List of practice Questions

If a plane meets the coordinate axes at $A,B$ and $C$ such that the centroid of the triangle is $(1, 2, 4)$, then the equation of the plane is

The value of $c$ from the Lagrange�s mean value theorem for which $f(x) = \sqrt{25 - x^2}$ in $[1,5]$, is

The area in the first quadrant between $x^2 + y^2 = \pi^2$ and $y = \sin \, x$ is

The differential equation of the rectangular hyperbola hyperbola, where axes are the asymptotes of the hyperbola, is

The least positive integer n such that $1-\frac{2}{3}-\frac{2}{3^{2}}-.......-\frac{2}{3^{n-1}} < \frac{1}{100},$ is :

The statement $(p \Rightarrow q ) \Leftrightarrow ( \sim p \Lambda q)$ is a

The triangle formed by the tangent to the curve $f (x) = x2 + bx - b$ at the point $(1,1)$ and the coordinate axes lies in the first quadrant. If its area is $2$, then the value of b is

If $\frac{1}{\sqrt{\alpha}}$ and $\frac{1}{\sqrt{\beta}}$ are the roots of the equation, $ax^{2} + bx +1 = 0 \left(a ^{ }\ne 0, a, b \in R\right)$, then the equation, $x\left(x + b^{3}\right) + \left(a^{3} ? 3abx\right)$ = 0 has roots :

If $(10)^9 +2(11)^1 (10)^8 + 3(11)^2 (10)^7 +............+10(11)^9 = k (10)^9,$ then k is equal to

If $ 1 + x^4 + x^5 = \sum\limits^{5}_{i =0} a_i$ $(1 + x)^i$ , for all $x$ in $R$, then $a_2$ is:

Let the population of rabbits surviving at a time $t$ be governed by the differential equation $\frac {dp(t)}{dt}=\frac {1}{2} p(t)-200.$ If $p(0)=100,$ then $p(t)$ is equal to

The slope of the line touching both the parabolas $y^2 = 4x$ and $x^2 = - 32y$ is :

Three positive numbers form an increasing $G.P.$ If the middle term in this $G.P.$ is doubled, then new numbers are in $A.P.$ Then, the common ratio of the $G.P.$ is

If $X = \{4^n- 3 n - 1: n \in N \}$ and $Y = \{9 (n - 1): n \in N \}$ ,where $N$ is the set of natural numbers, then $ X \cup Y$ is equal to

The area (in sq units) of the region described by $A=\left\{(x, y): x^{2}+y^{2} \leq 1\right.$ and $\left.y^{2} \leq 1-x\right\}$ is:

The coefficient of $x^{50}$ in the binomial expansion of $(1 + x)^{1000} + x (1 + x)^{999} + x^2(1 + x)^{998} + .... + x^{1000}$ is:

Equation of the line of the shortest distance between the lines $\frac{x}{1} = \frac{y}{-1} = \frac{z}{1}$ and $\frac{x-1}{0} = \frac{y+1}{-2} = \frac{z}{1}$ is :

If a line intercepted between the coordinate axes is trisected at a point $A(4, 3)$, which is nearer to $x$-axis, then its equation is :

Let $f\left(n\right) = \left[\frac{1}{3} + \frac{3n}{100}\right]{n},$ where $\left[n\right]$ denotes the greatest integer less than or equal to n. Then $\sum\limits^{56}_{n = 1} \Delta_{r} f\left(n\right)$ is equal to :

The number of terms in an $A.P$. is even; the sum of the odd terms in it is $24$ and that the even terms is $30$. If the last term exceeds the first term by $10 \frac{1}{2},$ then the number of terms in the $A.P$. is :

A conductor lies along the $z$ -axis at $-1.5 \leq z<1.5 \,m$ and carries a fixed current of $10.0 A$ in $-\hat{a}_{z}$ direction (see figure). For a field $\vec{B}=3.0 \times 10^{-4} e^{-0.2 x} \hat{a}_{y} T$, find the power required to move the conductor at constant speed to $x=2.0 m , y=0 \,m$ in $5 \times 10^{-3} \,s$. Assume parallel motion along the $x$ -axis

Through the vertex $O$ of a parabola $y^2 = 4x$, chords $OP$ and $OQ$ are drawn at right angles to one another. The locus of the middle point of $PQ$ is

$3.92\,g$ of ferrous ammonium sulphate react completely with $50\, ml \frac{N}{10} KMnO_{4}$ solution. The percentage purity of the sample is

A projectile is fired with a velocity $u$ making an angle $\theta $ with the horizontal. What is the magnitude of change in velocity when it is at the highest point -

A small block of mass $m$ is kept on a rough inclined surface of inclination $\theta $ fixed in a elevator. The elevator goes up with a uniform velocity $v$ and the block does not slide on the wedge. The work done by the force of friction on the block in time $t$ will be :