List of practice Questions

The sides \( AB \), \( BC \), and \( CA \) of a triangle \( \triangle ABC \) have respectively 3, 4, and 5 points lying on them. The number of triangles that can be constructed using these points as vertices is

The vector \( \mathbf{r} = 3\hat{i} + 4\hat{k} \) can be written as the sum of a vector \( \mathbf{v} \), parallel to \( \hat{i} + \hat{k} \), and a vector \( \mathbf{u} \), perpendicular to \( \hat{i} + \hat{k} \). Then, the value of \( \mathbf{v} \) is

If the points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are collinear, then the rank of the matrix \[ \begin{bmatrix} x_1 & y_1 & 1
x_2 & y_2 & 1
x_3 & y_3 & 1 \end{bmatrix} \]

The value of the determinant \[ \begin{vmatrix} \cos(\alpha - \beta) & \cos \alpha & \cos \beta
\cos(\alpha - \beta) & 1 & \cos \beta
\cos \alpha & \cos \beta & 1 \end{vmatrix} \]

The number of integral values of \( K \), for which the equation \( 7 \cos x + 5 \sin x = 2K + 1 \) has a solution, is

The line joining two points \( A(2,0) \), \( B(3,1) \) is rotated about \( A \) in anti-clockwise direction through an angle of \( 15^\circ \). The equation of the line in the new position is

The line \( 2x + \sqrt{6}y = 2 \) is tangent to the curve \( x^2 - 2y^2 = 4 \). The point of contact is

The number of integral points (integral point means both the coordinates should be integers) exactly in the interior of the triangle with vertices \( (0, 0), (0, 21), (21, 0) \) is

A complex number \( z \) is such that \( \arg \left( \frac{-2}{3} + \frac{2i}{3} \right) = \frac{\pi}{3} \). The points representing this complex number will lie on

If \( a_1, a_2, a_3 \) be any positive real numbers, then which of the following statement is true?

If \( x^2 + 2x - 5 = 0 \), then the values of \( x \) are

The centres of a set of circles, each of radius 3, lie on the circle \( x^2 + y^2 = 25 \). The locus of any point in the set is

A tower \( A \) leans towards west making an angle \( \theta \) with the vertical. The angular elevation of \( B \), the topmost point of the tower is \( \beta \) as observed from a point \( C \) at a distance \( d' \) from \( B \). If the angular elevation of \( B \) from point \( D \) due east of \( C \) is the same and \( 2d \) from \( C \), then \( \theta \) can be given as

\( \theta \) and \( \gamma \) are the roots of the equation \( x^2 - \alpha x + \beta = 0 \) and if \( \theta + \gamma = \alpha \), then what is the value of \( \theta^2 + \gamma^2 \)?

The angle of intersection of the circles \( x^2 + y^2 - 8x - 9 = 0 \) and \( x^2 + y^2 + 2x - 4y - 11 = 0 \) is

Which of the following is the correct expansion of the series \[ \sum_{n=0}^{\infty} \left( \binom{C}{n} \right) \left( \frac{3}{5} \right)^n \left( \frac{2}{5} \right)^{n+1} \]

If two events A and B. If odds against A are 2:1 and those in favour of \( A \cup B \) are 3:1, then

The value of \( 2 \tan^{-1} x - \left( \text{cosec} \, \tan^{-1} x - \tan \, \cot \, x \right) \)

The proposition \( \neg (p \iff q) \) is equivalent to

If truth values of \( p \) be F and \( q \) be T, then truth value of \( \neg(p \vee q) \) is

The rate of change of the surface area of a sphere of radius \( r \), when the radius is increasing at the rate of \( 2 \, \text{cm/s} \), is proportional to

If \( N \) denote the set of all natural numbers and \( R \) the relation on \( N \times N \) defined by \( (a, b) R (c, d) \), if \( a(b + c) = b(a + d) \), then \( R \) is

What is the degeneracy of the level of H-atom that has energy \( \frac{R_H}{9} \)?

Match the following and choose the correct option given below. Compound/Type \quad Use A. Dry ice 1. Anti-knocking compound B. Semiconductor 2. Electronic diode or triode C. Solder 3. Joining circuits D. TEL 4. Refrigerant for preserving food

The relation \( R \) defined on set \( A = \{x : |x|<3, x \in \mathbb{R} \} \) by \( R = \{(x, y): y = |x|\} \) is