List of practice Questions

If \( g(x) \) is an antiderivative of \( f(x) = 1 + 2 \log 2 \) and the graph of \( y = g(x) \) passes through the point \( \left( -1, -\frac{1}{2} \right) \), then the curve meets the Y-axis at
The area bounded by the curves \( y = x^2 \) and \( y - 6 = -|x| \) is:
If \( \theta \) is the angle made by the normal drawn to the curve \[ x = e^t \cos t, \quad y = e^t \sin t \] at the point \( (1, 0) \) with the X-axis, then \( \theta = \):
If a normal drawn at a point \( P \) to the curve \( y = \sin x \) passes through the origin, then the locus of \( P \) is:
An extreme value of \( f(x) = \frac{4}{\sin x} + \frac{1}{1 - \sin x} \) in \( (0, \frac{\pi}{2}) \) is:
If the tangent drawn to the curve \( y = x^3 \) at point \( (\alpha, \beta) \) cuts again the curve at another point \( (\alpha_1, \beta_1) \), then \( \frac{\beta_1}{\beta} = \):
Evaluate the integral \[ \int \frac{dx}{(2ax + x^2)^{3/2}} \]
If a normal drawn to the ellipse \[ \frac{x^2}{4} + \frac{y^2}{3} = 1 \] touches the hyperbola \[ \frac{x^2}{4} - \frac{y^2}{3} = 1, \] then the square of the slope of that normal is:
If \( f(x) = 3x + \frac{12}{x} \) is continuous on \( \mathbb{R} - \{0\} \) and \( M \) is its maximum value, then \( \displaystyle \lim_{x \to M} f(x) = \):
The distance of the origin from the external centre of similitude for the circles \[ x^2 + y^2 - 8x - 10y - 8 = 0 \quad \text{and} \quad x^2 + y^2 + 2x - 2y - 2 = 0 \] is:
Let \( \ell \) be the directrix of the parabola \( 9y^2 + 12y + 9x - 14 = 0 \), and \( \ell_1 \) be the line passing through the vertex of this parabola and the origin. If \( (h, k) \) is the point of intersection of \( \ell \) and \( \ell_1 \), then \( h + k = \):
Let the point \( L \) lying in the first quadrant be one end of a latus rectum of the ellipse \[ \frac{x^2}{4} + \frac{y^2}{3} = 1 \] Let \( P \) and \( Q \) be the points where the normal drawn at \( L \) meets the major and minor axes. Then the distance between \( P \) and \( Q \) is:
Let X-axis be the transverse axis and Y-axis be the conjugate axis of a hyperbola \( H \). Let the eccentricity of \( H \) be the reciprocal of the eccentricity of the ellipse \[ \frac{x^2}{4} + \frac{y^2}{2} = 1 \] If \( (5, 4) \) lies on \( H \), then the length of the transverse axis is:
The equation of the line common to the pair of lines \[ (p^2 - q^2)x^2 + (q^2 - r^2)xy + (r^2 - p^2)y^2 = 0 \] and \[ (l - m)x^2 + (m - n)xy + (n - l)y^2 = 0 \] is:
The lines \( L_1: y - x = 0 \) and \( L_2: 2x + y = 0 \) intersect the line \( L_3: y + 2 = 0 \) at points \( P \) and \( Q \) respectively. The bisector of the acute angle between \( L_1 \) and \( L_2 \) intersects \( L_3 \) at \( R \). Statement-1: \( PR : RQ = 2\sqrt{2} : \sqrt{5} \) Statement-2: In any triangle, the bisector of an angle divides the triangle into two similar triangles.
In \( \triangle ABC \), if the midpoints of the sides \( AB, BC, CA \) are respectively \( (l, 0, 0), (0, m, 0), (0, 0, n) \), then: \[ \frac{AB^2 + BC^2 + CA^2}{l^2 + m^2 + n^2} = ? \]
If \( A(2, 3, 5), B(\alpha, 3, 3), C(7, 5, \beta) \) are the vertices of a triangle. If the median through \( A \) is equally inclined with the coordinate axes, then \( \frac{\beta}{\alpha} = \):
The equation of the plane passing through the point \( (1, 2, 2) \) and perpendicular to the planes \[ x - y + 2z = 3 \quad \text{and} \quad 2x - 2y + z + 12 = 0 \] is:
Let the locus of the point of intersection of the perpendicular tangents drawn to the circle \[ x^2 + y^2 + 6x - 4y - 12 = 0 \] be the circle \( S \). Then the equation of the tangent drawn to \( S \) which is perpendicular to the line \( 6x - 4y + k = 0 \) is:
If \( t \in \mathbb{R} \setminus \{-1\} \), then the locus of the point \[ \left( \frac{3at}{1 + t^3}, \frac{3at^2}{1 + t^3} \right) \] is:
The centre of a square of side 4 units length is \( (3, 7) \), and one of the diagonals is parallel to the line \( y = x \). If \( (x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4) \) are the vertices of this square, then \[ \frac{y_1 y_2 y_3 y_4}{x_1 x_2 x_3 x_4} = ? \]
An analysis of monthly wages paid to the workers of two jute mills A and B gives the following data:
\begin{tabular}{|c|c|c|} \hline & Mill - A & Mill - B
\hline No. of workers & 500 & 600
Average daily wage (in rupees) & 186 & 175
Variance of distribution of wages & 81 & 100
\hline \end{tabular} Then:
A, B, C are three horses participating in a race. The probability of horse A to win is twice that of B, and the probability of B to win is twice that of C. Then the probabilities of A, B, and C winning the race are respectively:
Let ‘O’ be the origin. A and B be two points with position vectors \( -3\vec{i} - 3\vec{j} + 4\vec{k} \) and \( 4\vec{i} - 4\vec{j} - 3\vec{k} \) respectively. Let \( P \) be a point such that the line drawn through \( P \) parallel to \( OB \) meets \( OA \) in \( L \), and another line through \( P \) parallel to \( OA \) meets \( OB \) in \( M \). If \( L \) divides \( OA \) in the ratio 2:3 and \( M \) divides \( OB \) in the ratio 3:2, then the distance from \( O \) to \( P \) is:
In \( \triangle ABC \), if \( a, b, c \) are 5, 12, and 13 respectively, then \( b^2 \sin 2C + c^2 \sin 2B \) is: