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If volume of parallelepiped whose coterminous edges are given by….

If volume of parallelepiped whose coterminous edges are given by \(\begin{array}{l}\vec{u} = \hat{i}+\hat{j}+\lambda \hat{k}\end{array} \), \(\begin{array}{l}\hat{v}==\hat{i}+\hat{j}+3\hat{k} \end{array} \) and \(\begin{array}{l}\vec{w}=2\hat{i}+\hat{j}+\hat{k}\end{array}... View Article

Let A and B be two independent events such that P(A) =1/3 and P(B) = 1/6. Then, which of the following is TRUE ?

a. P(A/(A ∪ B)) = 1/4 b. P(A/B′) = 1/3 c. P(A/B) = 2/3 d. P(A′/B′) = 1/3 Solution: If... View Article

Find the value of λf(pi/3)

If \(\begin{array}{l}\int \frac{cos\ x}{sin^3x(1+sin^6x)^{2/3}}dx\end{array} \) = f(x)(1+sin6x)1/ +c, where c is a constant of intergartion, then λf(π/3) is equal to:... View Article

If the equation, x² + x + 45 = 0 ( ∈ ) has conjugate complex roots and they satisfy | + 1| = 2√10 , then

a. b2 + b = 12 b. b2 − b = 42 c. b2 − b = 30 d. 2... View Article

Let f(x) = (sin(tan⁻¹ x) + sin(cot⁻¹ x) )² − 1, where |x| > 1.If [latex]frac{dy}{dx}=frac{1}{2}frac{d}{dx}(sin^{-1}f(x)))[/latex] and y(√3) = π/6, then y(-(√3) is equal to

a. π/3 b. 2π/3 c. – π/6 d. 5 π/6 Solution: f(x) = [sin(tan−1 x) + sin(cot−1 x) ]2 -1... View Article

[latex]\lim_{x\rightarrow 0}\left ( \frac{3x^{2}+2}{7x^{2}+2} \right )^{1/x^{2}}[/latex] is equal to

a. e b. 1/e2 c. 1/e d. e2 Solution: Let,L=\(\begin{array}{l}\lim_{x\rightarrow0}\left(\frac{3x^{2}+2}{7x^{2}+2}\right)^{1/x^{2}}\end{array} \) (intermediate from 1∞) \(\begin{array}{l}L = e^{\lim_{x\rightarrow 0}\frac{1}{x^{2}}\left ( \frac{3x^{2}+2}{7x^{2}+2}... View Article

The inversion function of f(x) = [latex]frac{8^{2x}-8^{-2x}}{8^{2x}+8^{-2x}},x epsilon (-1,1), is[/latex]

Solution: \(\begin{array}{l}f(x) = \frac{8^{2x}-8^{-2x}}{8^{2x}+8^{-2x}} = \frac{8^{4x-1}}{8^{4x+1}}\end{array} \) Put, y = \(\begin{array}{l}\frac{8^{4x-1}}{8^{4x+1}}\end{array} \) Applying componendo-dividendo on both sides \(\begin{array}{l}\frac{y+1}{y-1}=\frac{2\times 8^{4x}}{-2}\end{array} \)... View Article

For > 0, let the curves ₁: ² = x and ₂ ∶ x² = intersect at origin and a point .

For a > 0, let the curves C1: y2 = ax and C2 ∶ x2 = ay intersect at origin... View Article

The locus of a point which divides the line segment joining the point (0, −1) and a point on the parabola, x² = 4 , internally in the ratio 1: 2, is

a. 9x2 − 12y = 8 b. 4x2 − 3y = 2 c. x2 − 3y = 2 d. 9x2... View Article

Let : → be such that for all x ∈ , (2^1+x + 2^1−x), f(x) and (3^x + 3^−x) are in A.P., then the minimum value of f(x) is:

a. 0 b. 4 c. 3 d. 2 Solution: 21−x + 21+x, (x), 3x + 3−x are in A.P. \(\begin{array}{l}f(x)=\frac{3^{x}+3^{-x}+2^{1+x}+2^{1-x}}{2}=\frac{3^{x}+3^{-x}}{2}+\frac{2^{1+x}+2^{1-x}}{2}\end{array}... View Article

Which of the following is a tautology?

a. ( ∧ ( ⟶ )) ⟶ b. ∧ ( ∨ ) c. ( → ( ∧ ( → ))... View Article

If , and are the greatest values of ¹⁹ , ²⁰ , ²¹ respectively, then:

a. (a/11) = (b/22) = (c/42) b. (a/10) = (b/11) = (c/42) c. (a/11) = (b/22) = (c/21) d. (a/10)... View Article

Let y=(x) be a solution of the differential equation. [latex]sqrt{1-x^{2}}frac{dy}{dx}+sqrt{1-y^{2}}=0,left | x right |< 1[/latex].If y (1/2) = √3/2, then y(-1/√2) is equal to

a. -1/√2 b. -√3/2 c. 1/√2 d. √3/2 Solution: \(\begin{array}{l}\sqrt{1-x^{2}}\frac{dy}{dx}+\sqrt{1-y^{2}}=0,\end{array} \) \(\begin{array}{l}\frac{dy}{\sqrt{1-y^{2}}}+ \frac{dx}{\sqrt{1-x^{2}}}=0\end{array} \) sin-1y + sin-1x = c If... View Article

For which of the following ordered pairs ( , ), the system of linear equations

For which of the following ordered pairs (μ, δ), the system of linear equations x + 2y + 3z =... View Article

A bakelite beaker has volume capacity of 500 cc at 30°C. When it is partially filled with V_m volume (at 30°C) of mercury, it is found that the unfilled volume of the beaker remains constant as temperature is varied. If ϒ_(beaker)= 6 × 10–6 °C^–1 and ϒ _(mercury)= 1.5 × 10^–4 °C^–1, where ϒ is the coefficient of volume expansion, then V_m (in cc) is close to

Solution: Answer. 20 For the volume to remain constant at all temperature Exp. of the liquid = Exp. of solid... View Article

When a long glass capillary tube of radius 0.015 cm is dipped in a liquid, the liquid rises to a height of 15 cm within it. If the contact angle between the liquid and glass to close to 0°, the surface tension of the liquid, in milliNewton m^–1, is [ρ_(liquid) =900 kgm^–3 , g = 10 ms^–2] (Give answer in closes integer)

Solution: Answer. 101 r = 0.015 T = 101

A cricket ball of mass 0.15 kg is thrown vertically up by a bowling machine so that it rises to a maximum height of 20 m after leaving the machine. If the part pushing the ball applies a constant force F on the ball and moves horizotally a distance of 0.2 m while launching the ball, the value of F(in N) is (g = 10 ms^–2)

Solution: F x 0.2 = mgh F x 0.2 = 0.15 x 10 x 20 F = (3/2)x [(10 x20)/2]... View Article

An observer can see through a small hole on the side of a jar (radius 15 cm) at a point at height of 15 cm from the bottom (see figure). The hole is at a height of 45 cm. When the jar is filled with a liquid up to a height of 30 cm the same observer can see the edge at the bottom of the jar. If the refractive index of the liquid is N/100, where N is an integer, the value of N is _____.

Solution: \(\begin{array}{l}\mu sini=1sin45\end{array} \), \(\begin{array}{l}\mu \frac{15}{\sqrt{1125}}=\frac{1}{\sqrt{2}}\end{array} \), \(\begin{array}{l}\mu =\frac{\sqrt{1125}}{15\sqrt{2}}\end{array} \), \(\begin{array}{l}\mu =\frac{\sqrt{1125}}{\sqrt{450}}\end{array} \) μ=1.58 μ = N/100 = 1.58 N... View Article

A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about its axis at 5 revolutions per minute (rpm).

A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating... View Article

In the circuit shown in the figure, the total charge is 750 μC and the voltage across capacitor C₂ is 20 V. Then the charge on capacitor C₂ is:

1) 650 μC 2) 450 μC 3) 590 μC 4) 160 μC Solution: Answer. 3 Potential difference across C2 &... View Article

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