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Let a line = ( > 0) intersect the parabola, ² = at a point , other than the origin. Let the tangent to it at meet the −axis at the point . If area (Δ ) = 4 sq. units, then is equal to

Solution: Let the co-ordinates of 𝑃 be (t 2, t) Equation of tangent at (t 2, t) is y − t = (1/2t) (x − t 2 ) Therefore, co-ordinates of 𝑄 will... View Article

If the 10 ℎ term of an A.P. is 1/20 and its 20^ℎ term is 1/10 , then the sum of its first 200 terms is:

a. b. 100 c. 50 d. Solution: T10 = 1/20 T20 = 1/10 T20 − T10 = 10d => (1/20) = 10d d = 1/200 Therefore, a = 1/200 s200 =... View Article

Solve the system of given linear equations

The system of linear equations λx + 2y + 2z = 5 2λx + 3y + 5z = 8 4x + λy + 6z = 10 has: a. no solution when λ = 2 b. infinitely many... View Article

The differential equation of the family of curves, ² = 4b( + ), ∈ , is:

a. ′′ = ′ b. x( ′) 2 = + 2 ′ c. x( ′ ) 2 = − 2 ′ d. x( ′ ) 2 = 2 ′ − Solution: 2 = 4( + ) … (1) Differentiating both the sides... View Article

Let S be the set of all functions :[0,1] → , which are continuous on [0, 1] and differentiable on (0, 1). Then for every in , there exists a ∈ (0,1), depending on , such that:

a. (f(1)− ( ))/ (1− ) = ′( ) b. |f( ) − (1)| < | ′( )| c. |f( ) + (1)| < (1 + c)| ′ ( )| d. |f( ) − (1)| < (1 − )| ′( )|... View Article

The area (in sq. units) of the region {( , ) ∈ : ² ≤ ≤ 3 − 2 }, is

a. 31/3 b. 32/3 c. 29/3 d. 34/3 Solution: We have 2 ≤ ≤ −2 + 3 For point of intersection of two curves 2 + 2 − 3 = 0 ⇒ =... View Article

If matrix A is given, then 10 ⁻¹ is equal to

If , then 10 −1 is equal to: a. 6I b. − 6I c. 4I d. -4I Solution: If , , ⇒ 10 −1 = − 6I Answer: ( b)... View Article

Solve the integral problem

If , then: a. 1/6 < I 2 < 1/2 b. 1/8 < I 2 < 1/4 c. 1/9 < I 2 < 1/8 d. 1/16 < I 2 < 1/9 Solution: Let ′ ( ) =... View Article

Which of the following statements is a tautology?

Which of the following statements is a tautology? a. ~( ∧ ~ ) → ( ∨ ) b. (~ ∨ ~ ) → ( ∧ ) c. ∨ (~ ) → ( ∧ ) d. ~( ∨ ~ ) → ( ∨ )... View Article

The length of the perpendicular from the origin, on the normal to the curve, ² + 2 − 3 ² = 0 at the point (2, 2) is:

a. 2 b. 2√2 c. 4√2 d. √2 Solution: Given curve: x 2 + 2xy − 3y 2 = 0 ⇒ x 2 + 3xy − xy − 3y 2 = 0 ⇒ (x + 3y)(x − y) = 0 Equating we get, x... View Article

The mirror image of the point (1, 2, 3) in a plane is (− 7/3, − 4/3 , − 1/ 3 ). Which of the following points lies on this plane

a. (1, −1, 1) b. (−1, −1, 1) c. (1, 1, 1) d. (−1, −1, −1) Solution: Image of point P(1, 2, 3) w.r.t. a plane x + y + + = 0 is (− 7/3... View Article

Find the roots of the quadratic equation

Let = (−1+ √3)/2 . If , then and are the roots of the quadratic equation: a. x 2 + 101x + 100 = 0 b. x 2 + 102x + 101 = 0 c. x 2 −... View Article

If a line, = + is a tangent to the circle

If a line, y = x + is a tangent to the circle, (x − 3) 2 + y 2 = 1 and it is perpendicular to a line 1, where 1 is the tangent to the... View Article

Given limit is equal to:

a. 0 b. 1/10 c. – 1/10 d. − 1/5 Solution: Applying L - Hospital's Rule: = Answer: (a)... View Article

If a hyperbola passes through the point (10, 16) and it has vertices at (±6, 0), then the equation of the normal at is:

a. 3x + 4y = 94 b. x + 2y = 42 c. 2x + 5y = 100 d. x + 3y = 58 Solution: Vertex of hyperbola is (± , 0) ≡ (±6, 0) ⇒ = 6 Equation of... View Article

If and be the coefficients of ⁴ and ²respectively in the expansion of [latex](x+sqrt{x^{2}-1})^{6}+(x-sqrt{x^{2}-1})^{6}[/latex] , then

a. α + β = −30 b. α − β = −132 c. α + β = 60 d. α − β = 60 Solution: = 2[6𝐶0x 6 + 6𝐶2x 4 (x2 − 1) + 6𝐶4x 2(x 2 − 1) 2 + 6𝐶6 (x2 − 1)3] =... View Article

Let : (1,3) → be a function defined by ( ) = [ ] /( ²+1) , where [ ] denotes the greatest integer ≤ . Then the range of is

a. ( 2/5 , 3/5 ] U ( 3/4 , 4/5 ) b. ( 2/5 , 4/5 ] c. ( 3/5 , 4/5 ) d. ( 2/5 , 1/2 ) U ( 3/5 , 4/5 ] Solution: f(x) = x[x]/(x2+1) Answer: (d)... View Article

Let a and b be two vectors. If c is a vector such that….

Let ⃗ = ̂− 2 ̂+ ̂, ⃗ = ̂− ̂+ ̂ be two vectors. If ⃗ is a vector such that ⃗× ⃗= ⃗× ⃗and ⃗. ⃗= 0 then ⃗. ⃗ is equal to: a. 1/2 b.... View Article

The mean and variance of 20 observations are found to be 10 and 4, respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11. Then the correct variance is:

a. 4.01 b. 3.99 c. 3.98 d. 4.02 Solution: Mean = 10 ⇒ ∑ x /20 = 10 ⇒ ∑ x = 200 Variance = 4 ⇒ ∑ x 2/20 − 100 = 4 ⇒ ∑ x 2 = 2080 New... View Article

Let be the set of all real roots of the equation, 3^x (3^x − 1) + 2 = |3^x − 1| + |3^x − 2|. Then S:

a. is a singleton. b. is an empty set. c. contains at least four elements d. contains exactly two elements. Solution: 3x (3x − 1) + 2 = |3 x... View Article

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