List of practice Questions

Let S be the sum of all solutions (in radians) of the equation \(\sin^4\theta + \cos^4\theta - \sin\theta \cos\theta = 0\) in [0, 4\(\pi\)]. Then \(\frac{8S}{\pi}\) is equal to _________.

An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 candidates is 15. If \(\mu\) is the average marks of girls and \(\sigma^2\) is the variance of marks of 50 candidates, then \(\mu + \sigma^2\) is equal to _________.

The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m - n = 0 and mn + nl + lm = 0, is :

The equation of the plane passing through the line of intersection of the planes \( \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1 \) and \( \vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) + 4 = 0 \) and parallel to the x-axis is :

The Boolean expression \((p \land q) \Rightarrow ((r \land q) \land p)\) is equivalent to :

The area of the region bounded by the parabola \((y-2)^2 = (x-1)\), the tangent to it at the point whose ordinate is 3 and the x-axis is :

If 0 \(<\) x \(<\) 1 and y = \(\frac{1}{2}x^2 + \frac{2}{3}x^3 + \frac{3}{4}x^4 + ...\), then the value of e\(^{1+y}\) at x = \(\frac{1}{2}\) is :

If \( \lim_{x \to \infty} (\sqrt{x^2 - x + 1} - ax) = b \), then the ordered pair (a, b) is :

A box open from top is made from a rectangular sheet of dimension a \(\times\) b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to:

The set of all values of k \(>\) -1, for which the equation
(3x\(^2\)+4x+3)\(^2\) - (k+1)(3x\(^2\)+4x+3)(3x\(^2\)+4x+2) + k(3x\(^2\)+4x+2)\(^2\) = 0 has real roots, is :

Let [\(\lambda\)] be the greatest integer less than or equal to \(\lambda\). The set of all values of \(\lambda\) for which the system of linear equations x+y+z=4, 3x+2y+5z=3, 9x + 4y + (28 + [\(\lambda\)])z = [\(\lambda\)] has a solution is :

The addition of dilute NaOH to Cr\(^{3+}\) salt solution will give :

100 g of propane is completely reacted with 1000 g of oxygen. The mole fraction of carbon dioxide in the resulting mixture is \(x \times 10^{-2}\). The value of x is _________. (Nearest integer)
[Atomic weight : H=1.008; C=12.00; O=16.00]

The number of photons emitted by a monochromatic (single frequency) infrared range finder of power 1 mW and wavelength of 1000 nm, in 0.1 second is \(x \times 10^{13}\). The value of x is _________. (Nearest integer)
(h = 6.63 \(\times\) 10\(^{-34}\) Js, c=3.00 \(\times\) 10\(^8\) ms\(^{-1}\))

When 5.1 g of solid NH\(_4\)HS is introduced into a two litre evacuated flask at 27\(^\circ\)C, 20% of the solid decomposes into gaseous ammonia and hydrogen sulphide. The K\(_p\) for the reaction at 27\(^\circ\)C is \(x \times 10^{-2}\). The value of x is _________. (Integer answer)
[Given R = 0.082 L atm K\(^{-1}\) mol\(^{-1}\)]

Given below are two statements:
Statement I : Ethyl pent-4-yn-oate on reaction with CH\(_3\)MgBr gives a 3\(^\circ\)-alcohol.
Statement II : In this reaction one mole of ethyl pent-4-yn-oate utilizes two moles of CH\(_3\)MgBr.
In the light of the above statements, choose the most appropriate answer from the options given below :

The major product (A) formed in the reaction given below is :
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Potassium permanganate on heating at 513 K gives a product which is :

Which one of the following is used to remove most of plutonium from spent nuclear fuel ?

In stratosphere most of the ozone formation is assisted by :

Which one of the following tests used for the identification of functional groups in organic compounds does not use copper reagent ?

Lyophilic sols are more stable than lyophobic sols because,

The correct order of ionic radii for the ions, P\(^{3-}\), S\(^{2-}\), Ca\(^{2+}\), K\(^{+}\), Cl\(^{-}\) is:

The oxide that gives H\(_2\)O\(_2\) most readily on treatment with H\(_2\)O is :

Choose the correct statement from the following: