TG EAMCET (EAPCET) Mathematic Multiple Choice Questions (MCQs)

211. The standard equation of a hyperbola is

a) \( x^2 + y^2 = 1 \)

b) \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)

c) \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \)

d) \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

Answer: b) \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)

212. The parametric equations of a hyperbola are

a) \( x = a \sec \theta, y = b \tan \theta \)

b) \( x = a \tan \theta, y = b \sec \theta \)

c) \( x = a \cos \theta, y = b \sin \theta \)

d) \( x = a \sin \theta, y = b \cos \theta \)

Answer: a) \( x = a \sec \theta, y = b \tan \theta \)

213. The equation of the tangent to the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) at the point \( (x_1, y_1) \) is

a) \( \frac{x_1 x}{a^2} - \frac{y_1 y}{b^2} = 1 \)

b) \( \frac{x_1 x}{a^2} + \frac{y_1 y}{b^2} = 1 \)

c) \( \frac{x_1 x}{b^2} - \frac{y_1 y}{a^2} = 1 \)

d) \( \frac{x_1 x}{a^2} - \frac{y_1 y}{b^2} = 0 \)

Answer: b) \( \frac{x_1 x}{a^2} + \frac{y_1 y}{b^2} = 1 \)

214. The asymptotes of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are

a) \( y = \pm \frac{b}{a} x \)

b) \( y = \pm \frac{a}{b} x \)

c) \( y = \pm x \)

d) \( y = \pm b x \)

Answer: a) \( y = \pm \frac{b}{a} x \)

215. The centroid of a triangle with vertices \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), and \( C(x_3, y_3, z_3) \) is

a) \( \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \)

b) \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \)

c) \( \left( \frac{x_1 + x_2 + x_3}{2}, \frac{y_1 + y_2 + y_3}{2}, \frac{z_1 + z_2 + z_3}{2} \right) \)

d) \( \left( \frac{x_1 + x_2 + x_3}{4}, \frac{y_1 + y_2 + y_3}{4}, \frac{z_1 + z_2 + z_3}{4} \right) \)

Answer: a) \( \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \)

216. The section formula divides a line joining two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) in the ratio \( m:n \). The coordinates of the point dividing the line are given by

a) \( \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}, \frac{mz_2 + nz_1}{m + n} \right) \)

b) \( \left( \frac{mx_1 + nx_2}{m + n}, \frac{my_1 + ny_2}{m + n}, \frac{mz_1 + nz_2}{m + n} \right) \)

c) \( \left( \frac{m + n}{mx_2 + nx_1}, \frac{m + n}{my_2 + ny_1}, \frac{m + n}{mz_2 + nz_1} \right) \)

d) \( \left( \frac{mx_2 - nx_1}{m + n}, \frac{my_2 - ny_1}{m + n}, \frac{mz_2 - nz_1}{m + n} \right) \)

Answer: b) \( \left( \frac{mx_1 + nx_2}{m + n}, \frac{my_1 + ny_2}{m + n}, \frac{mz_1 + nz_2}{m + n} \right) \)

217. If \( l, m, n \) are the direction cosines of a line, then

a) \( l^2 + m^2 + n^2 = 1 \)

b) \( l^2 + m^2 + n^2 = 0 \)

c) \( l + m + n = 0 \)

d) \( l^2 - m^2 + n^2 = 1 \)

Answer: a) \( l^2 + m^2 + n^2 = 1 \)

218. The direction ratios of a line are \( \langle 2, -3, 4 \rangle \). The direction cosines of the line are

a) \( \langle \frac{2}{5}, \frac{-3}{5}, \frac{4}{5} \rangle \)

b) \( \langle \frac{2}{3}, \frac{-3}{4}, \frac{4}{5} \rangle \)

c) \( \langle \frac{2}{4}, \frac{-3}{4}, \frac{4}{4} \rangle \)

d) \( \langle \frac{2}{6}, \frac{-3}{6}, \frac{4}{6} \rangle \)

Answer: a) \( \langle \frac{2}{5}, \frac{-3}{5}, \frac{4}{5} \rangle \)

219. The Cartesian equation of the plane passing through the point \( (x_1, y_1, z_1) \) with normal vector \( \langle a, b, c \rangle \) is

a) \( a(x - x_1) + b(y - y_1) + c(z - z_1) = 0 \)

b) \( a(x + x_1) + b(y + y_1) + c(z + z_1) = 0 \)

c) \( a(x - x_1) - b(y - y_1) - c(z - z_1) = 0 \)

d) \( a(x + x_1) - b(y + y_1) - c(z + z_1) = 0 \)

Answer: a) \( a(x - x_1) + b(y - y_1) + c(z - z_1) = 0 \)

220. The equation of the plane \( 2x - 3y + 4z = 5 \) represents a plane with normal vector

a) \( \langle 2, -3, 4 \rangle \)

b) \( \langle 1, -1, 1 \rangle \)

c) \( \langle 3, -2, 5 \rangle \)

d) \( \langle 5, 4, -3 \rangle \)

Answer: a) \( \langle 2, -3, 4 \rangle \)