# Solved papers for 10th Class Mathematics Solved Paper - Mathematics-2016

### done Solved Paper - Mathematics-2016

• question_answer1) In $\Delta \,ABC,D$and $E$ are points $AC$ and $BC$ respectively such that $DE\parallel AB$. If $AD=2x,\text{ }BE=2x-1,CD=x+1$ and$CE=x-1$, then find the value of $x$.

• question_answer2) In A, B and C are interior angles of $\Delta \text{ }ABC$, then prove that: $\sin \frac{(A+C)}{2}=\cos \frac{B}{2}$.

• question_answer3) If $x=3\text{ }sin\text{ }\theta$ and $y=4\text{ }cos\,\theta$, find the value of $\sqrt{16{{x}^{2}}+9{{y}^{2}}}$.

• question_answer4) If empirical relationship between mean, median and mode is expressed as mean $=k$(3 medianmode), then find the value of$k$.

• question_answer5) Express 23150 as product of its prime factors. Is it unique?

• question_answer6) State whether the real number 52.0521 is rational or not. If it is rational express it in the form $\frac{p}{q}$, where $p,q$ are co-prime, integers and $q\ne 0$. What can you say about prime factorisation of $q$?

 Given the linear equation $x-2y-6=0$, write another linear equation in these two variables, such that the geometrical representation of the pair so formed is: (i) coincident lines (ii) intersection lines

• question_answer8) In an isosceles $\Delta \text{ }ABC$ right angled at B, prove that $A{{C}^{2}}=2A{{B}^{2}}$.

 Prove the following identity: ${{\left[ \frac{1-\tan \,A}{1-\cot \,A} \right]}^{2}}={{\tan }^{2}}A:\angle A$ is acute

 Given below is a cumulative frequency distribution table. Corresponding to it, make an ordinary frequency distribution table. $x$ $cf$ More than or equal to 0 45 More than or equal to 10 38 More than or equal to 20 29 More than or equal to 30 17 More than or equal to 40 11 More than or equal to 50 6

• question_answer11) Find LCM and HCF of 3930 and 1800 by prime factorisation method.

• question_answer12) Using division algorithm, find the quotient and remainder on dividing $f(x)$ by $g(x)$ where $f(x)=6{{x}^{3}}+13{{x}^{2}}+x-2$ and$g(x)=2x+1$.

• question_answer13) If three zeroes of a polynomial ${{x}^{4}}-{{x}^{3}}-3{{x}^{2}}+3x$ are $0,\sqrt{3}$ and $-\sqrt{3}$, then find the fourth zero.

 Solve the following pair of equations by reducing them to a pair of linear equations: $\frac{1}{x}-\frac{4}{y}=2$ $\frac{1}{x}+\frac{3}{y}=9$

• question_answer15) $\Delta \text{ }ABC$ is a right angled triangle in which $\angle B=90{}^\circ$. D and E are any point on AB and BC respectively. Prove that$A{{E}^{2}}+C{{D}^{2}}=A{{C}^{2}}+D{{E}^{2}}$.

 In the given figure, RQ and TP are perpendicular to PQ, also $TS\bot PR$ prove that $\text{ST}\text{.RQ=PS}\text{.PQ}$.

• question_answer17) If sec$A=\frac{2}{\sqrt{3}}$, find the value of $\frac{\tan \,\,A}{\cos \,\,A}+\frac{1+\sin \,\,A}{\tan \,\,A}$

• question_answer18) Prove that: ${{\sec }^{2}}\theta -{{\cot }^{2}}(90{}^\circ -\theta )=co{{s}^{2}}(90{}^\circ -\theta )+co{{s}^{2}}\theta .$

 For the month of February, a class teacher of Class IX has the following absentee record for 45 students- Find the mean number of days, a student was absent. Number of days of absent 0 ? 4 4 ? 8 8 ? 12 12 ? 16 16 ? 20 20 ? 24 Number of students 18 3 6 2 0 1

 Find the missing frequency (x) of the following distribution, if mode is 34.5: Marks obtained 0 ? 10 10 ? 20 20 ? 30 30 ? 40 40 ? 50 Number of students 4 8 10 x 8

• question_answer21) Prove that $\sqrt{5}$ is an irrational number. Hence show that $3+2\sqrt{5}$ is also an irrational number.

• question_answer22) Obtain all other zeroes or the polynomial${{x}^{4}}+6{{x}^{3}}+{{x}^{2}}-24x-20$, if two of its zeroes are $+2$ and $-5$.

 Draw graph of following pair of linear equations: $y=2(x-1)$ $4x+y=4$ Also write the coordinate of the points where these lines meets x-axis and y-axis.

• question_answer24) A boat goes $30\text{ }km$ upstream and $44\text{ }km$ downstream in 10 hours. The same boat goes $40\text{ }km$ upstream information some student guessed the speed of the boat in still water as $8.5\text{ }km/h$ and speed of the stream as $3.8\text{ }km/h$. Do you agree with their guess? Explain what do we learn from the incident?

• question_answer25) In an equilateral $\Delta \text{ }ABC,\text{ }E$ is any point on $BC$ such that $BE=\frac{1}{4}BC$. Prove that $16\text{ }A{{E}^{2}}=13\text{ }A{{B}^{2}}$.

 In the figure if $\angle ABD=\angle XYD=\angle CDB=90{}^\circ .\text{ }AB=a,XY=c$ and $CD=b$, then prove that$c\text{ (}a+b)=ab$.

 In the $\Delta \,ABC$(see figure), $\angle A=$ right angle, $AB=\sqrt{x}$ and $BC=\sqrt{x+5}$. Evaluate $sin\text{ }C.\text{ }cos\text{ }C.\text{ }tan\text{ }C+co{{s}^{2}}C.\text{ }sin\text{ }A$

• question_answer28) If $\frac{\cos \,B}{\sin \,A}=n$ and $\frac{\cos \,B}{\cos \,A}=m$ then show that $({{m}^{2}}+{{n}^{2}})co{{s}^{2}}A={{n}^{2}}$.

• question_answer29) Prove that: $\frac{\sec \,A-1}{\sec \,A+1}={{\left( \frac{\sin \,A}{1+\cos \,A} \right)}^{2}}={{(cot\,A-cosec\,A)}^{2}}$

 Following table shows marks (out of 100) of students in a class test: Marks No. of students More than or equal to 0 80 More than or equal to 10 77 More than or equal to 20 72 More than or equal to 30 65 More than or equal to 40 55 More than or equal to 50 43 More than or equal to 60 28 More than or equal to 70 16 More than or equal to 80 10 More than or equal to 90 8 More than or equal to 100 0 Draw a ?more than type? ogive. From the curve, find the median. Also, check the value of the median by actual calculation.

 From the following data find the median age of 100 residents of a colony who took part in swachch bharat abhiyan: Age (in yrs.) More than or equal to No. of residents 0 50 10 46 20 40 30 20 40 10 50 3