JEE Main 2026 April 4 Shift 2 Mathematics Question Paper with Solutions PDF

If \( \sum_{i=1}^{10} (x_i + 2)^2 = 180 \) and \( \sum_{i=1^{10 (x_i - 1)^2 = 90 , then the Standard Deviation is equal to

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

If the system of equations
\[ x + y + z = 5 \]
\[ x + 2y + 3z = 9 \]
\[ x + 3y + \lambda z = \mu \]

has infinitely many solutions, then value of \( \lambda + \mu \) is

• (1) \(13\)
• (2) \(20\)
• (3) \(18\)
• (4) \(26\)

Let
\[ f(x)= \begin{cases} e^{x-1}, & x<0
x^2-5x+6, & x\ge0 \end{cases} \]

and \(g(x)=f(|x|)+|f(x)|\).

If \(\alpha\) = number of points of discontinuity of \(g(x)\) and
\(\beta\) = number of points of non-differentiability of \(g(x)\),

then \(\alpha+\beta=\)

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

Area bounded between the curves
\[ x = -2y^2 \]
\[ x = 1 - 4y^2 \]

is

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

From point \(B(4,8)\) on the parabola \(y^2 = 16x\), two perpendicular chords \(BA\) and \(BC\) are drawn. Given that the locus of the centroid of triangle \(BAC\) is another parabola with length of the latus rectum equal to \(\ell\), then \(3\ell\) is equal to

• (1) \(14\)
• (2) \(15\)
• (3) \(12\)
• (4) \(16\)

If the quadratic equation
\[ (\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0 \quad (\lambda \ne -2) \]

has two positive roots then the number of possible integral values of \(\lambda\) is

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

If function \(y(x)\) satisfies the differential equation
\[ \frac{dy}{dx}+\left[\frac{6x^2+e^{-2x}(3x^2+2x^3+4)}{(x^3+2)(2+e^{-2x})}\right]y = e^{-2x}+2 \]

such that \(y(0)=\frac{3}{2}\) and
\[ y(1)=\alpha(e^{-2}+2) \]

then \(\alpha\) is equal to

• (1) \(\frac{13}{12}\)
• (2) \(\frac{12}{13}\)
• (3) \(\frac{4}{3}\)
• (4) \(\frac{17}{13}\)

3 numbers are selected randomly from numbers \(1,2,3,\ldots,31\). The probability that they are in A.P. is

• (1) \(\frac{15}{31}\)
• (2) \(\frac{7}{31}\)
• (3) \(\frac{8}{17}\)
• (4) \(\frac{45}{899}\)

The maximum value of
\[ E = 16\sin\frac{x}{2}\cos^3\frac{x}{2} \]

where \(x\in[0,\pi]\), is

• (1) \(3\)
• (2) \(3\sqrt{3}\)
• (3) \(6\sqrt{3}\)
• (4) \(6\)

Evaluate
\[ \int_{0}^{1}\cot^{-1}(1+x+x^2)\,dx \]

• (1) \(2\tan^{-1}2-\frac{1}{2}\ln\left(\frac{5}{4}\right)-\frac{\pi}{2}\)
• (2) \(2\tan^{-1}2-\frac{1}{2}\ln\left(\frac{5}{4}\right)+\frac{\pi}{2}\)
• (3) \(2\tan^{-1}2+\frac{1}{2}\ln\left(\frac{5}{4}\right)-\frac{\pi}{2}\)
• (4) \(2\tan^{-1}2+\frac{1}{2}\ln\left(\frac{5}{4}\right)+\frac{\pi}{2}\)

Let
\[ S=\{z: z^2+4z+16=0,\; z\in\mathbb{C}\} \]

then the value of
\[ \sum_{z\in S}|z+\sqrt{3}i|^2 \]

is

• (1) \(34\)
• (2) \(35\)
• (3) \(38\)
• (4) \(41\)

If \(f(x)\) satisfies the functional equation
\[ f(x+y)=f(x)+2y^2+y+\alpha xy \]

where \(x,y\) are whole numbers, such that \(f(0)=-1\) and \(f(1)=2\), then the value of
\[ \sum_{i=1}^{5}\big(f(i)+\alpha\big) \]

is

• (1) \(130\)
• (2) \(145\)
• (3) \(120\)
• (4) \(140\)

If
\[ f(x)=\int_{0}^{x}\tan(t-x)\,dt+\int_{0}^{x}f(t)\tan t\,dt \]

then the value of
\[ f''\left(\frac{\pi}{6}\right)-12f'\left(-\frac{\pi}{6}\right)+f\left(\frac{\pi}{6}\right) \]

is

• (1) \(-7-\frac{16}{3\sqrt3}\)
• (2) \(7+\frac{5}{3\sqrt3}\)
• (3) \(7-\frac{16}{3\sqrt3}\)
• (4) \(\frac{1}{3\sqrt3}\)

Number of non-negative integral solutions of the equation
\[ a+b+2c=22 \]

is

• (1) \(144\)
• (2) \(121\)
• (3) \(168\)
• (4) \(99\)

If
\[ \alpha=\frac14+\frac18+\frac1{16}+\cdots up to infinity \]
\[ \beta=\frac13+\frac19+\frac1{27}+\cdots up to infinity \]

Then value of
\[ (0.2)^{\log_5\alpha}+(0.04)^{\log_5\beta} \]

is

• (1) \(\frac12\)
• (2) \(8\)
• (3) \(3\)
• (4) \(\frac34\)

If \(z_1,z_2,z_3\) are roots of
\[ x^3+ax^2+bx+c=0 \]

Let \(z_1=1,\; z_2=1+i\sqrt2\) and \(a,b,c\in\mathbb{R}\). Then the value of
\[ \int_{-1}^{1}(x^3+ax^2+bx+c)\,dx \]

is

• (1) \(-8\)
• (2) \(8\)
• (3) \(6\)
• (4) \(-4\)

The shortest distance between the lines
\[ \vec r=\frac13\hat i+2\hat j+\frac83\hat k+\lambda(2\hat i-5\hat j+6\hat k) \]
\[ \vec r=\left(-\frac23\hat i-\frac13\hat k\right)+\mu(\hat j-\hat k), \quad \lambda,\mu\in\mathbb R \]

is

• (1) \(2\sqrt3\)
• (2) \(3\)
• (3) \(\sqrt{15}\)
• (4) \(\sqrt5\)

If \(\hat u,\hat v\) are unit vectors and
\[ |\hat u\times \hat v|=\frac{\sqrt3}{2} \]

and
\[ \vec A=\lambda\hat u+\hat v+\hat u\times \hat v \]

then find \(\lambda\). (Angle between \(\hat u\) and \(\hat v\) is acute)

• (1) \(\lambda=\frac13\vec A\cdot\hat u-\frac13\vec A\cdot\hat v\)
• (2) \(\lambda=\frac43\vec A\cdot\hat u-\frac23\vec A\cdot\hat v\)
• (3) \(\lambda=\frac83\vec A\cdot\hat u-\frac23\vec A\cdot\hat v\)
• (4) \(\lambda=\frac23\vec A\cdot\hat u-\frac43\vec A\cdot\hat v\)

In the expansion of
\[ \left(9x-\frac{1}{3\sqrt{x}}\right)^{18} \]

if coefficient of the term independent of \(x\) is \(221k\), then the value of \(k\) is

• (1) \(42\)
• (2) \(64\)
• (3) \(72\)
• (4) \(84\)

If \(f(x)=(x-1)^4+1 \quad \forall x\in[1,\infty)\).

Statement–1 : \(f(x)=f^{-1}(x)\) has only two solutions.

Statement–2 : \(f^{-1}(x+1)=f(x)\) has no solution.

• (1) Statement 1 and Statement 2 both are true
• (2) Statement 1 is false and Statement 2 is true
• (3) Statement 1 is true and Statement 2 is false
• (4) Statement 1 and Statement 2 both are false

If \(\left(2\alpha+1,\;\alpha^2-3\alpha,\;\frac{\alpha-1}{2}\right)\) is the image of \((\alpha,2\alpha,1)\) in the line
\[ \frac{x-2}{3}=\frac{y-1}{2}=\frac{z}{1}, \]

then the value of \(\alpha\) is:

P is a point on
\[ \frac{x^2}{9}+\frac{y^2}{4}=1 \]

as \(P(3\cos\alpha,2\sin\alpha)\).

Q is a point on
\[ x^2+y^2-14x+14y+82=0 \]

R is a point on line
\[ x+y=5 \]

If the centroid of triangle \(PQR\) is
\[ \left(\cos\alpha+2,\;\frac{2\sin\alpha}{3}+3\right) \]

find the sum of possible ordinates of \(R\).

P is the point of intersection of the two half-lines
\[ x-\sqrt{3}y=\alpha, \quad \alpha>0 \]

Points \(A\) and \(B\) lie on these lines at a distance \(\alpha\) from \(P\).
If the length of perpendicular from \(P\) on \(AB\) is \(\frac{\alpha}{2}\) and the radius of the circumcircle of \(\triangle PAB\) is \(R\), then find
\[ \frac{\alpha^2}{R} \]