If \( A(1,0,2) \), \( B(2,1,0) \), \( C(2,-5,3) \), and \( D(0,3,2) \) are four points and the point of intersection of the lines \( AB \) and \( CD \) is \( P(a,b,c) \), then \( a + b + c = ? \)
If a tangent to the hyperbola \( x^2 - \frac{y^2}{3} = 1 \) is also a tangent to the parabola \( y^2 = 8x \), then the equation of such tangent with the positive slope is:
If a circle of radius 4 cm passes through the foci of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and is concentric with the hyperbola, then the eccentricity of the conjugate hyperbola of that hyperbola is:
Let \( F \) and \( F' \) be the foci of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (where \( b<2 \)), and let \( B \) be one end of the minor axis. If the area of the triangle \( FBF' \) is \( \sqrt{3} \) sq. units, then the eccentricity of the ellipse is:
A common tangent to the circle \( x^2 + y^2 = 9 \) and the parabola \( y^2 = 8x \) is
If the equation of the circle passing through the points of intersection of the circles \[ x^2 - 2x + y^2 - 4y - 4 = 0, \quad x^2 + y^2 + 4y - 4 = 0 \] and the point \( (3,3) \) is given by \[ x^2 + y^2 + \alpha x + \beta y + \gamma = 0, \] then \( 3(\alpha + \beta + \gamma) \) is:
If the circles \( x^2 + y^2 - 8x - 8y + 28 = 0 \) and \( x^2 + y^2 - 8x - 6y + 25 - a^2 = 0 \) have only one common tangent, then \( a \) is:
Let \( a \) be an integer multiple of 8. If \( S \) is the set of all possible values of \( a \) such that the line \( 6x + 8y + a = 0 \) intersects the circle \( x^2 + y^2 - 4x - 6y + 9 = 0 \) at two distinct points, then the number of elements in \( S \) is:
If power of a point \( (4,2) \) with respect to the circle \( x^2 + y^2 - 2x + 6y + a^2 - 16 = 0 \) is 9, then the sum of the lengths of all possible intercepts made by such circles on the coordinate axes is
Equation of the circle having its centre on the line \( 2x + y + 3 = 0 \) and having the lines \( 3x + 4y - 18 = 0 \) and \( 3x + 4y + 2 = 0 \) as tangents is:
If the slope of one line of the pair of lines \( 2x^2 + hxy + 6y^2 = 0 \) is thrice the slope of the other line, then \( h \) = ?
A line \( L \) passing through the point \( (2,0) \) makes an angle \( 60^\circ \) with the line \( 2x - y + 3 = 0 \). If \( L \) makes an acute angle with the positive X-axis in the anticlockwise direction, then the Y-intercept of the line \( L \) is?
A line \( L \) intersects the lines \( 3x - 2y - 1 = 0 \) and \( x + 2y + 1 = 0 \) at the points \( A \) and \( B \). If the point \( (1,2) \) bisects the line segment \( AB \) and \( \frac{a}{b} x + \frac{b}{a} y = 1 \) is the equation of the line \( L \), then \( a + 2b + 1 = ? \)
If the origin is shifted to a point \( P \) by the translation of axes to remove the \( y \)-term from the equation \( x^2 - y^2 + 2y - 1 = 0 \), then the transformed equation of it is:
If the line segment joining the points \( (1,0) \) and \( (0,1) \) subtends an angle of \( 45^\circ \) at a variable point \( P \), then the equation of the locus of \( P \) is:
The reactions which occur in blast furnace at 500 – 800 K during extraction of iron from haematite are i. \(3Fe_2O_3 + CO \rightarrow 2Fe_3O_4 + CO_2\)ii. \(Fe_2O_3 + 3C \rightarrow 2Fe + 3CO\)iii. \(FeO + 4CO \rightarrow 3Fe + 4CO_2\)iv. \(FeO + CO \rightarrow 2FeO + CO_2\)
Which sol is used as an intramuscular injection?
‘X’ is a protecting colloid. The following data is obtained for preventing the coagulation of 10 mL of gold sol to which 1 mL of 10% NaCl is added. What is the gold number of ‘X’?
For a first-order reaction, the concentration of reactant was reduced from 0.03 mol L\(^{-1}\) to 0.02 mol L\(^{-1}\) in 25 min. What is its rate (in mol L\(^{-1}\) s\(^{-1}\))?
Aqueous CuSO4 solution was electrolysed by passing 2 amp of current for 10 min. What is the weight (in g) of copper deposited at cathode? (Cu = 63 u; F = 96500 C mol-1)
The osmotic pressure of seawater is 1.05 atm. Four experiments were carried out as shown in the table. In which of the following experiments, pure water can be obtained in part-II of the vessel?
The density of nitric acid solution is 1.5 g mL\(^{-1}\). Its weight percentage is 68. What is the approximate concentration (in mol L\(^{-1}\)) of nitric acid? (N = 14 u; O = 16 u; H = 1 u)
A solid compound is formed by atoms of A (cations), B (cations), and O (anions). Atoms of O form an hcp lattice. Atoms of A occupy 25% of tetrahedral holes and atoms of B occupy 50% of octahedral holes. What is the molecular formula of the solid?
An alkene X (\( C_4H_8 \)) on reaction with HBr gave Y (\( C_4H_9Br \)). Reaction of Y with benzene in the presence of anhydrous \( AlCl_3 \) gave Z which is resistant to oxidation with \( KMnO_4 + KOH \). What are X, Y, Z respectively?
