Let \(f:\mathbb{R}\rightarrow \mathbb{R},\,g:\mathbb{R}\rightarrow \mathbb{R}\) be differentiable function such that \((fog)(x)=x\)  . If \(f(x)=2x+cosx+sin^2x\) , then the value of \( \sum_{n=1}^{99}g(1+(2n-1)\pi )\) is

\(1250\pi\)

\((99)^2\frac{\pi}{2}\)

\((99)^2{\pi}\)

\(2500\pi\)

If \(f:[1,\infty )\rightarrow [1,\infty ]\) is defined by \(f(x)=\frac{1+\sqrt{1+4log_2\,x}}{1}\) then \( \,\,f^{-1}(3)=\)

\(0\)

\(1\)

\(64\)

\(\frac{1+\sqrt{5}}{2}\)

If \(α\) and \(β\) are the greatest divisors of \(n(n^2-1)\) and \(2n(n^2+2)\) respectively for all \(n\,\epsilon\, N\) then \(αβ=\)

\(18\)

\(36 \)

\(27 \)

\(9\)

Let \(A=\begin{bmatrix} \frac{1}{6} & \frac{-1}{3} &\frac{-1}{6} \\ \frac{-1}{3}& \frac{2}{3} &\frac{1}{3} \\ \frac{-1}6{} & \frac{1}{3} & \frac{1}{6} \end{bmatrix}\) . If \(A^{2016l}+A^{2017m}+A^{2018n}=\frac{l}{\alpha }A \,\, \) for every \(l,m,n\,\epsilon\, N\) , then the value of \(α\) is

\(\frac{1}{6}\)

\(\frac{1}{3}\)

\(\frac{1}{2}\)

\(\frac{2}{3}\)

Let \(l,m,n\,\epsilon\, \mathbb{R}\) and \(A=\begin{bmatrix} 1 & r& r^2 &1 \\ r& r^2 &1 & m\\ r^2& 1 &r & n \end{bmatrix}\) . Then the set of all real values of r for which the rank of A is 3, is

\((0,\infty)\)

\(R\)

\(R-\left \{ 1 \right \}\)

\(R-\left \{ 0 \right \}\)

The following system of equations

 \(x+y+z=9 \)

\(2x+5y+7z=52\)

\(x+7y+11z=77\)

has

no solution 

exactly 2 solutions 

only one solution 

infinitely many solutions

\(Z\) is a complex number such that \(\left | Z \right |\leq 2\) and \( \,\,-\frac{\pi }{3}\leq \,amp\,\,Z\leq \frac{\pi }{3}\) .The area of the region formed by locus of \(Z\) is

\(\frac{2\pi}{3}\)

\(\frac{\pi}{3}\)

\(\frac{4\pi}{3}\)

\(\frac{8\pi}{3}\)

The points on the argand plane is given by \(Z_1=-3+5i,\,Z_2=-1+6i,\,Z_3=-2+8i,\,Z_4=-4+7i\) form a

parallelogram 

rectangle 

rhombus 

square

When \(n=8,\,(\sqrt{3}+i)^n+(\sqrt{3}-i)^n=\)

\(−256 \)

\(−128 \)

\(256i \)

\(128i\)

If \(2\,cos\frac{7\pi }{5}\) is one of the value of \(z^\frac{1}{5}\) , then \( z =\)

\(32+ 32i \)

\(−32 \)

\(−1 \)

\(32\)

The set of real values of \(x\) for which the inequality \(\left | x-1 \right |+\left | x+1 \right |< 4\) always holds good is

\((−2,2) \)

\(\left ( -\infty ,-2 \right )\cup \left ( 2,\infty \right )\)

\(\left ( -\infty ,-1\right ]\cup \left ( 1,\infty \right )\)

\(\left ( -2 ,-1\right )\cup \left ( 1,2\right )\)

If the roots of the equation \(x^2+x+a=0\) exceed a , then

\(a>2\)

\(a<-2\)

\(2<a<3\)

\(-2<a<-1\)

If the roots of the equation \(\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{5}{2}\) are p and \(q\left ( p>q \right )\) and the roots of the equation \(\left ( p+q \right )x^4-pqx^2+\frac{p}{q}=0\) are \(α,\, β\,, γ\, , δ\) then \(\left ( \sum \alpha \right )^2-\sum \alpha \beta +\alpha \beta \gamma \delta=\)

\(0\)

\(\frac{104}{25}\)

\(\frac{25}{4}\)

\(\frac{16}{5}\)

 The equation \(x^5-5x^3+5x^2-1=0\) has three equal roots. If \(α\) and \(β\) are the two other roots of this equation, then \(α + β + αβ =\)

\(−4 \)

\(3 \)

\(−2 \)

\( −5\)

If all possible numbers formed by using the digits \( 1,2,3,5,7 \) without repetition and they are arranged in descending order, then the rank of the number \(327\) is

\(31 \)

\(175 \)

\(149 \)

\(271\)

If \(a\) is the number of all even divisors and \(b\) is the number of all odd divisors of the number \(10800\), then \(2a+3b=\)

\(72 \)

\(132 \)

\(96 \)

\(136\)

If the coefficient of \(x^5\) in the expansion of \(\left ( ax^2+\frac{1}{bx} \right )^{13}\) is equal to the coefficient of \(x^{-5}\) in the expansion of \(\left ( ax-\frac{1}{bx^2} \right )^{13}\) , then \(ab =\)

\(1\)

\(\frac{1}{6}\)

\(\frac{7}{6}\)

\(\frac{4}{2}\)

For \(n\,\epsilon \,N\) , in the expansion of \(\left ( \sqrt[4]{x^{-3}}+a\sqrt[4]{x^5} \right )^n\), the sum of all binomial coefficients lies between \(200\) and \(400\) and the term independent of \(x\) is \(448\).Then the value of \(a\) is

\(1\)

\(2\)

\(\frac{1}{2}\)

\(0\)

  If \(\frac{x^4+x^3+2x^2-2x+1}{x^3+x^2}=P(x)+\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}\) , then \(A+B+C=\)

\(P(0)\)

\(P(2)\)

\(P(3)\)

\(P(4)\)

If \(A(n)=sin^n\alpha +cos^n\alpha \) then \(A(1)A(4)+A(2)A(5)=\)

\(A(1)A(2)+A(4)A(5)\)

\(A(1)A(6)+A(2)A(3)\)

\(A(1)A(3)+A(2)A(6)\)

\(A(1)A(2)+A(3)A(6)\)

When \(\frac{sin\,9\theta}{cos\,27\theta}+\frac{sin\,3\theta}{cos\,9\theta}+\frac{sin\,\theta}{cos\,3\theta}=k\,(tan\,27\theta-tan\,\theta)\) is defined, then \(k=\)

\(\frac{\pi}{2}\)

\(-\frac{1}{2}\)

\(\frac{1}{2}\)

\(\frac{\pi}{4}\)

If \(x=\sum_{n=0}^{\infty }cos^{2n}\,\theta,\,\,\,y=\sum_{n=0}^{\infty }sin^{2n}\,\theta,\,\,\,z=\sum_{n=0}^{\infty }cos^{2n}\,\theta sin^{2n}\,\theta\)  and \(0<\theta<\frac{\pi }{2}\) , then

\(xz+yz=xy+z\)

\(xyz=yz+x\)

\(xy+z=xy+zx\)

\(x+y+z=xyz+z\)

Number of solutions of the equation \(sin\,x-sin\,2x+sin\,3x=2\,cos^2\,x-2\,cos\,x\) in \( (0,Ï€)\) is

\(1\)

\(3\)

\(2\)

\(4\)

\(2Tan^{-1}\frac{1}{5}+Sec^{-1}\frac{5\sqrt{2}}{7}+2Tan^{-1}\frac{1}{8}=\)

\(\frac{\pi}{6}\)

\(\frac{\pi}{4}\)

\(\frac{\pi}{3}\)

\(\frac{\pi}{8}\)

If \(cosh\,x=\frac{\sqrt{14}}{3},sinh\,x=cos\,\theta \) and \(\,\,-\pi <\theta<-\frac{\pi}{2}\) , then \(\,\,sin\,\theta=\)

\(\frac{1}{3}\)

\(\frac{2}{3}\)

\(-\frac{1}{3}\)

\(-\frac{2}{3}\)

In \(\Delta ABC \,\,\), if \(a=5\,\,\) and \(tan\frac{A-B}{2}=\frac{1}{4}tan\frac{A+B}{2}\), then \(\,\,\sqrt{a^2-b^2}=\)

\(2\)

\(3\)

\(4\)

\(5\)

In a triangle \(ABC\)  if \(A=2B\) and the sides opposite to the angles \(A,B,C \) are \(α+ 1,α− 1\) and \(\alpha\) respectively then \(α =\)

\(3\)

\(4\)

\(5\)

\(6\)

In \(∆ABC\) , right angled at \(A\) , the circumradius, inradius and radius of the excircle opposite to \(A\) are respectively in the ratio \(2:5:λ\) , then the roots of the equation \(x^2-(\lambda -5)x+(\lambda -6)=0\) are

\(3,4 \)

\( 5,13 \)

\( 1,3 \)

\(8,13\)

Let \(3\bar{i}+\bar{j}-\bar{k}\) be the position vector of a point \(B\) . Let \(A\) be a point on the line which is passing through \(B\) and parallel to the vector \(2\bar{i}-\bar{j}+2\bar{k}\) . If \(\left | \overline{BA} \right |=18\) , then the position vector of \(A\) is

\(-9\bar{i}+7\bar{j}-13\bar{k}\)

\(-9\bar{i}+3\bar{j}+12\bar{k}\)

\(9\bar{i}-3\bar{j}+2\bar{k}\)

\(3\bar{i}-\bar{j}+7\bar{k}\)

The vector that is parallel to the vector \(2\bar{i}-2\bar{j}-4\bar{k}\) and coplanar with the vectors \(\bar{i}+\bar{j}\) and \(\bar{j}+\bar{k}\) is

\(\bar{i}-\bar{k}\)

\(\bar{i}+\bar{j}-\bar{k}\)

\(\bar{i}-\bar{j}-2\bar{k}\)

\(3\bar{i}+3\bar{j}+6\bar{k}\)

A line \(L\) is passing through the point \(A\) whose position vector is \(\bar{i}+2\bar{j}-3\bar{k}\) and parallel to vector \(2\bar{i}+\bar{j}+2\bar{k}\) . A plane \(Ï€\) is passing through the points \(\bar{i}+\bar{j}+\bar{k},\bar{i}-\bar{j}-\bar{k}\)  and parallel to the vector \(\bar{i}-2\bar{j}\) . Then the point where this plane \(Ï€\) meets the line \(L\) is

\(\frac{1}{3}\left ( -7\bar{i}+\bar{j}-19\bar{k} \right )\)

\( -7\bar{i}+\bar{j}-19\bar{k} \)

\( 3\bar{i}+3\bar{j}-\bar{k} \)

\(​​ 2\bar{i}-\bar{j}+\bar{k}\)

If the position vector of three points \(A,B,C\) respectively are \(\bar{i}+2\bar{j}+\bar{k}, 2\bar{i}-\bar{j}+2\bar{k}\) and \(\bar{i}+\bar{j}+2\bar{k}\) , then the perpendicular distance of the point \(C\)from the line \(AB\) is,

\( \sqrt{\frac{3}{11}}\)

\( \sqrt{\frac{4}{11}}\)

\( \sqrt{\frac{6}{11}}\)

\( \sqrt{\frac{8}{11}}\)

The volume of a tetrahedron whose vertices are \(4\bar{i}+5\bar{j}+\bar{k},-\bar{j}+\bar{k},3\bar{i}+9\bar{j}+4\bar{k}\) and \(-2\bar{i}+4\bar{j}+4\bar{k}\) is (in cubic units)

\(\frac{14}{3}\)

\(5\)

\(6\)

\(30\)

If the vectors \(\bar{b},\bar{c},\bar{d}\) are not coplanar, then the vector \(\left [ \left ( \bar{a} \times \bar{b} \right ) \times \left ( \bar{c} \times \bar{d} \right )+\left ( \bar{a} \times \bar{c} \right ) \times \left ( \bar{d} \times \bar{b} \right ) +\left ( \bar{a} \times \bar{d} \right ) \times \left ( \bar{b} \times \bar{c} \right ) \right ]\) is

parallel to \(\bar{a}\)

parallel to \(\bar{b}\)

parallel to \(\bar{c}\)

perpendicular to \(\bar{a}\)

\(x_1,x_2,...,x_n\) are \(n\) observations with mean \(\bar{x}\) and standard deviation \(σ\) , Match the items of \(List-I \) with those of \(List-II\)

(a) (b)  (c)  (d)

(i)  (iv)  (ii) (iii)

(a) (b) (c) (d)

(i) (iv) (iii) (ii)

(a) (b) (c) (d)

(iii) (v) (iv) (ii)

(a) (b) (c) (d)

(iii) (v) (ii)  (i)

The variance of \(50\) observation is \(7\). If each observation is multiplied by \(6\) and then \(5\) is subtracted from it, then the variance of the new data is

\(37 \)

\(42 \)

\(247 \)

\(252\)

Two dice are thrown and two coins are tossed simultaneously. The probability of getting prime numbers on both the dice along with a head and a tail on the two coins is

\(\frac{1}{8}\)

\(\frac{1}{2}\)

\(\frac{3}{16}\)

\(\frac{1}{4}\)

\(5\) persons entered a lift cabin on the ground floor of a \(7\) floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first. The probability of all the \(5\) persons leaving the cabin at different floors, is

\(\frac{360}{2401}\)

\(\frac{5}{54}\)

\(\frac{5}{18}\)

\(\frac{5!}{7!}\)

A company produces 10,000 items per day. On a particular day 2500 items were produced on machine A, 3500 on machine B and 4000 on machine C. The probability that an item produced by the machines A, B, C to be defective is respectively 2%, 3% and 5%. If one item is selected at random from the output and is found to be defective, then the probability that it was produced by machine C, is

\(\frac{10}{71}\)

\(\frac{16}{71}\)

\(\frac{40}{71}\)

\(\frac{21}{71}\)

A random variable \(X\) takes the value \(1,2,3\) and \(4\) such that \(2P\left ( X=1 \right )=3P\left ( X=2 \right )=P\left ( X=3 \right )=5P\left ( X=4 \right )\) . If \( σ^2\) is the variance and \(µ\) is the mean of \(X\) then \( σ ^2+µ^2 =\)

\(\frac{421}{61}\)

\(\frac{570}{61}\)

\(\frac{149}{61}\)

\(\frac{3480}{3721}\)

An executive in company makes an average 5 telephone call per hour at a cost of Rs 2 per cell. The probability that in any hour the cost of the calls exceeds a sum of Rs.4 is

\(\frac{2e^4-35}{2e^5}\)

\(\frac{2e^5-37}{2e^5}\)

\(1-\frac{37}{e^4}\)

\(1-(18.5)e^5\)

A quadrilateral \(ABCD\) is divided by the diagonal \(AC\) in two triangles of equal areas. If \(A,B, C\) are respectively \((3,4),(-3,6),(-5,1),\) then the locus of \(D\) is

\((x-8y-57)(x-8y+11)=0\)

\((x-8y-57)(x-8y-11)=0\)

\((3x-8y-57)(3x-8y+11)=0\)

\((3x-8y-11)(3x-8y+57)=0\)

By rotating the coordinate axes in the positive direction about the origin by an angle \(α \), if the point \((1,2)\) is transformed to \(\left ( \frac{3\sqrt{3}-1}{2\sqrt{2}},\frac{\sqrt{3}+3}{2\sqrt{2}} \right )\) in new coordinate system then \( α =\)

\(\frac{\pi}{3}\)

\(\frac{\pi}{6}\)

\(\frac{\pi}{9}\)

\(\frac{\pi}{12}\)

Let a \(a\neq 0,b\neq 0,c\) be three real number and \(L(p,q)=\frac{ap+bq+c}{\sqrt{a^2+b^2}},\forall\, p,q\,\epsilon \,\mathbb{R}\),. If \(L\left ( \frac{2}{3},\frac{1}{3} \right )+L\left ( \frac{1}{3},\frac{2}{3} \right )+L(2,2)=0\) , then the line \(ax+by+c=0\) always passes through the fixed point

\((0,1)\)

\((1,1)\)

\((2,2)\)

\((-1,-1)\)

The incentre of the triangle formed by the straight line having \(3\) X-intercept and \(4\) as Y-intercept, together with the coordinate axes is

\((2,2)\)

\(\left ( \frac{3}{2},\frac{3}{2} \right )\)

\((1,2)\)

\((1,1)\)

The equation of the straight line in the normal form which is parallel to the lines \(x+2y+3=0\) and \(x+2y+8=0\) and dividing the distance between these two lines in the ratio \(1:2\) internally is,

\(x\,cos\,\alpha+y\,sin\,\alpha=\frac{10}{\sqrt{45}},\,\alpha=tan^{-1}\sqrt{2}\)

\(x\,cos\,\alpha+y\,sin\,\alpha=\frac{14}{\sqrt{45}},\,\alpha=\pi+tan^{-1}{2}\)

\(x\,cos\,\alpha+y\,sin\,\alpha=\frac{14}{\sqrt{45}},\,\alpha=tan^{-1}{2}\)

\(x\,cos\,\alpha+y\,sin\,\alpha=\frac{10}{\sqrt{45}},\,\alpha=\pi+tan^{-1}\sqrt{2}\)

A pair of straight lines is passing through the point \((1,1)\) . One of the lines makes an angle \( θ\) with the positive direction of \(X-\)axis and the other makes the same angle with the positive direction of Y-axis. If the equation of the pair of straight lines is \(x^2-(a+2)xy+y^2+a(x+y-1)=0,a\neq 2\) , then the value of \( θ\) is

\(\frac{1}{2}sin^{-1}\left ( \frac{2}{a+2} \right )\)

\(\frac{1}{2}sin\left ( \frac{2}{a+2} \right )\)

\((-\infty ,-1]\cup [1,\infty )\)

\(\frac{1}{2}tan\left ( \frac{2}{a+2} \right )\)

If the pair of lines  \(6x^2+xy-y^2=0\)  and  \(3x^2-axy-y^2=0\) , \(a>0\) have a common line, then \(a =\)

\(\frac{1}{2}\)

\(1\)

\(2\)

\(4\)

If the chord \(L\equiv y-mx-1=0\) of the circle \(S\equiv x^2+y^2-1=0\) touches the circle \(\,\,S_1\equiv x^2+y^2-4x+1=0\) , then the possible points for which \(\,\,L=0\) is a chord of contact of \(\,\,S=0\) are

\(\left ( 2\pm \sqrt{6} ,0\right )\)

\(\,\,\left ( 2\pm \sqrt{6} ,1\right )\)

\(\,\,(2,0)\)

\(\,\,(\sqrt{6},1)\)

If  \(y+c=0\)  is a tangent to the circle \(\,\,x^2+y^2-6x-2y+1=0\)  at \(\,\,(a,4)\) then

\(ac=360\)

\(\,\,ac=-12\)

\(\,\,a+c=0\)

\(\,\,4a=c\)

If the circles \(S\equiv x^2+y^2-14x+6y+33=0\) and \(\,\,S'\equiv x^2+y^2-a^2=0(a\,\epsilon\, N)\) have \(4\) common tangents, then the possible number of circles \(\,\,S'=0\) is

\(1\)

\(2\)

\(0\)

infinite

The center of the circle passing through the point \((1,0)\) and cutting the circles \(x^2+y^2-2x+4y+1=0\) and \(\,\,x^2+y^2+6x-2y+1=0 \) orthogonally is

\(\left ( -\frac{2}{3} ,\frac{2}{3}\right ) \)

\(\left ( \frac{1}{2}, \frac{1}{2}\right ) \)

\((0,1) \)

\((0,0)\)

The equation of the tangent at the point \((0,3)\) on the circle which cuts the circles \(x^2+y^2-2x+6y=0\)  ,  \(\,\,x^2+y^2-4x-2y+6=0\)    and\(\,\,x^2+y^2-12x+2y+3=0 \) orthogonally is

\(y=3 \)

\(x=0 \)

\(3x+y-3=0 \)

\(x+3y-9=0\)

If two tangents to the parabola \(y^2=8x\) meet the tangent at its vertex in \(M\) and \(N\) such that \(MN=4\) , then the locus of the point of intersection of those two tangents is

\(y^2=8(x+3)\)

\(y^2=8(x-2)\)

\(y^2=8(x+2)\)

\(y^2=4(x+2)\)

Three normals are drawn from the point \((c,0)\) to the curve \(y^2=x\) , If one of the normals is \(X-\)axis then the value of \(c\) for which the other two normals are perpendicular to each other is

\(\frac{1}{4}\)

\(\frac{1}{2}\)

\(\frac{3}{4}\)

\(\frac{5}{8}\)

If the normal drawn at one end of the latus rectum of the ellipse \(b^2x^2+a^2y^2=a^2b^2 \) with eccentricity \('e'\) passes through one end of the minor axis, then,

\(e^4+e^2=2 \)

\(e^4-e^2=1 \)

\(e^4+e^2=1 \)

\(e^4+e=1\)

A variable tangent to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \) makes intercept on both the axes. The locus of the middle point of the portion of the tangent between the coordinate axes is

\(\frac{x^2}{b^2}+\frac{y^2}{a^2}=1 \)

\(\frac{a^2}{x^2}+\frac{b^2}{y^2}=1 \)

\(b^2x^2+a^2y^2=4 \)

\(\frac{a^2}{x^2}+\frac{b^2}{y^2}=4 \)

If the eccentricity of a conic satisfies the equation \(2x^3+10x-13=0\) , then that conic is

a circle 

a parabola

an ellipse 

a hyperbola

Assertion (A): If \((−1,3,2)\) and \((5,3,2)\) are respectively the orthocenter and circumcentre of a triangle, then \((3,3,2) \) it its centroid.

Reason (R): Centroid of the triangle divides the line segment joining the orthocenter and the circumcentre in the ratio \(1:2\)

(A) and (R) are true and (R) is the correct explanation to (A)

(A) and (R) are true but (R) is not the correct explanation to (A)

(A) is true, (R) is false.

(A) is false, (R) is true

The lines whose direction cosine are given by the relations \(al+bm+cn=0\) and \(mn+nl+lm=0\) are

perpendicular if  \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\)

perpendicular if  \(\sqrt{a}+\sqrt{b}+\sqrt{c}=0\)

parallel if \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\)

parallel if \(a+b+c=0\)

If the plane passing through the points \( (1,2,3) ,( 2,3,1 )\)  and \((3,1,2)\) is \(ax+by+cz=1\) then \(\,\,a+2b+3c= \)

\(0\)

\(1\)

\(6\)

\(18\)

\(\lim_{x\rightarrow -\infty }\frac{3\left | x \right |-x}{\left | x \right |-2x}-\lim_{x\rightarrow 0}\frac{log(1+x^3)}{sin^3x}=\)

\(1\)

\(\frac{1}{3}\)

\(\frac{4}{3}\)

\(0\)

If \(f(x)=\left\{\begin{matrix} \frac{x-2}{\left | x-2 \right |}+a,& x<2\\ a+b, &x=2 \\ \frac{x-2}{\left | x-2 \right |}+b, & x>2 \end{matrix}\right.\) is continuous \(x=2\), then \(a+b=\)

\(2\)

\(1\)

\(0\)

\(-1\)

If \(f(x)=\left\{\begin{matrix} \frac{x^2\,log(cos\,x)}{log(1+x^2)},& x\neq 0\\ 0, &x=0 \\ \end{matrix}\right.\) then f is

discontinuous at zero 

continuous but not differentiable at zero 

differentiable at zero 

not continuous and not differentiable at zero

Match the items in \(List-A\) with those of the items of \(List-B\)

(a) (b) (c) (d)

(v) (iii) (i) (ii)

(a) (b) (c)  (d)

(iv) (ii) (i)  (iii)

(a) (b) (c) (d)

(iv) (i) (ii) (iii)

(a) (b) (c) (d)

(i) (iii) (iv) (ii)

If \(y=\frac{\left ( sin^{-1}\,x\right )^2}{2}\) , then \(\,\,(1-x^2)y_2-xy_1 \)

\(y\)

\(2y\)

\(1\)

\(2\)

If the relative errors in the base and radius and the height of a cone are same and equal to \(0.02\), then the percentage error in the volume of that cone is

\(2\)

\(4\)

\(6\)

\(8\)

The normal at a point \(\theta\) to the curve \(x=a(1+cos\,\theta),y=a\,sin\,\theta\) always passes through the fixed point

\((0,a)\)

\( (2a,0) \)

\((a,0)\)

\((a,a)\)

Let \(f(x)\) be continuous on\([0,6]\) and differentiable on \((0,6)\). Let \(f(0)=12\) and \(f(6)=-4\) . If \(g(x)=\frac{f(x)}{x+1}\) , then for some Lagrange’s constant \(c \in (0,6).{g}'(c)=\)

\(-\frac{44}{3}\)

\(-\frac{22}{21}\)

\(\frac{32}{21}\)

\(-\frac{44}{21}\)

If \((α ,β )\) and \(( γ ,δ ) \) where \( α<γ\) are the turning points of \(f(x)=2x^3-15x^2+36x-8\) then \(α−γ−β+δ=\)

0

-2

2

1

The height of the cylinder of the greatest volume that can be inscribed in a sphere of radius \(3\) is

\(3\sqrt{3}\)

\(2\sqrt{3}\)

\(\sqrt{3}\)

\(\sqrt{2}\)

\(\int \frac{dx}{\left ( e^x +e^{-x}\right )^2}=\)

\(\frac{1}{2\left ( e^{2x} +1\right )}+c\)

\(-\frac{1}{2\left ( e^{2x} +1\right )}+c\)

\(\frac{1}{3\left ( e^{2x} +1\right )}+c\)

\(-\frac{1}{\left ( e^{2x} +1\right )}+c\)

\(\int_{0}^{\pi/2}\frac{dx}{1+\left ( tan\,x \right )^\sqrt{2018}}=\)

\(\pi\)

\(\frac{3\pi}{4}\)

\(\frac{\pi}{2}\)

\(\frac{\pi}{4}\)

If \(\int \frac{x}{(x^2+1)(x-1)}dx=A\,log\left | x^2+1 \right |+B\,tan^{-1}\,x+C\,log\left | x-1 \right |+d,\) , then \(A+B+C =\)

\(\frac{1}{4}\)

\(\frac{1}{2}\)

\(\frac{3}{4}\)

\(\frac{5}{4}\)

\(\int_{0}^{\pi/2}\frac{cos^3\,x}{sin\,x+cos\,x}dx=\)

\(\frac{\pi-1}{2}\)

\(\frac{\pi-1}{4}\)

\(\frac{1+\pi}{4}\)

\(\frac{\pi-3}{3}\)

\(\int_{0}^{3}(2+x^2)dx=\)

\(\lim_{n\rightarrow \infty }\frac{1}{n}\left [ 2n+\frac{1^2+2^2+3^2+...+(3n)^2}{n^2} \right ]\)

\(\lim_{n\rightarrow \infty }\frac{1}{n}\left [ 3n+\frac{1^2+2^2+3^2+...+6n^2}{n^2} \right ]\)

\(\lim_{n\rightarrow \infty }\frac{1}{n}\left [ 6n+\frac{1^2+2^2+3^2+...+9n^2}{n^2} \right ]\)

\(\lim_{n\rightarrow \infty }\frac{1}{n}\left [ 3n+\frac{1^2+2^2+3^2+...+3n^2}{n^2} \right ]\)

The area enclosed (in square units) by the curve \(y=x^4-x^2\) , the \(X-\)axis and the vertical line passing through the two minimum points of the curve is

\(\frac{48\sqrt{2}}{5}\)

\(\frac{5}{48\sqrt2}\)

\(\frac{7}{60\sqrt{2}}\)

\(\frac{7}{30 \sqrt{2}}\)

The differential equation corresponding to the family of circles having centres on \(X-\)axis and passing through the origin is

\(y^2+x^2+\frac{dy}{dx}=0 \)

\(y^2-x^2+\frac{dy}{dx}=0 \)

\(y^2+x^2+2xy\,\frac{dy}{dx}=0 \)

\(y^2-x^2-2xy\frac{dy}{dx}=0\)

The general solution of the differential equation \((x^2+xy)y'=y^2 \) is

\(e^{\frac{y}{x}} =cx \)

\(e^{-\frac{y}{x}} =cy \)

\(e^{-\frac{y}{x}} =cxy \)

\(e^{-\frac{2y}{x}} =cy\)

At any point on a curve, the slope of the tangent is equal to the sum of abscissa and the product of ordinate and abscissa of that point. If the curve passes through \( (0,1)\), then the equation of the curve is

\(y=2e^\frac{x^2}{2}-1 \)

\(y=2e^{x^{2}} \)

\(y=e^{-x^{2}} \)

\(y=2e^{-x^2}-1\)