JEE Main 2023 Mathematics April 11 Shift 2 Question Paper with Solution PDF- Download Here

Let \(a\), \(b\), \(c\), and \(d\) be positive real numbers such that \(a + b + c + d = 11\). If the maximum value of \(a^5 b^3 c^2 d\) is \(3750\beta\), then the value of \(\beta\) is:

• (1) \(55\)
• (2) \(108\)
• (3) \(90\)
• (4) \(110\)

If \(f : \mathbb{R} \to \mathbb{R}\) is a continuous function satisfying
\[ \int_{0}^{\frac{\pi}{2}} f(\sin 2x) \sin x \, dx + \alpha \int_{0}^{\frac{\pi}{4}} f(\cos 2x) \cos x \, dx = 0, \]
then the value of \(\alpha\) is:

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

Let \(f\) and \(g\) be two functions defined by:
\[ f(x) = \begin{cases} x + 1, & x < 0
|x - 1|, & x \geq 0 \end{cases}, \quad g(x) = \begin{cases} x + 1, & x < 0
1, & x \geq 0 \end{cases}. \]
Then \((g \circ f)(x)\) is:

• (1) continuous everywhere but not differentiable at \(x = 1\)
• (2) continuous everywhere but not differentiable exactly at one point
• (3) differentiable everywhere
• (4) not continuous at \(x = -1\)

If the radius of the largest circle with centre \((2, 0)\) inscribed in the ellipse \(x^2 + 4y^2 = 36\) is \(r\), then \(12r^2\) is equal to:

• (1) \(69\)
• (2) \(72\)
• (3) \(115\)
• (4) \(92\)

Let the mean of 6 observations \(1, 2, 4, 5, x,\) and \(y\) and their variance be 10. Then their mean deviation about the mean is equal to:

• (1) \(\frac{7}{3}\)
• (2) \(\frac{10}{3}\)
• (3) \(\frac{8}{3}\)
• (4) 3

Let \( A = \{1, 3, 4, 6, 9\} \) and \( B = \{2, 4, 5, 8, 10\} \). Let \( R \) be a relation defined on \( A \times B \) such that \( R = \{((a_1, b_1), (a_2, b_2)): a_1 \leq b_2 and b_1 \leq a_2\} \). Then the number of elements in the set \( R \) is:

• (1) \(52\)
• (2) \(160\)
• (3) \(26\)
• (4) \(180\)

 Let \( P \) be the plane passing through the points \( (5, 3, 0) \), \( (13, 3, -2) \), and \( (1, 6, 2) \). For \( \alpha \in \mathbb{N} \), if the distances of the points \( A(3, 4, \alpha) \) and \( B(2, \alpha, a) \) from the plane \( P \) are \( 2 \) and \( 3 \) respectively, then the positive value of \( a \) is:

• (1) \(5\)
• (2) \(6\)
• (3) \(4\)
• (4) \(3\)

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial number, then the serial number of the word THAMS is:

• (1) \(102\)
• (2) \(103\)
• (3) \(101\)
• (4) \(104\)

If four distinct points with position vectors \(\vec{a}, \vec{b}, \vec{c}, and \; \vec{d}\) are coplanar, then \([\vec{a}\vec{b}\vec{c}]\) is equal to:

• (1) \([\vec{d}\vec{c}\vec{a}] + [\vec{b}\vec{d}\vec{a}] + [\vec{c}\vec{d}\vec{b}]\)
• (2) \([\vec{d}\vec{b}\vec{a}] + [\vec{a}\vec{c}\vec{d}] + [\vec{d}\vec{b}\vec{c}]\)
• (3) \([\vec{a}\vec{d}\vec{b}] + [\vec{d}\vec{c}\vec{a}] + [\vec{d}\vec{b}\vec{c}]\)
• (4) \([\vec{b}\vec{c}\vec{d}] + [\vec{d}\vec{a}\vec{c}] + [\vec{d}\vec{b}\vec{a}]\)

The sum of the coefficients of three consecutive terms in the binomial expansion of \((1 + x)^{n+2}\), which are in the ratio \(1 : 3 : 5\), is equal to:

• (1) \(63\)
• (2) \(92\)
• (3) \(25\)
• (4) \(41\)

Let \(y = y(x)\) be the solution of the differential equation \(\frac{dy}{dx} + \frac{5}{x(x^5 + 1)}y = \frac{(x^5 + 1)^2}{x^7}, \, x > 0.\) If \(y(1) = 2\), then \(y(2)\) is equal to:

• (1) \(\frac{693}{128}\)
• (2) \(\frac{637}{128}\)
• (3) \(\frac{697}{128}\)
• (4) \(\frac{679}{128}\)

The converse of \(((\sim p) \land q) \Rightarrow r\) is:

• (1) \((p \lor (\sim q)) \Rightarrow (\sim r)\)
• (2) \(((\sim p) \lor q) \Rightarrow r\)
• (3) \((\sim r) \Rightarrow ((\sim p) \land q)\)
• (4) \((\sim r) \Rightarrow (p \land q)\)

If the 1011th term from the end in the binomial expansion of \(\left(\frac{4x}{5} - \frac{5}{2x}\right)^{2022}\) is 1024 times the 1011th term from the beginning, the \(|x|\) is equal to:

• (1) \(8\)
• (2) \(12\)
• (3) \(\frac{5}{16}\)
• (4) \(15\)

If the system of linear equations:
\[ 7x + 11y + \alpha z = 13, \quad 5x + 4y + 7z = \beta, \quad 175x + 194y + 57z = 361, \]
has infinitely many solutions, then \(\alpha + \beta + 2\) is equal to:

• (1) \(3\)
• (2) \(6\)
• (3) \(5\)
• (4) \(4\)

Let the line passing through the point \(P(2, -1, 2)\) and \(Q(5, 3, 4)\) meet the plane \(x - y + z = 4\) at the point \(T\). Then the distance of the point \(R\) from the plane \(x + 2y + 3z + 2 = 0\), measured parallel to the line \(\frac{x - 7}{2} = \frac{y + 3}{2} = \frac{z - 2}{1}\), is equal to:

• (1) 3
• (2) \(\sqrt{61}\)
• (3) \(\sqrt{31}\)
• (4) \(\sqrt{189}\)

Let the function \(f : [0, 2] \to \mathbb{R}\) be defined as:
\[ f(x) = \begin{cases} e^{\min\{x^2, x - \lfloor x \rfloor\}}, & x \in [0, 1),
e^{x - \log_e x}, & x \in [1, 2), \end{cases} \]
where \(\lfloor t \rfloor\) denotes the greatest integer less than or equal to \(t\).

Then the value of the integral \(\int_0^2 x f(x) \, dx\) is:

• (1) \((e - 1) \left(e^2 + \frac{1}{2}\right)\)
• (2) \(1 + \frac{3e}{2}\)
• (3) \(2e - \frac{1}{2}\)
• (4) \(2e - 1\)

For \(a \in \mathbb{C}\), let:
\[ A = \{ z \in \mathbb{C} : Re(a + \overline{z}) > Im(\overline{a} + z) \}, \textbf{and} \quad B = \{ z \in \mathbb{C} : Re(a + \overline{z}) < Im(\overline{a} + z) \}. \]
Among the two statements:

(S1): If \(Re(a), Im(a) > 0\), then the set \(A\) contains all the real numbers.

(S2): If \(Re(a), Im(a) < 0\), then the set \(B\) contains all the real numbers.

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

If:
\[ \begin{vmatrix} x + 1 & x & x
x & x + \lambda & x
x & x & x + \lambda^2 \end{vmatrix} = \frac{9}{8}(103x + 81), \]
then \(\lambda, \frac{\lambda}{3}\) are the roots of the equation:

• (1) \(4x^2 - 24x - 27 = 0\)
• (2) \(4x^2 + 24x + 27 = 0\)
• (3) \(4x^2 - 24x + 27 = 0\)
• (4) \(4x^2 + 24x - 27 = 0\)

The domain of the function \(f(x) = \frac{1}{\sqrt{\lfloor x \rfloor^2 - 3\lfloor x \rfloor - 10}}\), where \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\), is:

• (1) \((-\infty, -3] \cup [6, \infty)\)
• (2) \((-\infty, -2) \cup (5, \infty)\)
• (3) \((-\infty, -3] \cup (5, \infty)\)
• (4) \((-\infty, -2) \cup [6, \infty)\)

If \(A\) is the area in the first quadrant enclosed by the curve \(C : 2x^2 - y + 1 = 0\), the tangent to \(C\) at the point \((1, 3)\), and the line \(x + y = 1\), then the value of \(60A\) is _____:

Let \(A = \{1, 2, 3, 4, 5\}\) and \(B = \{1, 2, 3, 4, 5, 6\}\). Then the number of functions \(f : A \to B\) satisfying \(f(1) + f(2) = f(4) - 1\) is equal to ____.

Let the tangent to the parabola \(y^2 = 12x\) at the point \((3, \alpha)\) be perpendicular to the line \(2x + 2y = 3\). Then the square of the distance of the point \((6, -4)\) from the normal to the hyperbola \(\alpha^2 x^2 - 9y^2 = 9\alpha^2\) at its point \((\alpha - 1, \alpha + 2)\) is equal to _____.

For \(k \in \mathbb{N}\), if the sum of the series \(1 + \frac{4}{k} + \frac{8}{k^2} + \frac{13}{k^3} + \frac{19}{k^4} + \ldots\) is 10, then the value of \(k\) is _____.

Let the line \(\ell : x = \frac{1 - y}{-2} = \frac{z - 3}{\lambda}, \, \lambda \in \mathbb{R}\) meet the plane \(P : x + 2y + 3z = 4\) at the point \((\alpha, \beta, \gamma)\). If the angle between the line \(\ell\) and the plane \(P\) is \(\cos^{-1}\left(\frac{\sqrt{5}}{\sqrt{14}}\right)\), then \(\alpha + 2\beta + 6\gamma\) is equal to ____.

The number of points where the curve \(f(x) = e^{8x} - e^{6x} - 3e^{4x} - e^{2x} + 1, x \in \mathbb{R}\) cuts the \(x\)-axis, is equal to ____.

If the line \(l_1 : 3y - 2x = 3\) is the angular bisector of the line \(l_2 : x - y + 1 = 0\) and \(l_3 : \alpha x + \beta y + 17 = 0\), then \(\alpha^2 + \beta^2 - \alpha - \beta\) is equal to _____.

Let the probability of getting a head for a biased coin be \(\frac{1}{4}\). It is tossed repeatedly until a head appears. Let \(N\) be the number of tosses required. If the probability that the equation \(64x^2 + 5Nx + 1 = 0\) has no real root is \(\frac{p}{q}\), where \(p\) and \(q\) are co-prime, then \(q - p\) is equal to ____.

Let \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\) and \(\vec{b} = \hat{i} + \hat{j} - \hat{k}\). If \(\vec{c}\) is a vector such that \(\vec{a} \cdot \vec{c} = 11\), \(\vec{b} \cdot (\vec{a} \times \vec{c}) = 27\) and \(\vec{b} \cdot \vec{c} = -\sqrt{3}|\vec{b}|\), then \(|\vec{a} \times \vec{c}|^2\) is equal to ____:

Let \(S = \left\{z \in \mathbb{C} - \{i, 2i\}: \frac{z^2 + 8iz - 15}{z^2 - 3iz - 2} \in \mathbb{R} \right\}\). If \(\alpha - \frac{13}{11}i \in S\), \(\alpha \in \mathbb{R} - \{0\}\), then \(242\alpha^2\) is equal to ________.