In this page we have created 10 no. of questions on Mathematics which can be handy the exam you are preparing for. 

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12:00

1. If `a+\frac{1}{a} = 3` then find the value of `\sqrt{a}+\frac{1}{\sqrt{a}}`?





`(sqrt{a}+\frac{1}{\sqrt{a}}) ^2 = a+2\times\sqrt{a}\times\frac{1}{\sqrt{a}}+\frac{1}{a}`

Or, `(sqrt{a}+\frac{1}{\sqrt{a}}) ^2 = a+2+\frac{1}{a}`

Or, `(sqrt{a}+\frac{1}{\sqrt{a}}) ^2 = a+frac{1}{a}+2`

Or, `(sqrt{a}+\frac{1}{\sqrt{a}}) ^2 = 3+2 `

So, `(\sqrt{a}+\frac{1}{\sqrt{a}}) =\sqrt{5} `

2. Travelling at`54` km/h, a train crosses a pole in 20 seconds. How much time in seconds will it take to cross a bridge `630` m. long?





To cross a pole means to cross the length of the train itself.
Speed of the train = `54\times\frac{5}{18}` m/s = `15` m/s.
Length of the train = `15\times20` m = `300` m.
To cross a bridge of length `630`m, the train has to cross `(630+300)` m. =`930`m.
Required time=`\frac{930}{15}` sec.=`62` sec.

4. What is the simplified value of `(7+1)(7^{2}+1)(7^{4}+1)(7^{8}+1)`?





`(7+1)(7^{2}+1)(7^{4}+1)(7^{8}+1)`

=`\frac{(7-1)(7+1)(7^{2}+1)(7^{4}+1)(7^{8}+1)}{(7-1)}`

=`\frac{(7^{2}-1)(7^{2}+1)(7^{4}+1)(7^{8}+1)}{6}`

=`\frac{(7^{4}-1)(7^{4}+1)(7^{8}+1)}{6}`

=`\frac{(7^{8}-1)(7^{8}+1)}{6}`

=`\frac{7^{16}-1}{6}`

5. A tradesman allows a discount of 19% on the marked price. How much above the cost price must he mark his goods as to gain 29% ?





Let the cost price be` x`

So, the selling price = `\frac{x\times129}{100}`

`(100-19)% = 89%` of marked price = `\frac{129x}{100}`

`100%` of marked price =` \frac{129x}{100}\times\frac{100}{89}`

= `\frac{129x}{89}`

So, Increase in C.P = `\frac{129x}{89}-x`

= `\frac{40x}{89}`

% increase = `\frac{40x}{89x}\times100`

=`44\frac{84}{89}%`

6. If the simple and compound interest on a certain sum is $350 and $364.25 respectively for two years, find the rate of interest?





The interest for the first year for both compound interest and simple interest will be the same and the amount is `$\frac{350}{2}=$175`

Now, the difference in the `2nd` year arises due to interest levied on the interest of the `1st` year.

The difference between C.I & S.I = `$(364.25-350) = $14.75`

So, the rate of interest = `\frac{14.75}{175}\times100 =7`

7. Iron is `12` times and Platinum is `20` times as heavy as water respectively. In what ratio they should be mixed so that the alloy is `1.2` times as heavy as Iron?





New alloy is `1.2` times heavier than Iron.

So, it is `12\times1.2 = 14.4` times heavier than water.

Now, apply the allegation rule.


So, the required ratio = `5.6:2.4 = 7:3`

8. What is the minimum value of `4Cosec^{2}\theta+5Sin^{2}\theta`?





When square of `cosec` function and `sin` function is added. The minimum value is `\sqrt{Product. of. co-efficient}`

So, in this case the minimum value will be

`\sqrt{4\times5}=\sqrt{20} = 2\sqrt{5}`

9. In the following figure, find the length of AB?






In this case remember: `AB+DC = AD+BC`

So, `AB+5 = 7+9`

So, ` AB = 16-5 = 11`cm

10. If` 5\sqrt{5}+\sqrt{125} = 21.43`, find the value of `13\sqrt{5}+\sqrt{245}`?





`5\sqrt{5}+\sqrt{125} = 21.43`

Or, `5\sqrt{5}+5\sqrt{5} = 21.43`

Or, `10\sqrt{5} = 21.43`

Now, `13\sqrt{5}+\sqrt{245}`

= `13\sqrt{5}+7\sqrt{5}`

=`20\sqrt{5}`

=`2\times10\sqrt{5}`

=`2\times21.43`

=`42.86`

11. What will be the remainder when `21^{23}+23^{21}` will be divided by `11`?





`21^{23}+23^{21}`
`(22-1)^{23}+(22+1)^{21}`
Applying binomial theory
All terms of `(22-1)^{23}` will be divided by `11` except the last term, `-1` and in case of `(22+1)^{21}` is `+1`
So, the remainder will be `-1+1=0`

12. What will be the difference between the sum of “base and height” and circumcenter of a right angled triangle having base and height `6` and `8` cm respectively?





Hypotenuse ` = \sqrt{base^{2}+height^{2}}`
`=\sqrt{6^{2}+8^{2}}`
`=\sqrt{100}`
`=10cm`
Circumcenter `= \frac{10}{2}=5cm`
Difference `= (6+8)-5cm`
`= 9cm`

13. A tank can be filled in `6` hours by three pipes `P, Q and R`. Pipe `R` is twice fast as `Q` and `Q` is twice fast as `P`. In how much time pipe P can fill the tank?





Efficiency
`P:Q=1:2`
`Q:R=1:2`
`P:Q:R=1:2:4`
Total work `= (1+2+4)\times6 = 42 unit.`
Time taken by `P` alone= `\frac{42}{1}=42 hours`

14.`(sec\theta-tan\theta)^{2}(1+sin\theta)^{2}\div sin^{2}\theta`=?





`(sec\theta-tan\theta)^{2}(1+sin\theta)^{2}\div sin^{2}\theta`
`=\{\frac{(sec\theta-tan\theta)(1+sin\theta)}{sin\theta}}^{2}`
`=(\frac{\frac{1-sin^{2}\theta}{cos\theta}}{sin\theta})^{2}`
`=cot^{2}\theta`

15. If `a=3+2\sqrt{2},` find the value of `\sqrt{a}-\frac{1}{\sqrt{a}} ?`





`a=3+2\sqrt{2}`
So,`\frac{1}{a}=\frac{1}{3+2\sqrt{2}}`
or, `\frac{1}{a}=\frac{3-2\sqrt{2}}{(3-2\sqrt{2})(3+2\sqrt{2})}`
or, `\frac{1}{a}=\frac{3-2\sqrt{2}}{9-8}`
So, `\frac{1}{a}=3-2\sqrt{2}`
Now,`(\sqrt{a}-\frac{1}{\sqrt{a}})^{2}=a-2\times\sqrt{a}\times\frac{1} {\sqrt{a}}+\frac{1}{a}`
or, `(\sqrt{a}-\frac{1}{\sqrt{a}})^{2}=a+\frac{1}{a}-2`
or, `(\sqrt{a}-\frac{1}{\sqrt{a}})^{2}=3+2\sqrt{2}+3-2\sqrt{2}-2`
so, `\sqrt{a}-\frac{1}{\sqrt{a}}=\sqrt{4}=2`

16. In what proportion water must be added with milk to gain `10%` by selling the mixture at cost price?





Required ratio `= 10% : 100%`
`=1 : 10`

17. The sides `BA` and `D`E of a regular pentagon `ABCDE` are produced to meet at `F`. What is the measure of `\angleEFA?`





`\angle EAB=\angle AED=\frac{540^{\circ}}{5}=108^{\circ}`
(as the sum of internal angle of a pentagon is `540^{\circ}`)
`So, \angle FEA=\angle FAE=(180^{\circ}-108^{\circ})=72^{\circ}`
`So, \angle EFA=(180^{\circ}-2\times 72^{\circ})=36^{\circ}`

18. In `\triangle ABC,\angle A=90^{\circ}`, `D` and `E` are mid points of sides `AC` and `BC` respectively. What is the value of `\frac{BC^{2}}{BD^{2}+CE^{2}}` ?





`BD` and `CE` are medians.
So, `5BC^{2}=4(BD^{2}+CE^{2})`
So, `\frac{BC^2}{(BD^2+CE^2)}=\frac{4}{5}`

19. If `cot\alpha+tan\alpha=2,` then the value of `tan^{89}\alpha +cot^{91}\alpha=?`





`cot\alpha+tan\alpha=2`
or, `cot\alpha+\frac{1}{cot\alpha}=2`
or, `\frac{cot^{2}\alpha+1}{cot\alpha}=2`
or, `cot^{2}\alpha+1=2cot\alpha`
or, `cot^{2}\alpha-2cot\alpha+1=0`
or, `(cot\alpha-1)^{2}=0`
or, `cot\alpha=1 and tan\alpha=1`
So,` tan^{89}\alpha +cot^{91}\alpha`
`=(1)^{89}+(1)^{91}`
`=1+1=2`

20. If the perimeter of an isosceles right angled triangle is `4x` m. its area is --





Let the equal sides of the triangle be `a`.
Then the hypotenuse will be `\sqrt{a^{2}+a^{2}}=a\sqrt{2}`
Now, perimeter `= a+a+a\sqrt{2} = 4x`
or, `2a+a\sqrt{2}=4x`
or, `a=\frac{4x}{\sqrt{2}(\sqrt{2}+1)}`
or, `a=\frac{2\sqrt{2}x}{\sqrt{2}+1}`
so, the area of triangle =`\frac{1}{2}\times(\frac{2\sqrt{2}x}{\sqrt{2}+1})^{2}`
`=\frac{1}{2}\times\frac{8x^{2}}{(\sqrt{2}+1)^{2}}`
`=(\frac{2x}{\sqrt{2}+1})^{2}`

21. If the ratio of areas of two circles is `p:q` what will be the ratio of their radii?





As the radius is proportional to `sqrt{area}`
So, the required ratio will be `sqrt{p}:\sqrt{q}`

22. The perimeter of a rhombus (non-square) is `40 cm` , one of its diagonal is `16 cm.` The other diagonal is -- `





Let ABCD is a rhombus (non-square) as shown bellow.
`AB=BC=CD=AD= \frac{40}{4}cm = 10 cm`
`AC=16 cm`
`So, AO= \frac{16}{2}=8cm`
`and \angle AOD=90^{\circ}`
`So, OD= \sqrt{10^{2}-8^{2}}=6cm`
`So, BD= 6\times2=12cm`

23. Two pipes `X` and `Y` can fill a cistern in `8` hours and `12` hours respectively. Pipe `Z` can empty the whole cistern in `5` hours. If all the pipes are open, in how many hours the cistern will be full?





Let the capacity of the cistern is `120` liter. (L.C.M of `8,12` and `5`).
No. of litter filled by pipe `X` and `Y` together = `(15+10)` liter= `25` litter.
No. of litter emptied by pipe `Z=24` litter.
No. of litter actually getting in `=(25-24)` litter = `1` litter.
Required time = `\frac{120}{1} = 120` hours.