List of practice Questions

The value of $2 \,tan^{-1} (cosec \,tan^{-1} x - tan \,cot^{-1} x))$ is

The value of the integral $\int\limits ^{1/2}_{-1/2}\left[\left(\frac{x+1}{x-1}\right)^{^2}+\left(\frac{x+1}{x-1}\right)^{^2}-2\right]^{^{1/2}}\:\:dx$ is

If | $x^2 - x - 6 | = x + 2$, then the values of $x$ are

A fluid is in streamline flow across a horizontal pipe of variable area of cross section. For this which of the following statements is correct?

If the three lines $x - 3y = p, \,ax + 2y = q$ and $ax + y = r$ form a right-angled triangle then :

A light ray emerging from the point source placed at $P( 1,3)$ is reflected at a point $Q$ in the axis of $x$. If the reflected ray passes through the point $R (6, 7)$, then the abscissa of $Q$ is:

Let $R=\left\{ \left(3,3\right) \left(5,5\right), \left(9,9\right), \left(12,12\right), \left(5,12\right), \left(3,9\right), \left(3,12\right), \left(3,5\right)\right\}$ be a relation on the set $A=\left\{3,5,9,12\right\}$. Then, $R$ is:

The term independent of $x$ in expansion of $\bigg(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\bigg)^{10}$ is

The integral $\int \frac{xdx}{2-x^{2}+\sqrt{2-x^{2}}}$ equals

The real number $k$ for which the equation $2x^3 + 3x + k = 0$ has two distinct real roots in $[0, 1]$

If the x-intercept of some line $L$ is double as that of the line, $3x+4y = 12$ and the $y$-intercept of $L$, is half as that of the same line, then the slope of $L$ is:

If $f(x) = sin\, (sin\, x)$ and $f'' (x) + tan \,x\,f'(x)+ g(x) = 0,$ then $g(x) $ is :

Let $a=Im\left(\frac{1+z^{2}}{2iz}\right),$ where $z$ is any non-zero complex number. The set $A=\left\{a:\left|z\right|=1\,and\,z\ne\pm1\right\}$ is equal to:

If $\vec{a}$ and $\vec{b}$ are non-collinear vectors, then the value of a for which the vectors $\vec{u}=\left(\alpha-2\right)\vec{a}+\vec{b}$ and $\vec{v}=\left(2+3\alpha\right)\vec{a}-3\vec{b}$ are collinear is :

The line $x - 2y = 2$ meets the parabola, $y^2 + 2x = 0$ only at the point $(- 2,-2)$. The line $ y=mx-\frac{1}{2m}\left(m\ne0\right)$ is tangent to the parabola, $y^{2} = - 2x$ at the point $\left(-\frac{1}{2m^{2}}, -\frac{1}{m}\right).$

A class has $175$ students. The following data shows the number of students obtaining one or more subjects. Mathematics $100$, Physics $70$, Chemistry $40$; Mathematics and Physics $30$, Mathematics and Chemistry $28$, Physics and Chemistry $23$; Mathematics, Physics and Chemistry $18$. How many students have offered Mathematics alone?

For the reaction $CO ( g )+\left(\frac{1}{2}\right) O _{2}( g ) \longrightarrow CO _{2}( g ), K _{ p } / K _{ c }$ is

Equal weights of $NaCl$ and $KCl$ are dissolved separately in equal volumes of solutions. Molarity of the two solutions will be :

How many grams of concentrated nitric acid solution should be used to prepare $250\, mL$ of $2 \cdot 0\, M HNO _{3}$ ? The concentrated acid is $70 \% HNO _{3}$.

The coefficient of the middle term in the expansion of $(2 + 3x)^4$ is :

Which of the following is not the product of dehydration of

The temperature of an ideal gas is increased from $27^{\circ} C$ to $127^{\circ} C$, then percentage increase in $v_{r m s}$ is

A table fan rotating at a speed of 2400 rpm is switched off and the resulting variation of revolution/minute with time is shown in figure. The total number of revolutions of the fan before it, comes to rest is

Each question has a group of sentences marked A, B, C, D and E. Arrange these to form a logical sentences.
A. Environment Education Unit of center for Science.
B. Environment has always been working towards provoiding easy to understand reading material.
C. Their new publication on this subject is an attempt to lend teachers a helping hand.
D. It unfolds in two sections first, Climate Change, how to make sense of it all and second, Natural Resources, how to share and care.
E. However, they are introduced to students not as a paragraph to memorise but as an activity to do.

Each question has a group of sentences marked A, B, C, D and E. Arrange these to form a logical sentences.
A. A famous Japanese rock garden at Ryoan ji in North-West Kyoto, Japan.
B. The rocks of various sizes are arranged on small white pebbles in five groups, each comprising, five, two, three, two and three rocks.
C. The garden is 30m long from East to West and 10m from North to South.
D. The garden contains 15 rocks arranged on the surface of white pebbles in such a manner that visitors can see only 14 of them at once from whichever angle the garden is viewed.
E. There are no trees, just around 15 irregularly shaped rocks of varying sizes, some around by rocks, arranged in a bad of white grancelsard that is ranked everyday.