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Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy sin^-1 (3x / 5) + sin^-1 (4x / 5) = sin^-1 x =

a. 1 b. 2 c. 3 d. 0 Answer: (c) Taking sin on both sides, (3x / 5) √[1 - (16x2 / 25)] + (4x / 5) √[1 - (9x2 / 25)] = x 3x √(25... View Article

The maximum value of f (x) = [latex]\begin{vmatrix} sin^2x &1+cos^2x &cos2x \\ 1+sin^2x&cos^2x &cos2x \\ sin^2x&cos^2x &sin2x \end{vmatrix}[/latex], x ∈ R is:

a. √7 b. √5 c. 5 d. 3 / 4 Answer: (b) C1 → C1 + C2 R1 → R1 – R2 = (–1)[2sin2x – cos2x] = cos2x... View Article

Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let α be the number of triangles having these points from different sides as vertices and β be the number of quadrilaterals having these points from different sides as vertices. Then (β – α) is equal to:

a. 1890 b. 795 c. 717 d. 1173 Answer: (c) α = 6C1 7C1 9C1 + 5C1 7C1 9C1 + 5C1 6C1 9C1 + 5C1 6C1 7C1 = 378 + 315 + 270 + 210 = 1173 β =... View Article

Let C be the locus of the mirror image of a point on the parabola y² = 4x with respect to the line y = x. Then the equation of tangent to C at P(2, 1) is:

a. 2x + y = 5 b. x + 2y = 4 c. x + 3y = 5 d. x – y = 1 Answer: (d) Image of y2 = 4x w.r.t. y = x is x2 = 4y Tangent from (2, 1) xx1 = 2(y... View Article

[latex]\lim_{x \rightarrow 0} \frac{\int_0^2(sin \sqrt{t})dt}{x^3}[/latex] is equal to

a. 2/3b. 0c. 1/15d. 3/2Answer: (a)... View Article

The distance of the point (1, 1, 9) from the point of intersection of the line (x – 3)/1 = (y – 4)/2 = (z – 5)/2 and the plane x + y + z = 17 is:

a. √38b. 19√2c. 2√19d. 38Answer: (a)(x - 3)/1 = (y - 4)/2 = (z - 5)/2 = λx = λ + 3, y = 2λ+ 4, z =... View Article

If f:R→ R is a function defined by f(x) = [x – 1] cos ((2x – 1)/2) , where [.] denotes the greatest integer function, then f is:

a. discontinuous only at x = 1b. discontinuous at all integral values of x except at x = 1c. continuous only at x = 1d. continuous for every real... View Article

Let f :R→R be defined as f(x) = 2x-1 and g:R – {1} →R be defined as g(x) = (x-½)/(x-1). Then the composition function f(g(x)) is:

a. Both one-one and ontob. onto but not one-onec. Neither one-one nor ontod. one-one but not ontoAnswer: (d)f(g(x)) = 2g(x) - 1= 2(x - 1/2)/(x -... View Article

The statement among the following that is a tautology is:

a. A∧(A∨B)b. B→[A∧(A→B)]c. A∨(A∧B)d. [A∧(A→B)]→BAnswer: (d)A∧ (~ A∨B)→B=... View Article

The equation of the plane passing through the point (1,2,–3) and perpendicular to the planes 3x + y – 2z = 5 and 2x – 5y – z = 7, is:

a. 3x – 10y – 2z + 11 = 0b. 6x – 5y – 2z – 2 = 0c. 11x + y + 17z + 38 = 0d. 6x – 5y + 2z + 10 = 0Answer: (c)... View Article

The locus of the mid-point of the line segment joining the focus of the parabola y²= 4ax to a moving point of the parabola, is another parabola whose directrix is:

a. x = ab. x = 0c. x = -a/2d. x = a/2Answer: (b)h = (at2+a)/2, k = (2at+0)/2⇒ t2 =(2h-a)/a and t=k/a⇒ k2/a2 =(2h-a)/a⇒ Locus of... View Article

The photoelectric current from Na (Work function, w₀ = 2.3 eV) is stopped by the output voltage of the cell

The photoelectric current from Na (Work function, w0 = 2.3 eV) is stopped by the output voltage of the cell Pt(s) H2 (g, 1 Bar) HCl (aq. pH =1)... View Article

The total number of monohalogenated organic products

The total number of monohalogenated organic products in the following (including stereoisomers) reaction is ______. Solution: 8... View Article

An element with molar mass 2.7 10^-2 kg mol^-1 forms a cubic unit cell with edge length 405 pm. If its density is 2.7 10³ kg m^-3, the radius of the element is approximately ______ 10^-12 m (to the nearest integer).

Solution:Density = [Z * GMM] / [NA * a3]2.7 × 103 = [Z * 2.7 * 10-2] / [6.023 * 1023 * (405 * 10-12)]3Z = 6.023 × 405 × 405 × 405 × 1023–36+3+2Z... View Article

The volume strength of 8.9 M H2O2 solution calculated at 273 K and 1 atm is ______. (R = 0.0821 L atm K^-1 mol^-1) (rounded off to the nearest integer)

Volume strength = [8.9 / 2] * [0.821 * 273] / 1= 99.73= 100Solution: 100... View Article

The mole fraction of glucose (C₆H₁₂O₆) in an aqueous binary solution is 0.1. The mass percentage of water in it, to the nearest integer, is

Solution: 47 %xGlucose = 0.1mass% of glucose = {[0.1* 180] / [0.1 * 180 + 0.9 * 18]} * 100= 1800 / [18 + 16.2]= [1800 / 34.2] %= 52.63%= 53%The... View Article