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JEE Mains | PYQ

and $h(x) = f(|x|) + |f(x)|$. Then $\int_{-2}^{2} h(x) \, dx$ is equal to:

If the solution \( y=y(x) \) of the differential equation \( (x^4+2x^3+3x^2+2x+2)\,dy-(2x^2+2x+3)\,dx=0 \) satisfies \( y(-1)=-\dfrac{\pi}{4} \), then \( y(0) \) is equal to:

If \( 2 \) and \( 6 \) are the roots of the equation \( ax^2+bx+1=0 \), then the quadratic equation, whose roots are \( \dfrac{1}{2a+b} \) and \( \dfrac{1}{6a+b} \), is:

Let a unit vector which makes an angle of $60^\circ$ with $2\hat{i} + 2\hat{j} - \hat{k}$ and an angle of $45^\circ$ with $\hat{i} - \hat{k}$ be $\vec{c}$. Then $\vec{c} + \left( -\frac{1}{2}\hat{i} + \frac{1}{3\sqrt{2}}\hat{j} - \frac{\sqrt{2}}{3}\hat{k} \right)$ is:

The vertices of a triangle are \( A(-1,3) \), \( B(-2,2) \) and \( C(3,-1) \). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is:

Let \( \alpha \in (0,\infty) \) and . If \( \det\!\big(\operatorname{adj}(2A-A^{T})\cdot \operatorname{adj}(A-2A^{T})\big)=2^{8} \), then \( (\det(A))^{2} \) is equal to:

Let the sum of the maximum and the minimum values of the function \( f(x)=\dfrac{2x^{2}-3x+8}{2x^{2}+3x+8} \) be \( \dfrac{m}{n} \), where \( \gcd(m,n)=1 \). Then \( m+n \) is equal to:

If the system of equations
\( x+\big(\sqrt{2}\sin\alpha\big)y+\big(\sqrt{2}\cos\alpha\big)z=0 \)
\( x+(\cos\alpha)y+(\sin\alpha)z=0 \)
\( x+(\sin\alpha)y-(\cos\alpha)z=0 \)
has a non-trivial solution, then \( \alpha\in\left(0,\dfrac{\pi}{2}\right) \) is equal to:

There are \( 5 \) points \( P_{1},P_{2},P_{3},P_{4},P_{5} \) on the side \( AB \), excluding \( A \) and \( B \), of a triangle \( ABC \). Similarly there are \( 6 \) points \( P_{6},P_{7},\ldots,P_{11} \) on the side \( BC \) and \( 7 \) points \( P_{12},P_{13},\ldots,P_{18} \) on the side \( CA \) of the triangle. The number of Triangles, that can be formed using the points \( P_{1},P_{2},\ldots,P_{18} \) as vertices, is:

Let \( \alpha,\beta \in \mathbb{R} \). Let the mean and the variance of \( 6 \) observations \( -3,4,7,-6,\alpha,\beta \) be \( 2 \) and \( 23 \), respectively. The mean deviation about the mean of these \( 6 \) observations is:

Let \( \alpha \) and \( \beta \) be the sum and the product of all the non-zero solutions of the equation \( (\bar{z})^{2}+|z|=0,\ z\in \mathbb{C} \). Then \( 4(\alpha^{2}+\beta^{2}) \) is equal to:

Let \( f(x) = x^5 + 2e^{x/4} \) for all \( x \in \mathbb{R} \). Consider a function \( g(x) \) such that \( (gof)(x) = x \) for all \( x \in \mathbb{R} \). Then the value of \( 8g'(2) \) is:

Let the point, on the line passing through the points \( P(1, -2, 3) \) and \( Q(5, -4, 7) \), farther from the origin and at a distance of \( 9 \) units from the point \( P \), be \( (\alpha, \beta, \gamma) \). Then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

Three urns \( A \), \( B \) and \( C \) contain \( 7 \) red, \( 5 \) black; \( 5 \) red, \( 7 \) black and \( 6 \) red, \( 6 \) black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn \( A \) is:

Let the first three terms \( 2 \), \( p \) and \( q \), with \( q \neq 2 \), of a G.P. be respectively the \( 7^{\text{th}} \), \( 8^{\text{th}} \) and \( 13^{\text{th}} \) terms of an A.P. If the \( 5^{\text{th}} \) term of the G.P. is the \( n^{\text{th}} \) term of the A.P., then \( n \) is equal to:

The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}} + 5^{\frac{1}{3}}\right)^{15}$ is equal to:

One of the points of intersection of the curves $y = 1 + 3x - 2x^2$ and $y = \frac{1}{x}$ is $\left(\frac{1}{2}, 2\right)$. Let the area of the region enclosed by these curves be $\frac{1}{24}\left(l\sqrt{5} + m\right) - n \log_e(1 + \sqrt{5})$, where $l, m, n \in \mathbb{N}$. Then $l + m + n$ is equal to:

Let \( f : R \to R \) be a function given by

where \( \alpha, \beta \in R \). If \( f \) is continuous at \( x = 0 \), then \( \alpha^2 + \beta^2 \) is equal to:

A square is inscribed in the circle \( x^2 + y^2 - 10x - 6y + 30 = 0 \). One side of this square is parallel to \( y = x + 3 \). If \( (x_i, y_i) \) are the vertices of the square, then \( \sum (x_i^2 + y_i^2 )\) is equal to:

In a survey of \( 220 \) students of a higher secondary school, it was found that at least \( 125 \) and at most \( 130 \) students studied Mathematics; at least \( 85 \) and at most \( 95 \) studied Physics; at least \( 75 \) and at most \( 90 \) studied Chemistry; \( 30 \) studied both Physics and Chemistry; \( 50 \) studied both Chemistry and Mathematics; \( 40 \) studied both Mathematics and Physics and \( 10 \) studied none of these subjects. Let \( m \) and \( n \) respectively be the least and the most number of students who studied all the three subjects. Then \( m+n \) is equal to \( \_\_\_\_\_\_\_\_\_\_ \).

Let the length of the focal chord \( PQ \) of the parabola \( y^2 = 12x \) be \( 15 \) units. If the distance of \( PQ \) from the origin is \( p \), then \( 10p^2 \) is equal to \( \_\_\_\_\_\_\_\_\_\_ \).

Let \( A \) be a square matrix of order \( 2 \) such that \( |A| = 2 \) and the sum of its diagonal elements is \( -3 \). If the points \( (x, y) \) satisfying \( A^2 + xA + yI = 0 \) lie on a hyperbola, whose transverse axis is parallel to the \( x \)-axis, eccentricity is \( e \) and the length of the latus rectum is \( l \), then \( e^4 + l^4 \) is equal to \( \_\_\_\_\_\_\_\_\_\_ \).

Let \( ABC \) be a triangle of area \( 15\sqrt{2} \) and the vectors \( \overrightarrow{AB} = \hat{i} + 2\hat{j} - 7\hat{k} \), \( \overrightarrow{BC} = a\hat{i} + b\hat{j} + c\hat{k} \)  and \( \overrightarrow{AC} = 6\hat{i} + d\hat{j} - 2\hat{k},\ d > 0 \). Then the square of the length of the largest side of the triangle \( ABC \) is \( \_\_\_\_\_\_\_\_\_\_ \).

Let the solution \( y = y(x) \) of the differential equation \( \frac{dy}{dx} - y = 1 + 4\sin x \) satisfy \( y(\pi) = 1 \). Then \( y\!\left(\frac{\pi}{2}\right) + 10 \) is equal to __________.

If the shortest distance between the lines \( \frac{x+2}{2} = \frac{y+3}{3} = \frac{z-5}{4} \) and \( \frac{x-3}{1} = \frac{y-2}{-3} = \frac{z+4}{2} \) is \( \frac{38}{3\sqrt{5}} \, k \), and \( \int_{0}^{k} [ x^{2} ]dx = \alpha - \sqrt{\alpha} \), where \( [x] \) denotes the greatest integer function, then \( 6\alpha^{3} \) is equal to ________.

Let \( a = 1 + \frac{{}^{2}C_{2}}{3!} + \frac{{}^{3}C_{2}}{4!} - \frac{{}^{4}C_{2}}{5!} \ldots \), \( b = 1 + \frac{{}^{1}C_{0} + {}^{1}C_{1}}{1!} + \frac{{}^{2}C_{0} + {}^{2}C_{1} + {}^{2}C_{2}}{2!} + \frac{{}^{3}C_{0} + {}^{3}C_{1} + {}^{3}C_{2} + {}^{3}C_{3}}{3!} + \ldots \). Then \( \dfrac{2b}{a^{2}} \) is equal to ________.

If \( \int_{0}^{\pi/4} \frac{\sin^{2} x}{1 + \sin x \cos x} \, dx = \frac{1}{a} \log_{e}\!\left(\frac{a}{3}\right) + \frac{\pi}{b\sqrt{3}} \) where \( a, b \in \mathbb{N} \), then \( a + b \) is equal to __________.

If \( \displaystyle \lim_{x \to 1} \frac{(5x+1)^{1/3} - (x+5)^{1/3}}{(2x+3)^{1/2} - (x+4)^{1/2}} = \frac{m\sqrt{5}}{n(2n)^{2/3}} \), where \( \gcd(m,n) = 1 \), then \( 8m + 12n \) is equal to ________.

Let \( A \) be a \( 3 \times 3 \) matrix of non-negative real elements such that . Then the maximum value of \( \det(A) \) is _________.