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The statement (p → (q → p)) → (p → (p ˅ q)) is:

1) equivalent to (p˅q)˄ (~ p) 2) equivalent to (p˄q)˅(~ p) 3) a contradiction 4) a tautology Solution: p q q→ p... View Article

[latex]\lim _{x \rightarrow 0} \frac{x\left(e^{\left(\sqrt{1+x^{2}+x^{4}}-1\right) / x}-1\right)}{\sqrt{1+x^{2}+x^{4}}-1}[/latex]

1) is equal to √e 2) is equal to 1 3) is equal to 0 4) does not exist Solution: Hence option 2 is the answer.... View Article

If x = 1 is a critical point of the function f(x) = (3x²+ax-2-a)e^x, then:

1) x = 1 is a local minima and x = -2/3 is a local maxima of f. 2) x = 1 is a local maxima and x = -2/3 is a local minima of f. 3) x = 1 and x... View Article

The sum of distinct values of λ for which the system of equations

The sum of distinct values of λ for which the system of equations (λ-1)x+(3λ+1)y+2λz = 0... View Article

Suppose that function f : R→R satisfies f(x+y) = f(x)f(y) for all x, y∈R and f(1) = 3. If [latex]sum_{i=1}^{n}f(i)=363[/latex] , then n is equal to

Solution: f(x+y) = f(x) f(y) f(x) = ax f(1) = a = 3 So f(x) = 3x ⇒ 3+32+33+....+3n = 363 ⇒ 3(3n-1)/2 = 363 ⇒ n = 5 Answer: 5... View Article

Consider the data on x taking the values 0, 2, 4, 8, …..,2n with frequencies ⁿC₀, ⁿC₁, ⁿC_2…ⁿC_n, respectively.

Consider the data on x taking the values 0, 2, 4, 8, .....,2n with frequencies nC0, nC1, nC2… nCn, respectively. If the mean of this data... View Article

If [latex]\vec{x}[/latex] and [latex]\vec{y}[/latex] be two non-zero vectors such that [latex]\left | \vec{x}+\vec{y} \right |= \left | \vec{x} \right |[/latex] and [latex]2\vec{x}+\lambda \vec{y}[/latex] is perpendicular to [latex]\vec{y}[/latex], then the value of λ is

Solution: ⇒ ..(i) And ⇒ ..(ii) by comparing (i) and (ii) we get λ = 1 Answer: 1... View Article

The number of words (with or without meaning) that can be formed from all the letters of the word “LETTER” in which vowels never come together is

Solution: Consonants → LTTR Vowels → EE Total No of words = 6!/(2!2!) = 180 (T and E repeated) Total no of words if vowels are... View Article

Consider the statement : “For an integer n, if n³-1 is even, then n is odd.” The contrapositive statement of this statement is

1) For an integer n, if n is even, then n3-1 is even 2) For an integer n, if n is odd, then n3-1 is even 3) For an integer n, if n3-1 is not... View Article

Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (-1, -4) in this line is

1) (11/5, 28/5) 2) (8/5, 29/5) 3) (29/5, 11/5) 4) (29/5, 8/5) Solution: Let (h, k) is the image of the point (-1, -4) in the line x/3 +... View Article

Let z = x+iy be a non-zero complex number such that z² = i|z|², where i = √-1 , then z lies on the

1) line, y = x 2) real axis 3) imaginary axis 4) line, y = -x Solution: z2 = i |z|2 x2-y2+2i xy = i(x2+y2) equating real terms x2-y2 = 0 ⇒... View Article

If the constant term in the binomial expansion of (√x – k/x²)¹⁰ is 405, then |k| equals

1) 1 2) 9 3) 2 4) 3 Solution: 10Cr (-k/x2)r (√x)10-r 10Cr (-k)r (x)5-5r/2 For constant term 5-5r/2 = 0 Or r = 2 Now, T3 = 10C2 k2... View Article

The angle of elevation of the summit of a mountain from a point on the ground is 45⁰.

The angle of elevation of the summit of a mountain from a point on the ground is 450. After climbing up one km towards the summit at an... View Article

Let f : R→ R be a function defined by f(x) = max {x,x²}. Let S denote the set of all points in R, where f is not differentiable. Then

1) {0,1} 2) an empty set 3) {1} 4) {0} Solution: Function is not differentiable at two point {0,1}. Answer: (1)... View Article

A plane P meets the coordinate axes at A, B and C respectively.

A plane P meets the coordinate axes at A, B and C respectively. The centroid of a triangle ABC is given to be (1,1,2). Then the equation of the... View Article

The centre of the circle passing through the point (0,1) and touching the parabola y = x² at the point (2,4) is

1) (3/10, 16/5) 2) (6/5, 53/10) 3) (-16/5, 53/10) 4) (-53/10, 16/5) Solution: Circle passing through point (0,1) and touching curve y = x2... View Article

Let θ = π/5 and A = [latex]\begin{bmatrix} \cos \theta & \sin \theta \\ – \sin \theta & \cos \theta \end{bmatrix}[/latex]. If B = A+A⁴, then det (B):

1) is one 2) lies in (1,2) 3) lies in (2,3) 4) is zero Solution: B = A+A4 A2 = = Similarly A4 = B = A4+A = = det B =... View Article

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