JEE Mathematics Practice Questions

Medium•Jee Main 2025

The value of

(\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)

is

Easy•3d Geometry

The distance between points

(1, 2, 3)

and

(4, 6, 3)

is:

Easy•Coordinate Geometry

The distance between the parallel lines 3x + 4y - 5 = 0 and 3x + 4y + 10 = 0 is:

Hard•Algebra

Let

P=\left[\begin{array}{ccc}1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1\end{array}\right]

and

I

be the identity matrix of order 3. If

Q=\left[q_{i j}\right]

is a matrix such that

P^{50}-Q=I

, then

\frac{q_{31}+q_{32}}{q_{21}}

equals

Hard•Algebra

Let

M=\left[\begin{array}{cc} \sin ^{4} \theta & -1-\sin ^{2} \theta \\ 1+\cos ^{2} \theta & \cos ^{4} \theta \end{array}\right]=\alpha I+\beta M^{-1}

where

\alpha=\alpha(\theta)

and

\beta=\beta(\theta)

are real numbers, and

I

is the

2 \times 2

identity matrix. If

\alpha^{*}

is the minimum of the set

\{\alpha(\theta): \theta \in[0,2 \pi)\}

and

\beta^{*}

is the minimum of the set

\{\beta(\theta): \theta \in[0,2 \pi)\}

then the value of

\alpha^{*}+\beta^{*}

is

Easy•Application Of Derivatives

The minimum value of

f(x) = x^2 + 4x + 5

is:

Easy•Area Under Curves

The area bounded by

y = x^2

,

x

-axis,

x = 0

and

x = 2

is:

Easy•Arithmetic Progression

The sum of the first 20 terms of an AP with first term 3 and common difference 2 is:

Medium•Binomial Theorem

The coefficient of

x^3

in the expansion of

(1+x)^{10}

is:

Medium•Calculus

The value of

\int_0^{\pi/2} \sin^2 x \, dx

is:

Easy•Circles

The center of the circle

x^2 + y^2 - 4x + 6y - 12 = 0

is:

Easy•Complex Numbers

The value of

i^{25}

is:

Easy•Complex Numbers

If

z = \frac{1 + i}{1 - i}

, then

|z|

and

\arg(z)

are respectively:

Hard•Complex Numbers

[JEE Mains 2026] If complex numbers z₁, z₂, ..., zₙ satisfy the equation zⁿ + z + 1 = 0, then |z₁| + |z₂| + ... + |zₙ| is equal to:

Easy•Conic Sections

The eccentricity of the ellipse

\frac{x^2}{25} + \frac{y^2}{16} = 1

is:

Hard•Conic Sections

[JEE Mains 2026] The value of α for which the line αx + 2y = 1 never touches the hyperbola x²/9 - y²/4 = 1 is:

Medium•Conic Sections

[JEE Mains 2026] Let one end of a focal chord of the parabola y² = 16x be at point P. If the focus F divides this focal chord internally in the ratio m:n, what is the minimum value of m + n (where m, n are positive integers with gcd(m,n) = 1)?

Medium•Continuity

The function

f(x) = \frac{x^2 - 4}{x - 2}

is discontinuous at:

Easy•Coordinate Geometry

A line

y=m x+1

intersects the circle

(x-3)^{2}+(y+2)^{2}=25

at the points

P

and

Q

. If the midpoint of the line segment

P Q

has

x

-coordinate

-\frac{3}{5}

, then which one of the following options is correct?

Easy•Coordinate Geometry

The distance between the parallel lines

3x + 4y - 5 = 0

and

6x + 8y + 15 = 0

is:

Hard•Coordinate Geometry

[JEE Mains 2026] The image of point P(1, 2, a) with respect to the line (x-1)/2 = (y-1)/1 = (z-1)/1 is point Q(5, b, c). Find the value of a + b + c.

Hard•Coordinate Geometry

[JEE Mains 2026] If two circles x² + y² - 4x + 2y - 4 = 0 and (x-1)² + (y-4)² = r² intersect at two distinct points, what is the range of r?

Hard•Coordinate Geometry

[JEE Mains 2026] The locus of the point of intersection of tangents drawn to the circle x² + y² = 9 which subtend an angle of 60° at the center is:

Easy•Definite Integration

\int_0^1 x^3 dx

equals:

Easy•Determinants

The value of

\begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix}

is:

Easy•Differential Equations

The order of the differential equation

\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0

is:

Easy•Differential Calculus

If y = sin(x²), then dy/dx is:

Medium•Differential Equations

The solution of the differential equation

\frac{dy}{dx} + y = e^{-x}

, given

y(0) = 1

, is:

Easy•Differentiation

If

y = x^3 + 2x

, then

\frac{dy}{dx}

is:

Easy•Relations Functions

The domain of

f(x) = \sqrt{x-2}

is:

Medium•Sets And Functions

[JEE Mains 2026] If A = {1, 2, 3, 4, 5, 6} and B = {1, 2, 3, ..., 9}, then the number of strictly increasing functions f: A → B such that f(x) ≥ x for all x ∈ A is:

Easy•General

How many

3 \times 3

matrices

M

with entries from

\{0,1,2\}

are there, for which the sum of the diagonal entries of

M^{T} M

is

5 ?

Hard•General

Let the functions

f: \mathbb{R} \rightarrow \mathbb{R}

and

g: \mathbb{R} \rightarrow \mathbb{R}

be defined by

f(x)=e^{x-1}-e^{-|x-1|} \quad \text { and } \quad g(x)=\frac{1}{2}\left(e^{x-1}+e^{1-x}\right)

Then the area of the region in the first quadrant bounded by the curves

y=f(x), y=g(x)

and

x=0

is

Easy•Mathematical Reasoning

The negation of "All birds can fly" is:

Hard•General

Let

O

be the origin and let

P Q R

be an arbitrary triangle. The point

S

is such that

\overrightarrow{O P} \cdot \overrightarrow{O Q}+\overrightarrow{O R} \cdot \overrightarrow{O S}=\overrightarrow{O R} \cdot \overrightarrow{O P}+\overrightarrow{O Q} \cdot \overrightarrow{O S}=\overrightarrow{O Q} \cdot \overrightarrow{O R}+\overrightarrow{O P} \cdot \overrightarrow{O S}

Then the triangle

P Q R

has

S

as its

Hard•General

The area of the region

\left\{(x, y): x y \leq 8,1 \leq y \leq x^{2}\right\}

is

Hard•General

Suppose

a, b

denote the distinct real roots of the quadratic polynomial

x^{2}+20 x-2020

and suppose

c, d

denote the distinct complex roots of the quadratic polynomial

x^{2}-20 x+2020

. Then the value of

a c(a-c)+a d(a-d)+b c(b-c)+b d(b-d)

is

Hard•General

Let

S

be the set of all complex numbers

Z

satisfying

|z-2+i| \geq \sqrt{5}

. If the complex number

Z_{0}

is such that

\frac{1}{\left|Z_{0}-1\right|}

is the maximum of the set

\left\{\frac{1}{|z-1|}: z \in S\right\}

, then the principal argument of

\frac{4-z_{0}-\overline{z_{0}}}{Z_{0}-\overline{z_{0}}+2 i}

is

Hard•General

If

y=y(x)

satisfies the differential equation

8 \sqrt{x}(\sqrt{9+\sqrt{x}}) d y=(\sqrt{4+\sqrt{9+\sqrt{x}}})^{-1} d x, \quad x>0

and

y(0)=\sqrt{7}

, then

y(256)=

Hard•General

If

f: \mathbb{R} \rightarrow \mathbb{R}

is a twice differentiable function such that

f^{\prime \prime}(x)>0

for all

x \in \mathbb{R}

, and

f\left(\frac{1}{2}\right)=\frac{1}{2}, f(1)=1

, then

Medium•General

Let

f:(0, \infty) \rightarrow \mathbb{R}

be a differentiable function such that

f^{\prime}(x)=2-\frac{f(x)}{x}

for all

x \in(0, \infty)

and

f(1) \neq 1

. Then

Medium•General

Let

b_{i}>1

for

i=1,2, \ldots, 101

. Suppose

\log _{e} b_{1}, \log _{e} b_{2}, \ldots, \log _{e} b_{101}

are in Arithmetic Progression (A.P.) with the common difference

\log _{e} 2

. Suppose

a_{1}, a_{2}, \ldots, a_{101}

are in A.P. such that

a_{1}=b_{1}

and

a_{51}=b_{51}

. If

t=b_{1}+b_{2}+\cdots+b_{51}

and

s=a_{1}+a_{2}+\cdots+a_{51}

, then

Medium•General

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

Medium•General

Let

S=\{1,2,3, \ldots, 9\}

. For

k=1,2, \ldots, 5

, let

N_{k}

be the number of subsets of

S

, each containing five elements out of which exactly

k

are odd. Then

N_{1}+N_{2}+N_{3}+N_{4}+N_{5}=

Medium•General

The least value of

\alpha \in \mathbb{R}

for which

4 \alpha x^{2}+\frac{1}{x} \geq 1

, for all

x>0

, is

Easy•Height And Distance

The angle of elevation of the top of a tower from a point

50

m away from its base is

60°

. The height of the tower is:

Easy•Integration

\int x^2 dx

equals:

Easy•Definite Integrals

The value of ∫₀² x dx is:

Hard•Integration

[JEE Mains 2026] If ∫ cos^(5/2)x × sin^(11/2)x dx = (1/p) cot

^q

x + C, where C is the constant of integration, and p, q are in lowest terms, find the value of p + q.

Easy•Inverse Trigonometry

The principal value of

\sin^{-1}\left(\frac{1}{2}\right)

is:

Easy•Limits

\lim_{x \to 2} (3x + 1)

equals:

Medium•Quadratic Equations

[JEE Mains 2026] If 4x² + y² = 52 where x, y ∈ ℤ (integers), then the number of ordered pairs (x, y) is:

Medium•Limits And Continuity

The value of

\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}

is:

Medium•Calculus

[JEE Mains 2026] If f(3) = 18, f'(3) = 0, and f''(3) = 4, then the value of lim(x→3) [f(x) - 18]/(x - 3)² is:

Easy•Linear Programming

The corner points of a feasible region are

(0, 0)

,

(4, 0)

,

(2, 3)

,

(0, 4)

. The maximum value of

Z = 3x + 2y

is:

Easy•Straight Lines

The slope of the line

3x + 4y - 7 = 0

is:

Easy•Matrices

If

A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

and

B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}

, then

A + B

is:

Easy•Matrices

If

A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

, then

\det(3A)

equals:

Easy•Matrices

If A = [[2, 3], [1, 2]], then A⁻¹ is:

Hard•Matrices

[JEE Mains 2026] If A = [[2, 3], [3, 5]], then the value of det(A⁶ - 6A⁵ + 9A⁴) is:

Easy•Permutations Combinations

The value of

^5P_3

is:

Easy•Permutations And Combinations

The number of ways to arrange the letters of the word MATH is:

Easy•Probability

A die is thrown once. The probability of getting an even number is:

Easy•Probability

A bag contains

5

red balls and

3

blue balls. Two balls are drawn at random without replacement. The probability that both balls are red is:

Easy•Probability

A bag contains 4 red and 6 blue balls. Two balls are drawn without replacement. The probability that the second ball is red given that the first ball drawn was blue is:

Hard•Probability

A computer producing factory has only two plants

T_{1}

and

T_{2}

. Plant

T_{1}

produces

20 \%

and plant

T_{2}

produces

80 \%

of the total computers produced.

7 \%

of computers produced in the factory turn out to be defective. It is known that

P

(computer turns out to be defective given that it is produced in plant

T_{1}

)

=10 P\left(\right.

computer turns out to be defective given that it is produced in plant

\left.T_{2}\right)

, where

P(E)

denotes the probability of an event

E

. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant

T_{2}

is

Medium•Probability

Three randomly chosen nonnegative integers

x, y

and

z

are found to satisfy the equation

x+y+z=10

. Then the probability that

z

is even, is

Easy•Quadratic Equations

If

\alpha, \beta

are roots of

x^2 - 5x + 6 = 0

, then

\alpha + \beta

is:

Easy•Quadratic Equations

If the roots of the equation x² - 5x + 6 = 0 are α and β, then α + β is:

Easy•Sequences And Series

The sum of the infinite geometric series

1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots

is:

Easy•Sequences And Series

The sum of the series

\sum_{n=1}^{99} \frac{1}{n(n+1)}

is:

Easy•Sequences And Series

If the arithmetic mean and geometric mean of two positive numbers are 10 and 8 respectively, then the harmonic mean of these numbers is:

Easy•Sequences And Series

The sum

1^2 + 2^2 + 3^2 + \ldots + 15^2

equals:

Medium•Sequences And Series

[JEE Mains 2026] If the sum of the first 4 terms of an A.P. is 6 and the sum of the first 6 terms is 4, then the sum of the first 12 terms of the A.P. is:

Medium•Sequences And Series

The sum of the series

1 + 2 \cdot 2 + 3 \cdot 2^2 + 4 \cdot 2^3 + \ldots + 10 \cdot 2^9

is:

Easy•Sequences And Series

The sum of the first 20 terms of an arithmetic progression with first term 5 and common difference 3 is:

Easy•Sequences And Series

In a geometric progression, the first term is 3 and the common ratio is 2. The 8th term of the GP is:

Easy•Sequences Series

The 10th term of AP:

2, 5, 8, 11, ...

is:

Hard•Sequences And Series

If the sum of the first

n

terms of an AP is given by

S_n = 3n^2 + 5n

, then the 10th term of the AP is:

Medium•Sequences And Series

Let

a_1, a_2, a_3, \ldots

be a G.P. of positive terms. If

a_1 a_5 = 32

and

a_2 + a_4 = 12

, then

a_3

equals:

Easy•Sets

If

A = \{1, 2, 3\}

and

B = \{2, 3, 4\}

, then

A \cap B

is:

Easy•Statistics

The mean of

2, 4, 6, 8, 10

is:

Medium•Statistics

[JEE Mains 2026] If the mean and variance of observations x, y, 12, 14, 16 are 12 and 8 respectively, where x < y, then the value of x + y is:

Medium•Trigonometry

The equation of the plane passing through the point

(1,1,1)

and perpendicular to the planes

2 x+y-2 z=5

and

3 x-6 y-2 z=7

, is

Medium•Statistics

[JEE Mains 2026] Mean deviation about median for data k, 2k, 3k, ..., 1000k is 500. Find the value of k.

Easy•Trigonometry

The value of

\sin 30° + \cos 60°

is:

Easy•Trigonometry

The value of sin²30° + cos²30° is:

Hard•Trigonometry

Let

C_{1}

and

C_{2}

be two biased coins such that the probabilities of getting head in a single toss are

\frac{2}{3}

and

\frac{1}{3}

, respectively. Suppose

\alpha

is the number of heads that appear when

C_{1}

is tossed twice, independently, and suppose

\beta

is the number of heads that appear when

C_{2}

is tossed twice, independently. Then the probability that the roots of the quadratic polynomial

x^{2}-\alpha x+\beta

are real and equal, is

Hard•Trigonometry

The value of

\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{x^{2} \cos x}{1+e^{x}} d x

is equal to

Hard•Trigonometry

Consider all rectangles lying in the region

\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: 0 \leq x \leq \frac{\pi}{2} \text { and } 0 \leq y \leq 2 \sin (2 x)\right\}

and having one side on the

x

-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

Hard•Trigonometry

Let

a, b

and

\lambda

be positive real numbers. Suppose

P

is an end point of the latus rectum of the parabola

y^{2}=4 \lambda x

, and suppose the ellipse

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1

passes through the point

P

. If the tangents to the parabola and the ellipse at the point

P

are perpendicular to each other, then the eccentricity of the ellipse is

Hard•Trigonometry

Let

S=\left\{x \in(-\pi, \pi): x \neq 0, \pm \frac{\pi}{2}\right\}

. The sum of all distinct solutions of the equation

\sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0

in the set

S

is equal to

Hard•Trigonometry

For any positive integer

n

, define

f_{n}:(0, \infty) \rightarrow \mathbb{R}

as

f_{n}(x)=\sum_{j=1}^{n} \tan ^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right) \text { for all } x \in(0, \infty)

(Here, the inverse trigonometric function

\tan ^{-1} x

assumes values in

\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

. ) Then, which of the following statement(s) is (are) TRUE?

Hard•Trigonometry

Let

P

be the image of the point

(3,1,7)

with respect to the plane

x-y+z=3

. Then the equation of the plane passing through

P

and containing the straight line

\frac{x}{1}=\frac{y}{2}=\frac{z}{1}

is

Hard•Trigonometry

Consider a triangle

\Delta

whose two sides lie on the

x

-axis and the line

x+y+1=0

. If the orthocenter of

\Delta

is

(1,1)

, then the equation of the circle passing through the vertices of the triangle

\Delta

is

Hard•Trigonometry

The general solution of

\sin 2x = \cos 3x

is:

Medium•Trigonometry

[JEE Mains 2026] If (cos²48° - sin²12°)/(sin24° + cos6°) = p/q where p and q are coprime positive integers, then the value of p + q is:

Medium•Trigonometry

If a chord, which is not a tangent, of the parabola

y^{2}=16 x

has the equation

2 x+y=p

, and midpoint

(h, k)

, then which of the following is(are) possible value(s) of

p, h

and

k

?

Medium•Trigonometry

If the function

f: \mathbb{R} \rightarrow \mathbb{R}

is defined by

f(x)=|x|(x-\sin x)

, then which of the following statements is TRUE?

Medium•Trigonometry

Let

-\frac{\pi}{6}<\theta<-\frac{\pi}{12}

. Suppose

\alpha_{1}

and

\beta_{1}

are the roots of the equation

x^{2}-2 x \sec \theta+1=0

and

\alpha_{2}

and

\beta_{2}

are the roots of the equation

x^{2}+2 x \tan \theta-1=0

. If

\alpha_{1}>\beta_{1}

and

\alpha_{2}>\beta_{2}

, then

\alpha_{1}+\beta_{2}

equals

Easy•Vectors

The magnitude of vector

\vec{a} = 3\hat{i} + 4\hat{j}

is:

Easy•Vectors

If

\vec{a} = 2\hat{i} + 3\hat{j}

and

\vec{b} = \hat{i} - \hat{j}

, then

\vec{a} \cdot \vec{b}

is:

Medium•Vectors

If

\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}

and

\vec{b} = \hat{i} - 2\hat{j} + 2\hat{k}

, then the area of the parallelogram with adjacent sides

\vec{a}

and

\vec{b}

is:

Medium•Jee Main 2025

Let

a_1, a_2, a_3, \ldots

be a G.P. of increasing terms. If

a_1 a_5 = 28

and

a_2 + a_4 = 29

, then

a_6

is equal to

Hard•Jee Main 2025

Let

x = x(y)

be the solution of the differential equation

y^2 \, dx + (x - \frac{1}{y}) \, dy = 0

. If $x

Medium•Jee Main 2025

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is

\frac{m}{n}

, where

\gcd(m, n) = 1

, then

m + n

is equal to

Medium•Jee Main 2025

The product of all solutions of the equation

e^{5 \log x^2 + 3} = x^8, x > 0

, is

Medium•Jee Main 2025

Let the triangle PQR be the image of the triangle with vertices

(1, 3), (3, 1)

and

(2, 4)

in the line

x + 2y = 2

. If the centroid of

\triangle PQR

is the point

(\alpha, \beta)

, then

15(\alpha - \beta)

is equal to

Medium•Jee Main 2025

Let for

f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x

,

I_1 = \int_{0}^{\pi/4} f(x) \, dx

and

I_2 = \int_{0}^{\pi/4} x f(x) \, dx

. Then

7I_1 + 12I_2

is equal to

Medium•Jee Main 2025

Let the parabola

y = x^2 + px - 3

, meet the coordinate axes at the points P, Q and R. If the circle C with centre at

(\alpha, \beta)

passes through the points P, Q and R, then the area of

\triangle PQR

is

Medium•Jee Main 2025

Let

L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}

and

L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4}

be two lines. Then which of the following points lies on the line of the shortest distance between

L_1

and

L_2

?

Medium•Jee Main 2025

Let

f(x)

be a real differentiable function such that

f(0) = 1

and

f(x + y) = f(x)f(y) + f'(x)f(y)

for all

x, y \in \mathbb{R}

. Then

\sum_{n=1}^{100} \log_2 f(n)

is equal to

Medium•Jee Main 2025

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is

Medium•Jee Main 2025

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of

16 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2

is

Medium•Jee Main 2025

For positive integers

n

, if

4a_n = (n^2 + 5n + 6)

and

S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right)

, then the value of

507S_{2025}

is

Medium•Jee Main 2025

Let

f : \mathbb{R} \rightarrow \mathbb{R}

be a twice differentiable function such that

f(x + y) = f(x)f(y)

for all

x, y \in \mathbb{R}

. If

f'(0) = 4a

and

f

satisfies

f''(x) - 3af'(x) - f(x) = 0, a > 0

, then the area of the region

R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\}

is

Medium•Jee Main 2025

The area of the region, inside the circle

(x - 2\sqrt{3})^2 + y^2 = 12

and outside the parabola

y^2 = 2\sqrt{3}x

is

Medium•Jee Main 2025

Let the foci of a hyperbola be

(1, 14)

and

(1, -12)

. If it passes through the point

(1, 6)

, then the length of its latus-rectum is

Medium•Jee Main 2025

If

\sum_{r=1}^{n} T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}

, then

\lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{T_r} \right)

is equal to

Medium•Jee Main 2025

A coin is tossed three times. Let

X

denote the number of times a tail follows a head. If

\mu

and

\sigma^2

denote the mean and variance of

X

, then the value of

64(\mu + \sigma^2)

is

Medium•Jee Main 2025

The number of non-empty equivalence relations on the set

\{1, 2, 3\}

is

Medium•Jee Main 2025

A circle

C

of radius 2 lies in the second quadrant and touches both the coordinate axes. Let

r

be the radius of a circle that has centre at the point

(2, 5)

and intersects the circle

C

at exactly two points. If the set of all possible values of

r

is the interval

(\alpha, \beta)

, then

3\beta - 2\alpha

is equal to

Medium•Jee Main 2025

Let

A = \{1, 2, 3, \ldots, 10\}

and

B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}

. Then

n(B)

is equal to

Medium•Jee Main 2025

Let

z_1, z_2

and

z_3

be three complex numbers on the circle

|z| = 1

with

\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0

and

\arg(z_3) = \frac{\pi}{4}

. If

|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}

, then the value of

\alpha^2 + \beta^2

is

Medium•Jee Main 2025

Let

P(4, 4\sqrt{3})

be a point on the parabola

y^2 = 4ax

and

PQ

be a focal chord of the parabola. If

M

and

N

are the foot of perpendiculars drawn from

P

and

Q

respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to

Medium•Jee Main 2025

The sum of all values of

\theta \in [0, 2\pi]

satisfying

2\sin^2 \theta = \cos 2\theta

and

2\cos^2 \theta = 3\sin \theta

is

Medium•Jee Main 2025

Let the curve

z(1 + i) + \bar{z}(1 - i) = 4

,

z \in \mathbb{C}

, divide the region

|z - 3| \leq 1

into two parts of areas

\alpha

and

\beta

. Then

|\alpha - \beta|

equals

Medium•Jee Main 2025

Let

E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b

and

H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1

. Let the distance between the foci of

E

and the foci of

H

be

2\sqrt{3}

. If

a - A = 2

, and the ratio of the eccentricities of

E

and

H

is

\frac{1}{3}

, then the sum of the lengths of their latus rectums is equal to

Medium•Jee Main 2025

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

Medium•Jee Main 2025

One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is

Medium•Jee Main 2025

Let

\frac{x^2}{16} + \frac{y^2}{25} = 1

,

z \in C

, be the equation of a circle with center at

C

. If the area of the triangle, whose vertices are at the points

(0, 0)

,

C

and

(\alpha, 0)

is 11 square units, then

\alpha^2

equals:

Medium•Jee Main 2025

Let the position vectors of the vertices

A, B

and

C

of a tetrahedron

ABCD

be

\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}

and

2\mathbf{i} + \mathbf{j} - \mathbf{k}

respectively. The altitude from the vertex

D

to the opposite face

ABC

meets the median line segment through

A

of the triangle

ABC

at the point

E

. If the length of

AD

is

\frac{\sqrt{11}}{3}

and the volume of the tetrahedron is

\frac{\sqrt{805}}{6}

, then the position vector of

E

is

Medium•Jee Main 2025

If

A, B,

and

(\text{adj} (A^{-1}) + \text{adj} (B^{-1}))

are non-singular matrices of same order, then the inverse of

A (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B

, is equal to

Medium•Jee Main 2025

Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is

Hard•Jee Main 2025

Let a curve

y = f(x)

pass through the points

(0, 5)

and

(\log_e 2, k)

. If the curve satisfies the differential equation

2(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0

, then

k

is equal to

Medium•Jee Main 2025

If the function

f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}

is continuous at

x = 0

, then

k_1^2 + k_2^2

is equal to

Medium•Jee Main 2025

If the line

3x - 2y + 12 = 0

intersects the parabola

4y = 3x^2

at the points

A

and

B

, then at the vertex of the parabola, the line segment

AB

subtends an angle equal to

Medium•Jee Main 2025

Let

P

be the foot of the perpendicular from the point

Q(10, -3, -1)

on the line

\frac{x-3}{7} = \frac{y-2}{1} = \frac{z+1}{2}

. Then the area of the right angled triangle

PQR

, where

R

is the point

(3, -2, 1)

, is

\begin{align*}

Medium•Jee Main 2025

Let the arc

AC

of a circle subtend a right angle at the centre

O

. If the point

B

on the arc

AC

, divides the arc

AC

such that

\frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5}

, and

\overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB}

, then

\alpha + \sqrt{2(\sqrt{3} - 1)}\beta

is equal to

\begin{align*}

Medium•Jee Main 2025

Let

f(x) = \log_2 x

and

g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1}

. Then the domain of

f \circ g

is

\begin{align*}

Medium•Jee Main 2025

(\lambda - 1)x + (\lambda - 4)y + \lambda z = 5

If the system of equations

\lambda x + (\lambda - 1)y + (\lambda - 4)z = 7

has infinitely many solutions, then

\lambda^2 + \lambda

is equal to

\begin{align*}

Medium•Jee Main 2025

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is

\begin{align*}

Medium•Jee Main 2025

Let

R = \{(1, 2), (2, 3), (3, 3)\}

be a relation defined on the set

\{1, 2, 3, 4\}

. Then the minimum number of elements, needed to be added in

R

so that

R

becomes an equivalence relation, is

\begin{align*}

Medium•Jee Main 2025

Let the area of a

\triangle PQR

with vertices

P(5, 4), Q(-2, 4)

and

R(a, b)

be 35 square units. If its orthocenter and centroid are

O \left(2, \frac{12}{7}\right)

and

C(c, d)

respectively, then

c + 2d

is equal to

\begin{align*}

Medium•Jee Main 2025

The value of

\int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx

is

\begin{align*}

Medium•Jee Main 2025

Let

I(x) = \int \frac{dx}{(x-11)(x+15)}

. If

I(37) - I(24) = \frac{1}{4} \left( \frac{1}{\beta x} - \frac{1}{c x} \right)

,

b, c \in \mathbb{N}

, then

3(b + c)

is equal to

Medium•Jee Main 2025

If

\frac{\pi}{6} \leq x \leq \frac{3\pi}{4}

, then

\cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right)

is equal to

Medium•Jee Main 2025

The distance of the line

\frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4}

from the point

(1, 4, 0)

along the line

\frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3}

is

Medium•Jee Main 2025

Let

A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\}

and

B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\}

. If

C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\}

, then

\sum_{(x, y) \in C} |x + y|

is

Medium•Jee Main 2025

Let

X = \mathbb{R} \times \mathbb{R}

. Define a relation

R

on

X

as:

(a_1, b_1)R(a_2, b_2) \iff b_1 = b_2

Statement I :

R

is an equivalence relation. Statement II : For some

(a, b) \in X

, the set

S = \{(x, y) \in X : (x, y)R(a, b)\}

represents a line parallel to

y = x

. In the light of the above statements, choose the correct answer from the options given below

Medium•Jee Main 2025

Let

\int x^3 \sin x \, dx = g(x) + C

, where

C

is the constant of integration. If

8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z}

, then

\alpha + \beta - \gamma

equals

Medium•Jee Main 2025

A rod of length eight units moves such that its ends

A

and

B

always lie on the lines

x - y + 2 = 0

and

y + 2 = 0

, respectively. If the locus of the point

P

, that divides the rod

AB

internally in the ratio

2 : 1

is

9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0

, then

\alpha - \beta - \gamma

is equal to

Medium•Jee Main 2025

If the square of the shortest distance between the lines

\frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3}

and

\frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3}

is

\frac{m}{n}

, where

m, n

are coprime numbers, then

m + n

is equal to

Medium•Jee Main 2025

\lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}}

is equal to

Medium•Jee Main 2025

Let the point

A

divide the line segment joining the points

P(-1,-1,2)

and

Q(5,5,10)

internally in the ratio

r : 1(r > 0)

. If

O

is the origin and

\overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2

= 10, then the value of

r

is

Medium•Jee Main 2025

The length of the chord of the ellipse

\frac{x^2}{4} + \frac{y^2}{3} = 1

, whose mid-point is

(1, \frac{1}{2})

, is

Medium•Jee Main 2025

The system of equations

x + 2y + 5z = 9

, has no solution if: (x)

x + 5y + \lambda z = \mu

,

Medium•Jee Main 2025

Let the range of the function

f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R}

be

[\alpha, \beta]

. Then the distance of the point

(\alpha, \beta)

from the line

3x + 4y + 12 = 0

is

Hard•Jee Main 2025

Let

x = x(y)

be the solution of the differential equation \[ y =

\left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x

Medium•Jee Main 2025

The equation of the chord, of the ellipse

\frac{x^2}{25} + \frac{y^2}{16} = 1

, whose mid-point is

(3, 1)

, is:

25x + 101y = 176

Medium•Jee Main 2025

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of

\frac{1}{4}

cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is

Medium•Jee Main 2025

The number of complex numbers

z

, satisfying

|z| = 1

and

|\frac{z}{2} + \frac{\bar{z}}{2}| = 1

, is

Hard•Jee Main 2025

Let

A = [a_{ij}]

be a

3 \times 3

matrix such that

A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}

, then

a_{23}

equals

Medium•Jee Main 2025

If

I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx

, then

I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx

equals

Medium•Jee Main 2025

A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is

Medium•Jee Main 2025

Let the shortest distance from

(a, 0), a > 0

to the parabola

y^2 = 4x

be 4. Then the equation of the circle passing through the point

(a, 0)

and the focus of the parabola, and having its centre on the axis of the parabola is

Medium•Jee Main 2025

If in the expansion of

(1 + x)^p(1 - x)^q

, the coefficients of

x

and

x^2

are 1 and -2, respectively, then

p^2 + q^2

is equal to

Medium•Jee Main 2025

If the area of the region

\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}

is

\frac{e^{x+1} e^{x+1}}{e}

, then the value of

a

is

Medium•Jee Main 2025

Let circle

C

be the image of

x^2 + y^2 - 2x + 4y - 4 = 0

in the line

2x - 3y + 5 = 0

and

A

be the point on

C

such that

OA

is parallel to

x

-axis and

A

lies on the right hand side of the centre

O

of

C

. If

B(\alpha, \beta)

, with

\beta < 4

, lies on

C

such that the length of the arc

AB

is

(1/6)^{th}

of the perimeter of

C

, then

\beta - \sqrt{3}\alpha

is equal to

Medium•Jee Main 2025

Let in a

\triangle ABC

, the length of the side

AC

be

6

, the vertex

B

be

(1, 2, 3)

and the vertices

A, C

lie on the line

\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}

. Then the area (in sq. units) of

\triangle ABC

is

Medium•Jee Main 2025

Let the product of the focal distances of the point

\left(\sqrt{3}, \frac{1}{3}\right)

on the ellipse

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

,

(a > b)

, be

\frac{7}{4}

. Then the absolute difference of the eccentricities of two such ellipses is

Medium•Jee Main 2025

If the system of equations

5x + \lambda y + 3z = 12

and

100x - 47y + \mu z = 212

has infinitely many solutions, then

\mu - 2\lambda

is equal to

Medium•Jee Main 2025

For some

n \neq 10

, let the coefficients of the

5

th,

6

th and

7

th terms in the binomial expansion of

(1 + x)^{n+4}

be in A.P. Then the largest coefficient in the expansion of

(1 + x)^{n+4}

is

Medium•Jee Main 2025

The product of all the rational roots of the equation

\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3

, is equal to

Medium•Jee Main 2025

Let the line passing through the points

(-1, 2, 1)

and parallel to the line

\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}

intersect the line

\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}

at the point

P

. Then the distance of

P

from the point

Q(4, -5, 1)

is

Medium•Jee Main 2025

Let the lines

3x - 4y - \alpha = 0

,

8x - 11y - 33 = 0

, and

2x - 3y + \lambda = 0

be concurrent. If the image of the point

(1, 2)

in the line

2x - 3y + \lambda = 0

is

\left(\frac{57}{13}, \frac{-40}{13}\right)

, then

|\alpha\lambda|

is equal to

Medium•Jee Main 2025

If

\alpha

and

\beta

are the roots of the equation

2x^2 - 3x - 2i = 0

, where

i = \sqrt{-1}

, then

16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right)

is equal to

Medium•Jee Main 2025

For a statistical data

x_1, x_2, \ldots, x_{10}

of 10 values, a student obtained the mean as 5.5 and

\sum_{i=1}^{10} x_i^2 = 371

. He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is

Medium•Jee Main 2025

The area of the region

\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}

is equal to

Medium•Jee Main 2025

Let

S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots

upto

n

terms. If the sum of the first six terms of an A.P. with first term

-p

and common difference

p

is

\sqrt{2025} S_{2025}

, then the absolute difference between 20th and 15th terms of the A.P. is

Medium•Jee Main 2025

Let

f : R \to \{0\} \to R

be a function such that

f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}

. If the limit as

x \to 0

\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R

, then

\alpha + 2\beta

is equal to

Medium•Jee Main 2025

If

I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0

, then

I(9, 14) + I(10, 13)

is

Medium•Jee Main 2025

A

and

B

alternately throw a pair of dice.

A

wins if he throws a sum of 5 before

B

throws a sum of 8, and

B

wins if he throws a sum of 8 before

A

throws a sum of 5. The probability, that

A

wins if

A

makes the first throw, is

Medium•Jee Main 2025

Let

f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}

. Then the value of

8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)

is equal to

Hard•Jee Main 2025

Let

y = y(x)

be the solution of the differential equation

(xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0

. Then

y(\sqrt{3})

is equal to

Medium•Jee Main 2025

\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)

is

Medium•Jee Main 2025

Consider the region

R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \}

. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in

R

, is

Medium•Jee Main 2025

Let

\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k}

and

\vec{c}

be three vectors such that

\vec{c}

is coplanar with

\vec{a}

and

\vec{b}

. If the vector

\vec{C}

is perpendicular to

\vec{b}

and

\vec{a} \cdot \vec{c} = 5

, then

|\vec{c}|

is equal to

Medium•Jee Main 2025

Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to

Medium•Jee Main 2025

If the system of equations

2x + \lambda y + 5z = 5

has infinitely many solutions, then

\lambda + \mu

is equal to:

14x + 3y + \mu z = 33

Medium•Jee Main 2025

Let

A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\}

and

B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}

. Then

n(A \cup B)

is equal to

Medium•Jee Main 2025

The area of the region enclosed by the curves

y = e^x

,

y = |e^x - 1|

and y-axis is

Medium•Jee Main 2025

Let the points

(\frac{11}{2}, \alpha)

lie on or inside the triangle with sides

x + y = 11

,

x + 2y = 16

and

2x + 3y = 29

. Then the product of the smallest and the largest values of

\alpha

is equal to

Medium•Jee Main 2025

Let

f : (0, \infty) \rightarrow \mathbb{R}

be a function which is differentiable at all points of its domain and satisfies the condition

x^2 f'(x) = 2x f(x) + 3

, with $f

Medium•Jee Main 2025

If

7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty

, then the value of

\alpha

is

Medium•Jee Main 2025

Let

[x]

denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function

f(x) = [x] + [x - 2]

,

-2 < x < 3

, is not continuous and not differentiable. Then

m + n

is equal to

Hard•Jee Main 2025

Let

A = [a_{ij}]

be a square matrix of order 2 with entries either 0 or 1. Let

E

be the event that

A

is an invertible matrix. Then the probability

P(E)

is

Medium•Jee Main 2025

Let the position vectors of three vertices of a triangle be

4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}

and

2\mathbf{p} - \mathbf{q} + 2\mathbf{r}

. If the position vectors of the orthocenter and the circumcenter of the triangle are

\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}

and

\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}

respectively, then

\alpha + 2\beta + 5\gamma

is equal to

Medium•Jee Main 2025

Let

\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})

and

\mathbf{c} = \mathbf{b} \times \mathbf{k}

. Then the projection of

\mathbf{c} - 2\mathbf{j}

on

\mathbf{a}

is

Medium•Jee Main 2025

The number of real solution(s) of the equation

x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}

is

Medium•Jee Main 2025

The function

f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}

is

Medium•Jee Main 2025

In an arithmetic progression, if

S_{10} = 1030

and

S_{12} = 57

, then

S_{30} - S_{10}

is equal to

Medium•Jee Main 2025

Suppose

A

and

B

are the coefficients of 30th and 12th terms respectively in the binomial expansion of

(1 + x)^{2n-1}

. If

2A = 5B

, then

n

is equal to

Medium•Jee Main 2025

Let

(2, 3)

be the largest open interval in which the function

f(x) = 2\log_e(x - 2) - x^2 + ax + 1

is strictly increasing and

(b, c)

be the largest open interval, in which the function

g(x) = (x - 1)^3(x + 2 - a)^2

is strictly decreasing. Then

100(a + b - c)

is equal to

Hard•Jee Main 2025

For some

a, b

, let

f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b

. Then

(\lambda + \mu + \nu)^2

is equal to

Medium•Jee Main 2025

If the equation of the parabola with vertex

V\left(\frac{3}{2}, 3\right)

and the directrix

x + 2y = 0

is

\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0

, then

\alpha + \beta + \gamma

is equal to

Medium•Jee Main 2025

If

\alpha > \beta > \gamma > 0

, then the expression

\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}

is equal to

Medium•Jee Main 2025

Let

O

be the origin, the point

A

be

z_1 = \sqrt{3} + 2\sqrt{2}i

, the point

B(z_2)

be such that

\sqrt{3} |z_2| = |z_1|

and

\arg(z_2) = \arg(z_1) + \frac{\pi}{6}

. Then

Medium•Jee Main 2025

Let

f : \mathbb{R} \to \mathbb{R}

be a function defined by

f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1.

If

f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy

, then the value of

28 \sum_{i=1}^{5} |f(i)|

is

Medium•Jee Main 2025

Let

ABCD

be a trapezium whose vertices lie on the parabola

y^2 = 4x

. Let the sides

AD

and

BC

of the trapezium be parallel to

y

-axis. If the diagonal

AC

is of length

\frac{25}{4}

and it passes through the point

(1, 0)

, then the area of

ABCD

is

Medium•Jee Main 2025

The sum of all local minimum values of the function

f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\ \frac{1}{12}(x - 4)(x - 5), & x > 2 \end{cases}

is

Medium•Jee Main 2025

Let

^nC_{r-1} = 28, ^nC_r = 56

and

^nC_{r+1} = 70

. Let

A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t)

and

C(3r - n, r^2 - n - 1)

be the vertices of a triangle

ABC

, where

t

is a parameter. If

(3x - 1)^2 + (3y)^2 = \alpha

, is the locus of the centroid of triangle

ABC

, then

\alpha

equals

Medium•Jee Main 2025

Let the equation of the circle, which touches

x

-axis at the point

(a, 0)

,

a > 0

and cuts off an intercept of length

b

on

y

-axis be

x^2 + y^2 - \alpha x + \beta y + \gamma = 0

. If the circle lies below

x

-axis, then the ordered pair

(2a, b^2)

is equal to

Medium•Jee Main 2025

If

f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R},

then

\sum_{k=1}^{81} f \left( \frac{k}{82} \right)

is equal to

Medium•Jee Main 2025

Two number

k_1

and

k_2

are randomly chosen from the set of natural numbers. Then, the probability that the value of

i^{k_1} + j^{k_2}, (i = \sqrt{-1})

is non-zero, equals

Medium•Jee Main 2025

If the image of the point

(4, 4, 3)

in the line

\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3}

is

(\alpha, \beta, \gamma)

, then

\alpha + \beta + \gamma

is equal to \[

Medium•Jee Main 2025

\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)

is equal to: \[

Medium•Jee Main 2025

Let

A(x, y, z)

be a point in

xy

-plane, which is equidistant from three points

(0, 3, 2), (2, 0, 3)

and

(0, 0, 1)

. Let

B = (1, 4, -1)

and

C = (2, 0, -2)

. Then among the statements (S1) :

\triangle ABC

is an isosceles right angled triangle, and (S2) : the area of

\triangle ABC

is

\frac{9\sqrt{2}}{2}

, \[

Medium•Jee Main 2025

The area (in sq. units) of the region

\{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \}

is \[

Medium•Jee Main 2025

The sum of the squares of all the roots of the equation

x^2 + |2x - 3| - 4 = 0

, is \[

Medium•Jee Main 2025

Let

T_r

be the

r^{th}

term of an A.P. If for some

m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}

, and

20 \sum_{r=1}^{25} T_r = 13

, then

5m \sum_{r=m}^{m+2} T_r

is equal to \[

Medium•Jee Main 2025

Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If

x

denote the number of defective oranges, then the variance of

x

is \[

Medium•Jee Main 2025

Let for some function

y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0

and

f(2) = 3

. Then

f(6)

is equal to \[

Medium•Jee Main 2025

If

\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}

, then

(\alpha + \beta)^2

equals \[

Medium•Jee Main 2025

Let

\{a_n\}

be a sequence such that

a_0 = 0, a_1 = \frac{1}{2}

and

2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots

. Then

\sum_{k=1}^{100} a_k

is equal to

Medium•Jee Main 2025

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is

Medium•Jee Main 2025

The relation

R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \}

is

Hard•Jee Main 2025

Let

A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix}

and

P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0.

If

B = PAP^T, C = P^TB^TP

and the sum of the diagonal elements of

C

is

\frac{m}{n}

, where

\gcd(m, n) = 1

, then

m + n

is

Medium•Jee Main 2025

If the components of

\vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}

along and perpendicular to

\vec{b} = 3\hat{i} + \hat{j} - \hat{k}

respectively, are

\frac{16}{10}(3\hat{i} + \hat{j} - \hat{k})

and

\frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k})

, then

\alpha^2 + \beta^2 + \gamma^2

is equal to

Medium•Jee Main 2025

Let

A, B, C

be three points in

xy

-plane, whose position vectors are given by

\sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j}

and

\hat{i} + (1 - a)\hat{j}

respectively with respect to the origin

O

. If the distance of the point

C

from the line bisecting the angle between the vectors

\overrightarrow{OA}

and

\overrightarrow{OB}

is

\frac{a}{\sqrt{2}}

, then the sum of all the possible values of

a

is

Medium•Jee Main 2025

Let the coefficients of three consecutive terms

T_r, T_{r+1},

and

T_{r+2}

in the binomial expansion of

(a + b)^{\frac{1}{2}}

be in a G.P. and let

p

be the number of all possible values of

r

. Let

q

be the sum of all rational terms in the binomial expansion of

(\sqrt{3} + \sqrt{4})^{\frac{1}{2}}

. Then

p + q

is equal to

Medium•Jee Main 2025

Let

[x]

denote the greatest integer less than or equal to

x

. Then the domain of

f(x) = \sec^{-1}(2[x] + 1)

is

Medium•Jee Main 2025

Let

S

be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set

S

, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is

Medium•Jee Main 2025

If

\sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)}

=

a\sqrt{3} + b

,

a, b \in \mathbb{Z}

, then

a^2 + b^2

is equal to

Medium•Jee Main 2025

Let

f

be a real valued continuous function defined on the positive real axis such that

g(x) = \int_0^x t f(t) \, dt

. If

g(x^2) = x^6 + x^7

, then value of

\sum_{r=1}^{15} f(x^3)

is

Medium•Jee Main 2025

Let

f : [0, 3] \rightarrow A

be defined by

f(x) = 2x^3 - 15x^2 + 36x + 7

and

g : [0, \infty) \rightarrow B

be defined by

g(x) = \frac{x^{2025}}{x^{2025} + 1}

. If both the functions are onto and

S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \}

, then

n(S)

is equal to

Medium•Jee Main 2025

Bag

B_1

contains 6 white and 4 blue balls, Bag

B_2

contains 4 white and 6 blue balls, and Bag

B_3

contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag

B_2

, is

Medium•Jee Main 2025

Let

f : \mathbb{R} \to \mathbb{R}

be a twice differentiable function such that

f(2) = 1

. If

F(x) = xf(x)

for all

x \in \mathbb{R}

,

\int_{x}^{2} x F'(x)\,dx = 6

and

\int_{x}^{2} x^2 F''(x)\,dx = 40

, then

F'(2) + \int_{x}^{2} F(x)\,dx

is equal to

Medium•Jee Main 2025

Let

f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)

be a polynomial of degree 2, satisfying

f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)

. If

f(K) = -2K

, then the sum of squares of all possible values of

K

is

Medium•Jee Main 2025

If

A

and

B

are the points of intersection of the circle

x^2 + y^2 - 8x = 0

and the hyperbola

\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1

and a point

P

moves on the line

2x - 3y + 4 = 0

, then the centroid of

\triangle PAB

lies on the line

Medium•Jee Main 2025

If

f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}

,

f(0) = -6

, then $f

Medium•Jee Main 2025

The area of the region bounded by the curves

x \left( 1 + y^2 \right) = 1

and

y^2 = 2x

is

Medium•Jee Main 2025

The square of the distance of the point

\left( \frac{15}{7}, \frac{22}{7}, 7 \right)

from the line

\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}

in the direction of the vector

\hat{i} + 4\hat{j} + 7\hat{k}

is

Medium•Jee Main 2025

If the midpoint of a chord of the ellipse

\frac{x^2}{9} + \frac{y^2}{4} = 1

is

(\sqrt{2}, 4/3)

, and the length of the chord is

\frac{2\sqrt{5}}{3}

, then

\alpha

is

Medium•Jee Main 2025

If

\alpha + i\beta

and

\gamma + i\delta

are the roots of

x^2 - (3 - 2i)x - (2i - 2) = 0

,

i = \sqrt{-1}

, then

\alpha\gamma + \beta\delta

is equal to

Medium•Jee Main 2025

Two equal sides of an isosceles triangle are along

-x + 2y = 4

and

x + y = 4

. If

m

is the slope of its third side, then the sum, of all possible distinct values of

m

, is

Medium•Jee Main 2025

Let

x_1, x_2, \ldots, x_{10}

be ten observations such that

\sum_{i=1}^{10} (x_i - 2) = 30,

\sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2,

and their variance is

\frac{4}{5}.

If

\mu

and

\sigma^2

are respectively the mean and the variance of

2(x_1 - 1) + 4\beta,

2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta,

then

\frac{\partial \mu}{\partial \beta}

is equal to

Medium•Jee Main 2025

Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is

Medium•Jee Main 2025

The number of solutions of the equation

\left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0

is

Medium•Jee Main 2025

Define a relation

R

on the interval

[0, \frac{\pi}{4}]

by

xRy

if and only if

\sec^2 x - \tan^2 y = 1.

Then

R

is

Medium•Jee Main 2025

Two parabolas have the same focus

(4, 3)

and their directrices are the

x

-axis and the

y

-axis, respectively. If these parabolas intersect at the points

A

and

B,

then

(AB)^2

is equal to

Medium•Jee Main 2025

Let

P

be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in

P

are formed by using the digits 1, 2 and 3 only, then the number of elements in the set

P

is

Medium•Jee Main 2025

Let

\vec{a} = \hat{i} + 2\hat{j} + \hat{k}

and

\vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}.

Let

L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in \mathbb{R}

and

L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R}

be two lines. If the line

L_3

passes through the point of intersection of

L_1

and

L_2,

and is parallel to

\vec{a} + \vec{b},

then

L_3

passes through the point

Medium•Jee Main 2025

Let

\vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k}

and

\vec{c}

be a vector such that

\vec{a} \times \vec{c} = \vec{c} \times \vec{b}

and

(\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168.

Then the maximum value of

|\vec{c}|^2

is

Medium•Jee Main 2025

The integral

80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta

is equal to

Medium•Jee Main 2025

Let the ellipse

E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b

and

E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B

have same eccentricity

\frac{1}{\sqrt{3}}

. Let the product of their lengths of latus rectums be

\frac{32}{\sqrt{3}}

, and the distance between the foci of

E_1

be 4. If

E_1

and

E_2

meet at

A, B, C

and

D

, then the area of the quadrilateral

ABCD

equals:

Hard•Jee Main 2025

Let

A = [a_{ij}] = \begin{bmatrix} \log_5 128 \log_4 5 \\ \log_5 8 \log_4 25 \end{bmatrix}

. If

A_{ij}

is the cofactor of

a_{ij}

,

C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2

, and

C = [C_{ij}]

, then

8|C|

is equal to:

Medium•Jee Main 2025

Let

|z_1 - 8 - 2i| \leq 1

and

|z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C}

. Then the minimum value of

|z_1 - z_2|

is:

Medium•Jee Main 2025

Let

L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4}

and

L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1}

be two lines. Let

L_3

be a line passing through the point

(\alpha, \beta, \gamma)

and be perpendicular to both

L_1

and

L_2

. If

L_3

intersects

L_1

, then

|5\alpha - 11\beta - 8\gamma|

equals:

Hard•Jee Main 2025

Let

M

and

m

respectively be the maximum and the minimum values of

f(x) = \begin{bmatrix} 1 + \sin^2 x \cos^2 x 4\sin 4x \\ \sin^2 x 1 + \cos^2 x 4\sin 4x \\ \sin^2 x \cos^2 x 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R}

Then

M^4 - m^4

is equal to:

Medium•Jee Main 2025

Let

ABC

be a triangle formed by the lines

7x - 6y + 3 = 0, x + 2y - 31 = 0

and

9x - 2y - 19 = 0

. Let the point

(h, k)

be the image of the centroid of

\Delta ABC

in the line

3x + 6y - 53 = 0

. Then

h^2 + k^2 + hk

is equal to:

Medium•Jee Main 2025

The value of

\lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right)

is:

Medium•Jee Main 2025

The least value of

n

for which the number of integral terms in the Binomial expansion of

(\sqrt{7} + \sqrt{11})^n

is 183, is:

Hard•Jee Main 2025

Let

y = y(x)

be the solution of the differential equation

\cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).

If

y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2}

, then

y \left( \frac{\pi}{8} \right)

is equal to:

Medium•Jee Main 2025

Let the line

x + y = 1

meet the circle

x^2 + y^2 = 4

at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to:

Medium•Jee Main 2025

Let the area of the region

\{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\}

be A. Then

6A

is equal to:

Medium•Jee Main 2025

Let

f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5.

If the range of

f

is

[\alpha, \beta]

, then

4(\alpha + \beta)

equals

Medium•Jee Main 2025

Let

\vec{a}

be a unit vector perpendicular to the vectors

\vec{b} = \hat{i} - 2\hat{j} + 3\hat{k}

and

\vec{c} = 2\hat{i} + 3\hat{j} - \hat{k}

, and makes an angle of

\cos^{-1}\left(-\frac{1}{2}\right)

with the vector

\hat{i} + \hat{j} + \hat{k}

. If

\vec{a}

makes an angle of

\frac{\pi}{3}

with the vector

\hat{i} + \alpha\hat{j} + \hat{k}

, then the value of

\alpha

is

Medium•Jee Main 2025

Let

\alpha, \beta (\alpha \neq \beta)

be the values of

m

, for which the equations

x + y + z = 1, x + 2y + 4z = m

and

x + 4y + 10z = m^2

have infinitely many solutions. Then the value of

\sum_{n=1}^{10} (n^\alpha + n^\beta)

is equal to

Hard•Jee Main 2025

If for the solution curve

y = f(x)

of the differential equation

\frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2}

,

x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

, then

f\left(\frac{\pi}{4}\right)

is equal to

Medium•Jee Main 2025

Let

P

be the foot of the perpendicular from the point

(1, 2, 2)

on the line

L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2}

. Let the line

\vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R}

, intersect the line

L

at

Q

. Then

2(PQ)^2

is equal to

Hard•Jee Main 2025

Let

A = [a_{ij}]

be a matrix of order

3 \times 3

, with

a_{ij} = (\sqrt{2})^{i+j}

. If the sum of all the elements in the third row of

A^2

is

\alpha + \beta\sqrt{2}

,

\alpha, \beta \in \mathbb{Z}

, then

\alpha + \beta

is equal to

Medium•Jee Main 2025

Let the line

x + y = 1

meet the axes of

x

and

y

at

A

and

B

, respectively. A right angled triangle

AMN

is inscribed in the triangle

OAB

, where

O

is the origin and the points

M

and

N

lie on the lines

OB

and

AB

, respectively. If the area of the triangle

AMN

is

\frac{4}{5}

of the area of the triangle

OAB

and

AN : NB = \lambda : 1

, then the sum of all possible value(s) of

\lambda

is

Medium•Jee Main 2025

If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged in a dictionary, then the word at 440th position in this arrangement, is

Medium•Jee Main 2025

If the set of all

a \in \mathbb{R}

, for which the equation

2x^2 + (a - 5)x + 15 = 3a

has no real root, is the interval

(\alpha, \beta)

, and

X = \{x \in \mathbb{Z} : \alpha < x < \beta\}

, then

\sum_{x \in X} x^2

is equal to

Hard•Jee Main 2025

Let

A = [a_{ij}]

be a

2 \times 2

matrix such that

a_{ij} \in \{0, 1\}

for all

i

and

j

. Let the random variable

X

denote the possible values of the determinant of the matrix

A

. Then, the variance of

X

is

Medium•Jee Main 2025

Let the function

f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x|

be not differentiable at the two points

x = \alpha = 2

and

x = \beta

. Then the distance of the point

(\alpha, \beta)

from the line

12x + 5y + 10 = 0

is equal to

Medium•Jee Main 2025

Let the area enclosed between the curves

|y| = 1 - x^2

and

x^2 + y^2 = 1

be

\alpha

. If

9\alpha = \beta \pi + \gamma

,

\beta, \gamma

are integers, then the value of

|\beta - \gamma|

equals.

Medium•Jee Main 2025

The remainder, when

7^{10^3}

is divided by 23, is equal to

Medium•Jee Main 2025

If

\alpha x + \beta y = 109

is the equation of the chord of the ellipse

\frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1

, whose mid point is

\left( \frac{1}{2}, \frac{1}{4} \right)

, then

\alpha + \beta

is equal to

Medium•Jee Main 2025

If the domain of the function

\log_5 (18x - x^2 - 77)

is

(\alpha, \beta)

and the domain of the function

\log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right)

is

(\gamma, \delta)

, then

\alpha^2 + \beta^2 + \gamma^2

is equal to

Medium•Jee Main 2025

Let a circle

C

pass through the points

(4, 2)

and

(0, 2)

, and its centre lie on

3x + 2y + 2 = 0

. Then the length of the chord, of the circle

C

, whose mid-point is

(1, 2)

, is

Medium•Jee Main 2025

Let a straight line

L

pass through the point

P(2, -1, 3)

and be perpendicular to the lines

\frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2}

and

\frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4}

. If the line

L

intersects the

yz

-plane at the point

Q

, then the distance between the points

P

and

Q

is

Medium•Jee Main 2025

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains

n

white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is

\frac{29}{45}

, then

n

is equal to

Medium•Jee Main 2025

Let

S = N \cup \{0\}

. Define a relation

R

from

S

to

R

by

R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \}

Then, the sum of all the elements in the range of

R

is equal to

Medium•Jee Main 2025

If

\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right)

, then

(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)

is equal to

Browse by Topic

All Mathematics Questions