Let n be a positive integer. Let A = ∑k=0n (-1)knCk [(1 / 2)k + (3 / 4)k + (7 / 8)k + (15 / 16)k + (31 / 32)k]. If 63A = 1 – (1 / 230), then n is equal to
Answer: (6) A = (1 / 2)n + (1 / 4)n + (1 / 8)n + (1 / 16)n +... View Article
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Let [latex]S_{n}(x)=log_{a^{\frac{1}{2}}}x + log_{a^{\frac{1}{3}}}x + log_{a^{\frac{1}{6}}}x + log_{a^{\frac{1}{11}}}x + log_{a^{\frac{1}{18}}}x + log_{a^{\frac{1}{27}}}x +…[/latex] up to n-terms, where a >1. If S24(x) = 1093 and S12(2x) = 265, then value of a is equal to
Answer: (16) Sn (x) = 2 loga x + 3 loga x + 6 loga x + 11 loga x... View Article
Consider the statistics of two sets of observations as follows:
Consider the statistics of two sets of observations as follows: Size Mean Variance Observation I 10 2 2 Observation... View Article
In ΔABC, the lengths of sides AC and AB are 12 cm and 5 cm, respectively. If the area of ΔABC is 30 cm2 and R and r are respectively the radii of circumcircle and incircle of ΔABC, then the value of 2R + r (in cm) is equal to
Answer: (15) Area = (1 / 2) * 5 * 12 * sin θ = 30 sin θ = 1... View Article
Let [latex]vec{c}[/latex] be a vector perpendicular to the vectors a = i + j – k and b = i + 2j + k. If c . (i + j + 3k) = 8, then the value of c . (a x b) is equal to
Answer: (28) \(\begin{array}{l}\vec{c}.(\vec{a}\times\vec{b})=[\vec{c}\ \vec{a}\ \vec{b}]\\ (\vec{a}\times\vec{b})= \begin{vmatrix} i &j &k \\ 1&1 &-1 \\ 1&2 &1 \end{vmatrix}=(3,-2,1)\\ \vec{c}\perp \vec{a},\vec{c}\perp \vec{b}\Rightarrow... View Article
Let A = {2, 3, 4, 5, ……, 30} and ‘⩭’ be an equivalence relation on A×A, defined by (a, b) ⩭ (c, d), if and only if ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) is equal to :
a. 7 b. 5 c. 6 d. 8 Answer: (a) a = bc (a, b) R (4, 3) ⇒ 3a... View Article
If the points of intersections of the ellipse (x2 / 16) + (y2 / b2) = 1 and the circle x2 + y2 = 4b, b > 4 lie on the curve y2 = 3x2, then b is equal to :
a. 5 b. 6 c. 12 d. 10 Answer: (c) (x2 / 16) + (y2 / b2) = 1 …(1)... View Article
Let P(x) = x2 + bx + c be a quadratic polynomial with real coefficients such that ∫01 P (x) dx and P(x) leaves remainder 5 when it is divided by (x – 2). Then the value of 9 (b+c) is equal to :
a. 7 b. 11 c. 15 d. 9 Answer: (a) (x – 2) Q (x) + 5 = x2 +... View Article
Let a = i + 2j – 3k and b = 2i – 3j + 5k. If r x a = b x r, r . (ɑi + 2j + k) = 3 and r . (2i + 5j – ɑk) = – 1, ɑ ∈ R, then the value of ɑ + |r|2 =
a. 11 b. 15 c. 9 d. 13 Answer: (b) r x a = – r x b r x... View Article
Let α ∈ R be such that the function [latex]f(x) = \left\{\begin{matrix} \frac{cos^{-1}(1-\left \{ x \right \}^{2})sin^{-1}(1-\left \{ x \right \})}{\left \{ x \right \}-\left \{ x \right \}^{3}} & x\neq 0\\ \alpha & x=0 \end{matrix}\right.[/latex] is continuous at x = 0, where {x} = x – [x], [x] is the greatest integer less than or equal to x. Then:
1) α = π/4 2) No such α exists 3) α = 0 4) α = π/√2 Solution: RHL =... View Article
Let C1 be the curve obtained by the solution of the differential equation 2xy (dy / dx) = y2 – x2, x > 0. Let curve C2 be the solution of 2xy / (x2 – y2) = dy / dx. If both the curves pass through (1, 1), then the area enclosed by the curves C1 and C2 is equal to :
a. (π / 2) – 1 b. (π / 4) + 1 c. π – 1 d. π + 1 Answer: (a)... View Article
Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy sin-1 (3x / 5) + sin-1 (4x / 5) = sin-1 x =
a. 1 b. 2 c. 3 d. 0 Answer: (c) Taking sin on both sides, (3x / 5) √[1 –... View Article
The maximum value of f (x) = [latex]\begin{vmatrix} sin^2x &1+cos^2x &cos2x \\ 1+sin^2x&cos^2x &cos2x \\ sin^2x&cos^2x &sin2x \end{vmatrix}[/latex], x ∈ R is:
a. √7 b. √5 c. 5 d. 3 / 4 Answer: (b) C1 → C1 + C2 \(\begin{array}{l}\begin{vmatrix} 2 &1+cos^2x... View Article
Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let α be the number of triangles having these points from different sides as vertices and β be the number of quadrilaterals having these points from different sides as vertices. Then (β – α) is equal to:
a. 1890 b. 795 c. 717 d. 1173 Answer: (c) α = 6C1 7C1 9C1 + 5C1 7C1 9C1 +... View Article
Let C be the locus of the mirror image of a point on the parabola y2 = 4x with respect to the line y = x. Then the equation of tangent to C at P(2, 1) is:
a. 2x + y = 5 b. x + 2y = 4 c. x + 3y = 5 d. x... View Article
[latex]\lim_{x \rightarrow 0} \frac{\int_0^2(sin \sqrt{t})dt}{x^3}[/latex] is equal to
a. 2/3 b. 0 c. 1/15 d. 3/2 Answer: (a)
The distance of the point (1, 1, 9) from the point of intersection of the line (x – 3)/1 = (y – 4)/2 = (z – 5)/2 and the plane x + y + z = 17 is:
a. √38 b. 19√2 c. 2√19 d. 38 Answer: (a) (x – 3)/1 = (y – 4)/2 = (z –... View Article
If f:R→ R is a function defined by f(x) = [x – 1] cos ((2x – 1)/2) , where [.] denotes the greatest integer function, then f is:
a. discontinuous only at x = 1 b. discontinuous at all integral values of x except at x = 1... View Article
Let f :R→R be defined as f(x) = 2x-1 and g:R – {1} →R be defined as g(x) = (x-½)/(x-1). Then the composition function f(g(x)) is:
a. Both one-one and onto b. onto but not one-one c. Neither one-one nor onto d. one-one but not onto... View Article
The statement among the following that is a tautology is:
a. A∧(A∨B) b. B→[A∧(A→B)] c. A∨(A∧B) d. [A∧(A→B)]→B Answer: (d) A∧ (~ A∨B)→B = [(A∧~A)∨(A∧B)]→ B = (A∧ B)→ B... View Article
