If \[P=\sqrt[16]{7}+\sqrt[16]{5}\], \[q=\sqrt{7}+\sqrt{5}\], \[r=\sqrt[8]{7}+\sqrt[8]{5}\],\[s=\sqrt[16]{7}-\sqrt[16]{5}\] and \[t=\sqrt[4]{7}+\sqrt[4]{5}\] then which one of the following is a rational number?
In the figure given below, two line segments PQ and RS intersect each other at the point O such that PO = 4.8, OS = 4.2 cm, RO = 3.2 cm and \[\angle ORP=65{}^\circ ,\] \[\angle POR=80{}^\circ \] and \[\angle OSQ=35{}^\circ \]then find the value of OQ.
From a point p, inside of an equilateral triangle ABC, the perpendicular distances of the three sides \[2\,\sqrt{3}\,cm,\] are \[3\,\sqrt{3}\,cm\] and \[4\,\sqrt{3}\,cm,\]respectively. Find the semiperimeter of the triangle.
The length of the largest possible rod that can be placed in a cubical room is \[105\sqrt{3}\,m\]. The surface area (in sq metre) of the largest possible sphere that fits within the cubical room is _______. \[\left( Take\,\pi =\frac{22}{7} \right)\]
A cylindrical tank with radius 50 cm is being filled by a circular pipe with internal diameter of 3 cm at the rate of 10 m/sec. Find the height of the water column in 24 minutes.
A number p is selected from the numbers 1, 3, 5 and then a second number q is randomly selected from the numbers 1, 6, 9. What is the probability that the product p q of the two numbers will be less than 20?
A number consists of 3 digits whose sum is 10. The middle digit is equal to the sum of the other two and the number will be increased by 99 if its digits are reversed. What will be the number if 789 is added to the number?
A person standing on the top of a light house of height 300 m above sea level observes that the angle of depression of a ship sailing directly towards it changes from \[30{}^\circ \] to \[45{}^\circ \]. The distance travelled by the ship during this period is:\[[Take\sqrt{3}=1.73]\]
If \[m=a\,{{\cos }^{3}}\theta +3a\,\,\cos \theta {{\sin }^{2}}\theta ,\] \[n=a{{\sin }^{3}}\theta +3a\sin \theta .{{\cos }^{2}}\theta ,\] then the value of \[{{(m+n)}^{\frac{2}{3}}}+{{(m-n)}^{\frac{2}{3}}}\] is:
In the figure given below, OQRP is a quarter circle of radius 2 cm and another circle inscribed in it is touching this quarter circle at three points R, S and T. The radius of the inscribed circle (in cm) is ________.
In the figure given below PQ and RS are two parallel chords of respective lengths 16 cm and 12 cm on the same side of the centre of a circle. If the distance between the chords is 2 cm, then find the diameter of the circle.
In the figure given below, two identical circles are intersecting each other at A and B. If O and O? are their centres and OAO'B forms a square of side 1 cm, then find the area (in sq cm) of the shaded portion.
If a metallic cone of radius 90 cm and height 25 cm is melted and recasted into a metallic sphere of radius 15 cm. Find the number of spheres so obtained.
A hollow hemispherical bowl of thickness 1 cm has an inner radius of 8 cm. Find the volume of the metal required to make the bowl. \[\left( Take\pi =\frac{22}{7} \right)\]
A die has its six faces marked 0, 1, 1, 2, 2, 6. Two such dice are thrown together and the total score is recorded. Find the probability of getting a total of 8.
If \[\sin 2\alpha =2\sin \alpha \cdot \cos \alpha \] and \[\sin (360{}^\circ -\alpha )=-sin\alpha \] then find the value of \[\cos 40{}^\circ \cdot \cos 80{}^\circ \cdot \cos 160{}^\circ .\]
A, B, C, D, E, F and G are the members of a family consisting of four adults and three children, two of whom (F and G) are girls. A and D are brothers and A is a lawyer. E is a professor married to one of the brothers and has two children. B is married to D and G is their child. C is:
Among A, B, C, D and E each having a different height, B is taller than D. E is shorter than A. C is taller than B but shorter than E. Who among them is the tallest?
Direction: In these types of questions, the sets of numbers given in the alternatives are represented. The columns and rows of matrix I are numbered from 0 to 4 and of matrix II are numbered from 5 to 9. A letter from these matrices can be represented first by its row and next by its column, e.g., E can be represented by 13, 24, etc. and T can be represented by 58, 66, etc. similarly you have to identify the set for the word in each given question.
Direction: In these types of questions, the sets of numbers given in the alternatives are represented. The columns and rows of matrix I are numbered from 0 to 4 and of matrix II are numbered from 5 to 9. A letter from these matrices can be represented first by its row and next by its column, e.g., E can be represented by 13, 24, etc. and T can be represented by 58, 66, etc. similarly you have to identify the set for the word in each given question.
If '\[-\]' stands for 'division', '+' for 'multiplication',\[\div \]for 'subtraction' and '\[\times \]' for 'addition' then which one of the following equations is correct?
The integers 1 to 60 are written on a blackboard. The following operation is then repeated 59 times: In each repetition, any two numbers say p and q, currently on the blackboard are erased and a new number \[p+q-2\] is written. What will be the number left on the board at the end?
At t minutes past 4 PM, the time needed by the minute hand of a clock to show 5 PM was found to be 22 minutes less than \[\frac{{{(t-10)}^{2}}}{6}\] minutes. Find t.
A four - digit number is formed by using the digits 2, 3, 5, 7 and 8 without repetition. If one number is selected from those numbers, then find the probability that it will be an even number.
Consider four digit even natural numbers for which the first two digits are same and the last two digits are same. How many such numbers are perfect squares?