JEE Main 2023 Mathematics Question Paper Jan 25 Shift 2- Download Question Paper with Solution PDF

Let the function \( f(x) = 2x^3 + (2p - 7)x^2 + 3(2p - 9)x - 6 \) have a maxima for some value of \( x < 0 \) and a minima for some value of \( x > 0 \). Then, the set of all values of \( p \) is:

• (1) \( \left( \dfrac{9}{2}, \infty \right) \)
• (2) \( \left( 0, \dfrac{9}{2} \right) \)
• (3) \( \left( -\infty, \dfrac{9}{2} \right) \)
• (4) \( \left( -\dfrac{9}{2}, \dfrac{9}{2} \right) \)

Let \( z \) be a complex number such that \( \dfrac{|z - 2i|}{|z + i|} = 2 \), \( z \neq -i \). Then \( z \) lies on the circle of radius 2 and centre:

• (1) \( (2, 0) \)
• (2) \( (0, 0) \)
• (3) \( (0, 2) \)
• (4) \( (0, -2) \)

If the function \[ f(x) = \begin{cases} \left(1 + |\cos x|\right)^{\lambda / |\cos x|}, & 0 < x < \dfrac{\pi}{2}
\mu, & x = \dfrac{\pi}{2}
e^{\dfrac{\cot 6x}{\cot 4x}}, & \dfrac{\pi}{2} < x < \pi \end{cases} \]
is continuous at \( x = \dfrac{\pi}{2} \), then \( 9\lambda + 6\(\log_e\) \mu + \mu^6 - e^{6\lambda \) is equal to:

• (1) \( 11 \)
• (2) \( 8 \)
• (3) \( 2e^4 + 8 \)
• (4) \( 10 \)

Let \( f(x) = 2x^n + \lambda \), where \( \lambda \in \mathbb{R} \) and \( n \in \mathbb{N} \). Given that \( f(4) = 133 \) and \( f(5) = 255 \), Then the sum of all the positive integer divisors of \( f(3) - f(2) \)?

• (1) \( 61 \)
• (2) \( 60 \)
• (3) \( 58 \)
• (4) \( 59 \)

If the four points, whose position vectors are \( 3\hat{i} - 4\hat{j} + 2\hat{k} \), \( \hat{i} + 2\hat{j} - \hat{k} \), \( -2\hat{i} - \hat{j} + 3\hat{k} \), and \( 5\hat{i} - 2\alpha\hat{j} + 4\hat{k} \) are coplanar, then \( \alpha \) is equal to:

• (1) \( \dfrac{73}{17} \)
• (2) \( -\dfrac{107}{17} \)
• (3) \( -\dfrac{73}{17} \)
• (4) \( \dfrac{107}{17} \)

Let



, then the inverse of the matrix \( A M^{2023} A^T \) is:

• (1)
• (2)
• (3)
• (4)

Let \( \triangle \) and \( \bigtriangledown \in [\land, \lor] \) be such that the expression \( (p \to q) \land (p \bigtriangledown q) \) is a tautology. Then:

• (1) \( \triangle = \land, \bigtriangledown = \lor \)
• (2) \( \triangle = \lor, \bigtriangledown = \land \)
• (3) \( \triangle = \lor, \bigtriangledown = \lor \)
• (4) \( \triangle = \land, \bigtriangledown = \land \)

The number of numbers, strictly between 5000 and 10000, that can be formed using the digits 1, 3, 5, 7, 9 without repetition, is:

• (1) \( 6 \)
• (2) \( 12 \)
• (3) \( 120 \)
• (4) \( 72 \)

The number of functions \( f: \{1, 2, 3, 4\} \to \{a\in Z : |a| \leq 8\} \) satisfying \( f(n) + 1/n f(n+1) \) =1, for \( {A} n \in \{1, 2, 3\} \) is:

• (1) \( 3 \)
• (2) \( 4 \)
• (3) \( 1 \)
• (4) \( 2 \)

The equations of two sides of a variable triangle are \( x = 0 \) and \( y = 3 \), and its third side is a tangent to the parabola \( y^2 = 6x \). The locus of its circumcentre is:

• (1) \( 4y^2 - 18y-3x - 18 = 0 \)
• (2) \( 4y^2 + 18y + 3x + 18 = 0 \)
• (3) \( 4y^2 - 18y + 3x + 18 = 0 \)
• (4) \( 4y^2 - 18y - 3x + 18 = 0 \)

Let \( f: \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = \log_{\sqrt{m}}{\left(\sqrt{2}(\sin x - \cos x) + m - 2\right)} \), for some \( m \), such that the range of \( f \) is [0, 2]. Then the value of \( m \) is:

• (1) \( 5 \)
• (2) \( 3 \)
• (3) \( 2 \)
• (4) \( 4 \)

Let \( A, B, C \) be \( 3 \times 3 \) matrices such that \( A \) is symmetric and \( B \) and \( C \) are skew-symmetric. Consider the statements:

• (1) Only S2 is true
• (2) Only S1 is true
• (3) Both S1 and S2 are false
• (4) Both S1 and S2 are true

Let y= y(t) be a solution to the differential equation::

dy/dt + αy = γe^(-βt)

where α, β, γ > 0. We are interested in finding the value of:

lim(t → ∞) y(t)

then:

• (1) Is 0
• (2) does not exist
• (3) Is 1
• (4) Is -1

\( \sum_{k=0}^{6} {}^{(51-k)}C_3 \): is equal to

• (1) \( {}^{51}C_4 - {}^{45}C_4 \)
• (2) \( {}^{51}C_3 - {}^{45}C_3 \)
• (3) \( {}^{52}C_4 - {}^{45}C_4 \)
• (4) \( {}^{52}C_3 - {}^{45}C_3 \)

The shortest distance between the lines \( x + 1 = 2y = -12z \) and \( x = y + 2 = 6z - 6 \) is:

• (1) 2
• (2) 3
• (3) \( \frac{5}{2} \)
• (4) \( \frac{3}{2} \)

Let \( N \) be the sum of the numbers appeared when two fair dice are rolled and let the probability that \( N-2, \sqrt{3N}, N+2 \) are in geometric progression be \( \frac{k}{48} \). Then the value of \( k \) is:

• (1) \( 2 \)
• (2) \( 4 \)
• (3) \( 16 \)
• (4) \( 8 \)

The integral \( 16\int_{1}^{2} \frac{dx}{x^3(x^2+2)^2} \) is equal to:

• (1) \( \frac{11}{6} + log_e 4 \)
• (2) \( \frac{11}{12} + log_e 4 \)
• (3) \( \frac{11}{12} - log_e 4 \)
• (4) \( \frac{11}{6} - log_e 4 \)

Let \( T \) and \( C \) respectively be the transverse and conjugate axes of the hyperbola \( 16x^2 - y^2 + 64x+ 4y + 44 = 0 \). Then the area of the region above the parabola \( x^2 = y + 4 \), below the transverse axis \( T \) and on the right of the conjugate axis \( C \) is:

• (1) \( 4\sqrt{6} + \frac{44}{3} \)
• (2) \( 4\sqrt{6} + \frac{28}{3} \)
• (3) \( 4\sqrt{6} - \frac{44}{3} \)
• (4) \( 4\sqrt{6} - \frac{28}{3} \)

Let \( {\vec{a}} = -\hat{i} - \hat{j} + \hat{k} \), \( \vec{a} \cdot \vec{b} = 1 \) and \( \vec{a} \times \vec{b} = \hat{i} - \hat{j} \). Then \( \vec{a} - 6\vec{b} \) is equal to:

• (1) \( 3(\hat{i} - \hat{j} - \hat{k}) \)
• (2) \( 3(\hat{i} + \hat{j} + \hat{k}) \)
• (3) \( 3(\hat{i} - \hat{j} + \hat{k}) \)
• (4) \( 3(\hat{i} + \hat{j} - \hat{k}) \)

The foot of the perpendicular from the point \( (2, 0, 5) \) on the line \( \frac{x+1}{2} = \frac{y-1}{5} = \frac{z+1}{-1} \) is \( (\alpha, \beta, \gamma) \). Then, which of the following is NOT correct?

• (1) \( \frac{\alpha \beta}{\gamma} = \frac{4}{15} \)
• (2) \( \frac{\alpha}{\beta} = -8 \)
• (3) \( \frac{\beta}{\gamma} = -5 \)
• (4) \( \frac{\gamma}{\alpha} = \frac{5}{8} \)

For the two positive numbers \( a, b \), if \( a, b \) and \( \frac{1}{18} \) are in a geometric progression, while \( \frac{1}{a}, 10, \frac{1}{b} \) are in an arithmetic progression, then \( 16a + 12b \) is equal to:

Points \( P(-3, 2), Q(9, 10) \), and \( R(\alpha, 4) \) lie on a circle \( C \) with \( PR \) as its diameter. The tangents to \( C \) at \( Q \) and \( R \) intersect at point \( S \). If \( S \) lies on the line \( 2x - ky = 1 \), then \( k \) is equal to:

Let \( a \in \mathbb{R} \) and let \( \alpha, \beta \) be the roots of the equation \( x^2 + 60^{\frac{1}{4}} x + a = 0 \). If \( \alpha^4 + \beta^4 = -30 \), then the product of all possible values of \( a \) is:

Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples, and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple, and at least one white apple must be given, then the number of ways Anil’s mother can offer 5 fruits to Anil is:

If \( m \) and \( n \) respectively are the numbers of positive and negative values of \( \theta \) in the interval \( [-\pi, \pi] \) that satisfy the equation \( \cos 2\theta \cdot \cos\frac{\theta}{2} = \cos 3\theta \cdot \cos\frac{9\theta}{2} \), then \( mn \) is equal to:

If ∫(1/3 to 3) |ln(x)| dx = (m/n) ln(n^2/e)​, where m and n are coprime natural numbers, then m^2 + n^2 - 5 is equal to ____.

The remainder when \( (2023)^{2023} \) is divided by 35 is:

If the shortest distance between the line joining the points \( (1, 2, 3) \) and \( (2, 3, 4) \), and the line \( \frac{x-1}{2} = \frac{y+1}{-1} = \frac{z-2}{0} \) is \( \alpha \), then \( 28\alpha^2 \) is equal to:

25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non-smoker. If a person is diagnosed with lung cancer, and the probability that this person is a smoker is \( \frac{k}{10} \), then the value of \( k \) is:

A triangle is formed by the \( X \)-axis, \( Y \)-axis, and the line \( 3x + 4y = 60 \). Then the number of points \( P(a, b) \), where \( a \) is an integer and \( b \) is a multiple of \( a \), which lie strictly inside the triangle, is:____