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Questions and Answers
Given $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, and $I_1 = \int f(x) dx$ and $I_2 = \int x f(x) dx$, what is the value of $7I_1 + 12I_2$?
Given $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, and $I_1 = \int f(x) dx$ and $I_2 = \int x f(x) dx$, what is the value of $7I_1 + 12I_2$?
- 0
- A constant (correct)
- A function dependent on $x$
- A definite numerical value
A real differentiable function $f(x)$ satisfies $f(0) = 1$ and $f(x)f'(y) + f(y)f'(x) = 0$ for all $x, y \in R$. What is $\sum_{n=1}^{100} \log_e f(n)$?
A real differentiable function $f(x)$ satisfies $f(0) = 1$ and $f(x)f'(y) + f(y)f'(x) = 0$ for all $x, y \in R$. What is $\sum_{n=1}^{100} \log_e f(n)$?
- 0 (correct)
- 1
- e
- 100
A hyperbola has foci at $(1, 14)$ and $(1, -12)$ and passes through the point $(1, 6)$. What is the length of its latus rectum?
A hyperbola has foci at $(1, 14)$ and $(1, -12)$ and passes through the point $(1, 6)$. What is the length of its latus rectum?
- 13/2
- 13 (correct)
- 52/13
- 26
If $\sum_{r=0}^{5} \frac{{11 \choose m}}{2r + 2} = \frac{m}{n}$, where $\gcd(m, n) = 1$, then what is $m-n$?
If $\sum_{r=0}^{5} \frac{{11 \choose m}}{2r + 2} = \frac{m}{n}$, where $\gcd(m, n) = 1$, then what is $m-n$?
Given set $A = {1, 2, 3, ..., 10}$, and set $B = { \frac{m}{n} : n > m, m, n \in A, \gcd(m, n) = 1 }$. How many elements are in set B?
Given set $A = {1, 2, 3, ..., 10}$, and set $B = { \frac{m}{n} : n > m, m, n \in A, \gcd(m, n) = 1 }$. How many elements are in set B?
How many 5-letter words can be formed using distinct alphabets, with 'M' in the middle, and letters in increasing order?
How many 5-letter words can be formed using distinct alphabets, with 'M' in the middle, and letters in increasing order?
What is the shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{4}$ and $\frac{x+2}{7} = \frac{y-2}{8} = \frac{z+1}{2}$?
What is the shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{4}$ and $\frac{x+2}{7} = \frac{y-2}{8} = \frac{z+1}{2}$?
A bag contains 4 white and 6 black balls. Two balls are selected randomly one by one without replacement. What is the probability the first ball is black given the second ball is also black? If the probability is $\frac{m}{n}$ and gcd(m, n) = 1, what is $m + n$?
A bag contains 4 white and 6 black balls. Two balls are selected randomly one by one without replacement. What is the probability the first ball is black given the second ball is also black? If the probability is $\frac{m}{n}$ and gcd(m, n) = 1, what is $m + n$?
Given $s_n = \sum_{r=0}^{n} T_r = \frac{2^{n-1} + 2^{n+1} + 2^{n+3} + 2^{n+5}}{64}$, what is $\sum_{r=1}^{n} \frac{1}{T_r}$?
Given $s_n = \sum_{r=0}^{n} T_r = \frac{2^{n-1} + 2^{n+1} + 2^{n+3} + 2^{n+5}}{64}$, what is $\sum_{r=1}^{n} \frac{1}{T_r}$?
Given $y^2 dx + (\frac{1}{y} - x) dy = 0$ and $x(1) = 1$, find $x(\frac{1}{2})$
Given $y^2 dx + (\frac{1}{y} - x) dy = 0$ and $x(1) = 1$, find $x(\frac{1}{2})$
In the expansion of $(-3ax^2 - \frac{2}{x} )^n$, the sum of the coefficients of the terms is $m$. If $n=3$, find $4m + 3n$.
In the expansion of $(-3ax^2 - \frac{2}{x} )^n$, the sum of the coefficients of the terms is $m$. If $n=3$, find $4m + 3n$.
Triangle PQR is the image of the triangle with vertices (1, 3), (3, 1), and (2, 4) reflected across the line x + 2y = 2. If the centroid of ΔPQR is $(\alpha, \beta)$, what is the value of $15 \alpha - \beta$?
Triangle PQR is the image of the triangle with vertices (1, 3), (3, 1), and (2, 4) reflected across the line x + 2y = 2. If the centroid of ΔPQR is $(\alpha, \beta)$, what is the value of $15 \alpha - \beta$?
If A = {1, 2, 3}, how many non-empty equivalence relations are possible on set A?
If A = {1, 2, 3}, how many non-empty equivalence relations are possible on set A?
Flashcards
Shortest distance between lines
Shortest distance between lines
To find the shortest distance between two lines, we need to find a line that is perpendicular to both and passes through a point on one of the lines.
Conditional Probability
Conditional Probability
The probability of an event A given that event B has already occurred is calculated by dividing the probability of both events happening (A and B) by the probability of event B.
Geometric Mean
Geometric Mean
The geometric mean (GM) of a set of numbers is found by multiplying the numbers together and then taking the nth root of the product, where n is the number of terms in the set.
Value of a variable given initial value and rate of change
Value of a variable given initial value and rate of change
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Centroid of a Triangle
Centroid of a Triangle
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Equivalence Relation
Equivalence Relation
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Mean and Variance
Mean and Variance
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Geometric Progression
Geometric Progression
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What is the latus rectum of a hyperbola?
What is the latus rectum of a hyperbola?
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How do you find the latus rectum of a hyperbola given its foci and a point it passes through?
How do you find the latus rectum of a hyperbola given its foci and a point it passes through?
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How do you find the area between a parabola and a circle?
How do you find the area between a parabola and a circle?
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How do you find the sum of maximum and minimum values for a function like f(x) = 16sec^{-1}(x/2) + cosec^{-1}(x/2)?
How do you find the sum of maximum and minimum values for a function like f(x) = 16sec^{-1}(x/2) + cosec^{-1}(x/2)?
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How do you find the range of the radius of a circle given its intersection with another circle and the coordinates of its center?
How do you find the range of the radius of a circle given its intersection with another circle and the coordinates of its center?
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Study Notes
JEE Main 2025 Paper Discussion
- This is a discussion of problems from the JEE Main 2025 Mathematics paper.
- The problems cover various topics in mathematics.
- The discussion includes a 5-letter word problem, shortest distance between lines, probability, summation series, differential equations, area enclosed by functions, exponential equations, matrices, triangles, equivalence relations, and more.
Five-Letter Word
- Create a 5-letter word using 5 distinct letters.
- The middle letter must be 'M'.
- Letters must be in increasing order.
Shortest Distance Between Lines
- Calculate the shortest distance between two lines given by equations.
- Options for the answer are provided.
Probability Problem
- Two balls are picked from a bag, without replacement.
- The bag contains 4 white and 6 black balls.
- Find the probability that the first ball is black and the second ball is black.
- The problem involves conditional probability.
Summation Series
- Calculate the sum of a series using a formula.
- Series involve variables and factors.
Differential Equation
- Find the function by solving a differential equation.
- Initial condition is given.
Area Enclosed
- Calculate an area enclosed by a function and a straight line.
- Function is defined piecewise.
Exponential Equation
- Find the product of real values of x in the equation.
- The given function involves logarithms.
Matrices
- Given a matrix A with determinant -2
- Calculate a specific determinant that involves adj(A) and multiples of A.
- The question involves matrices and their properties.
Triangle Problem
- A triangle is given with vertices (x, y) values.
- The triangle is transformed by a line.
- Find the coordinates of the centroid of the transformed triangle.
Equivalence Relation
- Find the number of non-empty equivalence relations for a specific set.
- The set has 3 elements.
Coin Tossing
- A coin is tossed three times.
- Determine the mean and variance of a specific random variable.
- This variable represents tails occurring after heads.
Geometric Progression (G.P.)
- Series of numbers are a geometric progression (G.P.).
- Two specific equations involving terms in the progression are given.
- Find the first term of the G.P.
Trigonometric Identities
- Problem involves trigonometric functions and their reciprocals.
- Calculate the sum or range of values of the function given.
Circle and Parabola
- Find the area outside a parabola and inside a circle.
- Equations of parabola and circle are given.
Circles in Quadrants
- Two circles are given.
- One circle lies in a specific quadrant.
- Find the range of radius for the second circle that intersects the first circle at two points.
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Description
Join us for an in-depth discussion on selected problems from the JEE Main 2025 Mathematics paper. Tackle diverse topics including word problems, distance between lines, probability, and more, as we dissect each problem to enhance your understanding and preparation.
