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Find sum of all possible values of θ in the interval (-π/2, π) for which [latex]frac{3+2isin theta }{1-2isin theta}[/latex] is purely imaginary;

a) π/3 b) π c) 2π/3 d) π/2 Solution: \(\begin{array}{l}z=\frac{3+2i\sin \theta }{1-2i\sin \theta}\times \frac{1+2i\sin \theta}{1+2i\sin \theta}\end{array} \) = \(\begin{array}{l}\frac{3-4\sin ^{2}\theta+i(8\sin... View Article

There is a parabola having axis as x-axis, vertex is at a distance of 2 unit from origin & focus is at (4, 0). Which of the following point does not lie on the parabola.

a) (6, 8) b) (5, 2√6) c) (8, 4√3) d) (4,-4) Solution: The equation of parabola is y2 = 8(x-2)... View Article

[latex]\lim_{y\to 0}\frac{\sqrt{1+\sqrt{1+y^{4}}}-\sqrt{2}}{y^{4}}[/latex] =

a) 1/4√2 b) 1/2√2 c) 1/2√2(1+√2) d) does not exist Solution: Rationalising numerator, Answer: (a)

{2⁴⁰³/15} = k/15 then find k. (where {.} denotes fractional part function)

a) 2 b) 8 c) 1 d) 4 Solution: Solution: 24 ≡ 1 (mod 15) 2400 ≡ 1 (mod 15)... View Article

a,b,c are in G.P. a+b+c = bx , then x cannot be;

a) 2 b) -2 c) 3 d) 4 Solution: Let the terms of G.P be a/r, a, ar (a/r) +... View Article

Average height and variance of 5 students in a class is 150 and 18 respectively. A new student whose height is 156 cm is added to the group. Find new variance.

a) 20 b) 22 c) 16 d) 14 Solution: Let 5 students are x1 , x2 , x3 , x4... View Article

Check if function is continuous or discontinuous

Question: If then: a) f (x) is discontinuous ∀ a ∈ R, b ∈ R b) f (x) is continuous... View Article

If [latex]\vec{a}=i -j[/latex] , [latex]\vec{b}=i +j+k[/latex] are two vectors, and c is another vector such that [latex]\vec{a}\times \vec{c}+\vec{b}=\vec{0}[/latex] and [latex]\vec{a}. \vec{c}=4[/latex] then [latex]\left | \vec{c}\right |^{2} =[/latex]

a) 9 b) 8 c) 19/2 d) 17/2 Solution: \(\begin{array}{l}\vec{a}\times \vec{c}+\vec{b}=\vec{0}\end{array} \) \(\begin{array}{l}\vec{a}\times \vec{c}=-\vec{b}\end{array} \) \(\begin{array}{l}\vec{a\times }\left ( \vec{a}\times \vec{c}\right... View Article

If x² ≠ nπ+1, n ∈N, then [latex]\int x \sqrt{\frac{2 \sin \left(x^{2}-1\right)-\sin 2\left(x^{2}-1\right)}{2 \sin \left(x^{2}-1\right)+\sin 2\left(x^{2}-1\right)}} d x[/latex] is equal to

a) \(\begin{array}{l}ln \cos \left ( \frac{x^{2}-1}{2} \right )+c\end{array} \) b) \(\begin{array}{l}\frac{1}{2}ln \cos \left ( \frac{x^{2}-1}{2} \right )+c\end{array} \) c) \(\begin{array}{l}ln... View Article

[latex]\int_{0}^{\pi }\left | \cos x \right |^{3}dx[/latex] is equal to;

a) 4/3 b) 2/3 c) 0 d) 8/3 Solution: Answer: (a)

The equation x²+2x+2 = 0 has roots α and β. Then value of α¹⁵ + β¹⁵is

a) 512 b) 256 c) -512 d) -256 Solution: Solution: (x+1)2 = -1 ⇒ x+1 = ±i x = -1+I,... View Article

There are 5 girls and 7 boys. A team of 3 boys and 2 girls is to be formed such that no two specific boys are in the same team. Number of way to do so

a) 400 b) 250 c) 200 d) 300 Solution: Total number of teams = 7C3× 5C2 = 35×10 = 350... View Article

The sum [latex]sum _{k=1}^{20}(1+2+3+ ……+k)[/latex] is

Solution: \(\begin{array}{l}\sum_{k=1}^{20}\frac{k(k+1)}{2}\end{array} \) \(\begin{array}{l}=\frac{1}{2}\sum_{k=1}^{20}k^{2} + k\end{array} \) \(\begin{array}{l}=\frac{1}{2}\left [ \frac{20(21)(41)}{6}+\frac{20(21)}{2} \right ]\end{array} \) =(1/2)[2870+210]=1540

The least positive value of ′ ′ for which the equation, 2 ²+( −10) +(33/2) =2 has real roots is

Solution: ∵2x2+(a−10)x+33/2=2a, a ∈ 𝐙+ has real roots =>D ≥ 0 =>(a−10)2−4×2×((33/2)−2a) ≥ 0 =>(a−10)2−4(33−4a) ≥ 0 =>a2−4a−32 ≥ 0... View Article

Let the normal at a on the curve ² − 3 ² + + 10 = 0 intersect the y-axis at (0,3/2). If is the slope of the tangent at to the curve, then | | is equal to

Let co-ordinate of P be (x1, y1) Differentiating the curve w.r.t x 2yy′ − 6x + y′ = 0 Slope... View Article

An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that at most three of them are red is

Solution: Number of ways to select at most 3 red balls = P(0 red balls) +P(1 red ball) + (2... View Article

Let f(x) = x cos⁻¹ (sin(−|x|)) , x ∈ (− /2 , /2 ), then which of the following is true ?

a. f ′ (0) = − π/2 b. f ′ is decreasing in (− π/2 , 0) and increasing in... View Article

If C is a point at which Rolle’s theorem holds for the function, f(x) = [latex]log_{e}\left ( \frac{x^{2} + \alpha }{7x} \right )[/latex] in the interval [3, 4], where α ∈ 𝐑, then f′′(c) is equal to :

a. −1/24 b. −1/12 c. √3/7 d. 1/12 Solution: Rolle’s theorem is applicable on f(x) in [3, 4] ⇒ f(3)... View Article

Let the line y = mx and the ellipse 2x² + y² = 1 intersect a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at (−1/3√2,0) and (0, ), then is equal to

a. 2/√3 b. 2/3 c. 2√2/3 d. √2/3 Solution: Let ≡ (x1, y1) 2x2 + y2 = 1 is given... View Article

Let two points be A(1, −1) and B(0, 2). If a point P(x′, ′) be such that the area of ΔPAB = 5 sq. units and it lies on the line, 3x + − 4 = 0, then the value of lambda is

a. 4 b. 1 c. −3 d. 3 Solution: Area of triangle is \(\begin{array}{l}A=\frac{1}{2}\begin{vmatrix} 1 & -1 & 1\\ 0... View Article

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