Equation of a Plane : Intercept Form

Q. If {x} represents the least integer, not less than x, then total number of solutions of the equation (x−1)2+{x}=4, is equal to

Q. Let a denote the number of non negative values of p for which the equation p(2^x) + 2^-x = 5. Find the value of a.

Q. 4.The number of solutions of the equation tanx + secx = 2cosx , x [0, ] is

Q. Find the number of non-zero integral solutions of the equation 1-i =2\ast.If (a + ib) (c + id) (e + if) (g ih) A + iB, then show that(a+b2) (c2 + d) (e2 +f2) (g+h) = A2 + B2

Q. Let n_K be the number of real solutions of the equation \vert x+1\vert+\vert x-3\vert=K, then (1) n_K=0, if K<4 (2) n_K=2, if K>4 (3) n_K is infinitely many if K=4 (4) Minimum value of f(x)=\vert x+1\vert+\vert x-3\vert is 2

Q. If a is any real number, the number of roots of $\cot x-\tan x=a$ in the first quadrant is (are).
(a) 2
(b) 0
(c) 1
(d) none of these

Q. 8. Prove that the points (1, 1, 1), (-2, 4, 1), (- 1, 5, 5) and (2, 2, 5) are thesquare.

Q. Points (1, 2, 3), (9, 5, 8), (-2, 5, 6) and (-3, -8, 8) are in the same plane.

1. True
2. False

Q. 12. The number of solutions of equation \vert\sqrt x-2\vert+\sqrt x(\sqrt x-4)+2=0 , is

Q. 3x^4+5x^3+6x^2+7x+5+cos x=0 What is the number of real solutions of x?

Q. The number of solutions of the equation z^2+\overline{z =}0is

Q. Let $\pi$ be the plane parallel to $y-$axis and containing the points $(1,0,1)$ and $(3,2,-1).$ Also, $A\equiv (4,0,0)$ and $B\equiv (6,0,-2)$ are two points and $P\equiv (x_0,y_0,z_0)$ is a variable point on the plane $\pi .$ Then which of the following is/are CORRECT?

1. The equation of the plane $\pi$ is $x+z=2$
2. If $(PA+PB)$ is minimum, then $|4x_0+y_0+2z_0|$ is $12$
3. If $|PA-PB|\in [0,\sqrt{N}],$ then $N$ is $8$
4. If the reflection of the line $AB$ in the plane $\pi$ is $\frac{x-2}{1}=\frac{y-\alpha }{0}=\frac{z+\beta }{-1},$ then $({\alpha }^4+{\beta }^4)$ is $16$

Q. If $z=\frac{-2}{1+i\sqrt{3}}$, then the value of arg (z) is
(a) π
(b) $\frac{\mathrm{\pi }}{3}$
(c) $\frac{2\mathrm{\pi }}{3}$
(d) $\frac{\mathrm{\pi }}{4}$

Q. 3x^4+5x^3+6x^2+7x+5+cos x=0 What is the number of real solutions of x?

Q. If a plane meets the co-ordinate axes in A, B, C such that the centroid of the triangle ABC is the point $(1,r,r^2),$ then equation of the plane is

1. $x+ry+r^{2}z=3r^2$
2. $r^{2}x+ry+z=3r^2$
3. $x+ry+r^{2}z=3$
4. $r^{2}x+ry+z=3$

Q. The plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ meets the coordinate axes at A, B, C respectively. D and E are the mid-points of AB and AC respectively. Coordinates of the mid-point of DE are

1. $(a,\frac{b}{4},\frac{c}{4})$
2. $(\frac{a}{4},b,\frac{c}{4})$
3. $(\frac{a}{4},\frac{b}{4},c)$
4. $(\frac{a}{2},\frac{b}{4},\frac{c}{4})$.