are (A) \$(0,3)\$ (B) \$(0,2)\$ (C) \$(0,5)\$ (D) none of these...
##### The distance between two parallel lines is unity. A point \$P\$ lies between the lines at a distance \$a\$ from one of them. The length of a side of an equilateral triangle \$P Q R\$, vertex \$Q\$ of which lies on one of the parallel lines and vertex \$R\$ lies on the other line, is(A) \$frac{2}{sqrt{3}} cdot sqrt{a^{2}+a+1}\$(B) \$frac{2}{sqrt{3}} sqrt{a^{2}-a+1}\$(C) \$frac{1}{sqrt{3}} sqrt{a^{2}+a+1}\$(D) \$frac{1}{sqrt{3}} sqrt{a^{2}-a+1}\$
The distance between two parallel lines is unity. A point \$P\$ lies between the lines at a distance \$a\$ from one of them. The length of a side of an equilateral triangle \$P Q R\$, vertex \$Q\$ of which lies on one of the parallel lines and vertex \$R\$ lies on the other line, is (A) \$frac{2}{sqrt{3}} cdot...
##### Two points \$A\$ and \$B\$ are given. \$P\$ is a moving point on one side of the line \$A B\$ such that \$angle P A B-angle P B A\$ is a positive constant \$2 heta\$. The locus of the point \$P\$ is(A) \$x^{2}+y^{2}+2 x y cot 2 heta=a^{2}\$(B) \$x^{2}+y^{2}-2 x y cot 2 heta=a^{2}\$(C) \$x^{2}+y^{2}+2 x y an 2 heta=a^{2}\$(D) \$x^{2}-y^{2}+2 x y cot 2 heta=a^{2}\$.
Two points \$A\$ and \$B\$ are given. \$P\$ is a moving point on one side of the line \$A B\$ such that \$angle P A B-angle P B A\$ is a positive constant \$2 heta\$. The locus of the point \$P\$ is (A) \$x^{2}+y^{2}+2 x y cot 2 heta=a^{2}\$ (B) \$x^{2}+y^{2}-2 x y cot 2 heta=a^{2}\$ (C) \$x^{2}+y^{2}+2 x y an 2 ...
##### The four points \$A(p, 0), B(q, 0), C(r, 0)\$ and \$D(s, 0)\$ are such that \$p, q\$ are the roots of the equation \$a x^{2}+2 h x+\$ \$b=0\$ and \$r, s\$ are those of equation \$a^{prime} x^{2}+2 h^{prime} x+b^{prime}=0\$. If the sum of the ratios in which \$C\$ and \$D\$ divide \$A B\$ is zero, then(A) \$a b^{prime}+a^{prime} b=2 h h^{prime}\$(B) \$a b^{prime}+a^{prime} b=h h^{prime}\$(C) \$a b^{prime}-a^{prime} b=2 h h^{prime}\$(D) none of these
The four points \$A(p, 0), B(q, 0), C(r, 0)\$ and \$D(s, 0)\$ are such that \$p, q\$ are the roots of the equation \$a x^{2}+2 h x+\$ \$b=0\$ and \$r, s\$ are those of equation \$a^{prime} x^{2}+2 h^{prime} x+b^{prime}=0\$. If the sum of the ratios in which \$C\$ and \$D\$ divide \$A B\$ is zero, then (A) \$a b^{prime}+...