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Question Number 20131 Answers: 1 Comments: 4

In rectangle ABCD,AB=8, BC=20.P is a point on AD so that ∠BPC=90°.If r_1 ,r_2 ,r_3 are the radii of the incircles of APB, BPC, and CPD. find r_1 +r_2 +r_3

$${In}\:{rectangle}\:{ABCD},{AB}=\mathrm{8}, \\ $$$${BC}=\mathrm{20}.{P}\:{is}\:{a}\:{point}\:{on}\:{AD}\:{so} \\ $$$${that}\:\angle{BPC}=\mathrm{90}°.{If}\:{r}_{\mathrm{1}} ,{r}_{\mathrm{2}} ,{r}_{\mathrm{3}} \:{are}\:{the} \\ $$$${radii}\:{of}\:{the}\:{incircles}\:{of}\:{APB}, \\ $$$${BPC},\:{and}\:{CPD}.\:{find}\:{r}_{\mathrm{1}} +{r}_{\mathrm{2}} +{r}_{\mathrm{3}} \\ $$

Question Number 20001 Answers: 1 Comments: 0

The number of the roots of the quadratic equation 8sec^2 θ − 6secθ + 1 = 0 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{quadratic} \\ $$$$\mathrm{equation}\:\mathrm{8sec}^{\mathrm{2}} \theta\:−\:\mathrm{6sec}\theta\:+\:\mathrm{1}\:=\:\mathrm{0}\:\mathrm{is} \\ $$

Question Number 20045 Answers: 0 Comments: 0

Question Number 19986 Answers: 1 Comments: 0

An aeroplane has to go from a point A to point B, 500 km away due 30° east of north. A wind is blowing due north at a speed of 20 ms^(−1) . The air speed of the plane is 150 ms^(−1) . Find the direction in which the pilot should head the plane to reach point B.

$$\mathrm{An}\:\mathrm{aeroplane}\:\mathrm{has}\:\mathrm{to}\:\mathrm{go}\:\mathrm{from}\:\mathrm{a}\:\mathrm{point}\:{A} \\ $$$$\mathrm{to}\:\mathrm{point}\:{B},\:\mathrm{500}\:\mathrm{km}\:\mathrm{away}\:\mathrm{due}\:\mathrm{30}°\:\mathrm{east} \\ $$$$\mathrm{of}\:\mathrm{north}.\:\mathrm{A}\:\mathrm{wind}\:\mathrm{is}\:\mathrm{blowing}\:\mathrm{due}\:\mathrm{north} \\ $$$$\mathrm{at}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{20}\:\mathrm{ms}^{−\mathrm{1}} .\:\mathrm{The}\:\mathrm{air}\:\mathrm{speed}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{plane}\:\mathrm{is}\:\mathrm{150}\:\mathrm{ms}^{−\mathrm{1}} .\:\mathrm{Find}\:\mathrm{the}\:\mathrm{direction} \\ $$$$\mathrm{in}\:\mathrm{which}\:\mathrm{the}\:\mathrm{pilot}\:\mathrm{should}\:\mathrm{head}\:\mathrm{the} \\ $$$$\mathrm{plane}\:\mathrm{to}\:\mathrm{reach}\:\mathrm{point}\:{B}. \\ $$

Question Number 19982 Answers: 1 Comments: 1

What is the number of ordered pairs (A, B) where A and B are subsets of {1, 2, ..., 5} such that neither A ⊆ B nor B ⊆ A?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ordered}\:\mathrm{pairs} \\ $$$$\left({A},\:{B}\right)\:\mathrm{where}\:{A}\:\mathrm{and}\:{B}\:\mathrm{are}\:\mathrm{subsets}\:\mathrm{of} \\ $$$$\left\{\mathrm{1},\:\mathrm{2},\:...,\:\mathrm{5}\right\}\:\mathrm{such}\:\mathrm{that}\:\mathrm{neither}\:{A}\:\subseteq\:{B} \\ $$$$\mathrm{nor}\:{B}\:\subseteq\:{A}? \\ $$

Question Number 19970 Answers: 0 Comments: 4

The velocity-time graph of a body is shown in figure. The displacement covered by the body in 8 seconds is

$$\mathrm{The}\:\mathrm{velocity}-\mathrm{time}\:\mathrm{graph}\:\mathrm{of}\:\mathrm{a}\:\mathrm{body}\:\mathrm{is} \\ $$$$\mathrm{shown}\:\mathrm{in}\:\mathrm{figure}.\:\mathrm{The}\:\mathrm{displacement} \\ $$$$\mathrm{covered}\:\mathrm{by}\:\mathrm{the}\:\mathrm{body}\:\mathrm{in}\:\mathrm{8}\:\mathrm{seconds}\:\mathrm{is} \\ $$

Question Number 19976 Answers: 1 Comments: 0

Three vectors A^(→) , B^(→) and C^(→) add up to zero. Find which is false. (a) (A^(→) ×B^(→) )×C^(→) is not zero unless B^(→) , C^(→) are parallel (b) (A^(→) ×B^(→) )∙C^(→) is not zero unless B^(→) , C^(→) are parallel (c) If A^(→) , B^(→) , C^(→) define a plane, (A^(→) ×B^(→) ×C^(→) ) is in that plane (d) (A^(→) ×B^(→) ).C^(→) = ∣A^(→) ∣∣B^(→) ∣∣C^(→) ∣ → C^2 = A^2 + B^2

Question Number 19964 Answers: 1 Comments: 1

Question Number 19955 Answers: 1 Comments: 2

Question Number 19948 Answers: 0 Comments: 0

Question Number 19945 Answers: 1 Comments: 0

If α and β are the roots of equation x^2 + px + q = 0 and α^2 , β^2 are roots of the equation x^2 − rx + s = 0, show that the equation x^2 − 4qx + 2q^2 − r = 0 has real roots.

$$\mathrm{If}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{equation} \\ $$$${x}^{\mathrm{2}} \:+\:{px}\:+\:{q}\:=\:\mathrm{0}\:\mathrm{and}\:\alpha^{\mathrm{2}} ,\:\beta^{\mathrm{2}} \:\mathrm{are}\:\mathrm{roots}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{2}} \:−\:{rx}\:+\:{s}\:=\:\mathrm{0},\:\mathrm{show} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{2}} \:−\:\mathrm{4}{qx}\:+\:\mathrm{2}{q}^{\mathrm{2}} \:−\:{r}\:=\:\mathrm{0} \\ $$$$\mathrm{has}\:\mathrm{real}\:\mathrm{roots}. \\ $$

Question Number 19940 Answers: 0 Comments: 2

A circle is inscribed in an isosceles trapezium. Prove that the ratio of the area of the circle to the area of the trapezium is equal to the ratio of the circum- ference of the circle to the perimeter of the trapezium.

$${A}\:{circle}\:{is}\:{inscribed}\:{in}\:{an} \\ $$$${isosceles}\:{trapezium}.\:{Prove}\:{that} \\ $$$${the}\:{ratio}\:{of}\:{the}\:{area}\:{of}\:{the}\:{circle} \\ $$$${to}\:{the}\:{area}\:{of}\:{the}\:{trapezium}\:{is} \\ $$$${equal}\:{to}\:{the}\:{ratio}\:{of}\:{the}\:{circum}- \\ $$$${ference}\:{of}\:{the}\:{circle}\:{to}\:{the}\: \\ $$$${perimeter}\:{of}\:{the}\:{trapezium}. \\ $$

Question Number 19939 Answers: 0 Comments: 0

f(x)=lnx

$${f}\left({x}\right)={lnx} \\ $$

Question Number 19936 Answers: 1 Comments: 1

Find the pricipal value of (1−i)^(1+i) .

$${Find}\:{the}\:{pricipal}\:{value}\:{of}\: \\ $$$$\:\:\:\:\:\left(\mathrm{1}−{i}\right)^{\mathrm{1}+{i}} \:. \\ $$

Question Number 19935 Answers: 0 Comments: 0

Which of the following points is a convex combination of (2, − 5, 0) and and (− 4, 2, 4) in R^3 (a) (0, 6, 1) (b) (− 4, − 2, 5) (c) (− 1, 0, 4) (d) (− 2, − (1/3), (8/3)) (e) None of the above

$$\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{points}\:\mathrm{is}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{combination}\:\mathrm{of}\:\left(\mathrm{2},\:−\:\mathrm{5},\:\mathrm{0}\right)\:\mathrm{and} \\ $$$$\mathrm{and}\:\left(−\:\mathrm{4},\:\mathrm{2},\:\mathrm{4}\right)\:\mathrm{in}\:\mathbb{R}^{\mathrm{3}} \\ $$$$\left(\mathrm{a}\right)\:\:\left(\mathrm{0},\:\mathrm{6},\:\mathrm{1}\right) \\ $$$$\left(\mathrm{b}\right)\:\left(−\:\mathrm{4},\:−\:\mathrm{2},\:\mathrm{5}\right) \\ $$$$\left(\mathrm{c}\right)\:\left(−\:\mathrm{1},\:\mathrm{0},\:\mathrm{4}\right) \\ $$$$\left(\mathrm{d}\right)\:\left(−\:\mathrm{2},\:−\:\frac{\mathrm{1}}{\mathrm{3}},\:\frac{\mathrm{8}}{\mathrm{3}}\right) \\ $$$$\left(\mathrm{e}\right)\:\mathrm{None}\:\mathrm{of}\:\mathrm{the}\:\mathrm{above} \\ $$

Question Number 20132 Answers: 1 Comments: 0