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If [latex]\lim_{x \rightarrow 0} \frac{ax-(e^{4x}-1)}{ax(e^{4x}-1)}[/latex] exists and is equal to b, then the value of a – 2b is

\(\begin{array}{l}\lim_{x \rightarrow 0} \frac{ax-(e^{4x}-1)}{ax(e^{4x}-1)}\end{array} \) Applying L’Hospital Rule \(\begin{array}{l}\lim_{x \rightarrow 0} \frac{a-4e^{4x}}{a(e^{4x}-1)+ax(4e^{4x})}\end{array} \) So for limit to exist,a=4 Applying L’Hospital... View Article

If 0 < x, y < π and cos x + cos y – cos (x + y) = 3/2, then sin x + cos y is equal to:

a. (1+√3)/2 b. (1–√3)/2 c. √3/2 d. 1/2 Answer: (a)

cosec[2 cot^–1( 5) + cos^–1(4/5) ] is equal to

a. 75/56 b. 65/56 c. 56/33 d. 65/33 Answer: (b) cosec[2 cot–1( 5) + cos–1(4/5) ]

The shortest distance between the line x – y = 1 and the curve x² = 2y is:

a. 1/2 b. 0 c. 1/(2√2) d. 1/√2 Answer: (c) Shortest distance must be along common normal m1 (slope of... View Article

Let A be a 3 × 3 matrix with det(A) = 4. Let R_i denote the i^th row of A. If a matrix B is obtained by performing the operation R₂→ 2R₂ + 5R₃ on 2A, then det(B) is equal to:

a. 64 b. 16 c. 80 d. 128 Answer: (a)

Let α and β be the roots of x² – 6x – 2 = 0. If a_n = αⁿ – βⁿ for n ≥ 1, then the value of (a₁₀–2a₈)/(3a₉ ) is

a. 4 b. 1 c. 2 d. 3 Answer: (c) α and β be the roots of x2–6x–2=0. So, α2–6α–2=0⇒α2–2=6α... View Article

A function f(x) is given by f(x) = 5^x/(5^x+5), then the sum of the series:f(1/20) + f(2/20) + f(3/20) + …… + f(39/20) is equal to:

a. 19/2 b. 49/2 c. 39/2 d. 29/2 Answer: (c) f(x) = 5x/(5 x + 5)….(i) ⇒ f(2–x) = 52–x/(52–x... View Article

The minimum value of [latex]f(x) = a^{a^{x}} + a^{1-a^{x}}[/latex] where a, x ∈ R and a > 0, is equal to

a. a + 1/a b. a + 1 c. 2a d. 2√a Answer: (d) Using AM ≥ GM inequality, we... View Article

The following system of linear equations: 2x + 3y + 2z = 9, 3x + 2y + 2z = 9 and x – y + 4z = 8

a. does not have any solution b. has a unique solution c. has a solution (α, β, γ) satisfying α... View Article

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: Dipole-dipole interactions are the only non-covalent interactions, resulting in hydrogen bond formation Reason R: Fluorine is the most... View Article

Find A, B and C in the following reaction:

Find A, B and C in the following reaction: NH3 + A + CO2 → (NH4)2CO3 (NH4)2CO3 + H2O... View Article

On treating a compound with warm dil. H₂SO₄, gas X is evolved which turns K₂Cr₂O₇ paper acidified with dil. H₂SO₄ to a green compound Y. X and Y respectively are:

a. X = SO2, Y = Cr2 (SO4)3 b. X = SO2, Y = Cr2O3 c. X = SO3, Y... View Article

The presence of ozone in troposphere:

The presence of ozone in troposphere: a. Generates photochemical smog b. Protects us from the UV radiation c. Protects us... View Article

Which of the following vitamin is helpful in delaying the blood clotting?

a. Vitamin B b. Vitamin C c. Vitamin K d. Vitamin E Answer: (c) Vitamin K is used by the... View Article

Given below are two statements: Statement I: A mixture of chloroform and aniline can be separated by simple distillation Statement II: When separating aniline from a mixture of aniline and water by steam distillation aniline boils below its boiling point

In the light of the above statements, choose the most appropriate answer from the options given below: a. Statement I... View Article

Which of the following is ‘a’ FALSE statement?

Which of the following is ‘a’ FALSE statement? a. Carius tube used in the estimation of sulphur in an organic... View Article

Statement about heavy water are given below

A. Heavy water is used in exchange reactions for the study of reaction mechanisms B. Heavy water is prepared by... View Article

For the given reaction: What is A?

For the given reaction: What is A? Answer: (c) Hence the answer is c

An amine on reaction with benzenesulfonyl chloride produces a compound insoluble in alkaline solution. This amine can be prepared by ammonolysis of ethyl chloride. The correct structure of amine is:

An amine on reaction with benzenesulfonyl chloride produces a compound insoluble in alkaline solution. This amine can be prepared by... View Article

Given below are two statement: Statement I: o-Nitrophenol is steam volatile due to intramolecular hydrogen bonding Statement II: o-Nitrophenol has high melting point due to hydrogen bonding.

In the light of the above statements, choose the most appropriate answer from the options given below: a. Both Statement... View Article

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