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Let the vectors [latex]vec{a}[/latex], [latex]vec{b}[/latex], [latex]vec{c}[/latex] such that [latex]left |vec{a} right | = 2[/latex], [latex]left |vec{b} right | = 4[/latex], and [latex]left |vec{c} right | = 4[/latex].

Let the vectors \(\begin{array}{l}\vec{a}\end{array} \), \(\begin{array}{l}\vec{b}\end{array} \), \(\begin{array}{l}\vec{c}\end{array} \) such that \(\begin{array}{l}\left |\vec{a} \right | = 2\end{array} \), \(\begin{array}{l}\left |\vec{b}... View Article

The coefficient of x⁴ in the expansion of (1+x+x²+x³)⁶ in powers of x, is

Solution: (1+x+x2+x3)n = (1+x)n(1+x2)n (1+x+x2+x3)6= (1+x)6(1+x2)6 Coefficient of x4 ⇒ nC0 nC2+ nC2. nC1+ nC4. nC6 = 6C0 6C2+ 6C2.... View Article

Let A = {a,b,c} and B = {1,2,3,4}. Then the number of elements in the set C = {f : A → B 2∈ f(A)and f is not one-one} is

Solution: C = {f : A → B 2 ∈ f(A) and f is not one-one} Case-I : If f(x)... View Article

Which of the following points lies on the tangent to the curve 4x³e^y+x⁴e^y+2√(y+1) = 3 at the point (1,0)?

1) (2,6) 2) (2,2) 3) (-2,6) 4) (-2,4) Solution: 4x3ey+x4eyy’ +2y’/2√(y+1) = 0 At (1,0) 4+y’+(2y’/2) = 0 2y’ =... View Article

If the system of linear equations

If the system of linear equations x+y+3z = 0 x+3y+k2z = 0 3x+y+3z = 0 has a non-zero solution (x,y,z)... View Article

Let y = y(x) be the solution of the differential equation cos x (dy/dx)+2y sinx = sin2x, x∈(0, π/2). If y(π/3) = 0, then y(π/4) is equal to:

1) 2+√2 2) √2-2 3) (1/√2)-1 4) 2-√2 Solution: (dy/dx) +(2tanx) y = 2sinx I.F = e2ln(sec x) = sec2x... View Article

The value of [((-1+i√3)/(1-i))]³⁰ is

1) 215i 2) -215 3) -215i 4) 65 Solution: [((-1+i√3)/(1-i))]30 ⇒ [((-1+i√3)/2)(1+i)]30 ω30(1+i)30 = 215(-i) Answer: (3)

If for some α∈ R , the lines L₁ : (x+1)/2 = (y-2)/-1 = (z-1)/1 and L₂ : (x+2)/α = (y+1)/(5-α) = (z+1)/1 are coplanar, then the line L₂ passes through the point

1) (2, -10, -2) 2) (10, -2, -2) 3) (10, 2, 2) 4) (-2, 10, 2) Solution: A(-1,2,1), B(-2,-1,-1) \(\begin{array}{l}\left... View Article

If a+x = b+y = c+z+1, where a,b,c,x,y,z are non-zero distinct real numbers, then [latex]\begin{vmatrix} x &a+y & x+a\\ y & b+y & y+b\\ z & c+y & z+c \end{vmatrix}[/latex] is equal to:

1) y(a-b) 2) 0 3) y(b-a) 4) y(a-c) Solution: Given a+x = b+y = c+z+1 Now, \(\begin{array}{l}\begin{vmatrix} x &a+y &... View Article

The area (in sq. units) of the region A = {(x,y): (x-1)[x] ≤y ≤2√x, 0 ≤x ≤2} where [t] denotes the greatest integer function, is

1) (4/3)√2-(1/2) 2) (8/3)√2-(1/2) 3) (8/3)√2-1 4) (4/3)√2+1 Solution: y = f(x) = (x-1) [x] = y2 ≤ 4x =... View Article

If the line y = mx+c is a common tangent to the hyperbola (x²/100)-(y²/64) = 1 and the circle x²+y²= 36, then which one of the following is true?

1) 4c2 = 369 2) c2 = 369 3) 8m+5 = 0 4) 5m = 4 Solution: c = √(a2m2-b2)... View Article

If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is

1) (2/13)(350-1) 2) (1/26)(349-1) 3) (1/13)(350-1) 4) (1/26)(350-1) Solution: (ar+ar2+ar3)/(ar5+ar6+ar7) = 3/243 (1+r+r2)/r4(1+r+r2) = 1/81 r = 3 a(3+9+27) =... View Article

If α and β are the roots of the equation, 7x²-3x-2 = 0, then the value of α/1-α² and β/1-β² is equal to

1) 27/32 2) 1/24 3) 27/16 4) 3/8 Solution: α+β = 3/7 αβ = -2/7 [(α+β)-αβ(α+β)]/[1-(α2+β2)+(αβ)2] = [(3/7)+(2/7)×(3/7)](1-{(9/49)+(4/7)}+(4/49)) = (21+6)/49(16/49)... View Article

If the length of the chord of the circle, x²+y² = r²(r>0) along the line, y-2x = 3 is r, then r² is equal to:

1) 12 2) 24/5 3) 9/5 4) 12/5 Solution: AB = 2√(r2-9/5) = r r2-9/5 = r2/4 3r2/4 = 9/5... View Article

If ∫ cos θ/(5+7sinθ-2+2sin²θ)dθ = A log_e |B(θ)| + C where C is a constant of integration, then B(θ)/A can be:

1) 5(2sinθ+1)/(sinθ+3) 2) 5(sinθ+3)/(2sinθ+1) 3) (2sinθ+1)/(sinθ+3) 4) (2sinθ+1)/5(sinθ+3) Solution: ∫ cos θ/(5+7sinθ-2+2sin2θ)dθ Put sinθ = t, cosθ dθ = dt... View Article

The derivative of tan^-1(√(1+x²)-1)/x with respect to tan^-1(2x√(1-x²))/(1-2x²)) at x = 1/2 is

a) 2√3/3 b) 2√3/5 c) √3/12 d) √3/10 Solution: x = tan θ u = tan-1(sec θ-1)/tan θ = tan-1... View Article

If the mean and the standard deviation of the data 3,5,7,a,b are 5 and 2 respectively, then a and b are the roots of the equation

1) x2-20x+18 = 0 2) x2-10x+19 = 0 3) 2x2-20x+19 = 0 4) x2-10x+18 = 0 Solution: S.D = \(\begin{array}{l}\sqrt{\frac{\sum... View Article

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