Higher Mathematics Sample Question for Class-IX

Class -IX

Subject :Higher Mathematics

Time : 2 hrs.                                                                                                                                         Marks  60

[N.B. Answer any six questions by taking three questions from group A and also three questions from group -B.]

Group -A ( Algebra) : 3x10 =30

1.         Given that A = { a, b, c, d} and  B= { 1, 2, 3, 4 }

            a)         Define powes set with example                                                                         2+4+4=10

            b)         Find the value of P(A) and P(B)

            c)         Show that P(A) U P(B) ¹ P ( AUB)

2.         If A = { a, b, c, d },  B = { a, c, e, } and C = { b, d, f}

            a)         Find the value of AUB  and AÇC                                                                                 2+4+4=10

            b)         Show that A (BUC)  = ( AB) U ( A C)

            c)         Prove that AÇ (BUC) = ( AÇB) U ( AÇC)

3.         If P (x) = 6x - 5x² + 2x³-3 is a polynomial and Q(x) = 2x⁴ - x² -1 is another polynomial.            2+4+4=10

            a)         Express P(x)  as a polynomial in x in standard form.

            b)         What will be the remainders if the both polynomials are divided by x+2.

            c)         Find out the common factor of the two polynomials.

4.         If    =K,  K¹O, then

            a)         x²+y²+z² =?

            b)         Show that ( a+ b+c) ( x + y +z) = ax + by + cz.

            c)         If x = b+c- a,  y = c+a-b,  z = a +b -c then again prove that x³ +y³ +z³-3xyz = 4

                        ( a³+b³+c³ -3abc)

Group -B ( Geometry) : 3x10=30

5.         In an obtuse angled triangle, the square on the side opposite to the obtuse angle is equal to the sum of    the squares on the other two sides together with twice the projection of the other side upon it.

            a)         From above statement, draw signed figure and write down the particular enunciation.

            b)         Prove the statement.

            c)         When that obtuse angle is 120⁰, then again establish the proof.

6.         a)         State the theorem of Apollonious.

            b)         Prove the theorem.

            c)         Using the theorem, Prove that three times the sum of the squares of the sides of a triangle is        equal to four times the sum of the squares of the medians.

7.         If AB=4 cm. and PQ is a given line, then

            a)         Draw a rough geometric figure ie, a circle which passes through two given points A and B          and whose centre lies on the given straight line PQ.

            b)         Construct a circle which passes through two given points A and B and whose center lies on       the given straight line PQ. (Description and sign of construction are needed.)

            c)         Construction another circle which passes through two given points, A, B and whose radius is     equal to r.  ( Description and sign of the construction are essential)