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If ∑⁹_i=1 (x_i-5) = 9 and ∑⁹_i=1 (x_i-5)² = 45, then the standard deviation of the 9 items x₁, x₂,…x₉ is:

a) 2b) 3c) 9d) 4Solution:Variance = (1/n) Ʃi=1n xi2-[(1/n) Ʃi=1n xi]2= (1/9)×45-((1/9)×9)2= 5-1= 4S.D = √4 = 2Answer: (a)... View Article

For each t ∈ R, let [t] be the greatest integer less than or equal to t.

For each t ∈ R, let [t] be the greatest integer less than or equal to t. Then a) is equal to 120b) does not exist (in R)c) is equal to 0d) is... View Article

Let g(x) = cos x², f(x) = √x and α, β(α< β) be the roots

Let g(x) = cos x2, f(x) = √x and α, β(α< β) be the roots of the quadratic equation 18x2-9πx+π2 = 0. Then the area... View Article

The value of [latex]\int_{\frac{-\pi }{2}}^{\frac{\pi }{2}}\frac{\sin ^{2}x}{1+2^{x}}dx[/latex] is:

a) 4π b) π/4 c) π/8 d) π/2 Solution: Answer: (b)... View Article

A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is

a) 3x+2y = xyb) 3x+2y = 6xyc) 3x+2y = 6d) 2x+3y = xySolution: line y-3 = m(x-2)y-3 = mx-2mP = ((2m-3)/m, 0)Q = (0, 3-2m)Let R(α, β)So... View Article

Let a₁, a₂, a₃,..a₄₉ be in A.P

Let a1, a2, a3,..a49 be in A.P such that Ʃk=012 a4k+1 = 416 and a9+a43 = 66. If a12+a22+…+a172 = 140m, then m is equal to:a) 34b) 33c) 66d)... View Article

The sum of the coefficients of all odd degree terms in the expansion of (x+√(x³-1))⁵+(x-√(x³-1))⁵, (x>) is

a) 1b) 2c) -1d) 0Solution:(a+b)n+(a-b)n = 2(nC0an + nC2an-2b2+ nC4an-4b4…)(x+√(x3-1))5+(x-√(x3-1))5, (x>)=... View Article

If [latex]\begin{vmatrix} x-4 & 2x &2x \\ 2x& x-4 & 2x\\ 2x& 2x & x-4 \end{vmatrix}[/latex] = (A+Bx)(x-A)² then the ordered pair (A, B) is equal to

a) (-4, 5)b) (4, 5)c) (-4, -5)d) (-4, 3)Solution: = (5x-4)(-1)(x+4)(x-4-2x)= -1(5x-4)(x+4)(-x-4)= (5x-4)(x+4)(x+4)2 = (-4+5x)(x-(-4))2 Hence A =... View Article

Tangent and normal are drawn at P(16, 16) on the parabola y² = 16x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and ∠CPB = θ, then a value of tan θ is

a) 3b) 4/3c) 1/2d) 2Solution: Tangent and normal are drawn at P (16, 16) on y2 = 16xy2 = 16xSlope of tangent dy/dx = 16/2y= 8/y(dy/dx)(16,16) =... View Article

The Boolean expression : ~(p ˅q) ˅(~p˄q) is equivalent to:

The Boolean expression : ~(p ˅q) ˅(~p˄q) is equivalent to:a) qb) ~qc) ~pd) pSolution:p(1)q(2)~p(3)(p ˅q)(4)~(p ˅q)(5)~p˄q(6)~(p ˅q)... View Article

The distance between an object and a screen is 100 cm.

The distance between an object and a screen is 100 cm. A lens can produce real image of the object on the screen for two different positions... View Article

Orange light of wavelength 6000×10^–10 m illuminates a single slit of width 0.6 × 10^–4 m.

Orange light of wavelength 6000×10–10 m illuminates a single slit of width 0.6 × 10–4 m. The maximum possible number of... View Article

The change in the magnitude of the volume of an ideal gas

The change in the magnitude of the volume of an ideal gas when a small additional pressure ∆P is applied at a constant temperature, is the same... View Article

Four resistances 40 Ω, 60 Ω, 90 Ω and 110 Ω make the arms of a quadrilateral ABCD.

Four resistances 40 Ω, 60 Ω, 90 Ω and 110 Ω make the arms of a quadrilateral ABCD. Across AC is a battery of emf 40 V and... View Article

SIPRI Yearbook 2022 – Latest SIPRI Report

SIPRI Yearbook is an annual publication on the current state of armament, disarmament and international security released by the Swedish think... View Article

Sant Kabir Das Jayanti

Sant Kabir Das was one among the foremost influential saints. He was a 15th-century Indian mystic poet, whose writings influenced the Bhakti... View Article

01 July 2021: The Hindu Analysis

UPSC Current Affairs Preparation: The Hindu Analysis Watch expert analysis of ‘The Hindu’ dated 01st Jul 2021. Important news and views to boost... View Article

01 July 2021: UPSC Exam Comprehensive News Analysis

CNA 01st July 2021:- Download PDF Here TABLE OF CONTENTS A. GS 1 Related SOCIAL ISSUES 1. Two arrested in Dwarka honour killing incident B. GS 2... View Article

30 June 2021: PIB Summary & Analysis

June 30th, 2021, PIB:- Download PDF Here TABLE OF CONTENTS 1. Goods and Services Tax (GST) 2. MoU approved by the Cabinet 3. Atmanirbhar Bharat... View Article