How Cheenta works to ensure student success?
Explore the Back-Story

Bhaskara Contest (NMTC JUNIOR LEVEL-IX and X Grades) 2023 - Problems and Solutions

Join Trial or Access Free Resources
Problem 1

If $a, b, c$ are real numbers such that the polynomial $x^3+6 x^2+a x+b$ is the cube of $(x+c)$ then
a) $(a+b+c)$ is divisible by 13
b) $a+b=11 c$
c) $a>b$ and $b<c$.
(4) $(a+b+c)$ is divisible by 11 .

Problem 2

In the adjoining figure, $A B=9 \mathrm{~cm}, A C=7 \mathrm{~cm}$, $B C=8 \mathrm{~cm}, A D$ is the median and $\angle C=40^{\circ}$. Then measure of $\angle A D B$ (in degrees) is

a) 100
b) 140
c) 45
d) 120

Problem 3

If $x^2+6 x+1=0$ and $\frac{x^3+k x^2+1}{3 x^3+k x^2+3 x}=2$ then the value of $k$ is

a) 68
b) 72
c) 65
d) 70

Problem 4

If $x=\sqrt[3]{49}+\sqrt[3]{42}+\sqrt{36}$, then the value of $x-\frac{1}{x^2}$ is

a) $2 \sqrt[3]{42}$
b) $3 \sqrt[4]{42}$
c) $\sqrt[4]{42}$
d) $4 \sqrt[3]{42}$

Problem 5

In the adjoining figure,
AB = BC = CD

$P$ is the midpoint of $AQ$. If $CR = 4, QC=12$, then $PQ$ is equal to

a) $4 \sqrt{3}$
b) $6 \sqrt{3}$
c) $8 \sqrt{3}$
d) $2 \sqrt{3}$

Problem 6

In the adjoining figure, $A$ is the midpoint of the are $B A C$.
Given that $A B=15$ and $A D=10$.

Then the value of $A B$ is

a) 22
b) 23
c) 22.5
d) 23.5

Problem 7

The number of real numbers $x$ which satisfy the equation $\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$ is

a) 1
b) 2
c) 0
d) 4

Problem 8

$a, b$ are real numbers such that $2 a^2+5 b^2=20$. Then the maximum value of $a^4 b^6$ is

a) 256
b) 1024
c) 1262
d) 16

Problem 9

The number of ordered pairs $(x, y)$ of integers such that $x-y^2=4$ and $x^2+y^4=26$ is

a) 4
b) 3
c) 2
d) 1

Problem 10

In the adjoining figure, three equal squares are placed. The squares are unit squares. The area of the shaded region $\left(\mathrm{in} \mathrm{cm}^2\right)$ is

a) $\frac{5}{4}$
b) $\frac{4}{5}$
c) $\frac{3}{2}$
d) $\frac{3}{4}$

Problem 11

In the adjoining figure, $A B$ is a diameter of the circle. Given $\angle B A C=20^{\circ}, \angle A E B=56^{\circ}$, Then the measure (in degrees) of $\angle B C D$ is

a) 12
b) 10
c) 14
d) 16

Problem 12

The number of ordered pairs $((m, n)$ of integers such that $1 \leq m, n \leq 100$ and $m^n n^*$ leaves a remainder 1 when divided by 4 is

a) 2250
b) 1000
c) 1125
d) 1250

Problem 13

The number of ordered pairs of positive integers $(x, y)$ satisfying the equation $x^2+4 y=3 x+16$ is

a) 1
b) 2
c) 3
d) 4

Problem 14

The algebraic expression $(a+b+a b+2)^2+(a-a b+2-b)^2-2 b^2\left(1+a^2\right)$ reduces to

a) $4(a+2)^{2}$
b) $2(a+2)^2+4 a b^2$
c) $(a-2)^2$
d) $2(a-2)^2+4 a b^2$

Problem 15

The sum of $(1 \times 4)+(2 \times 7)+(3 \times 10)+(4 \times 13)+\ldots 49$ terms is equal to

a) 122500
b) 116800
c) 11800
d) 117600

Problem 16

If the equations $x^3+\alpha x+1=0$ and $x^4-\alpha x^2+1=0$ have a common root, then the value of $a^2$ is \(\rule{1cm}{0.15mm}\) .

Problem 17

If $a, b, c, d$ are positive reals such that abcd=1 then the maximum value of $a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd$ is \(\rule{1cm}{0.15mm}\) .

Problem 18

The sum of all natural numbers ' $n$ ' for which $n(n+1)$ is a perfect square is \(\rule{1cm}{0.15mm}\) .

Problem 19

$P$ is a point inside the square $A B C D$ such that $P A=P B=$ Distance of $P$ from $C D$.
The ratio of the areas of the triangle $P A B$ to the area of the square $A B C D$ is $\frac{m}{n}$ where $m, n$ are relatively prime integers. Then the value of $m+n=$ \(\rule{1cm}{0.15mm}\).

Problem 20

The sum of roots of the simultuneous equations
$ \sqrt[y]{4^x}=32 \sqrt[x]{8^y} , \sqrt[y]{3^x}=3 \sqrt[y]{9^{1-y}}$ is \(\rule{1cm}{0.15mm}\) .

Problem 21

If $2 \sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}=\sqrt{a}+\sqrt{b}$ where $a, b$ are natural numbers, then the value of $a+b$ is \(\rule{1cm}{0.15mm}\) .

Problem 22

In the adjoining figure, $\angle A C D=38^{\circ}$.
Then the measure (in degrees) of angle $x$ is \(\rule{1cm}{0.15mm}\) .

Problem 23

If $\frac{a}{b+c}+\frac{c}{a+b}=\frac{2 b}{c+a}$ (where $a+b, b+c, c+a, a+b+c$ are all not zero), then the numerical value of $\frac{a^2+c^2}{b^2}$ is \(\rule{1cm}{0.15mm}\) .

Problem 24

The geometric and arithmetic means of two positive numbers are respectively 8 and 17 . The larger among the two numbers is \(\rule{1cm}{0.15mm}\) .

Problem 25

The number of two-digit numbers in which the tens and the units digit are different and odd is \(\rule{1cm}{0.15mm}\) .

Problem 26

The value of $(5 \sqrt[3]{4}-3 \sqrt[3]{\frac{1}{2}})(12 \sqrt[3]{2}+\sqrt[3]{16}-2 \sqrt[3]{2})$ is equal to \(\rule{1cm}{0.15mm}\) .

Problem 27

If $\frac{x y}{x+y}=1, \frac{y z}{y+z}=2, \frac{z x}{z+x}=3$, then the numerical value of $15 x-7 y-z$ is \(\rule{1cm}{0.15mm}\) .

Problem 28

The sum of all natural numbers which satisfy the simultaneous inequations $x+3<4+2 x$ and $5 x-3<4 x-1$ is \(\rule{1cm}{0.15mm}\) .

Problem 29

In an increasing geometric progression (with $1^{\text {st }}$ term $a$ and $n^{\text {th }}$ term $t_n$ ). the difference between the fourth and the first term is 52 and the sum of the first three terms is 26. Then the numerical value of $\frac{t_{2024}}{t_{2023}}+\frac{a^{2024}}{a^{2023}}$ is \(\rule{1cm}{0.15mm}\) .

Problem 30

The base of a triangle is 4 units less than the altitude drawn to it. The area of the triangle is $96\left(\right.$ unit $\left.^2\right)$. The ratio of the base to height is $\frac{p}{q}$ where $p , q$ are relatively prime to each other. Then the value of $p+q$ is \(\rule{1cm}{0.15mm}\) .

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
Trial
Math Olympiad Program
magic-wandrockethighlight