Two projectiles P1 and P2 thrown with speed in the ratio \( \sqrt3 \) : \( \sqrt2 \), attain the same height during their motion. If P2 is thrown at an angle of 60\( ^\circ \) with the horizontal, the angle of projection of P1 with horizontal will be :
Two bodies A and B of masses 5 kg and 8 kg are moving such that the momentum of body B is twice that of the body A. The ratio of their kinetic energies will be :
A 2 kg block is pushed against a vertical wall by applying a horizontal force of 50 N. The coefficient of static friction between the block and the wall is 0.5. A force F is also applied on the block vertically upward (as shown in figure). The maximum value of F applied, so that the block does not move upward, will be :
(Given : g = 10 ms\( - \)2)
An air bubble of negligible weight having radius r rises steadily through a solution of density \( \sigma \) at speed v. The coefficient of viscosity of the solution is given by :
The conditional statement
\( ((p \wedge q) \to (( \sim p) \vee r)) \vee ((( \sim p) \vee r) \to (p \wedge q)) \) is :
If n main scale divisions coincide with (n + 1) vernier scale divisions. The least count of vernier callipers, when each centimetre on the main scale is divided into five equal parts, will be :
Let for \( f(x) = {a_0}{x^2} + {a_1}x + {a_2},\,f'(0) = 1 \) and \( f'(1) = 0 \). If a0, a1, a2 are in an arithmatico-geometric progression, whose corresponding A.P. has common difference 1 and corresponding G.P. has common ratio 2, then f(4) is equal to _____________.
Let \( {S_1} = \{ x \in [0,12\pi ]:{\sin ^5}x + {\cos ^5}x = 1\} \)
and \( {S_2} = \{ x \in [0,8\pi ]:{\sin ^7}x + {\cos ^7}x = 1\} \)
Then \( n({S_1}) - n({S_2}) \) is equal to ______________.
At t = 0, truck, starting from rest, moves in the positive x-direction at uniform acceleration of 5 ms\( - \)2. At t = 20 s, a ball is released from the top of the truck. The ball strikes the ground in 1 s after the release. The velocity of the ball, when it strikes the ground, will be :
(Given g = 10 ms\( - \)2)
Let a vector \( \overrightarrow c \) be coplanar with the vectors \( \overrightarrow a = - \widehat i + \widehat j + \widehat k \) and \( \overrightarrow b = 2\widehat i + \widehat j - \widehat k \). If the vector \( \overrightarrow c \) also satisfies the conditions \( \overrightarrow c \,.\,\left[ {\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\overrightarrow a \times \overrightarrow b } \right)} \right] = - 42 \) and \( \left( {\overrightarrow c \times \left( {\overrightarrow a - \overrightarrow b } \right)} \right)\,.\,\widehat k = 3 \), then the value of \( |\overrightarrow c {|^2} \) is equal to :
Consider three circles:
\( {C_1}:{x^2} + {y^2} = {r^2} \)
\( {C_2}:{(x - 1)^2} + {(y - 1)^2} = {r^2} \)
\( {C_3}:{(x - 2)^2} + {(y - 1)^2} = {r^2} \)
If a line L : y = mx + c be a common tangent to C1, C2 and C3 such that C1 and C3 lie on one side of line L while C2 lies on other side, then the value of \( 20({r^2} + c) \) is equal to :
If a random variable X follows the Binomial distribution B(5, p) such that P(X = 0) = P(X = 1), then \( {{P(X = 2)} \over {P(X = 3)}} \) is equal to :
The distance of the point (3, 2, \( - \)1) from the plane \( 3x - y + 4z + 1 = 0 \) along the line \( {{2 - x} \over 2} = {{y - 3} \over 2} = {{z + 1} \over 1} \) is equal to :
\( \mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{r \over {2{r^2} - 7rn + 6{n^2}}}} \) is equal to :
Let \( {{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}},\,a,b,c \in R \), represents a circle with center (\( \alpha \), \( \beta \)). Then, \( \alpha \) + 2\( \beta \) is equal to :
Let A and B be two square matrices of order 2. If \( det\,(A) = 2 \), \( det\,(B) = 3 \) and \( \det \left( {(\det \,5(det\,A)B){A^2}} \right) = {2^a}{3^b}{5^c} \) for some a, b, c, \( \in \) N, then a + b + c is equal to :
Consider a triangle ABC whose vertices are A(0, \( \alpha \), \( \alpha \)), B(\( \alpha \), 0, \( \alpha \)) and C(\( \alpha \), \( \alpha \), 0), \( \alpha \) > 0. Let D be a point moving on the line x + z \( - \) 3 = 0 = y and G be the centroid of \( \Delta \)ABC. If the minimum length of GD is \( \sqrt {{{57} \over 2}} \), then \( \alpha \) is equal to ____________.
The value of \( 1 + {1 \over {1 + 2}} + {1 \over {1 + 2 + 3}} + \,\,....\,\, + \,\,{1 \over {1 + 2 + 3 + \,\,.....\,\, + \,\,11}} \) is equal to:
The probability distribution of X is :
For the minimum possible value of d, sixty times the mean of X is equal to _______________.
Let PQ be a focal chord of length 6.25 units of the parabola y2 = 4x. If O is the vertex of the parabola, then 10 times the area (in sq. units) of \( \Delta \)POQ is equal to ___________.
A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected students on 4th day is 30, then number of infected students on 8th day will be __________.
If for some \( \alpha \) > 0, the area of the region \( \{ (x,y):|x + \alpha | \le y \le 2 - |x|\} \) is equal to \( {3 \over 2} \), then the area of the region \( \{ (x,y):0 \le y \le x + 2\alpha ,\,|x| \le 1\} \) is equal to ____________.
Let \( f(t) = \int\limits_0^t {{e^{{x^3}}}\left( {{{{x^8}} \over {{{({x^6} + 2{x^3} + 2)}^2}}}} \right)dx} \). If \( f(1) + f'(1) = \alpha e - {1 \over 6} \), then the value of 150\( \alpha \) is equal to ___________.
The number of 6-digit numbers made by using the digits 1, 2, 3, 4, 5, 6, 7, without repetition and which are multiple of 15 is ____________.
Suppose \( \mathop {\lim }\limits_{x \to 0} {{F(x)} \over {{x^3}}} \) exists and is equal to L, where
\( F(x) = \left| {\matrix{ {a + \sin {x \over 2}} & { - b\cos x} & 0 \cr { - b\cos x} & 0 & {a + \sin {x \over 2}} \cr 0 & {a + \sin {x \over 2}} & { - b\cos x} \cr } } \right| \).
Then, \( - \)112 L is equal to ___________.
