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When neopentyl alcohol is heated with an acid

When neopentyl alcohol is heated with an acid, it is slowly converted into an 85:15 mixture of alkenes A and B, respectively. What are these... View Article

The ionic radii of O^2–, F^–, Na⁺ and Mg²⁺ are in the order:

1) F– > O2– > Na+ > Mg2+ 2) Mg2+ > Na+ > F– > O2– 3) O2– > F– > Na+ > Mg2+ 4) O2– > F– > Mg2+ > Na+... View Article

On heating, lead(II) nitrate gives a brown gas (A). The gas (A) on cooling changes to a colourless solid/liquid (B). (B) on heating with NO changes to a blue solid (C). The oxidation number of nitrogen in solid (C) is:

1) +3 2) +4 3) +2 4) +5 Answer: (1) Pb(NO3)2 → PbO + NO2 + O2 (A) Brown gas 2NO2 → N2O4 [on cooling] (B) colourless solid NO2 + NO à... View Article

The IUPAC name of the following compound is:

The IUPAC name of the following compound is: 1) 3-Bromo-5-methyl cyclopentane carboxylic acid 2) 4-Bromo-2-methyl cyclopentane carboxylic acid... View Article

For real numbers alpha, beta, gamma and delta the value of 10 (α + βγ + δ) is equal to

For real numbers α, β, γ and δ, if where C is an arbitrary constant, then the value of 10 (α + βγ +... View Article

Let [latex]A=\begin{bmatrix} a_{1}\\ a_{2} \end{bmatrix}[/latex] and [latex]B=\begin{bmatrix} b_{1}\\ b_{2} \end{bmatrix}[/latex] be two 2 × 1 matrices with real entries such that A = XB, where X =[latex]\frac{1}{\sqrt3}\begin{bmatrix} 1 &-1 \\ 1&k \end{bmatrix}[/latex], and k ∈ R. If (a₁² + a₂²) = (2 / 3) (b₁² + b₂²) and (k² + 1) b₂²≠ – 2b₁b₂, then the value of k is

Answer: (1) XB = A (1 / √3) b1 - b2 = √3a1 3a12 = b12 + b22 - 2b1b2 b1 + kb2 = √3a2 3a22 = b12 + k2b22 + 2kb1b2 Adding both... View Article

Let 1 / 16, a and b be in G.P. and 1/a, 1/b, 6 be in A.P., where a, b, > 0. Then 72 (a + b) is equal to

Answer: (14) a2 = b / 16 and 2 / b = (1 / a) + 6 Solving, we get a = 1 / 12 or a = - 1 / 4 [rejected] If a = 1 / 12 ⇒ b = 1 / 9 72 (a + b) =... View Article

If the distance of the point (1, – 2, 3) from the plane x + 2y – 3z + 10 = 0 measured parallel to the line, [x – 1] / 3 = [2 – y] / m = [z + 3] / 1 is √7 / 2, then the value of |m| is equal to

Answer: (2) [x - 1] / 3 = [y + 2] / - m = [z - 3] / 1 = λ Point Q (3λ + 1, - mλ - 2, λ + 3) lies on the plane.... View Article

Let f : R → R and g : R → R be defined as f (x) = [latex]left{begin{matrix} x+a, &x<0 \ |x-1|, &xgeq 0 end{matrix}right.[/latex] and g (x) = [latex]left{begin{matrix} x+1, &x<0 \ (x-1)^2+b, &xgeq 0 end{matrix}right.[/latex] where a, b are non-negative real numbers. If (gof) (x) is continuous for all x ∈ R, then a + b is equal to

Given (gof) (x) is continuous for all x ∈ R Answer: (1) Given g(f(x)) is continuous. So, at x = –a & at x = 0 1 = b + 1 &... View Article

Let n be a positive integer. Let A = ∑_k=0ⁿ (-1)^kⁿC_k [(1 / 2)^k + (3 / 4)^k + (7 / 8)^k + (15 / 16)^k + (31 / 32)^k]. If 63A = 1 – (1 / 2³⁰), then n is equal to

Answer: (6) A = (1 / 2)n + (1 / 4)n + (1 / 8)n + (1 / 16)n + (1 / 32)n = (1 / 2n) + (1 / 22n) + (1 / 23n) + (1 / 24n) + (1 / 25n) = (1 / 2n)... View Article

Let [latex]S_{n}(x)=log_{a^{\frac{1}{2}}}x + log_{a^{\frac{1}{3}}}x + log_{a^{\frac{1}{6}}}x + log_{a^{\frac{1}{11}}}x + log_{a^{\frac{1}{18}}}x + log_{a^{\frac{1}{27}}}x +…[/latex] up to n-terms, where a >1. If S₂₄(x) = 1093 and S₁₂(2x) = 265, then value of a is equal to

Answer: (16) Sn (x) = 2 loga x + 3 loga x + 6 loga x + 11 loga x + …….. Sn (x) = loga x (2 + 3 + 6 + 11 + …… ) Sr... View Article

Consider the statistics of two sets of observations as follows:

Consider the statistics of two sets of observations as follows: Size Mean Variance Observation I 10 2 2 Observation... View Article

In ΔABC, the lengths of sides AC and AB are 12 cm and 5 cm, respectively. If the area of ΔABC is 30 cm² and R and r are respectively the radii of circumcircle and incircle of ΔABC, then the value of 2R + r (in cm) is equal to

Answer: (15) Area = (1 / 2) * 5 * 12 * sin θ = 30 sin θ = 1 ⇒ θ = π / 2 ABC is right-angled at A. r = (s... View Article

Let [latex]vec{c}[/latex] be a vector perpendicular to the vectors a = i + j – k and b = i + 2j + k. If c . (i + j + 3k) = 8, then the value of c . (a x b) is equal to

Answer: (28)... View Article

Let A = {2, 3, 4, 5, ……, 30} and ‘⩭’ be an equivalence relation on A×A, defined by (a, b) ⩭ (c, d), if and only if ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) is equal to :

a. 7 b. 5 c. 6 d. 8 Answer: (a) a = bc (a, b) R (4, 3) ⇒ 3a = 4b a = (4 / 3) b b must be multiple of 3 b can be 3, 6, 9,... View Article

If the points of intersections of the ellipse (x² / 16) + (y² / b²) = 1 and the circle x² + y² = 4b, b > 4 lie on the curve y² = 3x², then b is equal to :

a. 5 b. 6 c. 12 d. 10 Answer: (c) (x2 / 16) + (y2 / b2) = 1 …(1) x2 + y2 = 4b …(2) y2 = 3x2 …(3) From eqns. (2) and... View Article

Let P(x) = x² + bx + c be a quadratic polynomial with real coefficients such that ∫₀¹ P (x) dx and P(x) leaves remainder 5 when it is divided by (x – 2). Then the value of 9 (b+c) is equal to :

a. 7 b. 11 c. 15 d. 9 Answer: (a) (x – 2) Q (x) + 5 = x2 + bx + c Put x = 2 5 = 2b + c + 4 …(1) ∫01 (x2 + bx + c) dx = 1... View Article

Let a = i + 2j – 3k and b = 2i – 3j + 5k. If r x a = b x r, r . (ɑi + 2j + k) = 3 and r . (2i + 5j – ɑk) = – 1, ɑ ∈ R, then the value of ɑ + |r|² =

a. 11 b. 15 c. 9 d. 13 Answer: (b) r x a = - r x b r x (a + b) = 0 (a + b = 3i - j + 2k) r || (a + b) r = λ (a + b) ∵ r . (2i + 5j -... View Article

Let α ∈ R be such that the function [latex]f(x) = \left\{\begin{matrix} \frac{cos^{-1}(1-\left \{ x \right \}^{2})sin^{-1}(1-\left \{ x \right \})}{\left \{ x \right \}-\left \{ x \right \}^{3}} & x\neq 0\\ \alpha & x=0 \end{matrix}\right.[/latex] is continuous at x = 0, where {x} = x – [x], [x] is the greatest integer less than or equal to x. Then:

1) α = π/4 2) No such α exists 3) α = 0 4) α = π/√2 Solution: RHL = = = (L'Hospital's rule) = LHL =... View Article

Let C₁ be the curve obtained by the solution of the differential equation 2xy (dy / dx) = y² – x², x > 0. Let curve C₂ be the solution of 2xy / (x² – y²) = dy / dx. If both the curves pass through (1, 1), then the area enclosed by the curves C₁ and C₂ is equal to :

a. (π / 2) - 1 b. (π / 4) + 1 c. π – 1 d. π + 1 Answer: (a) dy / dx = [y2 - x2] / 2xy Put y = vx v + x (dv /... View Article