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If ∫₀^ae^x-[x] dx = 10e – 9, then the value of a is (where [.] is greatest integer function)

a. 9 + ln⁡ 2b. 10 + ln⁡ 2c. 10d. 9 Solution: Answer: (b)Let a = K + λ, 0 ≤ λ < 1∫0a e{x} dx = ∫0K e{x} dx +... View Article

The probability of selecting integers a ∈ [-5,30], such that x²+ 2(a+4)x – 5a + 64 >0 for all x∈ R is:

Answer: 1/2 x2+ 2(a+4)x - (5a - 64)>0 D<0 Therefore 4(a+4)2 + 4(5a - 64) < 0 ⇒ a2 + 13a - 48<0 ⇒ (a + 16)(a - 3)<0... View Article

If an invertible function f(x) is defined as f(x) = 3x – 2, g(x) is also an invertible function such that f^-1 (g^-1 (x) )= x – 2, then g(x) is

a. (x-8)/3b. (x+8)/3c. (x-3)/8d. (x+3)/8Solution: Answer: (b)f-1 (g-1(x) ) = x-2⇒ f(x-2) = g-1 (x)3(x - 2) - 2 = g-1(x)3x - 8 = g-1 (x)g-1... View Article

If z and ω are complex numbers such that |zω| = 1, arg⁡(z) – arg⁡(ω) = (3π )/2 then find arg [latex]left ( frac{1-2bar{z}omega }{1+3bar{z}omega } right )[/latex]

a. π/4b. -π/4c. 3π/4d. -3π/4Solution: Answer:= (1-2i)/(1+3i)= -½ - ½ iarg( -½ - ½ i) = -3π/4Hence... View Article

A continuous differentiable function f(x) is increasing in (-∞, 3/2) and decreasing in (3/2, ∞). Then x = 3/2 is

a. point of local maximab. point of local minimac. point of inflectiond. None of theseSolution: Answer: (a)A rough graph of f(x) can be drawn... View Article

Let y = mx+c, m>0 be the focal chord of y² = −64x which is tangent to (x+10)²+ y²= 4. Then the value of 4√2(m+c) is equal to

Answer: 34Focus of parabola is (−16,0)So, −16m + c=0⇒c = 16m …(i)Now slope form of tangent to the circle is given byy =... View Article

If the shortest distance between the lines [latex]r_{1}=alphahat{i}+2hat{j}+2hat{k}+lambda (hat{i}-2hat{j}+2hat{k})[/latex], λ∈R, α>0 and [latex]r_{2}=-4hat{i}-hat{k}+mu (3hat{i}-2hat{j}-2hat{k})[/latex] , µ∈R is 9, then the value of α is

a. 2b. 4c. 6d. √6Solution: Answer: (c)⇒ |(α + 4) ×8 + 16 + 12| = 108⇒ (α + 4) ×8 = 80 (since α... View Article

In ∆ABC, if AB = 5, ∠B=cos^-1 (3/5) and the radius of circumcircle of triangle is 5. Then the area of ∆ABC is

a. 6+8√3b. 3+4√3c. 3+8√3d. 6+4√3Solution: Answer: (a)cos ⁡B = ⅗⇒ sin⁡B = 4/5, R = 5b = 2R sin⁡B = 8AB = c = 5By... View Article

If the roots of the quadratic equation x²+ 3^1/4x + 3^1/2= 0 are α and β, then the value of α⁹⁶(α¹² – 1) + β⁹⁶(β¹² −1)

a. 50⋅324b. 51⋅324c. 52⋅324d. 104⋅324Solution: Answer: (c)x2 + 31/2 = - 31/4x⇒ x4 + 3 = -31/2x2⇒ x8 + 9 + 6x4 =... View Article

If for vectors A and B, A.B = |A×B|, then |A-B| is

a. √(A2 + B2√(2AB))b. √(A2 + B2 - √(2AB))c. √(A2 + B2 - √2 AB)d. √(A2 + B2 + √2 AB)Solution:... View Article

Consider the curve y² = 2x and point A(2, 2). If the normal at A intersects the curve again at point B and the tangent at A intersects the x-axis at C, then area of ∆ABC is

Answer: 25/2Equation of tangent at Ayy1 = x+x1⇒2y = x+2Tangent at A intersects the x-axis at CC(-2, 0)Equation of normal at Ay+ 2x =... View Article

The logical equivalence of the Boolean expression ~(p∧~q)∨(q∨~p) is

Answer: ≡ p → q~(p∧~q) ∨ (q∨~p)≡ (~p∨q) ∨ (q∨~p)≡ ~p∨q≡ p → q... View Article

Evaluate ∫_-1¹ log (√(1 – x) + √(1 + x))dx

Answer: 2 ln(2) - (1 - π/2)f(-x) = f(x)⇒ f is even∴ I = 2∫01 log ((1 - x) + √(1 + x))dx…(i)We will solve it by... View Article

If [latex]\lim_{x\to0}\left \{ 2-\cos x\sqrt{\cos 2x} \right \}^{\frac{x+2}{x^{2}}}=e^{\alpha }[/latex], then α = ?

Answer: 3 (1∞ form)⇒ e(2+1)2/(1+1)= eα⇒ e3 = eα= α = 3... View Article

Find the number of integral terms in the expansion of [latex](4^{\frac{1}{4}} + 5^{\frac{1}{6}})^{120}[/latex]

a. 1b. 21c. 20d. 11 Solution: Answer: (b)T(r+1) = 120Cr 4(120 - r)/45r/6= 120Cr 2(120 - r)/25r/6For integral terms, r should be even and... View Article

All the letters of the word EXAMINATION are arranged. Find the probability that ‘M’ is at the 4^th position.

a. 2/11b. 1/11c. 4/11d. 8/11 Solution: Answer: (b)EXAMINATIONLetters: AAIINNE T O X MTotal possible wordsn(S) = 11!/2!3Now when M is at 4th... View Article

There are 6 batsmen, 7 bowlers and 2 wickets keepers. Find the number of ways of forming a team of 11 players having at least 4 batsmen, at least 5 bowlers and at least 1 wicket keeper.

a. 567b. 525c. 462d. 777 Solution: Answer: (d)Team Size =11 playersCase 1: 4 batsmen and 5 bowlers and 2 wicket-keepers⇒ 6C4 × 7C5... View Article

The number of solutions of tan^-1√(x(x+1)) + sin^-1√(x²+x+1) = π/2 is

Answer:Given tan-1√(x(x+1)) + sin-1√(x2+x+1) = π/2⇒ tan-1 √(x(x+1)) = π/2 - sin-1√(x2+x+1)⇒ tan-1... View Article

What is the value of a+b when function f(x) is given

Question : If f(x) = is continuous in (-infinity,1], find the value of a+b. Solution: Given f(x) = = -2 f(1-) = a + (-1) = a - 1 f(1+)... View Article

If [latex]\vec{a}[/latex], [latex]\vec{b}[/latex], [latex]\vec{c}[/latex] are three mutually perpendicular vectors equally inclined to [latex]\vec{a}+\vec{b}+\vec{c}[/latex] at angle θ, then find the value of 36 cos² 2θ.

Answer: 4 ⇒ cos2θ = 1/31 + cos 2θ = 2cos2θ36 cos2 2θ = 36(2×1/3 - 1)2= 36/9= 4... View Article