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... View Article
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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 = 1 2) 4x + 2y = 7 3) x + 2y = 4 4) 2y... 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)... 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)... View Article
If (2 + sinx) (dy/dx) + (y + 1) cosx = 0 and y(0) = 1, then y(π/2) is equal to
1) -2/3 2) -1/3 3) 4/3 4) 1/3 Solution: Answer: (4) (2 + sinx) (dy/dx) + (y + 1) cos... 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 / 2 2) 7 / 3 3) 5 / 2 4) 59 / 12 Solution: Answer: (3) Area... View Article
The integral [latex]int_{frac{pi}{4}}^{frac{3pi}{4}}frac{dx}{1+cosx}[/latex] is equal to:
1) 2 2) 4 3) −1 4) −2 Solution: Answer: (1)
Let In = ∫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,... 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) 10 2) 25 3) 30 4) 12.5 Solution: Answer: (2) 2r + θr = 20 —- (i) A =... 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... 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:... View Article
[latex]\lim_{x\rightarrow \pi/2}\frac{cotx-cosx}{(\pi-2x)^{3}}[/latex] equals:
1) 1 / 16 2) 1 / 8 3) 1 / 4 4) 1 / 24 Solution: Answer: (1) \(\begin{array}{l}\lim_{x\rightarrow... 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) 165 2) 190 3) 255 4) 330 Solution: Answer: (4) As f (x + y) = f (x) +... View Article
For any three positive real numbers a, b and c, 9 (25a2 + b2) + 25 (c2 − 3ac) = 15b (3a + c). Then :
1) b, c and a are in A.P. 2) a, b and c are in A.P. 3) a, b and... View Article
The value of (21C1 – 10C1) + (21C2 – 10C2) + (21C3 – 10C3) + (21C4 – 10C4) + … (21C10 – 10C10) is
1) 221 − 210 2) 220 − 29 3) 220 − 210 4) 221 − 211 Solution: Answer: (3) 21C1... View Article
If S is the set of distinct values of ‘b’ for which the following system of linear equations
If S is the set of distinct values of ‘b’ for which the following system of linear equations x +... View Article
The adj (3A2 + 12A) is equal to
Ifthen adj (3A2 + 12A) is equal to : Solution: Answer: (1) = (2 – 2λ – λ + λ2)... View Article
Let ω be a complex number such that 2ω + 1 = z….
Let ω be a complex number such that 2ω + 1 = z where z = √-3. If then k... View Article
If, for a positive integer n, the quadratic equation,
If, for a positive integer n, the quadratic equation,has two consecutive integral solutions, then n is equal to: 1) 9... View Article
The function f : R → [(-1/2), (1/2)] is defined as f(x) = [x]/[1 + x2], is:
1) injective but not surjective. 2) surjective but not injective. 3) neither injective nor surjective. 4) invertible. Solution: Answer: (2)... View Article
