Tripura JEE 2024 Mathematics Set P Question Paper with Answer Key PDFs

Question 1:

Let \( \phi_1(x) = e^{\sin x}, \phi_2(x) = e^{\phi_1(x)}, \ldots, \phi_{n+1}(x) = e^{\phi_n(x)}, \forall n \geq 1.\) Then for any fixed \( n \), the expression \( \frac{d}{dx} \phi_n(x) \) is:

• (1) \( \phi_n(x) \cdot \phi_{n-1}(x) \)
• (2) \( \phi_n(x) \cdot \phi_{n-1}(x) \cdot \ldots \cdot \phi_1(x) \cos x \)
• (3) \( \phi_n(x) \cdot \phi_{n-1}(x) \cdot \ldots \cdot \varphi_1(x) \sin x \)
• (4) \( \phi_n(x) \cdot \phi_{n-1}(x) \cdot \ldots \cdot \varphi_1(x) e^{\sin x} \)

The value(s) of \( c \in (1, 2) \), where the conclusion of Lagrange’s M.V.T. is satisfied for the function \( f(x) = x^2 + 3x + 2 \) in \([1,2]\), is/are:

• (A) \( \left( -\frac{3}{2}, \frac{1}{2} \right) \)
• (B) \( \left( \frac{1}{2}, \frac{3}{2} \right) \)
• (C) \( \left( -\frac{1}{2} \right) \)
• (D) \( \left( \frac{3}{2} \right) \)

Let the function \( f(x) \) be defined as follows:
\[ f(x) = \begin{cases} (1 + | \sin x |)^{\frac{a}{|\sin x|}}, & -\frac{\pi}{6} < x < 0
b, & x = 0
\frac{\tan 2x}{\tan 3x}, & 0 < x < \frac{\pi}{6} \end{cases} \]

Then the values of \( a \) and \( b \) are:

• (A) \( a = -\frac{2}{3}, b = \frac{2}{3} \)
• (B) \( a = \frac{2}{3}, b = e^{\frac{2}{3}} \)
• (C) \( a = e^{\frac{2}{3}}, b = \frac{2}{3} \)
• (D) \( a = \frac{2}{3}, b = e^{-\frac{2}{3}} \)

If \[ \int \frac{3e^x + 5e^{-x}}{4e^x - 5e^{-x}} \, dx = Ax + B \ln |4e^{2x} - 5| + C \]
then, the values of \( A \), \( B \), and \( C \) are:

• (A) \( A = -1, B = -\frac{7}{8}, C = constant of integration \)
• (B) \( A = 1, B = \frac{7}{8}, C = constant of integration \)
• (C) \( A = -1, B = \frac{7}{8}, C = constant of integration \)
• (D) \( A = \frac{7}{8}, B = \frac{3}{8}, C = constant of integration \)

Evaluate the integral: \[ \int_0^2 |x^2 + x - 2| \, dx \]

• (A) \( \frac{11}{3} \)
• (B) \( \frac{-11}{3} \)
• (C) \( \frac{1}{3} \)
• (D) \( \frac{-1}{3} \)

The differential equation of all circles of radius \( a \) is:

• (A) \( \left( 1 + \frac{dy}{dx} \right)^3 = a^2 \left( \frac{d^2 y}{dx^2} \right)^2 \)
• (B) \( \left( 1 - \left( \frac{dy}{dx} \right)^2 \right)^3 = a^2 \left( \frac{d^2 y}{dx^2} \right)^2 \)
• (C) \( \left( 1 - \frac{dy}{dx} \right)^3 = a^2 \left( \frac{d^2 y}{dx^2} \right)^2 \)
• (D) \( \left( 1 + \frac{dy}{dx} \right)^3 = a^2 \left( \frac{d^2 y}{dx^2} \right)^2 \)

The range of the function \( f(x) = 7 - x\cdot P_{x-3} \) is:

• (A) \( \{1, 2, 3, 4, 5, 6\} \)
• (B) \( \{1, 2, 3, 4\} \)
• (C) \( \{1, 2\} \)
• (D) \( \{1, 2, 3\} \)

If \( f(x) \) and \( g(x) \) are two functions such that \( f(x) + g(x) = e^x \) and \( f(x) - g(x) = e^{-x} \), then:

• (A) \( f(x) \) is odd function, \( g(x) \) is odd function
• (B) \( f(x) \) is even function, \( g(x) \) is even function
• (C) \( f(x) \) is even function, \( g(x) \) is odd function
• (D) \( f(x) \) is odd function, \( g(x) \) is even function

The mapping \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(x) = |x - 1| \), \( x \in \mathbb{R} \) is:

• (A) one-one, onto
• (B) many-one, onto
• (C) one-one, into
• (D) neither one-one nor onto

The area bounded by the curves \( y = \log_e x \) and \( y = (\log_e x)^2 \) is:

• (A) \( (3 - e) \) sq. units
• (B) \( (e - 3) \) sq. units
• (C) \( \frac{1}{2} (3 - e) \) sq. units
• (D) \( \frac{1}{2} (e - 3) \) sq. units

For what value of \( n \), the curve \[ \left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2 touches the straight line \frac{x}{a} + \frac{y}{b} = 2 at the point (a, b)? \]

• (A) \( n = 3 \)
• (B) Any value of \( n \)
• (C) \( n = 2 \)
• (D) \( n = 4 \)

The function \( f(x) = x^3 - 3x \) is:

• (A) increasing in \( (-\infty, -1) \cup (1, \infty) \) and decreasing in \( (-1, 1) \)
• (B) decreasing in \( (-\infty, -1) \cup (1, \infty) \) and increasing in \( (-1, 1) \)
• (C) increasing in \( (0, \infty) \) and decreasing in \( (-\infty, 0) \)
• (D) decreasing in \( (0, \infty) \) and increasing in \( (-\infty, 0) \)

If \( A \) is the A.M. of the roots of the equation \( x^2 - 2ax + b = 0 \) and \( G \) is the G.M. of the roots of the equation \( x^2 - 2bx + a^2 = 0 \), \( a > 0 \), then:

• (A) \( A > G \)
• (B) \( A = G \)
• (C) \( A < G \)
• (D) None of these

Out of 6 boys and 4 girls, a committee of 5 members is to be formed. In how many ways can this be done, if at least 2 girls are included?

• (A) 126
• (B) 186
• (C) 140
• (D) 156

\( (666 \ldots up to n digits)^2 + (888 \ldots up to n digits) \) is:

• (A) \( \frac{9}{4}(10^n - 1) \)
• (B) \( \frac{9}{4}(10^n - 1)^2 \)
• (C) \( \frac{4}{9}(10^{2n} + 1) \)
• (D) \( \frac{4}{9}(10^{2n} - 1) \)

Sum of the last 40 coefficients in the expansion of \( (1 + x)^{79} \), when expanded in ascending power of \( x \) is:

• (A) \( 2^{79} \)
• (B) \( 2^{40} \)
• (C) \( 2^{39} \)
• (D) \( 2^{78} \)

• (A) \( -1 \)
• (B) \( 2 \)
• (C) \( 1 \)
• (D) \( 0 \)

For what values of \( \lambda \) and \( \mu \), the following system of equations has a unique solution?
\[ 2x + 3y + 5z = 9 \] \[ 7x + 3y - 2z = 8 \] \[ 2x + 3y + \lambda z = \mu \]

• (A) \( \lambda \neq 5 \), any value of \( \mu \)
• (B) \( \lambda = 5 \), \( \mu = 9 \)
• (C) \( \lambda \neq 5 \), \( \mu = 9 \)
• (D) \( \lambda = 5 \), any value of \( \mu \)

If \( 0 < \theta < \pi \) and \( \cos \theta + \sin \theta = \frac{1}{2} \), then the value of \( \tan \theta \) is:

• (A) \( \frac{1 - \sqrt{7}}{4} \)
• (B) \( \frac{4 - \sqrt{7}}{3} \)
• (C) - \( \frac{4 + \sqrt{7}}{3} \)
• (D) \( \frac{1 + \sqrt{7}}{4} \)

In a triangle \( ABC \), if angles \( A \), \( B \), and \( C \) are in A.P., then \( \frac{a + c}{b} \) is equal to:

• (A) \( \frac{2 \sin \frac{A - C}{2}}{2} \)
• (B) \( 2 \cos \frac{A - C}{2} \)
• (C) \( \frac{\cos \frac{A - C}{2}}{2} \)
• (D) \( \sin \frac{A - C}{2} \)

The value of \[ \tan^{-1} \left( \frac{\sin 2 - 1}{\cos 2} \right) \]
is:

• (A) \( 1 - \frac{\pi}{4} \)
• (B) \( \frac{\pi}{2} - 1 \)
• (C) \( 2 - \frac{\pi}{2} \)
• (D) \( \frac{\pi}{4} - 1 \)

The equation of the image of the line \( 2y - x = 1 \) obtained by the reflection on the line \( 4y - 2x = 5 \) is:

• (A) \( 2y - x = 4 \)
• (B) \( 2x - y = 4 \)
• (C) \( 2y + x = 4 \)
• (D) \( 2x + y = 4 \)

The equation of the circle of radius 3 units which touches the circles \( x^2 + y^2 - 6|x| = 0 \) is:

• (A) \( x^2 + y^2 + 6\sqrt{3}y - 18 = 0 \) or \( x^2 + y^2 - 6\sqrt{3}y - 18 = 0 \)
• (B) \( x^2 + y^2 + 4\sqrt{3}y + 18 = 0 \) or \( x^2 + y^2 - 4\sqrt{3}y + 18 = 0 \)
• (C) \( x^2 + y^2 + 6\sqrt{3}y + 18 = 0 \) or \( x^2 + y^2 - 6\sqrt{3}y + 18 = 0 \)
• (D) \( x^2 + y^2 + 4\sqrt{3}y - 18 = 0 \) or \( x^2 + y^2 - 4\sqrt{3}y - 18 = 0 \)

The angle between the lines joining the foci of an ellipse to one particular extremity of the minor axis is \( 90^\circ \). The eccentricity of the ellipse is:

• (A) \( \frac{1}{8} \)
• (B) \( \frac{1}{\sqrt{3}} \)
• (C) \( \frac{\sqrt{2}}{3} \)
• (D) \( \frac{1}{\sqrt{2}} \)

The direction ratios of the normal to the plane passing through the points \[ (1, 2, -3), \quad (1, -2, 1) \quad and parallel to the line \quad \frac{x - 2}{2} = \frac{y + 1}{3} = \frac{z}{4} is: \]

• (A) \( (-2, 0, -3) \)
• (B) \( (14, -8, -1) \)
• (C) \( (2, 3, 4) \)
• (D) \( (1, -2, -3) \)

The area of a parallelogram whose diagonals are given by \( \vec{u} + \vec{v} \) and \( \vec{v} + \vec{w} \), where: \[ \vec{u} = 2\hat{i} - 3\hat{j} + \hat{k}, \quad \vec{v} = -\hat{i} + \hat{k}, \quad \vec{w} = 2\hat{j} - \hat{k} \]
is:

• (A) \( \sqrt{14} \) sq. unit
• (B) \( \sqrt{21} \) sq. unit
• (C) \( \frac{1}{2} \sqrt{21} \) sq. unit
• (D) \( \frac{1}{2} \sqrt{14} \) sq. unit

The shortest distance between the lines \[ \frac{x + 1}{1} = \frac{y - 3}{-1} = \frac{z - 1}{-1} \quad and \quad \frac{x}{3} = \frac{y - 1}{2} = \frac{z + 1}{-1} \]
is:

• (A) \( \frac{3}{\sqrt{16}} \) unit
• (B) \( \frac{3}{\sqrt{14}} \) unit
• (C) \( \frac{3}{\sqrt{38}} \) unit
• (D) \( \frac{1}{\sqrt{3}} \) unit

If the probability for A to fail in an examination is 0.2 and that for B is 0.3, then the probability that either A or B fails is:

• (A) \( \leq 0.4 \)
• (B) \( \leq 0.25 \)
• (C) \( \leq 0.5 \)
• (D) \( \leq 0.7 \)

If the probability density function of a random variable is given by \[ f(x) = \begin{cases} 12x^2(1 - x), & 0 \leq x \leq 1
0, & elsewhere \end{cases} \]
then the mean and variance are respectively:

• (A) \( 0.6, 0.4 \)
• (B) \( 0.4, 0.6 \)
• (C) \( 0.2, 0.6 \)
• (D) \( 0.6, 0.2 \)

Let the two variables \( x \) and \( y \) satisfy the following conditions: \[ x + y \leq 50, \quad x + 2y \leq 80, \quad 2x + y \geq 20, \quad x, y \geq 0. \]
Then the maximum value of \( Z = 4x + 3y \) is:

• (A) 120
• (B) 170
• (C) 200
• (D) 210