List of practice Questions

Among the statements :
\((S1)\) \((( p \vee q ) \Rightarrow r ) \Leftrightarrow( p \Rightarrow r )\)
\((S2)\)\((( p \vee q ) \Rightarrow r ) \Leftrightarrow(( p \Rightarrow r ) \vee( q \Rightarrow r ))\)

Three dice are thrown. Then the probability that no outcomes is similar is p/q then q – p is (where p and q are co-prime)

The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to ______.

Three dice are rolled. If the probability of getting different numbers on the three dice is \(\frac{p}{q}\), where \(p\) and \(q\) are co-prime, then \(q - p\) is equal to:

Let the line \(\frac{x}{1}=\frac{6−y}{2}=\frac{z+8}{5}\) intersect the lines \(\frac{x-5}{1}=\frac{y-7}{3}=\frac{z+2}{1}\) and \(\frac{x+3}{6}=\frac{3−y}{3}=\frac{z-6}{1}\) at the points A and B respectively. Then the distance of the mid-point of the line segment AB from the plane 2x – 2y + z = 14 is

Let \(\overrightarrow{a}=2\^i+7\^j−\^k ,\overrightarrow{b} =3\^i+5 \^k\) and \(\overrightarrow{c}=\^i=\^j+2\^k\) . Let \(\overrightarrow{d}\) be a vector which is perpendicular to both \(\overrightarrow{a}\) and \(\overrightarrow{b}\) , and \(\overrightarrow{c}.\overrightarrow{d}=12\). Then \((-\^i+\^j=\^k).(\overrightarrow{c}\times\overrightarrow{d})\) is equal to

If the line x = y = z intersects the line
xsinA+ysinB+zsinC-18=0=xsin2A+ysin2B+zsin2C-9,
where A, B, C are the angles of a triangle ABC, then 80\(\left( sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2} \right)\) is equal to ________.

Let a circle of radius 4 be concentric to the ellipse \( 15x^2 + 19y^2 = 285 \). Then the common tangents are inclined to the minor axis of the ellipse at the angle:

A plane P contains the line of intersection of the plane \(\overrightarrow{r} .( \^i+\^j+\^k )=6\) and \(\overrightarrow{r }.( 2\^i+3\^j+4\^k) = − 5\). If P passes through the point (0, 2, –2), then the square of distance of the point (12, 12, 18) from the plane P is

Let the plane P contain the line 2x+y-z-3=0=5x-3y+4z+9 and be parallel to the line \(\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}\). Then the distance of the point
A(8, -1, -19) from the plane P measured parallel to the line \(\frac{x}{-3}=\frac{y-5}{4}=\frac{2-z}{-12}\) is equal to ___________.

Let a die be rolled \( n \) times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is \( \frac{k}{2^{15}} \), then \( k \) is equal to:

The area enclosed by \(y = |x-1| + |x-2|\) and \(y = 3\)

Let an ellipse with centre (1, 0) and latus rectum of length \(\frac{1}{2}\) have its major axis along x-axis. If its minor axis subtends an angle 60° at the foci, then the square of the sum of the lengths of its minor and major axes is equal to

Let f be a continuous function satisfying \(∫^{t ^2}_ 0 ( f ( x ) + x ^2 ) d x = \frac{4}{3} t^3 , ∀ t > 0.\) Then \(f(\frac{π^2}{4})\) is equal to

Orthocentre of triangle having vertices as A (1,2), B(3,-4), C(0,6) is

If the tangents at the points P and Q on the circle x² + y²– 2x + y = 5 meet at the point R \(( \frac{9}{4},2 ) \)then the area of the triangle PQR is

If the solution curve f(x, y) = 0 of the differential equation (1+log_e x) \(\frac{dx}{dy} \)- x log_e x = e^y , x > 0, passes through the points (1,0) and (α, 2) then α^a is equal to

If n \(\in\) [10,100] and n \(\in\) N, then how many such n are possible where 3n - 3 is divisible by 7?

Let A = {1, 2, 3, 4} and R be a relation on the set AxA defined by R = {((a, b), (c, d)): 2a+3b = 4c +5d}. Then the number of elements in R is

if A=\(\frac{1}{5! 6! 7!}\begin{bmatrix} 5! & 6! & 7!\\ 6! & 7! & 8! \\ 7! & 8! & 9! \end{bmatrix}\), then |adj(adj(2A))| is equal t

Let m be the mean and σ be the standard deviation of the distribution

xi	0	1	2	3	4	5
fi	k+2	2k	k²-1	k²-1	k²+1	k-3

where ∑f_i = 62. if [x] denotes the greatest integer ≤ x, then [μ² + σ²] is equal

Negation of \( p \land (q \land \neg (p \land q)) \) is:}

the points P and Q are respectively the circumcentre and the orthocentre of a ΔABC, then \( \overrightarrow{PA}+ \overrightarrow{PB}+ \overrightarrow{PC}\) is

The statement ~[p v (~(p ∧ q))] is equivalent to

Let the image of the point P(1, 2, 6) in the plane passing through the points A(1, 2, 0), B(1, 4, 1) and C(0, 5, 1) be Q (α, β, γ). Then (α² + β² + γ²) is equal to