Recent questions tagged past papers

If the lines $2x +3y+1=0$ and $3x-y-4=0$ lie along diameters of a circle of circumference $10 \pi$, then the equation of the circle is

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A variable circle passes through the fixed point $A(p, q)$ and touches x-axis. The locus of the other end of the diameter through A is

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If a circle passes through the point (a, b) and cuts the circle $x^2+y^2=4$ orthogonally, then the locus of its centre is

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If one of the lines given by $6x^2-xy+4cy^2=0$ is $3x+4y=0$, then $c$ equals

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If the sum of the slopes of the lines given by $x^2 - 2cxy -7y^2 =0$ is four times their product, then $c$ has the value

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The equation of the straight line passing through the point $(4, 3)$ and making intercepts on the co-ordinate axes whose sum is $-1$ is

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Let $A(2, -3) $ and $B(-2, 1)$ be vertices of a triangle ABC. If the centroid of this triangle moves on the line $2x+3y=1$, then the locus of the vertex C is the line

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The solution of the differential equation $ y \;dx + (x +x^2y) dy=0$ is

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The differential equation for the family of curves $x^2 + y^2 - 2ay = 0$, where $a$ is an arbitary constant is

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The area of the region bounded by the curves $y = |x-2|, \; x=1, \;x=3$ and the $x-axis$ is

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If $ \begin{align} f(x) = \frac{e^x}{1+e^x}, \; I_1 = \int_{f(-a)}^{f(a)} xg \{ x (1-x) \} dx \end{align} $ and $\begin{align}I_2 = \int_{f(-a)}^{f(a)} g \{x(1-x)\} \end{align}$ then the value of $\frac{I_2}{I_1}$ is

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If $\begin{align} \int_0^{\pi} x f (\sin x) dx = A \int_0^{\pi/2} f(\sin x) dx \end{align}$, then A is

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The value of $\begin{align} I = \int_0^{\pi/2} \frac{(\sin x + \cos x )^2}{\sqrt{1+\sin 2x}} \;dx \end{align}$ is

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The value of $\begin{align} \int_{-2}^{3} |1 - x^2| dx \end{align}$ is

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$\begin{align} \frac{dx}{\cos x - \sin x} \end{align}$ is equal to

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If $\begin{align} \int \frac{\sin x}{\sin(x-\alpha)} dx = Ax + B \log \sin (x-\alpha) + C \end{align}$, then value of $(A, B)$ is

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$\begin{align} \displaystyle \lim_{n \to \infty} \displaystyle\sum_{r=1}^{n} \frac{1}{n} e^{^{\frac{r}{n}}} \end{align}$ is

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If $2a + 3b + 6c = 0$, then at least one root of the equation $ax^2 + bx +c =0$ lies in the interval

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

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A function $y = f(x)$ has a second order derivative $f"(x) = 6 (x=1)$. If its graph passes through the point $(2, 1)$ and at that point the tangent to the graph is $y = 3x -5$, then the function is

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A point on the parabola $y^2 =18 x$ at which the ordinate increases at twice the rate of the abscissa is

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If $ \begin{align} x = e^{{y+e}^{^{y+...to\; \alpha}}}, \; x >0 \end{align}$, then $\frac{dy}{dx}$ is

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Let $f(x) = \frac{1- \tan x}{4x - \pi}, \; x \neq \frac{\pi}{4} , \; x \in [0, \frac{\pi}{2}]$. If $f(x)$ is continuous in $[0, \frac{\pi}{2}]$, then $f(\frac{\pi}{4})$ is

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If $\displaystyle\sum_{x \to \alpha} \begin{pmatrix}1 + \frac{a}{x} + \frac{b}{x^2} \end{pmatrix}^{2x} = e^2 $ then the values of $a$ and $b$, are

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The domain of the function $f(x) = \frac{ \sin^{-1} (x-3) }{\sqrt{9-x^2}}$ is

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The graph of the function $y = f(x)$ is symmetrical about the line $x=2$, then

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If $f : R \to S$, defined by $f(x) = \sin x - \sqrt{3} \cos x + 1$, is onto, then the interval of $S$ is

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A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is $60^{\circ}$ and when he retires $40$ meter away from the tree the angle of elevation becomes $30^{\circ}$. The breadth of the river is

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The sides of the triangle are $\sin \alpha, \cos \alpha$ and $\sqrt{1 + \sin \alpha \cos \alpha}$ for some $0 < \alpha < \frac{\pi}{2}$. Then the greatest angle of the triangle is

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If $u = \sqrt{a^2 \cos^2 \theta +b^2 \sin^2 \theta} + \sqrt{a^2 \sin^2 \theta + b^2 \cos^2 \theta}$, then the difference between the maximum and minimum values of $u^2$ is given by

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Let $\alpha, \beta$ be such that $\pi - \alpha - \beta < 3 \pi$. If $\sin \alpha + \sin \beta = -\frac{21}{65}$ and $\cos \alpha + \cos \beta = -\frac{27}{65}$, then the value of $\cos \frac{\alpha - \beta}{2}$ is

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The sum of series $\frac{1}{2!} + \frac{1}{4!} + \frac{1}{6!}+... $ is

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The sum of the first n terms of the series $1^2 + 2.2^2 + 3^2 + 2.4^2 + 5^2 + 2.6^2+...$ is $\frac{n(n+1)^2}{2}$ when $n$ is even. When $n$ is odd the sum is

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Let $T_f$ be the $r^{th}$ term of an A.P whose first term is a and common difference is $d$. If for some positive integers $m, \;n, \;m \neq n, \;T_m = \frac{1}{n}$ and $T_n = \frac{1}{m}$, then $a-d$ equals

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If $S_n =\displaystyle\sum_{r=0}^{n} \frac{1}{^nC_r}$ and $t_n= \displaystyle\sum_{r=0}^{n} \frac{r}{^nC_r}$, then $\frac{t_n}{S_n}$ is equal to

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The coefficient of $x^n$ in expansion of $(1+x)(1-x)^n$ is

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The coefficient of the middle term in the binomial expansion in powers of $(1 + \alpha x)^4$ and of $(1- \alpha x)^6$ is the same if $\alpha$ equals

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The one root of the equation $x^2 + px + 12 = 0$ is 4, while the equation $x^2+px+q=0$ has equal roots, then the value of 'q' is

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The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

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Hoe many ways are there to arrange the letters in the worf GARDEN with the vowels in alphabetical order ?

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Let $S(K) = 1 + 3 + 5 + ....+ (2K-1) = 3 + K^2$. Then which of the following is true ?

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If $(1-p)$ is a root of quadratic equation $x^2 + px + (1-p) =0$, then its roots are

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Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation-

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If $a_1, a_2, a_3, ....,a_n,....$ are in G.P., then the value of the determinant $\begin{vmatrix} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{vmatrix}$, is

asked Nov 5, 2018 by pady_1

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Let $A =\begin{pmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{pmatrix} \; (10) B = \begin{pmatrix} 4 &2 &2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3 \end{pmatrix} $. If $B$ is the inverse of matrix $A$, then $\alpha$ is

asked Nov 5, 2018 by pady_1

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Let $A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix}$. The only correct statement about the matrix $A$ is

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If $|z^2-1| = |z|^2 + 1 $, then $z$ lies on

asked Nov 5, 2018 by pady_1

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If $z = x -i y$ and $z^{\frac{1}{3}} = p + i q$, then $\frac{(\frac{x}{p} + \frac{y}{q})}{(p^2+q^2)}$ is equal to

asked Nov 5, 2018 by pady_1

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Let $z$, $w$ be complex numbers such that $\overline{z} +i \overline{w} =0$ and $arg \; zw = \pi$. Then $arg\; z$ equals

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The range of the function $f(x) = ^{7-x}P_{x-3}$ is

asked Nov 5, 2018 by pady_1

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