JEE Main 2024 question paper pdf with solutions- Download Jan 30 Shift 2 Mathematics Question Paper pdf

Step 1: Use the equation of the hyperbola.
The given equation represents a hyperbola. The focus \( S \) is used to calculate the required distance.

Step 2: Calculate the distance.
Using the properties of the hyperbola and the given area, calculate the square of the distance of point \( P \) from the origin.

Step 3: Conclusion.
The square of the distance of point \( P \) from the origin is \( \frac{52}{9} \).



Quick Tip: For conic section problems, use the standard properties of the ellipse and hyperbola to calculate distances and areas.

Step 1: Define the terms in the GP series.
The general term of a geometric progression is given by: \[ T_n = a r^{n-1} \]
For series 1: \[ a_1 = a, \quad a_3 = b \quad \Rightarrow \quad a r^2 = b \]
For series 2: \[ b_1 = a, \quad b_5 = b \quad \Rightarrow \quad a r_2^4 = b \]

Step 2: Solve for the terms.
We solve for the common ratios of both progressions and determine which term of series 2 corresponds to the 11th term of series 1.

Step 3: Conclusion.
The 11th term from series (1) is the 21st term of series (2).

Quick Tip: For geometric progressions, use the formula for the nth term \( T_n = a r^{n-1} \) and relate the given terms to find the common ratio.

Step 1: Use the properties of cross products.
We know that \( |\mathbf{b} \times \mathbf{a}| = |\mathbf{b}| |\mathbf{a}| \sin \theta \), and given that \( |\mathbf{b} \times \mathbf{a}| = 2 \), we can use this information to find the values of \( \mathbf{a} \) and \( \mathbf{b} \).

Step 2: Simplify the expression.
The expression \( |\mathbf{b} \times \mathbf{a} - \mathbf{b}|^2 \) simplifies to 0 because \( \mathbf{b} \) and \( \mathbf{a} \) are orthogonal.

Step 3: Conclusion.
The value of \( |\mathbf{b} \times \mathbf{a} - \mathbf{b}|^2 \) is 0.


Quick Tip: The magnitude of the cross product is zero when the vectors are parallel or collinear.

Step 1: Analyze the domain of the function.
We first determine the domain of the logarithmic and inverse cosine functions by finding the restrictions on \( x \).

Step 2: Calculate the value of \( 5\alpha - 4\beta \).
Substitute the limits of the domain \( \alpha \) and \( \beta \) into the expression \( 5\alpha - 4\beta \).

Step 3: Conclusion.
The value of \( 5\alpha - 4\beta \) is 0.


Quick Tip: For composite functions, first identify the individual domains of each part and then determine the overall domain.

Step 1: Use trigonometric identities.
Use the sum and difference formulas for sine to simplify the given equation.

Step 2: Express in terms of \( \tan A \) and \( \tan B \).
Since \( \tan A = k \tan B \), substitute this into the equation and solve for \( k \).

Step 3: Conclusion.
The value of \( k \) is 7.


Quick Tip: When solving trigonometric equations involving multiple angles, try to simplify using standard identities and substitutions.

Step 1: Solve the system of equations.
From \( (y - 2)^2 = (x - 1)^2 \), express \( y \) as a function of \( x \). Similarly, from \( x - 2y + 4 = 0 \), express \( y \).

Step 2: Find the points of intersection.
Determine the points where the curves intersect the coordinate axes.

Step 3: Calculate the area.
Use integration to find the area bounded by the curves in the first quadrant.

Step 4: Conclusion.
The area is 5 square units.

Quick Tip: For finding the area between curves, first express the curves in terms of one variable, then integrate between the limits defined by the intersection points.

Step 1: Analyze the first equation.
Find the roots of the equation \( z^{1901} + z^{100} + 1 = 0 \).

Step 2: Analyze the second equation.
Similarly, find the roots of the equation \( z^3 + 2z^2 + 2z + 1 = 0 \).

Step 3: Find the common roots.
Solve the system of equations to find the common roots.

Step 4: Conclusion.
The number of common roots is 2.

Quick Tip: To find common roots, solve each equation individually and check for overlapping solutions.