The shortest distances of the point (1, 2, 3) from the axis x, y and z respectively are
1) 1, 2, 3 2) √5, √13, √10 3) √10, √13, √5 4) √13, √10, √5 Solution: (4) √13, √10,... View Article
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If the length of a vector by 21 and direction ratios be 2, – 3, 6 then its direction cosines are
1) 2 / 21, – 1 / 7, 2 / 7 2) 2 / 7, – 3 / 7, 6... View Article
Direction ratios of the normal to the plane passing through the points (0, 1, 1), (1, 1, 2) and (-1, 2, – 2) are
1) (1, 1, 1) 2) (2, 1,-1) 3) (1, 2, -1) 4) (1,-2,-1) Solution: (4) (1,-2,-1) \(\begin{array}{l}\begin{array}{l} \begin{array}{l} l(1)+m(0)+n(2-1)=0 \Rightarrow... View Article
If a line makes angles of 30° and 45° with x-axis and y-axis, then the angle made by it with z-axis is
1) 45° 2) 60° 3) 120° 4) None of these Solution: (4) None of these cos = √1 – (3... View Article
Let = { ∈ : 1 ≤ ≤ 50}. If A = { ∈ : is a multiple of 2} and B = { ∈ : is a multiple of 7}, then the number of elements in the smallest subset of containing both A and B is
A = {x: x is multiple of 2} = {2,4,6,8, … } B = {x: x is multiple of 7}... View Article
If the mean and variance of eight numbers 3, 7, 9, 12, 13, 20, x and y be 10 and 25 respectively then xy is equal to
Mean = 10 (64 + x + y)/8 = 10 x + y = 80 – 64 = 16 Variance... View Article
If the function f defined on (-1/3, 1/3) by
If the function f defined on (-1/3, 1/3) by is continuous, then k is equal to Solution: As f(x) is... View Article
If the foot of perpendicular drawn from the point (1,0,3) on a line passing through (α, 7,1) is (5/3, 7/3, 17/3) then α is equal to:
Given points (1, 0, 3) and Q (5/3, 7/3, 17/3) Direction ratios of line L : (α-5/3, 7-7/3, 1-17/3) =... View Article
If system of linear equations: x + y + z = 6,x + 2y + 3z = 10 and 3x + 2y + z = has more than two solutions, then μ – λ2 is equal to
The system of equations has more than 2 solutions ∴ D = D3 = 0 \(\begin{array}{l}\begin{vmatrix} 1 & 1 &1... View Article
Let the tangents drawn from the origin to the circle, x2 + y2 − 8x − 4y + 16 = 0 touch it at the points A and B. The (AB)2 is equal to
a. 32/5 b. 64/5 c. 52/5 d. 56/5 Solution: x2 + y2 − 8x − 4y + 16 = 0... View Article
The area (in sq. units) of the region {(x, y)ϵR|4x2 ≤ y ≤ 8x + 12} is
a. 125/3 b. 128/3 c. 124/3 d. 127/3 Solution: For point of intersection, 4x2 = 8x + 12 ⇒ x2... View Article
Let α and β are the roots of the equation x2 − x − 1 = 0. If pk = (α)k + (β)k, k ≥ 1 then which one of the following statements is not true?
a. (p1 + p2 + p3 + p4 + p5) = 26 b. p5 = 11 c. p5 = p2... View Article
If θ1 and θ2 be respectively the smallest and the largest values of θ in (0,2π) − {π} which satisfy the equation [latex]2cot ^{2}theta -frac{5}{sin theta }+4 = 0[/latex] then [latex]int_{theta _{1}}^{theta _{2}}cos ^{2}3theta dtheta[/latex] is equal to:
a. 2π/3 b. π/3 c. (π/3) + (1/6) d. π/9 Solution: \(\begin{array}{l}2\cot ^{2}\theta -\frac{5}{\sin \theta }+4 = 0\end{array} \) ,... View Article
Let a, b,c be three unit vectors such that
Let a, b,c be three unit vectors such that \(\begin{array}{l}\vec{a}+\vec{b}+\vec{c} = 0\end{array} \) . If \(\begin{array}{l}\lambda =\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a}\end{array} \) and \(\begin{array}{l}\vec{d}... View Article
Let a1, a2, a3, … be a G.P. such that a1 < 0, a1 + a2 = 4 and a3 + a4 = 16. If [latex]sum_{i=1}^{9}a_{i} = 4lambda[/latex] then λ is equal to
a. 171 b. 511/3 c. -171 d. -513 Solution: a1 + a2 = 4 ⇒ a + ar = 4... View Article
The number of ordered pairs (r, k) for which 6 ⋅ (35)^C_r = (k_2 − 3) ⋅ (36)^C_(r+1), where k is an integer, is
a. 4 b. 6 c. 2 d. 3 Solution: Using 36Cr+1 = (36/r+1)×35Cr (36/r+1)×35Cr×(k2-3) = 35Cr×6 k2 – 3 =... View Article
Let f(x) be a polynomial of degree 5 such that x = ± 1 are its critical points
Let f(x) be a polynomial of degree 5 such that x = ± 1 are its critical points. If \(\begin{array}{l}\lim_{x\to... View Article
Let = [ ] and = [ ] be two 3 × 3 real matrices such that = (3)( + −2) , where , = 1, 2, 3. If the determinant of is 81, then the determinant of is :
a. 1/9 b. 1/81 c. 1/3 d. 3 Solution: 𝑏𝑖 = (3)(𝑖+𝑗−2)𝑎𝑗𝑖 B = \(\begin{array}{l}\begin{bmatrix} 3^{0}a_{11} & 3a_{21} &3^{2}a_{31} \\... View Article
If [latex]\frac{3+i\sin \theta }{4-i\cos \theta }, \theta \in [0,2\pi ][/latex] is a real number, then the argument of sin 𝜃 + 𝑖 cos 𝜃 is :
a. π- tan-1 (4/3) b. – tan-1 (3/4) c. π- tan-1 (4/3) d. tan-1 (4/3) Solution: Let \(\begin{array}{l}z=\frac{3+i\sin \theta }{4-i\cos... View Article
If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + ….. is (102)m, then m is equal to
a. 10 b. 25 c. 5 d. 20 Solution: S = 3 + 4 + 8 + 9 + 13... View Article
