Self Studies
Self Studies
Self Studies
Self Studies

Mathematics Test - 49

Result Self Studies

Mathematics Test - 49
  • Score

    -

    out of -
  • Rank

    -

    out of -
TIME Taken - -
Self Studies

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Self Studies Self Studies
Weekly Quiz Competition
  • Question 1
    1 / -0

    The sum of the first 16 terms of an AP whose first term and third term are 5 and 15 respectively is

    Solution
    \(1^{\text {st }}\) Method:
    \(1^{\text {st}}\) term \(=5\)
    \(3^{r d} \operatorname{term}=15\)
    Then, \(d=5\) \(16^{t h}\) term \(=a+15 d\)
    \(=5+15 \times 5=80\)
    \(\operatorname{Sum}=\left\{n \times \frac{(a+l)}{2}\right\}\)
    \(=\left\{\text { no. of terms } \times \frac{(\text { first term }+\text { last term })}{2}\right\}\) \(=\left\{16 \times \frac{(5+80)}{2}\right\}\)
    \(=16 \times \frac{85}{2}\)
    \(=8 \times 85\)
    \(=680\)
    \(2^{\text {nd }}\) Method(Thought Process):
    Sum = number of terms \(\times\) average of that \(\mathrm{AP}\)
    \(\begin{aligned} \operatorname{Sum} &=16 \times\left\{\frac{(5+80)}{2}\right\} \\ &=16 \times \frac{85}{2} \\ &=8 \times 85 \\ &=680 \end{aligned}\)
  • Question 2
    1 / -0

    What is the probability of getting a sum 9 from two throws of a dice?

    Solution
    In two throws of a dice, \(n(S)=(6 \times 6)=36\) Let \(E=\) event of getting a sum
    \(=\{(3,6),(4,5),(5,4),(6,3)\}\)
    \(\therefore P(E)=\frac{n(E)}{n(S)}\)
    \(=\frac{4}{36}=\frac{1}{9}\)
  • Question 3
    1 / -0

    Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even?

    Solution
    In a simultaneous throw of two dice, we have \(n(S)=(6 \times 6)=36\)
    Then, \(E=\{(1,2),(1,4),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,2),(3,\)
    4)\(,(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,2),(5,4),(5,6),(6,1),(6,2),\)
    (6,3),(6,4),(6,5),(6,6)\}
    \(\therefore n(E)=27\)
    \(\therefore P(E)=\frac{n(E)}{n(S)}\)
    \(=\frac{27}{36}=\frac{3}{4}\)
  • Question 4
    1 / -0
    If \(x y+y z+z x=0,\) then \(\left(\frac{1}{x^{2}-y z}+\frac{1}{y^{2}-z x}+\frac{1}{z^{2}-x y}\right)\)
    \((x, y, z \neq 0)=?\)
    Solution
    \(x y+y z+z x=0\)
    \(\therefore x y+z x=-y z\)
    \(\Rightarrow x y+y z=-z x\)
    \(\Rightarrow y z+z x=-x y\)
    \(\therefore \frac{1}{x^{2}-y z}+\frac{1}{y^{2}-z x}+\frac{1}{z^{2}-x y}\)
    \(\Rightarrow \frac{1}{x^{2}+(x y+z x)}+\frac{1}{y^{2}+(x y+y z)}+\frac{1}{z^{2}+(y z+z x)}\)
    \(\Rightarrow \frac{1}{x(x+y+z)}+\frac{1}{y(x+y+z)}+\frac{1}{z(x+y+z)}\)
    \(\Rightarrow \frac{1}{(x+y+z)}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\)
    \(\Rightarrow \frac{1}{(x+y+z)}\left(\frac{z y+x z+x y}{x y z}\right)\)
    \(\Rightarrow \frac{1}{x+y+z} \times 0\)
    \(\Rightarrow 0\)
  • Question 5
    1 / -0

    The speed of the train going from Nagpur to Allahabad is 100 km/h while when coming back from Allahabad to Nagpur, its speed is 150 km/h. find the average speed during whole journey.

    Solution
    Average speed,
    =(2×x×y)/(x+y)
    =(2×100×150)/(100+150)
    =(200×150)/250
    =120km/hr
  • Question 6
    1 / -0

    A pump can fill a tank with water in 2 hours. Because of a leak, it took 213hours to fill the tank. The leak can drain all the water of the tank in:

    Solution
    Work done by the leak in 1 hour
    =(1/2−3/7)=1/14
    ∴Leak will empty the tank in 14 hrs.
  • Question 7
    1 / -0

    d/dx (log tan x) =

    Solution
    \(\frac{d}{d x}(\log \tan x)=\frac{1}{\tan x} \sec ^{2} x=\frac{\cos x}{\cos ^{2} x \sin x}\)
    \(=\frac{2}{2} \frac{1}{\cos x \sin x}=2 \operatorname{cosec} 2 x\)
  • Question 8
    1 / -0
    If \(x=a^{2}+1 / a^{2}\) and \(y=a^{2}-1 / a^{2},\) then the value of \(x^{4}+y^{4}-2 x^{2} y^{2}\) is
    Solution
    \(x^{4}+y^{4}-2 x^{2} y^{2}=\left(x^{2}-y^{2}\right)^{2}=(x-y)^{2} \times(x+y)^{2}\)
    \(=\left(a^{2}+1 / a^{2}-a^{2}+1 / a^{2}\right)^{2} \times\left(a^{2}+1 / a^{2}+a^{2}-1 / a^{2}\right)^{2}\)
    \(\Rightarrow\left(2 / a^{2}\right)^{2} \times\left(2 a^{2}\right)^{2}=16\)
  • Question 9
    1 / -0

    If y=x sin x, then

    Solution
    \(y=x \sin x \quad \frac{d y}{d x}=x \cos x+\sin x\) \(\frac{1}{y} \frac{d y}{d x}=\cot x+\frac{1}{x}\)
  • Question 10
    1 / -0

    The average of a group of men is increased by 5 years when a person aged of 18 years is replaced by a new person of aged 38 years. How many men are there in the group?

    Solution
    Let N be the no. of persons in the group.
    Required number of person is given by;
    Member in group × aged increased = difference of replacement
    N × 5 = 38 - 18
    Or, 5N = 20
    Or, N = 4
Self Studies
User
Question Analysis

  • Correct -

  • Wrong -

  • Skipped -

My Perfomance
  • Score

    -

    out of -
  • Rank

    -

    out of -
Re-Attempt Weekly Quiz Competition
Self Studies Get latest Exam Updates
& Study Material Alerts!
No, Thanks
Self Studies
Click on Allow to receive notifications
Allow Notification
Self Studies
Self Studies Self Studies
To enable notifications follow this 2 steps:
  • First Click on Secure Icon Self Studies
  • Second click on the toggle icon
Allow Notification
Get latest Exam Updates & FREE Study Material Alerts!
Self Studies ×
Open Now