JEE Main 2023 Mathematics April 12 Shift 1 Question Paper with Solution PDF- Download Here

Question 1:

The number of five-digit numbers, greater than 40000 and divisible by 5, which can be formed using the digits 0, 1, 3, 5, 7, and 9 without repetition, is equal to:

• (1) 120
• (2) 132
• (3) 72
• (4) 96

Let \( \alpha, \beta \) be the roots of the quadratic equation \( x^2 + \sqrt{6}x + 3 = 0 \). Then, \( \frac{\alpha^{23} + \beta^{23} + \alpha^{14} + \beta^{14}}{\alpha^{15} + \beta^{15} + \alpha^{10} + \beta^{10}} \) is equal to:

• (1) 729
• (2) 72
• (3) 81
• (4) 9

Let < a_n > be a sequence such that \( a_1 + a_2 + \dots + a_n = \frac{n^2 + 3n}{(n+1)(n+2)} \). If \[ \sum_{k=1}^{10} \frac{1}{a_k} = p_1p_2p_3 \dots p_m \]
where \( p_1, p_2, \dots, p_m \) are the first \( m \) prime numbers, then \( m \) is equal to:

• (1) 7
• (2) 6
• (3) 5
• (4) 8

Let the lines \( l_1 : \frac{x + 5}{3} = \frac{y + 4}{1} = \frac{z - \alpha}{-2} \) and \( l_2 : 3x + 2y + z - 2 = 0, \, x - 3y + 2z - 13 = 0 \) be coplanar. If the point P(a, b, c) on \( l_1 \) is nearest to the point Q(-4, -3, 2), then \( |a| + |b| + |c| \) is equal to:

• (1) 12
• (2) 14
• (3) 10
• (4) 8

Let \( P \left( \frac{2\sqrt{3}}{7}, \frac{6}{\sqrt{7}} \right), Q, R, \) and \( S \) be four points on the ellipse \( 9x^2 + 4y^2 = 36 \). Let PQ and RS be mutually perpendicular and pass through the origin. If \[ \frac{1}{(PQ)^2} + \frac{1}{(RS)^2} = \frac{p}{q} \]
where \( p \) and \( q \) are coprime, then \( p + q \) is equal to:

• (1) 143
• (2) 137
• (3) 157
• (4) 147

Let \(a, b, c\) be three distinct real numbers, none equal to one. If the vectors \( a \hat{i} + \hat{j} + \hat{k}, \, \hat{i} + b\hat{j} + \hat{k}, \, a \hat{i} + \hat{j} + c \hat{k} \) are coplanar, then \( \frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} \) is equal to:

• (1) 1
• (2) -1
• (3) -2
• (4) 2

If the local maximum value of the function \[ f(x) = \left( \frac{\sqrt{3e}}{2 \sin x} \right)^{\sin^2 x}, \quad x \in \left( 0, \frac{\pi}{2} \right) \]
is \( \frac{k}{e} \), then \[ \left( \frac{k}{e} \right)^8 + \frac{k^8}{e^5} + k^8 \]
is equal to:

• (1) \( e^5 + e^6 + e^{11} \)
• (2) \( e^3 + e^5 + e^{11} \)
• (3) \( e^3 + e^6 + e^{11} \)
• (4) \( e^3 + e^6 + e^{10} \)

Let \( D \) be the domain of the function \( f(x) = \sin^{-1} \left( \log_{3x} \left( \frac{6 + 2 \log_3 x}{-5x} \right) \right) \). If the range of the function \( g: D \to \mathbb{R} \) defined by \( g(x) = x - \lfloor x \rfloor \), where \( \lfloor x \rfloor \) is the greatest integer function, is \( (\alpha, \beta) \), then \( \alpha^2 + \frac{5}{\beta} \) is equal to:

• (1) 46
• (2) 135
• (3) 136
• (4) 45

Let \( y = y(x), \, y > 0 \), be a solution curve of the differential equation \( (1 + x^2) \, dy = y(x - y) \, dx \). If \( y(0) = 1 \) and \( y(2\sqrt{2}) = \beta \), then:

• (1) \( e^{3\beta - 1} = e^{(3 + 2\sqrt{2})} \)
• (2) \( e^{\beta - 1} = e^{-2(5 + \sqrt{2})} \)
• (3) \( e^{\beta - 1} = e^{-2(3 + \sqrt{2})} \)
• (4) \( e^{3\beta - 1} = e^{(5 + \sqrt{2})} \)

Among the two statements

(S1): \( (p \Rightarrow q) \land (q \land (\sim q)) \) is a contradiction and

(S2): \( (p \land q) \lor (\sim p) \land (\sim q) \) is a tautology,

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

Let \( \lambda \in \mathbb{Z}, \vec{a} = \lambda \hat{i} + \hat{j} - \hat{k} \) and \( \vec{b} = 3\hat{i} - \hat{j} + 2\hat{k} \). Let \( \vec{c} \) be a vector such that \[ (\vec{a} + \vec{b} + \vec{c}) \times \vec{c} = 0, \quad \vec{a} \cdot \vec{c} = -17 \quad and \quad \vec{b} \cdot \vec{c} = -20. \]
Then \( \left| \vec{c} \times (\lambda \hat{i} + \hat{j} + \hat{k}) \right|^2 \) is equal to:

• (1) 62
• (2) 46
• (3) 53
• (4) 49

The sum of the coefficients of the first 50 terms in the binomial expansion of \((1-x)^{100}\), is equal to\

• (1) \( -^{101}C_{50} \)
• (2) \( ^{99}C_{49} \)
• (3) \( -^{99}C_{49} \)
• (4) \( ^{101}C_{50} \)

The area of the region enclosed by the curve \( y = x^3 \) and its tangent at the point \( (-1, -1) \) is:

• (1) \( \frac{27}{4} \)
• (2) \( \frac{19}{4} \)
• (3) \( \frac{23}{4} \)
• (4) \( \frac{31}{4} \)

Let \( A = \begin{bmatrix} 1 & \frac{1}{51}
0 & 1 \end{bmatrix} \). If \( B = \begin{bmatrix} 1 & 2
-1 & -1 \end{bmatrix} A \begin{bmatrix} -1 & -2
1 & 1 \end{bmatrix} \),
then the sum of all the elements of the matrix \[ \sum_{n=1}^{50} B^n \]
is equal to:

• (1) 100
• (2) 50
• (3) 75
• (4) 125

Let the plane \( P : 4x - y + z = 10 \) be rotated by an angle \( \frac{\pi}{2} \) about its line of intersection with the plane \( x + y - z = 4 \). If \( \alpha \) is the distance of the point \( (2, 3, -4) \) from the new position of the plane P, then \( 35\alpha \) is:

• (1) 90
• (2) 85
• (3) 105
• (4) 126

If \(\frac{1}{n+1}\) \({^{n}C_{n}}+\frac{1}{n}\) \({^{n}C_{n-1}}\) + \dots + \(\frac{1}{2}\) \({^{n}C_{1}}\) + \({^{n}C_{0}}\) = \(\frac{1023}{10}\) then n is equal to:

• (1) 6
• (2) 9
• (3) 8
• (4) 7

Let \( C \) be the circle in the complex plane with centre \( z_0 = \frac{1}{2}(1 + 3i) \) and radius \( r = 1 \). Let \( z_1 = 1 + i \) and the complex number \( z_2 \) be outside the circle \( C \) such that \( |z_1 - z_0| = |z_2 - z_0| = 1 \). If \( z_0, z_1 \) and \( z_2 \) are collinear, then the smaller value of \( |z_2|^2 \) is equal to:

• (1) \( \frac{13}{2} \)
• (2) \( \frac{5}{2} \)
• (3) \( \frac{3}{2} \)
• (4) \( \frac{7}{2} \)

• (1) 7
• (2) -7
• (3) \( -7\sqrt{3} \)
• (4) \( 7\sqrt{3} \)

Two dice A and B are rolled, Let the numbers obtained on A and B be \( \alpha \) and \( \beta \) respectively. If the variance of \( \alpha - \beta \) is \( \frac{p}{q} \), where \( p \) and \( q \) are coprime, then the sum of the positive divisors of \( p \) is equal to:

• (1) 36
• (2) 48
• (3) 31
• (4) 72

In a triangle ABC, if \( \cos A + 2 \cos B + \cos C = 2 \) and the lengths of the sides opposite to the angles A and C are 3 and 7 respectively, then \( \cos A - \cos C \) is equal to:

• (1) \( \frac{3}{7} \)
• (2) \( \frac{9}{7} \)
• (3) \( \frac{10}{7} \)
• (4) \( \frac{5}{7} \)

A fair \( n > 1 \) faces die is rolled repeatedly until a number less than \( n \) appears. If the mean of the number of tosses required is \( \frac{n}{9} \), then \( n \) is equal to:

Let the digits \( a, b, c \) be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Let \( [x] \) be the greatest integer \( \leq x \). Then the number of points in the interval \( (-2, 1) \), where the function \( f(x) = |[x]| + \sqrt{x - [x]} \) is discontinuous is:

Let the plane \( x + 3y - 2z + 6 = 0 \) meet the co-ordinate axes at the points A, B, C. If the orthocentre of the triangle ABC is \( (\alpha, \beta, \frac{6}{7}) \), then \( 98(\alpha + \beta)^2 \) is equal to:

Let I(x) = \(\int \sqrt{\frac{x+7}{x}}\) \, dx and I(9) = 12 + 7 \(\log_e\) 7.
If I(1) = \(\alpha + 7 \log_e (1+2\sqrt{2})\), then \(\alpha^4\) is equal to:

• (1) 64
• (2) 128
• (3) 32
• (4) 16

• (1) 6
• (2) 8
• (3) 4
• (4) 5

Let the positive numbers \( a_1, a_2, a_3, a_4 \), and \( a_5 \) be in a G.P. Let their mean and variance be \( \frac{31}{10} \) and \( \frac{m}{n} \), respectively, where \( m \) and \( n \) are co-prime. If the mean of their reciprocals is \( \frac{31}{40} \) and \( a_3 + a_4 + a_5 = 14 \), then \( m + n \) is equal to:

The number of relations, on the set \( \{1, 2, 3\} \) containing \( (1, 2) \) and \( (2, 3) \), which are reflexive and transitive but not symmetric, is:

If \[ \int_{-0.15}^{0.15} \left| 100x^2 - 1 \right| \, dx = \frac{k}{3000}, then k is equal to: \]

Two circles in the first quadrant of radii \( r_1 \) and \( r_2 \) touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line \( x + y = 2 \). Then \( r_1^2 + r_2^2 - r_1r_2 \) is equal to: