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A hyperbola passes through the point P (√2, √3) and has foci at (±2, 0). Then the tangent to this hyperbola at P also passes through the point

1) (2√2 , 3√3)2) (√3, √2)3) (− √2 , –√3)4) (3√2 , 2√3)Solution:Answer: (1) [x2 / a2] - [y2 / b2] = 1a2 + b2 = 4 and [2 / a2] - [3 / b2] = 1(2 /... View Article

The eccentricity of an ellipse whose centre is at the origin is 1 / 2. If one of its directrices is x = −4, then the equation of the normal to it at (1, (3 / 2)) is

1) 4x − 2y = 12) 4x + 2y = 73) x + 2y = 44) 2y − x = 2Solution:Answer: (1) x = –4e = 1 / 2(-a / e) = -4-a = -4 * ea = 2Now, b2 = a2 (1 - e2) =... View Article

The radius of a circle, having minimum area, which touches the curve y = 4 − x² and the lines, y = |x| is

1) 2 (√2 - 1)2) 4 (√2 - 1)3) 4 (√2 + 1)4) 2 (√2 + 1)Solution:Answer: (2) x2 = - (y - 4)Let a point on the parabola P [(t/2) , (4 -... View Article

Let k be an integer such that the triangle with vertices (k, −3k), (5, k) and (−k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point

1) (1, 3/4)2) (1, -3/4)3) (2, 1/2)4) (2, -1/2)Solution:Answer: (3) (k2 - 7k + 10) + 4k2 + 20k = ± 565k2 + 13k - 46 = 05k2 + 13k + 66 = 05k2... View Article

If (2 + sinx) (dy/dx) + (y + 1) cosx = 0 and y(0) = 1, then y(π/2) is equal to

1) -2/32) -1/33) 4/34) 1/3Solution:Answer: (4) (2 + sinx) (dy/dx) + (y + 1) cos x = 0y(0) = 1, y(π / 2) = ?(1/(y + 1)) dy + (cos x/[2 +... View Article

The area (in sq. units) of the region {(x, y) : x ≥ 0, x + y ≤ 3, x²≤ 4y and y ≤ 1 + x} is :

1) 3 / 22) 7 / 33) 5 / 24) 59 / 12Solution:Answer: (3) Area of the shaded region= sq units... View Article

The integral [latex]int_{frac{pi}{4}}^{frac{3pi}{4}}frac{dx}{1+cosx}[/latex] is equal to:

1) 22) 43) −14) −2Solution:Answer: (1)... View Article

Let I_n = ∫tanⁿ x dx, (n > 1). If I₄+ I₆= a tan⁵ x + bx⁵+ C, where C is a constant of integration, then the ordered pair (a, b) is equal to

1) (1/5, 0)2) (1/5, -1)3) (-1/5, 0)4) (-1/5, 1)Solution:Answer: (1) In = ∫tann x dx, (n > 1)I4 + I6 = ∫(tan4 x + tan6 x) dx= ∫tan4 x sec2 x... View Article

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is

1) 102) 253) 304) 12.5Solution:Answer: (2) 2r + θr = 20 ---- (i)A = Area = (θ / 2π) * (πr2) = (θr2 / 2) ---- (ii)A = (r2 / 2) ([20 - 2r]... View Article

The normal to the curve y (x − 2) (x − 3) = x + 6 at the point where the curve intersects the y-axis passes through the point

1) (1/2, 1/2)2) (1/2, -1/3)3) (1/2, 1/3)4) (-1/2, -1/2)Solution:Answer: (1) y (x - 2) (x - 3) = x + 6At y-axis, x = 0, y = 1Now, on... View Article

If for x ∈ (0, (1 / 4)), the derivative of tan^-1 (6x√x / (1 – 9x³)) is √x . g(x), then g(x) equals:

1) (3x√x / (1 - 9x3))2) (3x/(1 - 9x3))3) (3/(1 + 9x3))4) (9/(1 + 9x3))Solution:Answer: (4) f (x) = 2 tan-1 (3x√x) for x ∈ (0, (1/4))f ‘ (x)... View Article

[latex]\lim_{x\rightarrow \pi/2}\frac{cotx-cosx}{(\pi-2x)^{3}}[/latex] equals:

1) 1 / 162) 1 / 83) 1 / 44) 1 / 24Solution:Answer: (1)... View Article

Let a, b, c ∈ R. If f (x) = ax²+ bx + c is such that a + b + c = 3 and f (x + y) = f (x) + f (y) + xy, ∀ x, y ∈ R, then [latex]sum_{n=1}^{10}[/latex] f (n) = is equal to

1) 1652) 1903) 2554) 330Solution:Answer: (4)As f (x + y) = f (x) + f (y) + xyGiven, f (1) = 3Putting, x = y = 1⇒ f (2) = 2 f (1) + 1 =... View Article

Archives

A hyperbola passes through the point P (√2, √3) and has foci at (±2, 0). Then the tangent to this hyperbola at P also passes through the point

The eccentricity of an ellipse whose centre is at the origin is 1 / 2. If one of its directrices is x = −4, then the equation of the normal to it at (1, (3 / 2)) is

The radius of a circle, having minimum area, which touches the curve y = 4 − x² and the lines, y = |x| is

Let k be an integer such that the triangle with vertices (k, −3k), (5, k) and (−k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point

If (2 + sinx) (dy/dx) + (y + 1) cosx = 0 and y(0) = 1, then y(π/2) is equal to

The area (in sq. units) of the region {(x, y) : x ≥ 0, x + y ≤ 3, x²≤ 4y and y ≤ 1 + x} is :

The integral [latex]int_{frac{pi}{4}}^{frac{3pi}{4}}frac{dx}{1+cosx}[/latex] is equal to:

Let I_n = ∫tanⁿ x dx, (n > 1). If I₄+ I₆= a tan⁵ x + bx⁵+ C, where C is a constant of integration, then the ordered pair (a, b) is equal to

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is

The normal to the curve y (x − 2) (x − 3) = x + 6 at the point where the curve intersects the y-axis passes through the point

If for x ∈ (0, (1 / 4)), the derivative of tan^-1 (6x√x / (1 – 9x³)) is √x . g(x), then g(x) equals:

[latex]\lim_{x\rightarrow \pi/2}\frac{cotx-cosx}{(\pi-2x)^{3}}[/latex] equals:

Let a, b, c ∈ R. If f (x) = ax²+ bx + c is such that a + b + c = 3 and f (x + y) = f (x) + f (y) + xy, ∀ x, y ∈ R, then [latex]sum_{n=1}^{10}[/latex] f (n) = is equal to

For any three positive real numbers a, b and c, 9 (25a²+ b²) + 25 (c² − 3ac) = 15b (3a + c). Then :

The value of (²¹C₁ – ¹⁰C₁) + (²¹C₂ – ¹⁰C₂) + (²¹C₃ – ¹⁰C₃) + (²¹C₄ – ¹⁰C₄) + … (²¹C₁₀ – ¹⁰C₁₀) is

If S is the set of distinct values of ‘b’ for which the following system of linear equations

The adj (3A² + 12A) is equal to

Let ω be a complex number such that 2ω + 1 = z….

If, for a positive integer n, the quadratic equation,

The function f : R → [(-1/2), (1/2)] is defined as f(x) = [x]/[1 + x²], is:

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