WBJEE 2026 Physics and Chemistry Question Paper: Download Answer Key with Solution PDF

Question 1:

Given \(P(x)=x^{4}+ax^{3}+bx^{2}+cx+d\) such that \(x=0\) is the only real root of \(P^{\prime}(x)=0\). If \(P(-1)

• (A) \(P(-1)\) is the minimum but \(P(1)\) is not the maximum of P
• (B) \(P(-1)\) is not minimum but \(P(1)\) is the maximum of P
• (C) neither \(P(-1)\) is the minimum nor \(P(1)\) is the maximum of P
• (D) \(P(-1)\) is the minimum and \(P(1)\) is the maximum of P

If \(\alpha, \beta\) are the roots of the equation \(x^{2}-px+q=0\) and \(\alpha>0\), \(\beta>0\), then \(\alpha^{\frac{1}{4}}+\beta^{\frac{1}{4}}=(p+6\sqrt{q}+4q^{\frac{1}{4}}\sqrt{p+2\sqrt{q}})^{\kappa}\), where K is:

• (A) \(\frac{3}{2}\)
• (B) \(2\)
• (C) \(\frac{1}{3}\)
• (D) \(\frac{1}{4}\)

If \(\sum_{r=1}^{\infty}\tan^{-1}\left(\frac{1}{2r^{2}}\right)=a\), then \(\tan a\) is equal to:

• (A) \(1\)
• (B) \(0\)
• (C) \(\sqrt{3}\)
• (D) \(\frac{\pi}{4}\)

Consider a function \(f(x)\) which has exactly two roots at \(x=a\). If \(\lim_{x\rightarrow a}\left(\frac{\lambda f^{\prime}(x)}{f(x)}-\frac{1}{x-a}\right)=m \ (\ne0)\), then the value of \(\lambda\) is:

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

A vector given by \(\vec{P}=f(t)\hat{i}+g(t)\hat{j}+\hat{k}\) moves in such a way that it is always parallel to the vector \(\vec{Q}=-f^{\prime\prime}(t)\hat{i}+f^{\prime}(t)\hat{j}+\hat{k}\). The magnitude of \(\vec{P}\) is:

• (A) a linear function of time
• (B) a quadratic function of time
• (C) a cubic function of time
• (D) constant

The expression \(\sum_{K=1}^{32}(3K+2)\left\{\sum_{r=1}^{10}\left(\sin\frac{2r\pi}{11}-i \cos\frac{2r\pi}{11}\right)\right\}^{K}\) represents:

• (A) \(48 (1+i)\)
• (B) \(48 (1-i)\)
• (C) \(-\frac{48}{11}(1-i)\)
• (D) \(48(1-i)\)

\(\theta\) elimination from the equations \(x^{2}+y^{2}=\frac{x \cos 3\theta+y \sin 3\theta}{\cos^{3}\theta}=\frac{y \cos 3\theta-x \sin 3\theta}{\sin^{3}\theta}\) will be:

• (A) \(4(x^{4}+y^{4})=3x+4y\)
• (B) \((x^{2}+y^{2}+2x)(x^{2}+y^{2}-x)=2y^{2}\)
• (C) \((x^{2}+y^{2}-2x)(x^{2}+y^{2}+x)=9y\)
• (D) \(x^{2/3}+y^{2/3}=1\)

\(t_{n}\) denotes the nth term of an A.P. and \(t_{p}=\frac{1}{q}, \ t_{q}=\frac{1}{p}\). Then which one of the following options is a root of the equation \((p+2q-3r)x^{2}+(q+2r-3p)x+(r+2p-3q)=0\)?

• (A) \(t_{pq}\)
• (B) \(t_{p}\)
• (C) \(t_{q}\)
• (D) \(t_{p+q}\)

Consider the sequence of numbers \(\{1, 2, 3, \dots, 13\}\). A person chooses three numbers at random from the sequence. The probability that the chosen three numbers form an A.P. is:

• (A) \(\frac{21}{157}\)
• (B) \(\frac{18}{143}\)
• (C) \(\frac{29}{180}\)
• (D) \(\frac{24}{163}\)

If \(f(x)=\frac{1+x}{1-x}\) and \(A\) is a matrix such that \(A^{3}=0\), then \(f(A) =\)

• (A) \(I+2A+2A^{2}\)
• (B) \(I+2A+A^{2}\)
• (C) \(I-2A+A^{2}\)
• (D) \(I+A+A^{2}\)

Which of the following statements is always true?

• (A) If \(f(x)\) is decreasing, then \(\frac{1}{f(x)}\) is increasing
• (B) If \(f(x)\) is decreasing, then \(\frac{1}{f(x)}\) is also decreasing
• (C) If both \(f\) and \(g\) are positive functions such that \(f\) is decreasing and \(g\) is increasing, then \(\frac{f}{g}\) is a decreasing function
• (D) If both \(f\) and \(g\) are positive functions such that \(f\) is increasing and \(g\) is decreasing, then \(\frac{f}{g}\) is a decreasing function

If \(0<\alpha<\beta<\gamma<\frac{\pi}{2}\) then the equation \(\frac{1}{x-\sin \alpha}+\frac{1}{x-\sin \beta}+\frac{1}{x-\sin \gamma}=0\) has:

• (A) real and unequal roots
• (B) imaginary roots
• (C) real and equal roots
• (D) rational roots

On the set \(\mathbb{R}\) of real numbers the relation \(\rho\), defined by \(x\rho y\) \((x,y\in\mathbb{R})\) iff:

• (A) \(|x-y|<2\) is reflexive but neither symmetric nor transitive
• (B) \(|x|\ge y\) is reflexive and transitive but not symmetric
• (C) \(x>|y|\) is transitive but neither reflexive nor symmetric
• (D) \(x-y<2\) is reflexive and symmetric but not transitive

If \(\int\frac{\csc^{2}x-2010}{\cos^{2010}x}dx=-\frac{f(x)}{(g(x))^{2010}}+c\), where \(f(\frac{\pi}{4})=1\), then the number of solutions of the equation \(\frac{f(x)}{g(x)}=\{x\}\) in \([0, 2\pi]\) is/are (where \(\{\cdot\}\) represents fractional part function):

• (A) 3
• (B) 1
• (C) 0
• (D) 2

If the locus of mid point of any normal chord of the parabola \(y^{2}=4x\) is \(x-\lambda=\frac{\mu}{y^{2}}+\frac{y^{2}}{\nu}\), \(\lambda, \mu, \nu\in \mathbb{N}\), then \((\lambda+\mu+\nu)\) equals to:

• (A) 8
• (B) 16
• (C) 10
• (D) 17

The true set of values of 'K' for which \(\sin^{-1}\left(\frac{1}{1+\sin^{2}x}\right)=\frac{K\pi}{6}\) may have a solution is:

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

A mapping is selected at random from all mappings \(f:A\rightarrow A\) where set \(A=\{1,2,3,\dots,n\}\). If the probability that the mapping is injective is \(\frac{3}{32}\), then the value of \(n\) is:

• (A) 8
• (B) 14
• (C) 3
• (D) 4

Let \(A=[a,\infty)\) denotes the domain, then \(f:[a,\infty)\rightarrow B\) which is defined by \(f(x)=2x^{3}-3x^{2}+6\) will have an inverse for the smallest real value of 'a' if:

• (A) \(a=0, \ B=[6,\infty)\)
• (B) \(a=2, \ B=[10,\infty)\)
• (C) \(a=1, \ B=[5,\infty)\)
• (D) \(a=-1, \ B=[5,\infty)\)

If \(a=\lim_{n\rightarrow\infty}\cos^{2n}x\), \((x=n\pi)\) and \(b=\lim_{n\rightarrow\infty}\cos^{2n}x\), \((x\ne m\pi)\), then numerical value of the area of the triangle whose vertices are \((a, b)\), \((-2, 1)\) and \((2, 1)\) is:

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

The position vectors of two adjacent sides of a rectangle \(OACB\) are \(\vec{a}\) and \(\vec{b}\) respectively, where \(O\) is the origin. If \(16|\vec{a}\times\vec{b}|=3(|\vec{a}|+|\vec{b}|)^{2}\) and \(\theta\) be the acute angle between the diagonals \(OC\) and \(AB\), then the value of \(\tan(\frac{\theta}{2})\) is:

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

The point of intersection of \(\vec{r}\times\vec{a}=\vec{b}\times\vec{a}\) and \(\vec{r}\times\vec{b}=\vec{a}\times\vec{b}\), where \(\vec{a}=\hat{i}+\hat{j}\) and \(\vec{b}=2\hat{i}-\hat{k}\) is:

• (A) \(3\hat{i}+2\hat{j}+\hat{k}\)
• (B) \(\hat{i}-\hat{j}-\hat{k}\)
• (C) \(4\hat{i}+2\hat{j}-\hat{k}\)
• (D) \(3\hat{i}+\hat{j}-\hat{k}\)

Let \(a_{1},a_{2},a_{3},...\) are in G.P. such that \(n > m\), \(a_{n}>a_{m}\) and \(a_{1}+a_{n}=66\), \(a_{2}\cdot a_{n-1}=128\). If \(\sum_{r=1}^{n}a_{r}=126\), then \(n\) is:

• (A) 11
• (B) 8
• (C) 6
• (D) 64

The minimum length of intercept on any tangent to the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) cut by the circle \(x^{2}+y^{2}=25\) is:

• (A) 6
• (B) 9
• (C) 11
• (D) 8

Intercepts of the plane \(\vec{r}\cdot\vec{n}=d \ (\ne0)\) on the coordinate axes respectively are:

• (A) \(\frac{\hat{i}\cdot\vec{n}}{d}, \frac{\hat{j}\cdot\vec{n}}{d}, \frac{\hat{k}\cdot\vec{n}}{d}\)
• (B) \(\left|\frac{\hat{i}\cdot\hat{n}}{d}\right|, \left|\frac{\hat{j}\cdot\vec{n}}{d}\right|, \left|\frac{\hat{k}\cdot\vec{n}}{d}\right|\)
• (C) \(\frac{d}{\hat{i}\cdot\hat{n}}, \frac{d}{\hat{j}\cdot\hat{n}}, \frac{d}{\hat{k}\cdot\hat{n}}\)
• (D) \(\frac{d}{\hat{i}\cdot\vec{n}}, \frac{d}{\hat{j}\cdot\vec{n}}, \frac{d}{\hat{k}\cdot\vec{n}}\)

The general solution of the equation \(\sin^{100}x-\cos^{100}x=1\) is:

• (A) \(\{2n\pi+\frac{\pi}{3}:n\in I\}\)
• (B) \(\{n\pi\pm\frac{\pi}{2}:n\in I\}\)
• (C) \(\{n\pi+\frac{\pi}{4}:n\in I\}\)
• (D) \(\{2m\pi-\frac{\pi}{3}:n\in I\}\)

If \(\vec{a}=\hat{i}+\hat{j}+\hat{k}\), \(\vec{b}=\hat{i}-\hat{j}+\hat{k}\), \(\vec{c}=\hat{i}+2\hat{j}-\hat{k}\) then the value of \(\left|\begin{matrix}\vec{a}\cdot\vec{a}&\vec{a}\cdot\vec{b}&\vec{a}\cdot\vec{c}
\vec{b}\cdot\vec{a}&\vec{b}\cdot\vec{b}&\vec{b}\cdot\vec{c}
\vec{c}\cdot\vec{a}&\vec{c}\cdot\vec{b}&\vec{c}\cdot\vec{c}\end{matrix}\right|\) is equal to:

• (A) 64
• (B) 0
• (C) 14
• (D) 16

Number of elements in the range set of \(f(x)=\left[\frac{x}{15}\right]\left[-\frac{15}{x}\right]\), for all \(x \in (0,90)\); (where \([\cdot]\) denotes the greatest integer function) is:

• (A) 8
• (B) 7
• (C) 6
• (D) 5

Let 10 Bags \(B_{1},B_{2},...,B_{10}\) which contain 21, 22, ..., 30 different articles respectively. Then the total number of ways to bring out 10 articles from a Bag is:

• (A) \({}^{31}C_{20}+{}^{21}C_{10}\)
• (B) \({}^{31}C_{20}-{}^{21}C_{10}\)
• (C) \({}^{30}C_{20}-{}^{20}C_{10}\)
• (D) \({}^{30}C_{20}+{}^{20}C_{10}\)

Let domain and range of \(f(x)\) and \(g(x)\) is \([0, \infty)\). If \(f(x)\) is an increasing function, \(g(x)\) is a decreasing function, \(h(x)=f\{g(x)\}\), \(h(0)=0\) and \(p(x)=h(x^{3}-2x^{2}+2x)-h(4)\) then for all \(x\in(0,2)\):

• (A) \(p(x)=-3\)
• (B) \(p(x)=0\)
• (C) \(0
• (D) \(0\le p(x)\le-h(4)\)

Consider the following ellipse: \(\frac{x^{2}}{f(K^{2}+2K+5)}+\frac{y^{2}}{f(K+11)}=1\), where \(f(x)\) is a positive decreasing function. Then the value (values) of K for which the major axis coincides with x-axis is:

• (A) \(K=-5\)
• (B) \(K\in(-3,2)\)
• (C) \(K\in(-7,-5)\)
• (D) \(K=2\)

The solution of the differential equation \(2x^{2}y\frac{dy}{dx}=\tan(x^{2}y^{2})-2xy^{2}\),
given \(y(1)=\sqrt{\frac{\pi}{2}}\) is:

• (A) \(\sin(x^{2}y^{2})=e^{x-1}\)
• (B) \(\sin(x^{2}y^{2})=e^{2(x-1)}\)
• (C) \(\cos\left(\frac{\pi}{2}+x^{2}y^{2}\right)+x=0\)
• (D) \(\sin(x^{2}y^{2})=1\)

The value of the integral \(\int\frac{(\sqrt[3]{x+\sqrt{2-x^{2}}})(\sqrt[6]{1-x\sqrt{2-x^{2}}})}{\sqrt[3]{1-x^{2}}}\,dx\) for \(x\in(0,1)\) is:

• (A) \(2^{\frac{1}{12}}x+c\)
• (B) \(2^{\frac{3}{4}}x+c\)
• (C) \(2^{\frac{1}{3}}x+c\)
• (D) \(2^{\frac{1}{6}}x+c\)

Consider the function \(y=f(x)\) defined implicitly by the equation \(y^{3}-3y+x=0\) on the interval \((-\infty,-2)\cup(2,\infty)\). The area of the region bounded by the curve \(y=f(x)\), the x-axis and the lines \(x=a,x=b\), where \(-\infty

• (A) \(\int_{a}^{b}\frac{x\,dx}{3((f(x))^{2}-1)}-b\,f(b)+a\,f(a)\)
• (B) \(-\int_{a}^{b}\frac{x\,dx}{3((f(x))^{2}-1)}-b\,f(b)+a\,f(a)\)
• (C) \(\int_{a}^{b}\frac{x\,dx}{3((f(x))^{2}-1)}+b\,f(b)-a\,f(a)\)
• (D) \(-\int_{a}^{b}\frac{x\,dx}{3((f(x))^{2}-1)}+b\,f(b)-a\,f(a)\)

The total number of polynomials of the form \(x^{3}+ax^{2}+bx+c\) which are divisible by \(x^{2}+1\), where \(a,b,c\in\{1,2,3,...,10\}\) is:

• (A) 120
• (B) 45
• (C) 10
• (D) 15

The term independent of x in the expansion of \(\left(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}}\right)^{15}\) is equal to:

• (A) 5105
• (B) 5005
• (C) 1365
• (D) 105

For a real number y, consider \([y]\) denotes the greatest integer less than or equal to y. If \(f(x)=\frac{\tan(\pi[x-\pi])}{1+[x]^{2}}\), then:

• (A) \(f^{\prime}(x)\) exists for all x
• (B) \(f^{\prime}(x)\) does not exist
• (C) \(f^{\prime}(1)=\frac{\pi}{4}\)
• (D) \(f^{\prime}(1)=-\frac{\pi}{4}\)

If \(\int_{0}^{1}\left(\sum_{r=1}^{2013}\frac{x}{x^{2}+r^{2}}\right)\left(\prod_{r=1}^{2013}(x^{2}+r^{2})\right)dx=\frac{1}{2}\left[\left(\prod_{r=1}^{2013}(1+r^{2})\right)-K\right]\), then K is:

• (A) \(\frac{2013(2014)(4027)}{6}\)
• (B) \((2013)^{2013}\)
• (C) \((2013)!\)
• (D) \(((2013)!)^{2}\)

The least positive value of 'a' for which the equation \(\int_{0}^{x}(t^{2}-8t+13)dt=x \sin\frac{a}{x}\) has a solution is:

• (A) \(3\pi\)
• (B) \(4\pi\)
• (C) \(\pi\)
• (D) \(2\pi\)

Let all the points on the curve \(x^{2}+y^{2}-10x=0\) are reflected about the line \(y=x+3\). If the locus of the reflected points is in the form \(x^{2}+y^{2}+gx+f+c=0\), then the value of \((g+f+c)\) is:

• (A) 38
• (B) -28
• (C) 28
• (D) -38

The equation \(|x+1|^{\log_{x+1}(3+2x-x^{2})}=(x-3)|x|\) has:

• (A) no solution
• (B) two solutions
• (C) unique solution
• (D) infinite no. of solutions

If the domain of \(f(x)\) is \((0,1)\), then the domain of \(y=f(e^{x})+f(\ln|x|)\) is:

• (A) \((-1,-\frac{1}{e})\)
• (B) \((\frac{1}{e},1)\)
• (C) \((-e,-1)\)
• (D) \((-e,-1)\cup(1,e)\)

The number of 3-digit numbers of the form \(xyz\) with \(x

• (A) 284
• (B) 240
• (C) 44
• (D) 270

Suppose A is denoted the set of all numbers between 1 and 700 which are divisible by 3 and let B is denoted the set of all numbers between 1 and 300 which are divisible by 7. If \(C=\{(a,b)|a\in A,b\in B, a\ne b \text{ and } a+b=\text{even number}\}\), then order of C is:

• (A) 4879
• (B) 4789
• (C) 6789
• (D) 9876

Let us define the power of a matrix \(A\) as the maximum \(m\in \mathbb{Z}^{+}\) such that \(A^{m}=I\). For two matrices \(A\) and \(B\) if \(A^{5}=I\) and \(ABA^{-1}=B^{2}\), then the power of the matrix \(B\) is between:

• (A) 20 and 24
• (B) 28 and 32
• (C) 36 and 40
• (D) 4 and 8

If for two real numbers \(a, b\) with \(|a|\le1\) and \(|b|\le1\), \(\frac{1}{3}+\frac{\sin^{-1}a+\sin^{-1}b}{4}+\frac{(\sin^{-1}a+\sin^{-1}b)^{2}}{16}+\frac{(\sin^{-1}a+\sin^{-1}b)^{3}}{64}+...=\frac{2(8-3\pi)}{3(16+3\pi)}\), then the value of \(\sin^{-1}(a\sqrt{1-b^{2}}+b\sqrt{1-a^{2}})\) is:

• (A) \(\frac{2(32+15\pi)}{3\pi-8}\)
• (B) \(-\frac{\pi}{4}\)
• (C) \(-\frac{3\pi}{4}\)
• (D) \(\frac{1}{3}+\frac{\pi}{4}\)

Let \(\det A=\left|\begin{matrix}l&m&n
p&q&r
1&1&1\end{matrix}\right|\). If \((l-m)^{2}+(p-q)^{2}=9\), \((m-n)^{2}+(q-r)^{2}=16\), \((n-l)^{2}+(r-p)^{2}=25\), then the value of \((\det A)^{2}\) is:

• (A) 169
• (B) 144
• (C) 121
• (D) 100

Let \(f:(0,1)\rightarrow(0,1)\) be a bijective differentiable function such that \(f^{\prime}(x)\ne0 \ \forall x\in(0,1)\) and \(f\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}\). Suppose for all \(x\), \[ \lim_{t\rightarrow x} \frac{\int_{0}^{t}\sqrt{1-(f(s))^{2}}\,ds - \int_{0}^{x}\sqrt{1-(f(s))^{2}}\,ds}{f(t)-f(x)} = f(x) \]
Then the value of \(f\left(\frac{1}{4}\right)\) belongs to:

• (A) \(\{\sqrt{7},\sqrt{6}\}\)
• (B) \(\left\{\frac{\sqrt{7}}{2},\frac{\sqrt{15}}{2}\right\}\)
• (C) \(\left\{\frac{\sqrt{7}}{4},\frac{\sqrt{15}}{4}\right\}\)
• (D) \(\left\{\frac{\sqrt{7}}{3},\frac{\sqrt{15}}{3}\right\}\)

If 'a' is an integer lying in \([-5, 30]\), then the probability that the graph of \(y=x^{2}+2(a+4)x-5a+64\) lies above the x-axis is:

• (A) \(\frac{1}{6}\)
• (B) \(\frac{7}{36}\)
• (C) \(\frac{2}{9}\)
• (D) \(\frac{3}{5}\)

Consider a square \(ABCD\) of diagonal length \(2a\). The square is folded along the diagonal \(AC\) so that the plane of \(\Delta ABC\) is perpendicular to the plane of \(\Delta ADC\). In this case the shortest distance between \(AB\) and \(CD\) is:

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

If \(\int\frac{(1-x^{2})\,dx}{\sqrt{x}\sqrt{(1+x^{2})^{3}}}=\alpha\frac{x^{\beta}}{(1+x^{2})^{\gamma}}+C\), \(\alpha, \beta, \gamma\in\mathbb{R}\) and \(C\) is constant of integration, then \(\alpha:\beta:\gamma\) will be:

• (A) \(4:1:1\)
• (B) \(2:2:\frac{1}{2}\)
• (C) \(\frac{1}{6}:2:\frac{1}{2}\)
• (D) \(1:2:\frac{1}{2}\)

Let \(\vec{r}=\sin x(\vec{a}\times\vec{b})+\cos y(\vec{b}\times\vec{c})+2(\vec{c}\times\vec{a})\), where \(\vec{a},\vec{b}\) and \(\vec{c}\) are three non-coplanar vectors. It is given that \(\vec{r}\) is perpendicular to \((\vec{a}+\vec{b}+\vec{c})\). Then the possible value(s) of \((x^{2}+y^{2})\) is/are:

• (A) \(\frac{5\pi^{2}}{4}\)
• (B) \(\frac{35\pi^{2}}{4}\)
• (C) \(\frac{37\pi^{2}}{4}\)
• (D) \(\frac{\pi^{2}}{4}\)

Let \(A_{1},A_{2},...,A_{6}\) are six sets, each with four elements and \(B_{1},B_{2},...,B_{n}\) are \(n\) sets, each with two elements. Let \(S=A_{1}\cup A_{2}\cup...\cup A_{6}=B_{1}\cup B_{2}\cup...\cup B_{n}\). Given that each element of \(S\) belongs to exactly four of the \(A\)'s and to exactly three of the \(B\)'s. Then \(n\) is:

• (A) 12
• (B) 24
• (C) 6
• (D) 9

A figure is bounded by the curves \(y=x^{2}+1\), \(y=0\), \(x=0\) and \(x=1\). The point at which a tangent should be drawn to the curve \(y=x^{2}+1\) for it to cut off a trapezium of the greatest area from the figure is:

• (A) \((1,2)\)
• (B) \((-1,2)\)
• (C) \(\left(\frac{1}{2},\frac{5}{4}\right)\)
• (D) \((0,1)\)

The ends \(A, B\) of a straight line segment of constant length \(c\) slide upon the fixed rectangular axes \(OX, OY\) respectively. If the rectangle \(OAPB\) is completed, then the locus of the foot of the perpendicular drawn from \(P\) to \(AB\) is:

• (A) \(x^{2} + y^{2} = c^{2}\)
• (B) \(x^{2/3} + y^{2/3} = c^{2/3}\)
• (C) \(\sqrt{x} + \sqrt{y} = \sqrt{c}\)
• (D) \(xy = c^{2}\)

Let \(1\) lies between the roots of the equation \(y^{2}-my+1=0\) and \([x]\) denotes the greatest integer function. Then the value of \(\left[\left(\frac{4|x|}{x^{2}+16}\right)^{m}\right]\) is:

• (A) 5
• (B) 4
• (C) 0
• (D) 1

Let \(f(x)\) be a twice differentiable function in \([1,3]\) and \(f(1)=f(3)\). Further if \(|f^{\prime\prime}(x)|\le2\), then for all \(x\) in \([1, 3]\):

• (A) \(|f^{\prime}(x)|\ge4\)
• (B) \(|f^{\prime}(x)|\le-1\)
• (C) \(|f^{\prime}(x)|>2\)
• (D) \(|f^{\prime}(x)|<4\)

The quantities \(a_{1},a_{2},a_{3},......\) form an infinite decreasing G.P. If \(a_{1}=1\), then the common ratio of the progression for which the expression \(6a_{5}-16a_{4}-3a_{3}+12a_{2}\) is at a maximum is:

• (A) \(\frac{1}{4}\)
• (B) \(\frac{1}{2}\)
• (C) \(\frac{1}{3}\)
• (D) \(-\frac{1}{4}\)

If \(f\) be a real valued function defined for all real numbers \(x\) such that for some fixed \(a>0,\) it satisfies \(f(x+a)=\frac{1}{2}+\sqrt{f(x)-(f(x))^{2}}\ \forall x\>, then \(f(x)\) is periodic with period:

• (A) \(a\)
• (B) \(4a\)
• (C) \(\frac{a}{2}\)
• (D) \(2a\)

Four natural numbers selected at random are multiplied together, then the probability that the digit in the unit's place in the product be 1, 3, 7 or 9 is:

• (A) \(\frac{16}{625}\)
• (B) \(\frac{18}{625}\)
• (C) \(\frac{4}{625}\)
• (D) \(\frac{5}{625}\)

Let \(f(x)\) be a real valued function which is monotonic and differentiable. Then for any reals \(a\) and \(b,\int_{f(a)}^{f(b)}2x\{b-f^{-1}(x)\}dx=\)

• (A) \(\int_{a}^{b}(f^{2}(x)-f^{2}(a))dx\)
• (B) \(\int_{a}^{b}(f(x)-f(a))^{2}dx\)
• (C) \(\int_{a}^{b}(bf^{2}(x)-af^{2}(a))dx\)
• (D) \(bf^{2}(b)+f^{-1}(a)\)

Tangent at a point \(P_{1}\) (other than \((0, 0)\)) on the curve \(y=x^{3}\) meets the curve again at \(P_{2}\). The tangent at \(P_{2}\) meets the curve at \(P_{3}\) and so on. Then the abscissae of \(P_{1},P_{2},P_{3},...,P_{n}\) form:

• (A) an A.P. with common difference 1
• (B) an H.P. with common difference \(\frac{1}{2}\)
• (C) a G.P. with common ratio 2
• (D) a G.P. with common ratio \((-2)\)

The equation \(x^{3}+5x^{2}+px+q=0\) and \(x^{3}+7x^{2}+px+r=0\) have two roots in common. If the third root of each equation is represented by \(x_{1}\) and \(x_{2}\) respectively, then GCD of \(x_{1}\), \(x_{2}\) will be:

• (A) 3
• (B) 1
• (C) \(p\)
• (D) 2

Let \(a, b, c\) be non-zero real numbers, such that \(\int_{0}^{1}(1+\cos^{8}x)(ax^{2}+bx+c)dx=\int_{0}^{2}(1+\cos^{8}x)(ax^{2}+bx+c)dx\). Then \(ax^{2}+bx+c=0\) has:

• (A) no solution in \((0, 2)\)
• (B) at least one root in \((1, 2)\)
• (C) two imaginary roots
• (D) two roots in \((0, 2)\)

Let \(Z_{1}, Z_{2}\) be the roots of the equation \(Z^{2}+pZ+q=0\), where the coefficients \(p\) and \(q\) may be complex numbers and also let \(A, B\) represent \(Z_{1}, Z_{2}\) respectively in the complex plane. If \(\angle AOB=\alpha\ne0\) and \(OA=OB\), where \(O\) is the origin, then the value of \(\frac{p^{2}}{q}\sec^{2}\frac{\alpha}{2}\) will be:

• (A) \(\frac{1}{4}\)
• (B) \(\frac{3}{4}\)
• (C) 4
• (D) 1

Let \(g(x)=ax+b\), where \(a<0\) and \(g\) is defined from \([1, 3]\) onto \([0, 2]\). Then the value of \(\cot(\cos^{-1}(|\sin x|+|\cos x|)+\sin^{-1}(-|\cos x|-|\sin x|))\) is equal to:

• (A) \(g(2)+g(3)\)
• (B) \(g(2)\)
• (C) \(g(3)\)
• (D) \(g(1)+g(2)\)

If \(\sum_{r=0}^{2n}a_{r}(x-2)^{r}=\sum_{r=0}^{2n}b_{r}(x-3)^{r}\) and \(a_{k}=1\ \forall k\ge1\), then the value of \(\frac{b_{n}}{{}^{2n+1}C_{n+1}}\) is:

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

If \(f(x)\) is differentiable for all \(x\in\mathbb{R}\) and satisfies the relation \(x=\lim_{n\rightarrow\infty}\frac{[1^{2}(f(x))^{x}]+[2^{2}(f(x))^{x}]+\dots+[n^{2}(f(x))^{x}]}{n^{3}}\), where \([\cdot]\) denotes the greatest integer function, then \(f^{\prime}(x)\) is equal to:

• (A) \(\frac{1}{3x^{2}}\ln x\)
• (B) \(3x^{1/x}(1-\ln 3x)\)
• (C) \((3x)^{\frac{1}{x}}\left[\frac{1-\ln 3x}{x^{2}}\right]\)
• (D) \((3x)^{\frac{1}{x}}\frac{(\ln 3x+1)}{x^{2}}\)

If a differentiable function satisfies \[ (x-y)f(x+y) - (x+y)f(x-y) = 2(x^2y-y^3), \qquad \forall x,y\in\mathbb{R} \]
and \(f(1)=2\), then:

• (A) \(f(x)\) must be a polynomial function
• (B) \(f(3)=13\)
• (C) \(f(3)=12\)
• (D) \(f(0)=0\)

Let \(f(x)>0\) for all \(x\in\mathbb{R}\) and \(f(x)\) is bounded. If \(\lim_{n\rightarrow\infty}\sum_{r=1}^{n}a^{r-1}\int_{(r-1)a}^{ra}\frac{f(x)dx}{f(x)+f(2ra-a-x)}=\frac{3}{5}\), where \(0

• (A) \(\frac{5}{11}\)
• (B) \(\frac{7}{11}\)
• (C) \(\frac{6}{11}\)
• (D) \(\frac{1}{11}\)

Consider the curve \(x=1-3t^{2}, y=t-3t^{3}\). The tangent to the curve at the point is inclined at an angle \(\phi\) to \(OX\) and the tangent at \(P(-2,2)\) meets the curve again at \(Q\). Then:

• (A) the curve is symmetrical about \(x\)-axis
• (B) the curve is symmetrical about \(y\)-axis
• (C) \(3t=\tan\phi+\sec\phi\)
• (D) tangents at \(P\) and \(Q\) are at right angle

If \(f(x)=x(1331x^{2}-3630x+3300)\), then for \(a=\cos^{2}(\tan^{-1}(\sin(\cot^{-1}3)))\):

• (A) \(f(a+1)=2331\)
• (B) \(f^{\prime}(a)=11\)
• (C) \(\lim_{x\rightarrow a}f(x)=1000\)
• (D) \(\int_{0}^{a}(f(x)-1000)dx=\frac{2500}{11}\)

Let \(\vec{r}=\sin x(\vec{a}\times\vec{b})+\cos y(\vec{b}\times\vec{c})+2(\vec{c}\times\vec{a})\), where \(\vec{a},\vec{b}\) and \(\vec{c}\) are three non-coplanar vectors. It is given that \(\vec{r}\) is perpendicular to \((\vec{a}+\vec{b}+\vec{c})\). Then the possible value(s) of \((x^{2}+y^{2})\) is/are:

• (A) \(\frac{5\pi^{2}}{4}\)
• (B) \(\frac{35\pi^{2}}{4}\)
• (C) \(\frac{37\pi^{2}}{4}\)
• (D) \(\frac{\pi^{2}}{4}\)

The parabola \(y=4-x^{2}\) has vertex \(P\). It intersects the \(x\)-axis at \(A\) and \(B\). If the parabola is translated from its initial position to a new position by moving its vertex along the line \(y=x+4,\) so that it intersects the \(x\)-axis at \(B\) and \(C\), then the abscissa of \(C\) will be:

• (A) 12
• (B) 8
• (C) 6
• (D) \(\frac{7}{3}\)

If \(A_{1},A_{2},A_{3},...,A_{1006}\) be independent events such that \(P(A_{i})=\frac{1}{2i}\) \((i=1,2,...,1006)\) and the probability that none of the events occurs be \(\frac{\alpha!}{2^{\alpha}{(\beta!)}^{2}}\), then:

• (A) \(\beta\) is of the form \(4k+2, k\in I\)
• (B) \(\alpha=2\beta\)
• (C) \(\beta\) is of the form \(4k+1, k\in I\)
• (D) \(\beta\) is a prime number

If \((4^{\sec^{2}\alpha})x^{2}+2x+(\beta^{2}-\beta+\frac{1}{2})=0\) has real roots, then the value/values of \((\cos\alpha+\cos^{-1}\beta)\) is/are:

• (A) \(1+\frac{\pi}{3}\) if \(\alpha = 2n\pi\)
• (B) \(-1-\frac{\pi}{3}\) if \(\alpha = (2n+1)\pi\)
• (C) \(-1+\frac{\pi}{3}\) if \(\alpha = (2n+1)\pi\)
• (D) \(-1+\frac{\pi}{3}\) if \(\alpha = 2n\pi\)