Mathematics Tes...

If \(\underset{{x \rightarrow 1}}{\lim} \frac{x^{4}-1}{x-1}=\underset{{x \rightarrow k}}{\lim} \frac{x^{3}-k^{3}}{x^{2}-k^{2}}\), where \(k \neq 0\), then what is the value of \(k\)?

A student appears for tests I, II and III. The student is considered successful if the passes in tests I, II or I, III or all the three. The probabilities of the student passing in test I, II and III are m, n and \(\frac{1}{2}\) respectively. If the probability of the student to be successful is \(\frac{1}{2}\), then which one of the following is correct?

Find the area bounded by the curve \(y=\sqrt{x}\) and the line \(2 y=x\).

If \(\frac{\mathrm{a}^{\mathrm{n}+1}+\mathrm{b}^{\mathrm{n}+1}}{\mathrm{a}^{\mathrm{n}}+\mathrm{b}^{\mathrm{n}}}\) be the harmonic mean of \(\mathrm{a}\) and \(\mathrm{b}\) then value of \(\mathrm{n}\) is:

If \(f(x)=\sin ^{-1}\left[\frac{2^{x+1}}{1+4^{x}}\right]\), then \(f^{\prime}(0)=\)

A group of 80 candidates have their average height 148.5 cm with coefficient of variation 2.5%. What is the standard deviation of their height?

Find the general solution of the equation 4 sin 3x = 2?

Evaluate: \(\lim _{x \rightarrow \frac{\pi}{2}}(\sec x-\tan x)\)

For what value of α, the system of equation: x + y + z = 0, x – y + z = 0 and 2x + 3y + αz = 0 has infinitely many solutions.

Find the radius of the circle, \(5 x^{2}+5 y^{2}-20 x-6 y+15=0\)

Which region is described by the shade in the graph given below?

What is the general solution of \(\left(1+e^{x}\right) y d y=e^{x} d x\)?

Where \(\mathrm{c}\) is a constant of integration.

A merchant plans to sell two types of personal computers - a desktop model and a portable model, that will cost Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of the computers will not exceed 250 units. Determine the number of units of each type of computer which the merchant should stock to get the maximum profit, if he does not want to invest more than Rs. 70 lakhs and his profit on the desktop model is Rs. 4,500 and on the portable model is Rs. 5,000.

If \(\mathrm{A}+\mathrm{iB}=\frac{4+2 \mathrm{i}}{1-2 \mathrm{i}}\) where \(\mathrm{i}=\sqrt{-1}\), then what is the value of \(\mathrm{A} ?\)

Find the slope of the tangent to the curve \(x y^2-2 x=4\) at \(x=2\)