IIT-JEE 1978

The substance absorbs CO2 and violently reacts with water. The substance is

What weight of AgCl will be precipitated when a solution containing 4.77 g of NaCl is added to a solution of 5.77g of AgNO3?

What is the molarity and molality of a 13% solution (by weight) of sulphuric acid with a density of 1.02 g/ml? To what volume should 100 ml of this acid be diluted in order to prepare a 1.5 N solution?

One gram of an alloy of aluminium and magnetism when treated with excess of dil. HCl forms magnesium chloride, aluminium chloride and hydrogen. The evolved hydrogen, collected over mercury at \({0^o}C\) has a volume of 1.20 litres at 0.92 ...

Naturally occurring boron consists of two isotopes whose atomic weighs are 10.01 and 11.01. The atomic weight of natural oron is 10.81. Calculate the percentage of each isotope in natural boron.

Igniting MnO2 converts it quantitatively to Mn3O4. A sample of pyrolusite is of the following composition : MnO2 80%, SiO2 and other inert constituents 15%, rest being water. The sample is ignited in air to constant weight. What is the perc...

27 g of Al will react completely with how many grams if oxygen?

A compound was found to contain nitrogen and oxygen in the ratio 28 gm and 80 gm respectively. The formula of compound is

From a point \(O\) inside a triangle \(ABC,\) perpendiculars \(OD\), \(OE, OF\) are drawn to the sides \(BC, CA, AB\) respectively. Prove that the perpendiculars from \(A, B, C\) to the sides \(EF, FD, DE\) are concurrent.

Find the equation of the circle whose radius is 5 and which touches the circle \({x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,\) at the point (5, 5).

If x = a + b, y = a\(\gamma\) + b\(\beta\) and z = a\(\beta\) +b\(\gamma\) where \(\gamma\) and \(\beta\) are the complex cube roots of unity, show that xyz = \({a^3} + {b^3}\).

Express \({1 \over {1 - \cos \,\theta + 2i\sin \theta }}\) in the form x + iy.

Find the derivative of \(\sin \left( {{x^2} + 1} \right)\) with respect to \(x\) first principle.

Evaluate \(\int {{{\sin x} \over {\sin x - \cos x}}dx}\)

Six X' s have to be placed in the squares of figure below in such a way that each row contains at least one X. In how many different ways can this be done.

Balls are drawn one-by-one without replacement from a box containing \(2\) black, \(4\) white and \(3\) red balls till all the balls are drawn. Find the probability that the balls drawn are in the order \(2\) black, \(4\) white and \(3\) re...

A triangle \(ABC\) has sides \(AB=AC=5\) cm and \(BC=6\) cm Triangle \(A'B'C'\) is the reflection of the triangle \(ABC\) in a line parallel to \(AB\) placed at a distance \(2\) cm from \(AB\), outside the triangle \(ABC\). Triangle $$A''B'...

Show that the square of \(\,{{\sqrt {26 - 15\sqrt 3 } } \over {5\sqrt 2 - \sqrt {38 + 5\sqrt 3 } }}\) is a rational number.

Solve for \(x:\,\sqrt {x + 1} - \sqrt {x - 1} = 1.\)

Solve the following equation for \(x:\,\,2\,{\log _x}a + {\log _{ax}}a + 3\,\,{\log _{{a^2}x}}\,a = 0,a > 0\)

Solve for \(x:{4^x} - {3^{^{x - {1 \over 2}}}}\, = {3^{^{x + {1 \over 2}}}}\, - {2^{2x - 1}}\)

If \(\left( {m\,,\,n} \right) = {{\left( {1 - {x^m}} \right)\left( {1 - {x^{m - 1}}} \right).......\left( {1 - {x^{m - n + 1}}} \right)} \over {\left( {1 - x} \right)\left( {1 - {x^2}} \right).........\left( {1 - {x^n}} \right)}}\)
where $...

Sketch the solution set of the following system of inequalities:
\(${x^2} + {y^2} - 2x \ge 0;\,\,3x - y - 12 \le 0;\,\,y - x \le 0;\,\,y \ge 0.\)$

Find all integers \(x\) for which \(\left( {5x - 1} \right) < {\left( {x + 1} \right)^2} < \left( {7x - 3} \right).\)

The area of a triangle is \(5\). Two of its vertices are \(A\left( {2,1} \right)\) and \(B\left( {3, - 2} \right)\). The third vertex \(C\) lies on \(y = x + 3\). Find \(C\).

One side of rectangle lies along the line \(4x + 7y + 5 = 0.\) Two of its vertices are \((-3, 1)\) and \((1, 1).\) Find the equations of the other three sides.

A straight line segment of length \(\ell\) moves with its ends on two mutually perpendicular lines. Find the locus of the point which divides the line segment in the ratio \(1 : 2\)

If \(\tan \alpha = {m \over {m + 1}}\,\) and \(\tan \beta = {2 \over {2m + 1}},\) find the possible values of \(\left( {\alpha + \beta } \right).\)

A horizontal uniform rope of length L, resting on a frictionless horizontal surface, is pulled at one end by force F. What is the tension in the rope at a distance \(l\) from the end where the force is applied?

A car accelerates from rest at a constant rate \(\alpha\) for some time after which it decelerates at a constant rate \(\beta\) to come to rest. If the total time lapse is t seconds, evaluate.
(i) maximum velocity reached, and
(ii) the to...