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Let f : [–1,1] → R be defined as f (x) = ax² + bx + c

Let f : [–1,1] → R be defined as f (x) = ax2 + bx + c for all x ∈ [–1, 1], where a, b, c ∈ R such that f (–1) =... View Article

Let A = [latex]\begin{bmatrix} a &b \\ c&d \end{bmatrix}[/latex] and B= [latex]\begin{bmatrix} \alpha \\ \beta \end{bmatrix}\neq \begin{bmatrix} 0\\ 0 \end{bmatrix}[/latex] such that AB = B and a + d = 2021, then the value of ad-bc is equal to

Let A = such that AB = B and a + d = 2021, then the value of ad-bc is equal to ……………. Answer: (2020) ɑ (a... View Article

Let the coefficients of third, fourth and fifth terms in the expansion

Let the coefficients of third, fourth and fifth terms in the expansion of [x + (a / x2)]n, x ≠ 0 be in the ratio 12:8:3. Then the term... View Article

The number of solutions of the equation sin^-1 [x² + (1 / 3)] + cos^-1 [x² – (2 / 3)] = x², for x ∈ [- 1, 1], and [x] denotes the greatest integer less than or equal to x, is:

a. 0 b. 2 c. 4 d. Infinite Answer: (a) There are three cases possible for x ∈ [– 1, 1] Case I : x ∈ [- 1, - √(2/3) )... View Article

Two tangents are drawn from a point P to the circle

Two tangents are drawn from a point P to the circle x2 + y2 – 2x – 4y + 4 = 0, such that the angle between these tangents is tan-1... View Article

The value of ∑_r=0⁶ (⁶C_r . ⁶C_6-r) is equal to

a. 1124 b. 924 c. 1324 d. 1024 Answer: (b) ∑r=06 (6Cr . 6C6-r) = 6C0.6C6 + 6C1.6C5 + ....... + 6C6.6C0 Now, (1 + x)6 (1 + x)6 = (6C0 + 6C1x +... View Article

The value of lim_x→∞[r] + [2r] + ……. [nr] / [n²], where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to

a. 0 b. r c. r / 2 d. 2r Answer: (c) We know that r ≤ [r] < r + 1 and 2r ≤ [2r] < 2r + 1 3r ≤ [3r] < 3r + 1 nr ≤... View Article

If the sides AB, BC, and CA of a triangle ABC have, 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices is equal to:

a. 360 b. 240 c. 333 d. 364 Answer: (c) Total number of triangles = 3C1 × 5C1 × 6C1 + 3C1 × 5C2 + 5C1 × 3C2 +... View Article

Let S₁, S₂ and S₃ be three sets defined

Let S1, S2 and S3 be three sets defined as S1 = {z ∈ C : |z - 1| ≤ √2}, S2 = {z ∈ C : Re ((1 - i) z) ≥ 1}, S3 =... View Article

If the curve y = y(x) is the solution of the differential equation

If the curve y = y(x) is the solution of the differential equation 2 (x2 + x5/4) dy - y (x + x1/4) dx = 2x9/4 dx, x > 0 which passes through... View Article

The value of the limit lim_θ→0 [tan (π cos² θ)] / [sin (2π sin² θ)] is equal to

a. - 1 / 2 b. - 1 / 4 c. 0 d. 1 / 4 Answer: (a) Given, limθ→0 [tan (π cos2 θ)] / [sin (2π sin2 θ)] =... View Article

Let y = y (x) be the solution of the differential equation cosx (3 sinx + cosx + 3) dy = (1 + y sinx (3 sinx + cosx + 3)) dx, 0 ≤ x ≤ π / 2, y(0) = 0. Then, y (π / 3) is equal to

a. 2 loge [(2√3 + 10) / 11] b. 2 loge [(√3 + 7) / 2] c. 2 loge [(3√3 - 8)] / 4 d. 2 loge [(2√3 + 9) / 6] Answer: (a)... View Article

If the integral ∫₀¹⁰ [sin 2πx] / e^x-[x]dx = ɑe^-1 + βe^-½ + ,

If the integral ∫010 [sin 2πx] / ex-[x] dx = ɑe-1 + βe-½ + , where ɑ, β, are integers and [x] denotes the greatest... View Article

If x, y, z are in arithmetic progression with common difference d

If x, y, z are in arithmetic progression with common difference d, x ≠ 3d, and the determinant of the matrix is zero, then the value of k2... View Article

Let f : R → R be defined as f (x) = e^–xsinx.

Let f : R → R be defined as f (x) = e–x sinx. If F : [0,1] R→ is a differentiable function such that F (x) = ∫0x f (t) dt,... View Article

Let L be a tangent line to the parabola y² = 4x – 20 at (6, 2). If L is also a tangent to the ellipse (x² / 2) + (y² / b) = 1, then the value of b is equal to:

a. 20 b. 14 c. 16 d. 11 Answer: (b) Parabola y2 = 4x – 20 Tangent at P (6, 2) will be 2y = 4 [(x + 6) / 2] – 20 2y = 2x + 12... View Article

Let O be the origin. Let OP = xi + yj – k and OQ = – i + 2j + 3xk

Let O be the origin. Let OP = xi + yj - k and OQ = - i + 2j + 3xk, x, y ∈ R, x > 0, be such that |PQ| = √20 and the vector OP is... View Article

Consider the function f : R → R defined by f(x) = [latex]left{begin{matrix} (2-sin(1/x)|x|), &xneq 0 \ 0&x=0 end{matrix}right.[/latex]. Then f is

a. monotonic on (0, ∞) only b. Not monotonic on (–∞, 0) and (0, ∞) c. monotonic on (–∞, 0) only d. monotonic... View Article

If the equation of the plane passing through the mirror image of a point

If the equation of the plane passing through the mirror image of a point (2, 3, 1) with respect to line (x + 1) / 2 = (y - 3) / 1 = (z + 2) / -1... View Article

The number of solutions of the equation x + 2 tanx = π / 2 in the interval [0, 2π] is

a. 5 b. 2 c. 4 d. 3 Answer: (d) x + 2 tan x = π / 2 in [0, 2π] 2 tan x = π / 2 – x 2 tan x = π / 2 – x tan x = π / 4... View Article

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